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Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture Vincent A. Billock, Douglas W. Cunningham, and Paul R. Havig Logicon, Inc., U. S. Air Force Research Laboratory, P. O. Box 317258, Dayton, Ohio 45437-7258 Brian H. Tsou U. S. Air Force Research Laboratory, Building 146, Wright-Patterson Air Force Base, Ohio 45433 Received October 23, 2000; accepted April 4, 2001; revised manuscript received April 16, 2001 Recent work establishes that static and dynamic natural images have fractal-like 1/ faspatiotemporal spectra. Artiﬁcal textures, with randomized phase spectra, and 1/ faamplitude spectra are also used in studies of tex-ture and noise perception. Inﬂuenced by colorimetric principles and motivated by the ubiquity of 1/ faspatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions char-acterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporalappearance map and the spatiotemporal discrimination function (a map of Mac Adam-like just-noticeable-difference contours). © 2001 Optical Society of America OCIS codes: 330. 1730, 330. 5510, 330. 6100, 330. 6110, 330. 6790, 100. 6740. 1. GENERAL INTRODUCTION Early color science struggled with problems of quantify-ing, representing, and predicting color appearance. Thedevelopment of colorimetry solved many of these prob-lems with the creation of color spaces that quantify colors,relate them in appearance, and make predictions aboutdiscriminability. 1,2Texture perception suffers from simi-lar difﬁculties and seems in need of similar solutions. Ina pioneering study, Richards showed that colorimetrictexture and ﬂicker matches can be made to arbitrary spa-tial and temporal stimuli by mixing several spatial ortemporal ''primaries. '' 3,4This approach ﬁts well with both the multiscale structure of natural images and withthe multichannel visual processing of spatial and tempo-ral information, but the dimensionality of a spatiotempo-ral match would be cumbersomely large. This dimen-sionality can be reduced in some situations where there isa lawful relationship between the contributions at variousscales. Such an approach is suggested by fractal math-ematics and by a plethora of recent studies on fractal-likeproperties of natural images. Fractional Brownian tex-tures (favorite objects of study in recent textureexperiments 5-7), natural images,8-11and ''fractal forger-ies'' of natural images12-14all share 1/ fbspatial frequency spectra. Similarly, temporal sequences of naturalimages 15-18and the electronics used in imaging devices19,20produce 1/ fatemporal spectra. This suggests that a reasonable model for the amplitude spectra ofmany spatiotemporal textures is A ~fs,ft!5Kfs2bft2a. (1) where fsand ftare spatial and temporal frequency, re-spectively. Throughout this paper we will refer to dy-namic textures—with randomized phase spectra and power law [Eq. (1)] Fourier amplitude spectra—as spa-tiotemporal random fractals. 21We mapped the percep-tual space of such dynamic fractal textures, quantifyingappearance and discriminability. Speciﬁcally, we soughtto create an appearance map of the a,bdynamic texture space, analogous to a CIE space with labeled color bound-aries. We also sought to characterize discriminability ofdynamic textures, by measuring just noticable differences(JNDs) for the four cardinal directions in this space,analogous to the measurement of Mac Adam's ellipses incolorimetry. 2. EXPERIMENT 1: QUANTIFICATION OF APPEARANCE SPACE A. Introduction Every color researcher has a fairly clear notion of the ap-pearance of a speciﬁed aperture color based on its CIE co-ordinates. This expectation is based on a general aware-ness of the approximate CIE coordinates of many spectraland desaturated colors. Most CIE charts plot the spec-tral locus and the blackbody locus. Some reference CIEcharts are even helpfully colored over their entire surface. One can consider such a chart an ''appearance map'' of thecolorimetric space (see Fig. 1 for an example). 22-24We set out to produce an appearance map for spatiotemporalrandom-fractal space by varying the aandbparameters of Eq. (1) and asking several observers to characterize theappearance of these dynamic fractals. B. Methods 1. Participants The four observers (PH, DC, SF, and VB) were all myopes wearing their refractive prescriptions, yielding correc-2404 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al. 0740-3232/2001/102404-10$15. 00 © 2001 Optical Society of America	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
tions to at least 6/6 binocular acuity. All participants are published psychophysicists and highly experienced ob-servers, with prior work in the psychophysics of ''white''and ''colored'' (spectrally nonuniform) spatial noise. Three of the observers were also authors of this study; oneadditional observer (SF) was naı ¨ve to the purpose of the experiment. 2. Apparatus All stimuli were generated and presented on a Silicon Graphics O2 at 30 Hz. The lookup table was modiﬁed tolinearize the display. All stimuli were viewed binocu-larly with natural pupils in a well-lit room (the luminanceof a diffuse reﬂector in the plane of the monitor was 3. 5cd/m 2). Subjects were comfortably ﬁxed in place by a chin rest at a viewing distance of 40 cm from the monitor. 3. Stimuli The stimuli were 70 spatiotemporal, gray-scale, random-phase fractal-like image sequences. For each stimulusimage, the average luminance was constant at 8. 57 cd/m 2 Fig. 1. Appearance map of CIE 1931 color space. Reproduced from Ref. 22 by permission of the Optical Society of America. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2405	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
(not including the diffuse room illumination). Owing to computational constraints, each fractal was limited to64364 pixels (18 318 mm) in size and 64 frames in du-ration. Thus each stimulus subtended 2. 58 arc deg em-bedded in a 43. 9 (H) by 36. 4 (V) deg dark surround andlasted 2. 133 s. Each 64-frame stimulus sequence wascreated by ﬁrst producing a series of random white noiseimages. Gray-scale values are computed digitally withuse of 15-bit ﬂoating-point numbers and are displayedwith 8 bits (256-level gray scale). Each image in the se-ries was Fourier transformed, and the amplitudes of allspatial frequencies were equalized to ensure that thenoise was uniformly white. The resulting frequency dis-tributions were then ﬁltered so that the amplitude spec-trum varied over time following the power law relation-ship 1/ f taand over space (isotrophically over both Cartesian coordinates) following the power law relation-ship 1/ f sb@s5(x21y2)1/2#. Ten spatial exponents ( b Fig. 2. Appearance map of spatiotemporal fractal space. The appearance map is based on the responses of four observers. To help render the chart more readable to a diverse audience, the real-world descriptions provided by the subjects (which were highly individu-alistic and culturally based) have been replaced where possible with stimulus-based descriptions provided by the other subjects. 2406 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
50. 4, 0. 6, 0. 8, 1. 0, 1. 2, 1. 4, 1. 6, 1. 8, 2. 0, and 2. 2) were factorially combined with seven temporal exponents( a50. 2, 0. 4, 0. 6, 0. 8, 1. 0, 1. 2, and 1. 4) to create the 70 dynamic image sequences we employed. The resultingﬁltered spectra were inverse Fourier transformed to gray-scale levels for display. For noise stimuli the traditionalmeasures of contrast have proved inadequate. A usefulmetric of noise contrast is root mean square (RMS)contrast, 25which we express here in percentage terms as the standard deviation of the gray levels in the display di-vided by the mean. The standard deviation and mean ofthe gray-scale distribution in each stimulus were mea-sured. The gray levels in the ﬁnal image were thenrenormalized to a standard deviation of 28 and a meanlevel of 127, corresponding to a RMS contrast of 22%. 4. Procedure Observers were asked to provide a written description of the phenomenal appearance of each fractal sequence andwere free to use any terms they chose. They were givenan unlimited amount of time to respond. Each fractal se-quence was shown in a continuous loop until the observerindicated readiness to proceed to the next display. Thefractals were presented one at a time in order, starting with the lowest spatial and temporal exponents. All tem-poral exponents were shown for a given spatial exponentbefore the spatial exponent was increased. C. Results and Discussion There was considerable regularity across the four observ-ers' descriptions, so the responses were combined to pro-duce a consensus map of the appearance of the spatiotem-poral texture space (see Fig. 2). Very idiosyncratic termsprovided by some of the subjects were replaced by moregeneral terms consistent with that description. As ex-pected from previous work, 5-8,26,27the primary effect of increasing the spatial exponent bwas to increase the ap-parent coarseness of the fractal texture (see Fig. 3 for anillustration of the effect of bon static fractal textures). The primary effect of increasing the temporal exponent a was to decrease the apparent speed and jitter of the dy-namic textures. Perceptually, for aexponents of 0. 2 and 0. 4, the motion appeared very jittery; e. g., motion to theright was more often than not followed by motion to theleft. In the theory of fractional (biased) Brownian mo-tions, such behavior is called antipersistent and is differ-entiated from random (Brownian) motions and persistentmotions. 28-31Interestingly, when the spatial and tempo-ral exponents were both within the range of natural im-ages, the subjects tended to provide real-world descrip-tions of the stimuli. There were strong interactions between the spatial and temporal dimensions for the appearance of dynamic tex-ture. There are three primary examples of this interac-tion. First, the emergent texture regions were oftenmore apparent, larger, and more diffuse at higher tempo-ral exponents (slower speeds). Second, when the tempo-ral exponent was between 0. 6 and 1. 0, the texture oftenappeared to rotate, but only for smaller spatial exponents. Finally, when the temporal exponent was between 0. 4 and0. 8 and the spatial exponent was small, there was a ten-dency to see two transparent textures, with one moving infront of the other. 3. EXPERIMENT 2: DISCRIMINATION OF DYNAMIC TEXTURES A. Introduction In addition to being able to anticipate the appearance of an aperture color, given its CIE coordinates, color re-searchers are also able to gauge how tolerant subjectswould be to small perturbations in the CIE coordinates. Several studies, especially Wright's and Mac Adam's pio-neering work, 32-35have measured the just noticeable dif-ferences for perturbations of CIE coordinates. Becausethe CIE space is usually depicted as a plane, the locus of JNDs around a single point is usually an ellipse. A mapof such ''Mac Adam ellipses'' in the CIE space (Fig. 4) is auseful tool, which we wished to characterize for the spa-tiotemporal random-fractal space. So far as we areaware, prior fractal discrimination studies have been lim-ited to spatial discrimination. 26,27 Fig. 3. Static, random phase, fractal-like textures produced by 1/fbspatial-frequency ﬁltering of random white noise. Static snapshots of ﬁve of the ten fractal exponents (for one seed valueof the random number generator) are shown here. From top tobottom, the values of the exponent bare 0. 4, 0. 8, 1. 2, 1. 6, 2. 0. Note that as the spatial exponent increases, the apparent coarse-ness of the texture also increases. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2407	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
B. Methods 1. Participants and Apparatus These were the same as in Experiment 1. 2. Stimuli The stimuli were identical to those used in Experiment 1, with the sole exception that static fractals were also em-ployed. 3. Procedure The smallest detectable increase (Above JND) and de-crease (Below JND) in the spatial exponent band in the temporal exponent awere separately measured with an adaptive staircase procedure. For each of the four typesof discrimination, a reference and a comparison imagewere presented side by side (1. 1 mm apart) for 2. 133 s oruntil the observer responded, whichever came ﬁrst. Ineach case, the subject was given instructions that wereequivalent to identifying the reference stimulus. The lo-cation of the reference stimulus (left versus right) wasrandomized across trials. Immediately after the ob-server responded (using a computer keyboard), feedbackwas provided on the accuracy of this response. If the ob-server did not respond within the display time, the screenwas blanked and the observer was prompted for an an-swer. If the observer accurately identiﬁed the referencestimulus three times in a row (79% correct criteria), the difference between the exponents of the comparison andreference images was decreased by 10%. If the observerresponded incorrectly, the difference between the expo-nents was increased by 10%. Each staircase continuedfor eight reversals, and the mean of the last six reversalswas used as a measure of the threshold. When spatial JND's were measured, the reference and the comparison fractals were assigned the same temporalexponent and the spatial exponent of the comparisonstimulus was adaptively varied. The observers wereasked to identify the reference image. That is, they wereto ﬁnd the more ''ﬁne-grained'' texture for the Above dis-criminations and the less ﬁne-grained texture for the Be-low discriminations. When temporal JND's were measured, the spatial ex-ponent was held constant while the temporal exponentwas adaptively varied. The observers were again askedto identify the reference image. In other words, theywere to ﬁnd the faster and more jittery texture for the Above discriminations and the slower and less jittery tex-ture for Below discriminations. For all subjects, the 80 Above spatial discriminations were presented ﬁrst, followed by the 80 Below spatial dis-criminations, then the 70 Above temporal discrimina-tions, and ﬁnally the 70 Below temporal discriminations. Fig. 4. Mac Adam's ellipses. 33Discrimination thresholds (JNDs) for various directions in CIE 1931 color space. Ellipses are for Mac-Adam's observer PGN and are shown at ten times their actual size. Reproduced by permission of the Optical Society of America. 2408 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
Within each type of discrimination, the order of presenta-tion of the fractal exponents was randomized, with eachobserver receiving a different random order. We alsorandomized the ''seed'' values of the random number gen-erator used to generate the white noise before ﬁltering. The same seed values were used within each staircase butnot for subsequent staircases. Thus the data in Figs. 5-9, which are collapsed across subjects or conditions, represent discrimination for an ensemble of randomphase spectra rather than any particular pattern of ran-dom phases. Each subject required roughly 50 h to com-plete the measurements. C. Results and Analysis Figure 5 plots just-noticeable-difference contours (analo-gous to Mac Adam ellipses) for the dynamic fractals. It isinteresting to note that the ellipse size is fairly constantover much of the space. In color, much attention hasbeen devoted to obtaining colorimetric spaces in whichdiscrimination thresholds are relatively constant: the socalled Uniform Color Spaces (e. g., CIE 1976 Uniform Color Space). 1,2The spatial and temporal discrimina-tions will be discussed separately in more detail below. 1. Spatial Just-Noticeable Differences Overall, spatial discriminations were easiest when bwas between 1. 4 and 1. 8. This is consistent with previous re-search on static fractal discrimination. 26,27Spatial JND's are plotted in Fig. 6 as a function of spatial expo-nent. The data for both Above and Below JND's are re-markably similar across subjects. 2. Effect of Temporal Characteristics on Spatial Discriminations If the spatial and temporal dimensions are independent in dynamic fractals, as is often assumed, similar discrimi-nation thresholds should be found for dynamic fractals re-gardless of temporal exponent. Figure 7 helps to illus-trate the point that the spatial JND's did not vary muchas a function of the temporal nature of the stimuli. Al-though there is a slight suggestion of lower spatial JNDsfor the lowest temporal exponents, it seems that spatialdiscriminations are largely independent of the temporalcharacteristics in the stimuli. 3. Temporal Just-Noticeable Differences Figure 8 plots the temporal JND's as a function of tempo-ral exponent. For each subject, temporal discriminationswere easiest when awas between 0. 8 and 1. 0, a typical range of exponents for natural stimuli. 15-18The pattern of discriminations was remarkably consistent across sub-jects. Of particular interest is that the antipersistentstimuli were extremely difﬁcult to discriminate. More-over, for three of the four subjects, the Below JND's forthe antipersistent stimuli tended to be elevated forsmaller temporal exponents. Notice that for these three Fig. 5. Wright-Mac Adam-like contours for discrimination thresholds in four directions in a spatiotemporal fractal space. Average of four observers. The contours are simple quarter-circles ﬁtted to the JNDs that span their quadrants. These contours are not theo-retically motivated and are used to reduce the confusion that occurs for overlapping JND crosses. Note that over a wide range of thespace the discrimination thresholds vary only slightly; e. g., the space approaches the ideal of a uniform colorimetric space (such as the CIE 1976 uniform chromaticity space). 1,2Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2409	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
observers, the JND at a50. 2 was greater than 0. 2; i. e., the exponent a2Dabecomes negative, and the slope of the 1/ fa2Daspectra for the comparison image is slightlypositive. This can occur only for the Below discrimina-tions and is the most likely explanation of the asymmetrybetween Above and Below thresholds. Fig. 6. Spatial JND's collapsed across temporal exponents. Here the spatial discriminations are plotted as a function of spatial expo-nent. Each subﬁgure is for an observer named by his initials. Fig. 7. Spatial JND's collapsed across spatial exponents. Here the spatial discriminations are plotted as a function of temporal expo-nent. The temporal exponent labeled ''0'' is a place holder for the static condition. Observers are identiﬁed by initials. 2410 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
4. Effect of Spatial Characteristics on Temporal Discriminations Figure 9 plots the temporal JND's as a function of spatial exponent. For all four observers, thresholds increased asthe spatial exponent increased. In other words, discrimi-nating the temporal nature of the stimuli became more difﬁcult as the texture became coarser. This makes in-tuitive sense, as it would be easier to see a 1-mm object Fig. 8. Temporal JND's collapsed across spatial exponents. Here the temporal discriminations are plotted as a function of temporal exponent. Each subﬁgure is for an observer named by his initials. Fig. 9. Temporal JND's collapsed across temporal exponents. Here the temporal discriminations are plotted as a function of spatialexponent. Each subﬁgure is for an observer named by his initials. Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2411	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
move 2 mm (200% of its size) than it would be to see a 100-m object move the same distance (0. 002% of its size). 4. GENERAL DISCUSSION The fractal space described here was chosen not only forits multiscale elegance but for its potential applicabilityto real-world images and image sequences. Fractals—mathematical entities that are self-similar over manyspatial or temporal scales—are ideal tools for represent-ing natural phenomena that take place on many scalesand for studying vision, which contains multiple channelstuned on the spatial and temporal dimensions. Randomfractal textures (sometimes called fractional Browniantextures) have been widely studied because their 1/ f b spectra simulate the spectra of natural images without being confounded by phase information. If bis in the range of a typical natural object ( b50. 9-1. 2),11subjects will often spontaneously report its appearance12: a psy-chophysical version of the venerable children's game ofnaming the shapes in clouds. If phase is manipulated,fractal textures can take on the appearance of naturalscenes: Fractal forgeries of landscapes, coastlines andclouds are extremely compelling. 13,14In the temporal di-mension, sequences of events with 1/ faspectra are ubiq-uitous and fractal forgeries are especially striking; 1/ f acoustic sequences sound musical. 36In vision, 1/ faam-plitude spectra are found for dynamic images, where ais indicative of the character of the movement. 18,28-31The use of a common mathematical framework for character-izing both dynamic noise and dynamic images may alsofacilitate the study of masking of images by noise. ACKNOWLEDGMENTS We thank Jer Sen Chen, Steven Fullenkamp, and Eric Heft for technical support. Douglas Cunningham's current address is Max Planck Institute for Biological Cybernetics, Spemannstrasse 38,Tubingen 72076, Germany. REFERENCES AND NOTES 1. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry, 2nd ed. (Wiley, New York, 1963). 2. G. Wyzecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982). 3. W. Richards, ''Quantifying sensory channels: generalizing colorimetry to spatiotemporal texture, touch and tones,''Sens. Processes 3, 207-229 (1979). In general, Richards found that arbitrary textures could be matched by mixturesof four separated texture primaries and that temporalmodulations could be matched by mixtures of three sepa-rated ﬂicker primaries. 4. The analogy to colorimetry cannot be pushed too far here. Brill, in an interesting gedanken experiment, showed thatunder some circumstances Grassman's additivity law is vio-lated for some textures: M. H. Brill, ''Formalizing Grass-man's laws in a generalized colorimetry,'' Sens. Processes 3, 370-372 (1979). 5. J. E. Cutting and J. J. Garvin, ''Fractal curves and complex-ity,'' Percept. Psychophys. 42, 365-370 (1987). 6. T. Kumar, P. Zhou, and D. A. Glaser, ''Comparison of human performance with algorithms for estimating fractal dimen-sion of fractional Brownian statistics,'' J. Opt. Soc. Am. A 10,1 1 3 6-1146 (1993). 7. A. Pentland, ''Fractal-based description of surfaces,'' in Natural Computation, W. Richards, ed. (MIT Press, Cam-bridge, Mass., 1988), pp. 279-299. 8. A. P. Pentland, ''Fractal-based description of natural scenes,'' IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 661-674 (1984). 9. D. J. Field, ''Relations between the statistics of natural im-ages and the responses of cortical cells,'' J. Opt. Soc. Am. A4, 2379-2394 (1987). 10. D. J. Tolhurst, Y. Tadmor, and T. Chou, ''The amplitude spectrum of natural images,'' Ophthalmic Physiol. Opt. 12, 229-232 (1992). 11. V. A. Billock, ''Neural acclimation to 1/ fspatial frequency spectra in natural images transduced by the human visualsystem,'' Physica D 137, 379-391 (2000). 12. B. E. Rogowitz and R. F. Voss, ''Shape perception and low-dimensional fractal boundaries,'' Human Vision and Elec-tronic Imaging: Models, Methods, and Applications,J. P. Allebach and B. E. Rogowitz, eds., Proc. SPIE 1249, 387-394 (1990). 13. R. F. Voss, ''Random fractal forgeries,'' in Fundamental Al-gorithms for Computer Graphics, R. A. Earnshaw, ed. (Springer, Berlin, 1985), pp. 805-835. 14. B. B. Mandelbrot, The Fractal Geometry of Nature (Free-man, New York, 1983). 15. M. P. Eckert, G. Buchsbaum, and A. B. Watson, ''Separabil-ity of spatiotemporal spectra of image sequences,'' IEEETrans. Pattern Anal. Mach. Intell. 14, 1210-1213 (1992). 16. D. W. Dong and J. J. Atick, ''Statistics of time varying im-ages,'' Network Comput. Neural Syst. 6, 345-358 (1995). 17. J. H. van Hateren, ''Processing of natural time series by the blowﬂy visual system,'' Vision Res. 37, 3407-3416 (1997). 18. V. A. Billock, G. C. De Guzman, and J. A. S. Kelso, ''Fractal time and 1/ fspectra in dynamic images and human vision,'' Physica D 148, 136-146 (2001). 19. M. Savilli, G. Lecoy, and J. P. Nougier, Noise in Physical Systems and 1/f Noise (Elsevier, New York, 1983). 20. M. S. Keshner, ''1/ fnoise,'' Proc. IEEE 70, 212-218 (1982). 21. Technically, fractals have inﬁnite spectral bandwidths. Purists would designate textures physically obtainable ondisplays as prefractals or pseudofractals. 22. K. L. Kelly, ''Color designations for lights,'' J. Opt. Soc. Am. 33,6 2 7-632 (1943). 23. P. Keller, ''1976-UCS chromaticity diagram with color boundaries,'' Proc. Soc. Inf. Disp. 24, 317-321 (1983). 24. Color appearance maps of Kelly's CIE 1931 and Keller's 1976 spaces are available from Photo Research, 3000 N. Hollywood Way, Burbank, California 91505. 25. B. Moulden, F. Kingdom, and L. F. Gatley, ''The standard deviation of luminance as a metric for contrast in random-dot images,'' Perception 19,7 9-101 (1990). 26. D. C. Knill, D. Field, and D. Kersten, ''Human discrimina-tion of fractal textures,'' J. Opt. Soc. Am. A 7, 1113-1123 (1990). 27. Y. Tadmor and D. J. Tolhurst, ''Discrimination of changes in the second-order statistics of natural and synthetic images. ''Vision Res. 34, 541-554 (1994). 28. In the theory of fractional (biased) Brownian motion, the motion bias is quantiﬁed by the distance that a biasedwalker moves from the origin in unit time. A Brownian(random) walker's distance over unbroken ground is pro-portional to Atand has an amplitude spectrum exponent of 1. 0. Greater distance than this is covered if the motion ispersistently biased (if bias is perfect, distance is propor-tional to time), less distance is covered if the bias is anti-persistent. B. B. Mandelbrot and J. W. van Ness, ''Frac-tional Brownian motions, fractional noises andapplications,'' SIAM Rev. 10,4 2 2-437 (1968). 29. Over broken ground, a random walker covers less distance from the origin. The case closest to our stimuli that hasbeen studied in statistical physics is Brownian motion on afractal fractured surface. The temporal frequency spec-trum exponent afor this case is 0. 5. W. Lehr, J. Machta,2412 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al.	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
and M. Nelkin, ''Current noise and long time tails in biased disordered random walks,'' J. Stat. Phys. 36,1 5-29 (1984). This is in accord with our ﬁnding that stimuli with expo-nents below 0. 5 appear to move in a jittery fashion, whereasexponents above 0. 5 move smoothly. 30. In an interesting experiment Snippe and Koenderink stud-ied human ability to perceive correlations between mem-bers of a row of light sources. Positive correlations be-tween two lights were detected as apparent motion, but thecorresponding anticorrelation between the same two lightswas undetectable as motion. H. P. Snipe and J. J. Koen-derink, ''Detection of noise-like luminance functions,'' Per-cept. Psychophys. 55,2 8-41 (1994). 31. One oddity of random motion is that the visual system does not seem to be able to compensate for motion blur forstimuli undergoing Brownian motion. S. N. J. Watama-niuk, ''Visual persistence is reduced by ﬁxed-trajectory mo-tion but not random motion,'' Perception 21, 791-802 (1992). No one has yet studied deblurring for anticorre-lated (antipersistent) motion. 32. W. D. Wright, ''The graphical representation of small color differences,'' J. Opt. Soc. Am. 33, 632-636 (1943). 33. D. L. Mac Adam, ''Visual sensitivity to color differences in daylight,'' J. Opt. Soc. Am. 32,2 4 7-274 (1942). 34. L. Silberstein and D. L. Mac Adam, ''The distribution of color matchings around a color center,'' J. Opt. Soc. Am. 35, 32-39 (1945). 35. W. R. J. Brown and D. L. Mac Adam, ''Visual sensitivities to combined chromaticity and luminance differences,'' J. Opt. Soc. Am. 39, 808-834 (1949). 36. R. F. Voss and J. Clarke, ''1/ fnoise in music: music from 1/fnoise,'' J. Acoust. Soc. Am. 63, 258-263 (1978). Billock et al. Vol. 18, No. 10/October 2001/J. Opt. Soc. Am. A 2413	Perception of spatiotemporal random fractals_ an extension -- Billock Vincent A_ Cunningham Douglas W_ Havig Paul R_ -- Journal of the Optical -- 10_1364_JOSAA_18_002404 -- 9c79ac318129b904e04c3707d02
APPROVED FOR PUBLIC RELEASEPresented at the Human Factors Issues in Combat Identification Workshop, Gold Canyon, Arizona, May 13, 2008. WHAT VISUAL DISCRIMINATION OF FRACTAL TEXTURES CAN TELL US ABOUT DISCRIMINATION OF CAMOUFLAGED TARGETS Vincent A. Billock General Dynamics Advanced Information Systems Douglas W. Cunningham University of Tübingen Brian H. Tsou U. S. Air Force Research Laboratory Abstract Most natural images have 1/fȕ Fourier image statistics, a signature which is mimicked by fractals and which forms the basis for recent applications of fractals to camouflage. To distinguish a fractal camouflaged target (with 1/fȕ* statistics) from a 1/fȕ natural background (or another target), the exponents of target and background (or other target) must differ by a critical amount (dȕ ȕȕ*), which varies depending on experimental circumstances. The same constraint applies for discriminating between friendly and enemy camouflaged targets. Here, we present data for discrimination of both static and dynamic fractal images, and data on how discrimination varies as a function of experimental methods and circumstances. The discrimination function has a shallow minimum near ȕ=1. 6, which typifies images with less high spatial frequency content than the vast majority of natural images ( ȕ near 1. 1). This implies that discrimination between fractal camouflaged objects is somewhat more difficult when the camouflaged objects are sufficiently similar in statistics to the statistics of natural images (as any sensible camouflage scheme should be), compared to the less natural ȕ value of 1. 6. This applies regardless of the ȕ value of the background, which has implications for fratricide; friendlies and hostiles will be somewhat harder to tell apart for naturalistically camouflaged images, even when friendlies and hostiles are both visible against their backgrounds. The situation is even more perverse for “active camouflage”. Because of perceptual system nonlinearities (stochastic resonance), addition of dynamic noise to targets can actually enhance target detection and identification under some conditions.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 2 APPROVED FOR PUBLIC RELEASEIntroduction Discrimination tasks in combat target identification are legion. For example, operators need to discriminate a target against a background and to discriminate a set of similar targets from one another. The first task is a necessary, but not sufficient condition for targeting, while the second task is essential to solve decoy and friendly-fire problems. Both tasks are complicated by camouflage. If it were necessary to consider the set of all possible targets, backgrounds and camouflage, the combinatorial problem would be disheartening. However, a consideration of visual psychophysics, image science and fractal mathematics suggests that a particularly simple optical signature provides a low-dimensional solution. Figure 1. An Expeditionary Fighting Vehicle (General Dynamics, Inc. ) concealed against foliage with two different camouflage schemes. The rear of the vehicle is in standard single-scale NATO camouflage and pops-out from the foliage background. The front of the vehicle (see figure bottom) is concealed by a two-scale MARPAT camouflage pattern and is less conspicuous (O'Neill et al., 2004). If the number of scales increases, the perception of fractal-like camouflage is less distance dependent. Courtesy of the United States Marine Corps Systems Command. Background: Perceptual Popout, Fractals and Camouflage It is well known that humans effortlessly (and preattentively) discriminate images which differ significantly in their second-order statistics (the so-called "pop-out" phenomenon), while images	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 3 APPROVED FOR PUBLIC RELEASEthat have similar second-order statistics must usually be compared on a more laborious point-by-point basis (Julesz & Caelli, 1979; Caelli, 1981). Most natural (and many artificial) images have surprisingly regular 1/fȕ Fourier spatial amplitude spectra (Table 1; see Field & Brady, 1997 and Billock, 2000 for review), a signature which is mimicked by random fractals and which forms the basis for fractal forgeries (Voss, 1985) and digital camouflage. The exponent ȕ (the slope of the Fourier spectra when plotted on log-log coordinates) is a second-order statistic. A growing body of evidence suggests that humans are adapted to this statistical regularity in the environment and that this evolutionary/developmental adaptation forms the basis for neural image enhancement and debluring (Billock, 2000; Billock et al., 2001a,b; Campbell et al., 1978; Hammett & Bex, 1996). A hallmark of random fractal images is the presence of statistically similar features at every spatial scale. The lawful relationship between spatial scales is termed self-similarity and is one of the properties of natural images that give rise to 1/fȕ spectra. This property is what enables random fractals to mimic natural images and backgrounds. For example, a tree branch gives rise to several smaller branches, which give rise to many twigs-a random fractal that distributes and scales its features similarly can emulate foliage and act as camouflage. The ȕ value (or equivalently, fractal dimension) is often used as a mathematical measure of image texture and its perceptual correlates (Cutting & Garvin, 1987; Kumar et al., 1993; Pentland, 1988; Rogowitz & Voss, 1990; Taylor et al., 2005). It follows that some aspects of fractal image discrimination can emulate natural image discrimination (Hansen & Hess, 2006; Thomson & Foster, 1997; Parraga & Tolhurst, 2000; Tolhurst & Tadmor, 2000). Table 1. Second-order statistics of natural images Study Number of images ȕ±1sd Burton & Morehead (1987) 19 1. 05±. 12 Field & Brady (1997) 20 1. 10±0. 14 Parraga (1998) 29 1. 11±0. 13 Ruderman (1994) 45 0. 905 Webster & Miyahara (1997) 48 1. 13 Thomson & Foster (1997) 82 1. 19 Field (1993) 85 1. 10 van Hateren (1992) 117 1. 065±. 18 Tolhurst et al. (1992) 135 1. 20±. 13 Schaaf & Hateren (1996) 276 0. 94±0. 21 Dong & Atick (1995) 320 1. 15 Weighted average 1176 1. 08 If natural backgrounds are fractal-like, camouflage should be designed along similar principles. Newer camouflage schemes like MARPAT (U. S Marines) and CADPAT (Canadian Armed Forces) use a two-scale scheme which is noticeably better at blending into terrain and foliage than the older single-scale schemes. For example, detection times for MARPAT (Fig. 1) camouflaged targets are about 2. 5 times longer than detection of NATO single-scale camouflage	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 4 APPROVED FOR PUBLIC RELEASEand recognition times following detection increase by an additional 20% (O'Neill et al., 2004). Some newer camouflage schemes-inspired by fractals-have more than two scales. (True fractal camouflage would be defined by statistical similarity at every visible spatial scale, but limited size and printing resolution result in a restricted range of scales. ) More complicated schemes are possible, including the use of multi-fractals which mimic blends of particular textures that occur in natural images (e. g., plant growth on fractured rock). Here, we study human abilities to discriminate images based on small differences in the ȕ signature and place the results in context with camouflage and with earlier texture discrimination studies. Figure 2. Fractal textures like those used in the experiments. Each fractal in this figure has 1/fβ amplitude spectrum and identical phase spectra, and is synthesized by spatial frequency filtering the same set of random gray levels. The lack of weight in the higher spatial frequencies can easily be seen in the coarseness of the images as the exponent β increases.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 5 APPROVED FOR PUBLIC RELEASEMethods Participants The four observers were all myopes corrected to at least 20/20 binocular acuity. All are professional psychophysicists and highly experienced observers, with prior work in the psychophysics of "white" and "colored" (spatiotemporally non-uniform) visual noise. One subject (SF) was naïve to the purpose of the experiment. Another subject (VB) has a diagnosed, mild, congenital visual condition-optic nerve hypoplasia (a low density of neurons in the optic nerve). Although his vision is considered normal by standard clinical measures (including acuity), his contrast sensitivity is slightly depressed (about 1 sd) at all spatial frequencies relative to a large sample of age-matched-normals; this depression in sensitivity is worse for higher spatial frequencies. His vision is relevant here because it provides us a gauge of the effects of spatial under-sampling in an otherwise intact visual system. Apparatus All stimuli were generated and presented on a Silicon Graphics O 2 graphics workstation with a linearized 30 Hz display. Stimuli were viewed binocularly with natural pupils in a well-lit room (ambient luminance in the plane of the monitor was 3. 5 cd/m2). Subjects were comfortably fixed in place by a chin rest at two viewing distances, 40 cm and 100 cm. The far distance was a limiting case (e. g., each pixel subtends 0. 016 deg at 100 cm, matching the subjects' 1 arc min spatial resolution). The stimuli consisted of static, grayscale, random-phase fractals (e. g., Fig. 2) whose Fourier amplitude spectra were described by A(fs) = kf sȕ Eq. 1 Where k is a constant and f s is spatial frequency. (In visual psychophysics, amplitude rather than power spectra are used, because amplitude is proportional to perceptual contrast for each spatial frequency component. ) For each stimulus, the average luminance was constant at 8. 57 cd/m2and the Root Mean Square Contrast (a good measure of perceptual contrast for noise-like textures; Moulden et al., 1990; Peli, 1990, 1997) was 10. 98%. For consistency with another study, each fractal contained 64x64 pixels (18x18mm). Thus, at 40 cm, the stimuli subtended 2. 58O embedded in a 43. 9O horizontal by 36. 4O vertical dark surround. At 100 cm, each stimulus subtended 1. 03O embedded in a 21. 1O horizontal by 16. 4O vertical dark surround. Both a reference and a comparison image were generated for each trial. The images were created by filling a 64X64 array with random white noise (256 gray-levels). This white-noise image was Fourier-transformed and the amplitudes of all spatial frequencies were equalized to ensure that the noise was uniformly flat. The resulting amplitude spectra were filtered so that they followed a power law relationship (Eq. 1), and then inverse-Fourier transformed to produce the stimuli. Procedure Just noticeable discrimination thresholds (79% correct criterion) for fractal spatial exponents were measured using a two-alternative forced-choice adaptive staircase procedure with a 1 db step size (Mac Millan & Creelman, 1991). Ten β exponents were used (0. 4, 0. 6, 0. 8, 1. 0, 1. 2, 1. 4, 1. 6, 1. 8, 2. 0, and 2. 2) for the reference images. For the comparison image, the fractal exponent was equal to the exponent, β, of its reference image plus a small increment, Δβ. Observers were asked to identify the image with the lower spectral exponent, and were provided with immediate	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 6 APPROVED FOR PUBLIC RELEASEfeedback on the accuracy of their response. If the observer correctly identified the reference image three times in a row, the difference between the two images' exponents Δβ was decreased. In contrast, Δβ was increased after each incorrect response. Each staircase continued for 8 reversals, with the mean of the last 6 reversals being used as a measure of the threshold. Two presentation conditions (Sequential and Simultaneous) were used. In the Simultaneous condition, the reference and comparison stimuli were presented side by side (1. 1 mm apart) for 2. 133 seconds. The location of the reference image (left versus right) was randomized across trials. In the Sequential condition, the two stimulus images were sequentially presented in the center of the screen for 2. 133 seconds each. The screen was blanked for 500 ms between the two images to prevent masking effects. The order of presentation of the reference and comparison images was randomized across trials. Combining two viewing distances with two presentation modes yielded four experimental conditions. Two observers were presented with the Near conditions first, and two with the Far conditions first. For all observers, the presentation style (simultaneous vs. sequential) alternated after each threshold. The order of presentation of the 10 exponents was randomized for each of the 4 conditions. Each threshold was measured 3 times, with the thresholds in all 4 conditions being completed once before being re-measured. This required approximately 20 hours of data collection per subject, which was generally completed in 2 one-hour sessions each day, over a two-week period. Figure 3. Group averages for all 4 conditions. Results and Analysis General Findings Discrimination thresholds (d ȕ) are generally in the range of 0. 05-0. 20 for ȕ values of 0. 4-2. 2 (see Figs. 3, 4). The discrimination function is not flat; it has higher (worse) discrimination thresholds for both low and high values of ȕ, and lower (better) discrimination thresholds for in-between	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 7 APPROVED FOR PUBLIC RELEASEvalues of B. The minimum is near ȕ=1. 6, which typifies images with less high spatial frequency content than the vast majority of natural images ( ȕ near 1. 1). This implies that discrimination between fractal camouflaged objects is somewhat more difficult when the statistics of camouflaged objects are sufficiently similar to the statistics of natural images (as any sensible camouflage scheme should be), compared to the less natural B value of 1. 6. This applies regardless of the background's ȕ value, which has implications for fratricide; friendlies and hostiles will be somewhat harder to tell apart for naturalistically camouflaged images, even when friendlies and hostiles are both visible against their backgrounds. Figure 4. Individual data from all 4 observers for all 4 conditions.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 8 APPROVED FOR PUBLIC RELEASEEffect of Viewing Distance For ideal 1/f images, there should be little effect of viewing distance, because increasing viewing distance would simply shift a lower spatial frequency component into a higher spatial frequency, but the relationship between the spatial frequencies would be preserved. However, all physically obtainable fractals are limited to a range of spatial scales set at the lower end by the size of the image and at the upper end by the size of the pixels. Shifting the viewing distance from 40 to 100 cm therefore shifts the spatial frequency range of the fractal image by a factor of 2. 5, but no information is lost because the stimuli were designed so that the individual pixels were resolvable at the far viewing distance by all observers. Accordingly, for three of the observers (DC, PH, and SF), viewing distance had little effect, although there is a slight trend suggesting lower thresholds in the nearer viewing distances (see Figure 4a-c). For VB, however, the Far thresholds (and their variability) were noticeably elevated compared to either his Near thresholds or to the other observers' Far thresholds (see Fig. 4). VB's anomalous results may be due to sampling problems induced by a mild congenital defect-a developmental paucity of retinal ganglion cells (optic nerve hypoplasia). Electrophysiological studies in VB and other hypoplastic subjects and post-mortem histology in other hypoplastics indicate that both retinal pre-processing and cortical post-processing seem to be normal (Billock et al., 1994) and point to reduction in retinal ganglion cell numbers as the sole cause of abnormal vision in hypoplasia. In the case of VB, perimetric thresholds are flattened relative to normals, suggesting the subject did not gain a full measure of the elevated density of foveal ganglion cells that develops in normals. Since pixel size and stimulus size are fixed, any sampling problems would more likely manifest as a threshold elevation at the far viewing conditions. Moreover, if the reduced sampling is not homogeneous, then this could increase variability (because, from trial to trial, filtered noise features would fall on neighboring retinal locations with different retinal sampling densities). Effect of Presentation Style Simultaneous viewing simulates the task of making a side-by-side comparison of fractal camouflaged targets, while sequential viewing simulates the task of comparing a target to one that is in memory; in theory and experiment the two paradigms can lead to somewhat different results (Garcia-Perez et al., 2005; Hansen & Hess, 2006). The discrimination function is similar for both conditions (Figs. 3 and 4) but there is a small advantage for simultaneously-viewed images, relative to sequentially-viewed ones, especially for small values of ȕ. This tendency can be clearly seen when the data from the 4 subjects are pooled (Fig. 3). This is contrary to Hansen & Hess (2006), who found an advantage for sequential viewing, and attributed differences between the two conditions to differences in the portions of retina they cover. However, our near-sequential and far-simultaneous stimuli covered very similar regions of central retina (2. 6o and 2. 3o respectively), and yet simultaneous viewing yielded lower thresholds for nine of ten exponents (and tied for the tenth). This suggests that for our experiment, the memory demands of sequential viewing were disadvantageous, a design consideration for combat target displays. Discussion Comparison to Related Studies: Static Fractals Some prior studies of fractal discrimination overlap our work. Our discrimination functions resemble those of Knill et al. (1990), particularly their low-contrast near condition (17. 5% RMS	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 9 APPROVED FOR PUBLIC RELEASEcontrast at 1 meter with 64x64 pixel images), which is similar to our Far Sequential condition. While the average exponent of natural scenes is around 1. 1 (Table 1), the greatest sensitivity to changes in a fractal image's exponent are consistently found to be around 1. 6 across a wide range of conditions. We see no evidence for a second minimum at low ȕ (circa 0. 6) reported by Tadmor and Tolhurst, even when we used a simultaneous viewing condition similar to Tadmor and Tolhurst (1994). Nor is the discrepancy due to the angular size of the image, as all three studies had stimuli that were similar in size. Hansen & Hess (2006) note that the spatial presentation task uses two different parafoveal patches of retina, while the temporal task uses the same patch of central fovea; they find that fovea and parafovea yield somewhat different patterns of discriminability as a function of ȕ, but none of their data show a second minimum at low ȕ (rather, they find a maximum at ȕ=0. 8, with better thresholds on either side, similar to our findings). Another possible source of this difference in discriminability functions may be the specific nature of Tadmor and Tolhurst's task. In both the present study and Knill et al. (1990), standard two-Alternative Forced-Choice psychophysical procedures were used. In contrast, Tadmor and Tolhurst (1994) used an odd-one-out task (i. e., three images were presented simultaneously, two of which had identical exponents-the task was to choose the image that was different from the other two). In other words, Tadmor and Tolhurst's task was one of simple discrimination, while our task (and Knill's) requires discrimination and some form of identification (once the two images could be told apart, the subjects had to decide which had a lower exponent). These tasks coincide in difficulty only if all information required for identification is present at the discrimination threshold, which will most often take place when a single channel mediates performance of the task. Indeed, Tolhurst and Tadmor (1997) have shown that simple discrimination data is often consistent with a single channel mediating discrimination. However, since a comparison of channel outputs is required to estimate the spectral exponent of an image, discrimination plus identification would likely require a comparison of channel outputs, perhaps raising the JND for β near 0. 4 sufficiently to eliminate the second minimum that Tadmor and Tolhurst (1994) found. Figure 5. Camouflage can be dynamic in several ways, including simple movement. A Jordanian F-16 painted in Hyper Stealth Biotechnology Corp. 's fractal-like camouflage.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 10 APPROVED FOR PUBLIC RELEASEComparison to Related Studies: Dynamic Fractals So far we have discussed only perception of static fractals. But camouflaged images may move against their backgrounds and camouflage may be dynamic in other ways. In general, the effect of motion on such fractals is described by f t= v*f s (where v is velocity and f t is temporal frequency). Thus, if the spatial spectrum is a 1/f distribution, then for simple movement the temporal frequency distribution is a linear transform of spatial frequency, which is no more interesting than the viewing distance condition. There are however more interesting dynamic manipulations of fractals that are worth study. For example, it is possible to extend our study of discrimination to spatiotemporal fractals-fractals whose individual pixel intensities vary over time in a manner described by fractional Brownian motion. Such images have Fourier amplitude spectra of A(ft,fs) = kf tĮfsȕ Eq. 2 In general, as Į becomes larger, the motion of the texture becomes more coherent and can be used to mimic various biological motions (Billock et al., 2001a). Figure 6 shows the human perceptual discrimination space for spatiotemporal fractals (dynamic textures). Figure 6. Discrimination contours for spatiotemporal fractal textures (Eq. 2) using the same observers in Figures 3 and 4. The contours are estimated using JNDs for discrimination in four directions in the perceptual space and are fit using simple quarter-circles of no theoretical significance.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 11 APPROVED FOR PUBLIC RELEASETo define two-dimensional JNDs we measured discrimination in four directions: both increments and decrements for both spatial ( ȕ) and temporal ( Į) exponents. Not surprisingly, the JNDs in this space are ellipsoidal (they resemble color discrimination JNDs). Interestingly, the interior portion of the resulting two-dimensional discrimination space is remarkably flat, a feature that some psychophysicists have gone to great lengths to obtain in nonlinear mappings of other perceptual systems (e. g., color discrimination). Implications for Future Work It is worth noting that humans can become proficient at naming the spectral exponents of images (or equivalently, fractal dimension, which is a linear transform of the exponent; Cutting & Garvin, 1987; Kumar et al., 1993; Pentland, 1988). A neural ability to estimate the spectral drop-off and exploit it has been speculated on and deserves additional attention (Billock, 2000; Billock et al., 2001b; Campbell et al., 1978; Hammett & Bex, 1996; Rogowitz & Voss, 1990). Taken together with the natural image regularities and perceptual pop-out findings discussed earlier, this suggests that ȕ is a key signature, both for images and for the visual systems that evolved to transduce images. Of particular interest is the finding that, under some conditions (nonlinear systems near threshold), adding noise can facilitate detection and identification of some signals, including images (Repperger et al., 2001; Simonotto et al., 1997; Yang, 1998)-an example of stochastic resonance as an image enhancement mechanism. Dynamic noise is more effective than static (Simonotto et al, 1997). Because other studies of stochastic resonance show that 1/fȕ noise can be more efficient than white noise in inducing stochastic resonance effects (Billock & Tsou, 2007; Hangi et al., 1993; Nozaki et al., 1999), further studies of discrimination in spatiotemporal fractal noise (at various contrast levels) would be warranted and might uncover some practical applications. Implications of Fractal Discrimination for Camouflage and Combat ID Based on this and other work we can enumerate some implications for camouflage and combat ID: (i) Natural images have 1/fȕ spatial amplitude spectra. The most reasonable value of ȕ for general purpose camouflage is around 1. 1. Particular environments will vary in this statistic and in coloration. (ii) Keeping the difference between the ȕtarget and ȕbackground less than 0. 2 generally avoids preattentive popout, but discrimination will still be possible using a point-by-point search. (iii) Using many spatial scales makes camouflage effectiveness almost independent of distance. (iv) For IFF purposes, friendly camouflage schemes should have different ȕs than the unfriendly camouflage patterns, but this may conflict with concealment goals. The best outcome would be for hostile and friendly camouflage statistics to be on opposite sides of the ȕbackground value, with the friendly scheme not easily discriminable from background but discriminable from the hostile. (v) For identification purposes, side-by-side viewing of sensor and reference images is preferable. Sensor operators should be screened for spatial sampling problems (sub-clinical amblyopia) by measuring their contrast sensitivity functions. (vi) It may be possible to break many camouflage schemes by adding filtered noise to the sensor images. This seemingly perverse aspect of stochastic resonance should be exploited if possible. Since stochastic resonance's effectiveness is often dependent on the Fourier spectral qualities of the noise, fractal camouflage may be particularly vulnerable (because the spectral qualities of simple fractals are easily matched by varying one noise parameter). Multi-fractals may be less vulnerable in this regard. It would be ironic if the beautiful mathematical attributes of fractals (which give it so	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 12 APPROVED FOR PUBLIC RELEASEmuch utility in describing the natural environment and make it such an elegant solution to the problem of designing camouflage) also prove to be its Achilles' heel. Author Note and Acknowledgements Vincent A. Billock,General Dynamics, Inc., Suite 200, 5200 Springfield Pike, Dayton, OH 45431 ; vince. billock@gd-ais. com. Douglas W. Cunningham, University of Tübingen, Tübingen, Germany ;douglas. cunningham@gris. uni-tuebingen. de. Brian H. Tsou, AFRL/RHCI, Wright-Patterson Air Force Base, OH 45433 ;brian. tsou@afrl. af. mil. We thank Jer Sen Chen, Steven Fullenkamp, Paul Havig and Eric Heft for technical support.	Fractal_BCT20BILLOCK.pdf
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Billock, Cunningham & Tsou (2008) 14 APPROVED FOR PUBLIC RELEASEJulesz, B. & Caelli, T. (1979). On the limits of Fourier decompositions in visual texture perception. Perception, 8, 69-73. Knill, D. C., Field, D., & Kersten, D. (1990). Human discrimination of fractal images. Journal of the Optical Society of America A, 7, 1113-1123. Kumar, T., Zhou, P., & Glaser, D. A. (1993). A comparison of human performance with algorithms for estimating fractal dimension of fractional Brownian statistics. Journal of the Optical Society of America A, 10, 1136-1146. Mac Millan, N. A., & Creelman, C. D. (1991). Detection Theory: A User's Guide. Cambridge: Cambridge University Press. Moulden, B., Kingdom, F., & Gatley, L. F. (1990). The standard deviation of luminance as a metric for contrast in random-dot images. Perception, 19, 79-101. Nozaki, D., Collins, J. J., & Yamamota, Y. (1999). Mechanism of stochastic resonance enhancement in neuronal models driven by 1/f noise. Physical Review E, 60, 4637-4644. O'Neill, T., Matthews, M., & Swiergosz, M. (2004). Marine Corps innovative camouflage. Midyear meeting of the American Psychological Association, Divisions 19 & 21. Supplementary data at http://www. hyperstealth. com/digital-design/index. htm Parraga, C. A., Brelstaff, G., Troscianko, T., & Morehead, I. R. (1998). Color and luminance information in natural images. Journal of the Optical Society of America A, 15, 563-569. Parraga, C. A., & Tolhurst, D. J. (2000). The effect of contrast randomization on the discrimination of changes in the slopes of the amplitude spectra of natural scenes. Perception, 29, 1101-1116. Peli, E. (1990). Contrast in complex images. Journal of the Optical Society of America A, 7, 2032-2040. Peli, E. (1997). In search of a contrast metric: matching the perceived contrast of Gabor patches at different phases and bandwidths. Vision Research, 23, 3217-3224. Pentland, A. (1988). Fractal-based descriptions of surfaces, in W. Richards (Ed. ), Natural Computation (pp. 279-299). Cambridge: MIT Press. Repperger, D. W., Phyllips, C. A., Neidhard, A., & Haas, M. (2001). Designing human machine interfaces using principles of stochastic resonance. AFRL Technical Report ARRL-HE-WP-TR-2002-0187. DTIC# ADA412330. Rogowitz, B. E., & Voss, R. F. (1990). Shape perception and low-dimension fractal boundaries. Proceedings of the SPIE, 1249, 387-394. Ruderman, D. L. (1994). The statistics of natural images. Network: Computation in Neural Systems, 5, 517-548. van der Schaaf, A., & van Hateren, J. H. (1996). Modeling the power spectra of natural images: statistics and information. Vision Research, 36, 2759-2770. Simonotto, E., Riani, M., Seife, C., Roberts, M., Twitty, J., & Moss, F. (1997). Visual perception of stochastic resonance. Physical Review Letters, 78, 1186-1189. Tadmor, Y., & Tolhurst, D. J. (1994). Discrimination of changes in the second-order statistics of natural and synthetic images. Vision Research, 34, 541-554. Taylor, R. P., Spahar, B., Wise, J. A., Clifford, C. W. G., Newell, B. R., & Martin, T. P. (2005). Perceptual and physiological responses to the visual complexity of fractal patterns. Nonlinear Dynamics in Psychology and Life Sciences, 9, 89-114. Thomson, M. G. A., & Foster, D. H. (1997). Role of second-and third-order statistics in the discriminability of natural images. Journal of the Optical Society of America A, 14, 2081- 2090.	Fractal_BCT20BILLOCK.pdf
Billock, Cunningham & Tsou (2008) 15 APPROVED FOR PUBLIC RELEASETolhurst, D. J., & Tadmor, Y. (2000). Discrimination of spectrally blended natural images: Optimization of the human visual system for encoding natural images. Perception, 29, 1087-1100. Tolhurst, D. J., & Tadmor, Y. (1997). Band-limited contrast in natural images explains the detectability of changes in the amplitude spectra. Vision Research, 23, 3203-3215. Tolhurst, D. J., Tadmor, Y., & Chao, T. (1992). The amplitude spectra of natural images. Ophthalmic and Physiological Optics, 12, 229-232. Webster, M. A., & Miyahara, E. (1997). Contrast adaptation and the spatial structure of natural images. Journal of the Optical Society of America A, 14, 2355-2366. Voss, R. F. (1985). Random fractal forgeries, in R. A. Earnshaw, Ed., Fundamental Algorithms for Computer Graphics (pp. 805-835). Berlin: Springer. Yang, T. (1998). Adaptively optimizing stochastic resonance in visual system. Physics Letters A, 245, 79-86.	Fractal_BCT20BILLOCK.pdf
What do catastrophic visual binding failures look like? Vincent A. Billock1and Brian H. Tsou2 1General Dynamics, Suite 200, 5200 Springﬁeld Pike, Dayton, OH 45431, USA 2US Air Force Research Laboratory, Wright-Patterson Air Force Base, OH 45433, USA Ordinary vision is considered a binding success: all the pieces and aspects of an image are bound together,despite being processed by many different neurons in several different cortical areas. How this is accom-plished is a key problem in visual neuroscience. The study of visual binding might be facilitated if we had ways to induce binding failures. A particularly interest-ing failure would involve a loss of the physical integrity of the image. Here, we identify conditions that induce such perceptual failures (e. g. the melting together ofequiluminant colored images and the fragmentation of retinally stabilized images) and we suggest that these should studied using electrophysiological measuresof binding. In visual cortex, many neurons are specialized for detecting different visual attributes such as form, color, motion and depth. Often these cells are physically separated-in some cases, they seem to aggregate by stimulus preference in different cortical areas or at different spots of the same cortical area. So how does thevisual system combine these responses to create uniﬁed and veridical percepts? A leading theory suggests that the stimulus-based responses of widely spaced neurons can bebound together by the temporal synchronization of their gamma band (30-90 Hz) activity. Although controversial, an impressive body of theoretical and electrophysiologicalevidence for this theory has accumulated [1-12]. Some of the most compelling binding studies compare perceptual and electrophysiological measures in humans [10-12]. T o further such efforts, it is necessary to understand what perceptually constitutes a binding success or failure, so that its electrophysiological correlates can be sought. There are two especially interesting aspects of the 'binding problem': feature-binding and part-binding. Feature-bind-ing correctly associates various features of imagestogether, for example so that we see a yellow school bus moving towards us, rather than yellowness, curves and angles, and movement, as in a dynamic abstract painting. Part-binding refers to correct construction of spatially coherent percepts, which need to be extracted (segmented)from the noisy and often ambiguous retinal image. To study feature-binding, Treisman introduced the notion of feature-binding failures called illusory conjunctions-under some circumstances, especially in peripheral vision,two features such as shape and color can be incorrectly associated with one another-and this has been very useful for psychophysical studies of feature-binding [1]. However, there have been no comparable studies of part-binding failures (except in brain lesions [8]). For electro-physiological studies it would be useful to be able to produce part-binding failures in normal humans using reversible and noninvasive methods. So, what would a really spectacular part-binding failure look like? Has anyone ever seen one? Do they occur in normal (brain-undamaged) observers? In addressing these questions, we realized that we have seen candidates for such bind-ing failures. Recently, we studied the perception of retinally stabil-ized colored images [13]. We coupled a motorized mirror to an optical eye-tracker (which measures eye movements using infrared reﬂections from the cornea and lens). Subjects view the stimulus as a reﬂection in the mirror. By deﬂecting the mirror to compensate for eye movements, the reﬂected image remains stationary on the retina [13,14]. The textbook account of these stabilized images is that they fade (because of transient temporal responses of some retinal mechanisms), and fade they do. However, they also become unstable and lose their spatial coherence: their internal structure can ﬂow about like ﬁgures made from melting wax, or their internal borders can fade away altogether so that the colors of the image diffuse into oneanother. These and similar bizarre phenomena are not explicable in terms of peripheral neural mechanisms. But what central mechanism could exhibit such behaviors? It occurred to us that some of the behaviors that we (and many earlier investigators of equiluminant and stabilized images) found were good examples of part-binding fail-ures. Because the vast majority of these studies were made in an era innocent of binding theory, we have a rare opportunity to harness several decades of misunderstood research in the goal of advancing the current research in binding. Expected failure modes of part-binding There is an extensive literature of algorithms for part-binding. In general, these models need to bind things that belong together and segment them from backgrounds and other objects. At this level, one could imagine binding failures that result from an inability to link parts into a whole (such a binding failure will be discussed in a later section of this article). Additional possibilities can be Corresponding author: Vincent A. Billock (vince. billock@wpafb. af. mil). Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 www. sciencedirect. com 0166-2236/$-see front matter. Published by Elsevier Ltd. doi:10. 1016/j. tins. 2003. 12. 003	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
derived from segmentation models that rely on balancing cooperation and competition [15-18]. Competitive net-works deﬁne the locations of borders and edges; coopera-tive networks reinforce mechanisms that agree about thepresence of a border, and ﬁll-in areas between borders. Ifthe balance between these mechanisms breaks down, part-binding should fail in distinctive ways. If cooperativemechanisms are too strong, then the network will settleon a common value for all points in an image. If this occurs slowly enough in an already segmented image, the image would appear to fade, or gray-out, or melt. Conversely, if competition is too strong or cooperation istoo weak, the image should appear to fragment. Thisoccurs for some neural networks that use nearest neighborconnectivity [17,18]. Binding-like failures in retinal stabilization: fading and blackout Images can be mechanically stabilized from eye move-ments, either by mechanical coupling to a contact lens, orvia an eye-tracker [14,19]. When an image is stabilized, the most common perceptual effect is a gradual fading ofthe image. Similar effects can be obtained using steadyﬁxation, afterimages or ganzfelds. If the image starts outas a rather textureless surface (a ganzfeld), then gradually all aspects of the percept are lost and the subject perceives a uniform, depthless gray (eigengrau) even if the ganzfeldwas initially colored. Fading is usually attributed to thetransient properties of certain retinogeniculate units-thetransient M cells. This explanation is inadequate on atleast two grounds: (i),80% of retinogeniculate units belong to the P cell pathway, which has a sustained response to both achromatic form and color stimuli [20], and (ii) most visual mechanisms (including achromaticform, color and depth) receive inputs from P cells and canbe driven by P cells alone, if M cells are silenced [21]. And this information is utilized by higher mechanisms; forexample, the depth and multistability of a Necker cubeare just as salient when stabilized, so long as it remains visible [22]. Although P cells signal the attributes of stabilized images to cortex (a necessary condition for corticalprocessing), this is not always a sufﬁcient condition forveridical perception. Consider some properties of stabil-ized images, paradoxical to P cell function, implicatingcentral mechanisms in stabilized-image fading. For example, P cells have sustained responses for high spatial frequencies and high contrasts (unlike M cells, whichbecome more transient at high contrasts [23]); yet Purkinje's tree (the high-contrast pattern of blood vesselshadows on the retina) fades unusually rapidly, and fadesfastest (as fast as 80 ms) in the parafovea, where P cells arepresent in high densities [24]. Similarly, although P cells have strong sustained responses to chromatic stimuli, stabilized color images fade faster and more completelythan achromatic stimuli [25,26]. Actually, many attributes of visual fading are more consistent with central than peripheral mechanisms. Forexample, the fading of a stabilized image in one eye can beslowed or reversed by modulating the stimulation of theother eye [27]. Conversely, fading of a stabilized image inone eye makes images viewed by the other eye less visible. Remarkably, even if the stabilized image in one eye has completely disappeared, it can still be binocularly fused with a duplicate moving image in the other eye to yield aparadoxical percept of motion in depth [14]. Finally, fading of images can be slowed or reversed by stimulatingattention or stimulating the other senses [19]. Blackout is a related, albeit more dramatic effect. Sometimes imagestabilization, even if only in one eye, induces an extremely abrupt, binocularly simultaneous visual blackout. Recov-ery can be initiated by a blink. Descriptions of this 'blackﬁeld' as 'more black than black' do not do it justice [19]. Our subjects describe it as 'like someone cut my optic nerves,and I don't have a visual system anymore' [13]. Ditchburn attributes the black-ﬁeld effect to a fading mechanismcombined with a cortical inhibitory feedback loop that runs out of control [19]. Binding-like failures in retinal stabilization: image fragmentation From a binding failure perspective, the most interestingretinal stabilization effect is fragmentation [28-30] (Figures 1,2 ). In general, complex patterns are more likely to fragment than simple ones, angular patterns fragmentmore than round ones, and parallel lines tend to disappear and reappear together [31,32]. Sometimes the fragmenta-tion takes the form of a series of breaks in the image(Figure 1 ), whereas at other times, entire portions of the ﬁgure disappear and reappear ( Figure 2 ). The probability of fragmentation versus unitary fading grows with linelength and reaches equal probability for lines subtending45 0, suggesting a range for the neural processes involved [32]. Interestingly, if a stabilized image with gaps in its structure fades and then reappears, the gaps in the imageare often ﬁlled-in [19]. Moreover, ﬁlling-in can occur even as the overall image is fading [33]. When images fragment, they tend to fade and revive in clusters that are cognitivelymeaningful, and follow Gestalt-like rules of association(Figure 2 ). These processes act on the central image; fragmentation can eliminate contours present only in a binocular image and not in its retinal precursors [34]. Similar fragmentation can occur during migraine attacks[35] and in strabismic amblyopia (a cortical visual defect induced by poor coordination of the eyes during visualdevelopment), especially for high spatial frequency stimuli[36](Figure 1 ). The connection of fragmentation phenom-ena to amblyopia is especially interesting because, in cat, cortical cells driven by the amblyopic eye are poorlysynchronized compared with cell populations driven by thenormal eye-especially for high spatial frequency stimuli[37]. No studies of synchronization during stabilization have been made. However, some early electroencephalo-gram (EEG) studies report that alpha-rhythm (9-10 Hz) power increased,0. 7-1. 0 s before disappearances; higher frequencies (including the gamma band, which is nowimplicated in binding) were suppressed until,1 s before spontaneous reappearances [38,39]. Oddities of equiluminant images Like stabilized images, pictures in which hues have equalluminance are less perceptually salient than natural Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 85 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
images. Luminance differs slightly from brightness and is usually deﬁned operationally; for example, if two colors are shown in the same place in rapid sequence (a procedure known as heterochromatic ﬂicker photometry), the sen-sation of ﬂicker is drastically reduced for a certain ratio of the radiances of the colors-this is deﬁned as equilumi-nance. Slightly harder to use in practice, but of morerelevance to this discussion, is the minimum border criterion: at equiluminance, the apparent border contrast between differently colored image parts is minimized. Similarly, perception of form, depth, motion and color are all degraded for equiluminant stimuli [40-42]. Sometimes these phenomena are attributed to a lack of chromaticinput to these mechanisms, but several studies show thatthese systems can be driven by P cells in general and by hue-speciﬁc mechanisms in particular [43]. Indeed, even some putatively chromatic functions (e. g. chromaticrivalries) are disrupted by equiluminance [41]. Othershave attributed some effects to the lower acuity of chromatic mechanisms. This can explain some effects[42]but the most interesting effects are out of proportion to the acuity loss [43], or are seemingly unrelated to it [40,41]. From a binding point of view, the most signiﬁcant of these are linking failures and image melting. Binding-like failures at equiluminance: linking failures and perceptual melting A variety of symptoms suggest that there are part-binding failures in equiluminance. Included in these are failures tolink pieces of a surface. Consider random dot stereopsis. Itis possible to build a stereogram in which the two images contain only random dots. The two images are near duplicates; one image has a group of dots shifted overseveral pixels. Binocular fusion of the two images yields apercept of a group of random dots ﬂoating above or belowthe random dot background. In an ordinary random dot Figure 1. Examples of fragmentation percepts. (a)A square (left) breaks up during retinal stabilization, with fragmentary losses and fading starting at the corners. Repro-duced, with permission, from Ref. [29]. (b)The effects of stabilizing a long bar (left). The bar fragments in various places and the fragments fade. Reproduced from Ref. [19] by permission of Oxford University Press. The probability of such fragmentation increases with bar length [28]and some studies show a loss of high-frequency electro-encephalogram power during such stabilization-induced fragmentation and fading [38,39]. (c)Mosaic vision. An odd fragmentation phenomenon reported by a small min-ority of patents during epileptic seizures and migraine. According to Sacks, 'mosaic vision denotes the fracture of the visual image into irregular, crystalline, polygonal facets, dovetailed together as a mosaic. The size of the facets may vary greatly. If they are extremely ﬁne, the visual world presents an appearance of c rystalline iridescence or “graininess”, reminiscent of a pointillist painting... If the facets become larger, the visual image takes on the appearance of a classical mosaic... o r even a “cubist” appearance. ' Reproduced, with permission, from Ref. [35]. (d)A possibly related phenomenon, described by subjects with strabismic amblyopia. Vision is normal for low spatial frequency gratings but distorted at high spatial frequencies, as if the features were improperly bound by location. At the highest spatial fr equencies, breaks appear in the features. The phenomenon is not due to an inability to transduce the features because contrast sensitivity is normal (graph). 'c/d' (cycles per degree) refers to the periodicity of the coarse and ﬁne gratings (e. g. 1 c/d mean that one black and one white bar ﬁt into one degree of visual angle, which is roughly the angle s ubtended by the width of the thumb held at arms length; the best human acuity-20/10-is equivalent to being able to see a 60 c/d grating). Reproduced, with permission, f rom Ref. [36]. Normal Pointilliste Mosaic Cubist Vertical Horizontal Contrast sensitivity 110100300 0. 1 1 10 100 Spatial frequency (c/d)1c/d 4c/d 10c/d 15 c/d4o(a) (b) (c) (d)Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 86 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
stereogram, when a group of dots is segmented from the background, the pattern of dots deﬁnes a textured surfaceof common depth. Near equiluminance, depth is seen for the individual colored pixels but, unlike ordinary random dot stereopsis, the pixels seem to be separate rather thanpart of the same textured surface [41]. This is not a curiosity of stereopsis: 2D surfaces are not linked together when deﬁned only by hue. An image composed ofequiluminous patches does not appear uniﬁed but, rather, appears as patches of one color ﬂoating on the other [41,42]. In other cases, all sense of object and surface can be lost and the target is perceived as a 'jumble of lines' [41] or 'extremely confusing and hard to describe' [40]. More-over, the contrast of equiluminant images can seemunstable-Gregory [40]describes such images as 'jazzy'. In 1927, Liebmann (translated in Ref. [42]) reported that there is a 'critical zone [where] everything ﬂows... glim-mers... most everything is soft, jelly-like, colloidal. Often... parts which belong together in the normal ﬁgure now have nothing to do with one another. [It is] a world without ﬁrm things, without solidity. ' Such effects are mostsevere when the borders are deﬁned by tritan (S-cone) modulation: the borders of the images can appear to melt like hot wax and distort like pulled taffy (reminiscentof Liebmann's comment), and the colors diffuse into oneanother in the same way as spilled inks, blending into continuous color gradients [13,44]. There are interesting local boundary effects: fading is sometimes localized to particular features or regions [40]and when equiluminant features or edges fade away, they are sometimes replacedwith an illusory achromatic border [40,41] ). Cross-modal reinforcement: what is lost in stabilized and equiluminant images?We have discussed how a balance of cooperation and competition is required to produce reliable segmentation networks, and how these networks sometimes fail,especially for nearest-neighbor connectivity [17,18]. One way to avoid this is to allow the activity of a secondnetwork to reinforce the ﬁrst-in effect, the secondnetwork can act as a second set of nearest neighbors. Inthe segmentation literature, this reinforcement between different sensory modalities-called 'cross-modal con-struction' [45]-improves the precision and likelihood of both correct segmentation and feature binding [45-47]. I f part-binding mechanisms evolved to exploit this reinforce-ment, then its loss could compromise part-binding. Theevidence described in this article is consistent withreliance on cross-modal reinforcement of color segmenta-tion mechanisms by achromatic form mechanisms, and reliance on reinforcement of achromatic form segmenta-tion systems by mechanisms sensitive to motion ortemporal modulation. It takes two: combining equiluminance and stabilization makes everything worse If this cross-modal construction hypothesis is correct, thenwe would expect that compromising more feature systems would further worsen part-binding. This seems to be the case. A familiar example is the minimum border effect-the weakened border formed by an equiluminant bipartiteﬁeld can disappear completely when eye ﬁxation is steady[44]. A more dramatic example is our own experiment using stabilized opponent colors [13]; unlike the equilu-minant experiments already discussed, binding failures occur for any color combination. Perhaps the most estab-lished law in experimental psychology is that certain colorcombinations such as reddish green and bluish yellow arenever allowed; this observation formed the basis for Hering's theory of color opponency and much of moderncolor science. However, it had been reported that somesubjects saw these forbidden colors when colored bipartite ﬁelds such as those shown in Figure 3 were stabilized [14]. When we attempted to replicate these results, we foundthat the effect depended on the combination of equilumi-nance and stabilization. When we stabilized non-equilu-minant red-green or blue-yellow borders, we saw avariety of part-binding failures, which included effectswhere the border would disappear in places; the unbound colors both diffuse and compete, resulting in illusory textures such as red glitter on a ﬁeld of green or bluestreaks on yellow [13]. Sometimes, the two colored sides of the bipartite ﬁeld would appear to switch places or evenrotate 90 8(an effect reminiscent of Treisman's illusory conjunctions). Only when we made the two sides of thebipartite ﬁeld equiluminant did subjects report forbidden Figure 2. Some gestalt-like attributes of image fragmentation in retinal stabilization that implicate central mechanisms. (a-d) Fragmentation of monocular stabilized images. Reproduced, with permission, from Ref. [29]. Pictures on the left show unstabilized images and pictures to the right show stabilized percepts. Note that images tend to fragment and fade, such that the visible fragments follow rules. In (a) and (b), meaningful fragments are preserved. In (c), parallel fragments are pre-served. In (d), gestalt grouping rules are mostly followed. (e)Effects of stabilizing the monocular parts (left-eye image and right-eye image) of a binocular (fused) percept. Note that disappearance of image fragments is characteristic of the fused image, not the monocular forms, ruling out peripheral explanations. Reproduced,with permission, from Ref. [34]. (a) (b) (c) (d) (e) Left-eye image Right-eye image Fused image Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 87 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
colors. The border collapsed completely and the two colored sides appeared to run together, like diffusing inks or melting waxes, creating new color combinationsundreamt of in Hering's philosophy. In some cases, thesemanifest as a gradient which ran from red on one side togreen on the other, with every possible shade of reddishgreen in between. In other cases, subjects report seeing aspatially homogeneous reddish green, the red and greencomponents of which were as compelling as the red andblue components of a purple. We modeled this opponencyloss in terms of a cortical competition model whosecompetition (between color-labeled neurons) is disabled[13]. Why this should occur is not clear, but there is a model of cortical competition, whose competition is gated bygamma-band oscillations [48]; this is the frequency range that is implicated in many binding studies and thefrequency range in EEG that is lost during stabilization-induced fading and fragmentation [38,39]. Electrophysiological testing of part-binding failures We argue that phenomena such as border disappearance, fragmentation, diffusive color spreading, and linkingfailures of equiluminous and stabilized images are examples of part-binding failures, and that these particu-lar failure conditions are clues that part-binding relies onachromatic mechanisms reinforcing segmentation bycolor, and on motion-driven mechanisms reinforcingsegmentation by achromatic form. Testing these ideaswill require some notion of what the electrophysiologicalcorrelates of binding are and how they could be detectednon-invasively in intact responding humans. Becausegamma-band activity is implicated in binding processes,the loss of high-frequency EEG power that accompaniesstabilized-image fragmentation [38,39] is suggestive but not deﬁnitive. One potential problem is that some sourcesof gamma-band power in EEGs might be unrelated tobinding. Another potential problem is that our bindingfailures might actually be incorrect bindings and thereforenot distinguishable by EEG. Recent experiments have castlight on both questions, by using fragmented and camou-ﬂaged images in which subjects could eventually learn to see coherent embedded objects [10,12]. (These studies are especially relevant here because the fragmentation-to-order shift is the complement to the order-to-fragmenta-tion phenomena we studied. ) There was gamma-bandactivity whether subjects saw the coherent picture or not,but there were two differences in EEG when gestaltperception was obtained: the temporal output of widelyspaced electrodes became more correlated (consistent withsynchronization of widely spaced neural mechanisms [12]) and a second kind of gamma-band response, the 'inducedgamma response', was recorded. On every physicallyidentical stimulus presentation there is a gamma-bandresponse that is phase-locked to the camouﬂaged orfragmented stimulus, but only on trials where the gestaltpercept is seen does a second signature emerge thatconsists of gamma-band bursts in variable phase to theactual stimulus [10,12]. We therefore predict that EEG signatures of binding, such as the induced gamma response, will be reduced during binding-like failures induced by retinal stabilization or equiluminance, andthat these signatures will be further minimized oreliminated during the more dramatic perceptual effectscreated by combining stabilization and equiluminance[13]. A sophisticated version of this experiment could be done using the methods of Billock et al. [13], but the simpler methods of Buck et al. [44]might well sufﬁce. Acknowledgements We thank Kenneth Blum, Scott Kelso, Oliver Sacks, Wolf Singer and Anne Treisman for particularly helpful suggestions and remarks. References 1 Treisman, A. (1998) Feature binding, attention and object perception. Philos. Trans. R. Soc. Lond. B Biol. Sci. 353, 1295-1306 2 von der Malsburg, C. (1999) The what and why of binding: the modeler's perspective. Neuron 24, 95-104 3 Horn, D. and Opher, I. (2000) Temporal segmentation and binding in oscillatory neural systems. In Oscillations in Neural Systems (Levine, D. S. et al., eds), pp. 201-216, Erlbaum Figure 3. An image used in the stabilized-image studies of Billock et al. [13]. The image was stabilized on the retina and viewed through an unstabilized aperture. This arrangement keeps the whole image from fading [14]but does not protect the border between red and green ﬁelds. Typically, portions of the border melt or fade and the opponent colors tend to ﬂow and compete, creating complex illusory textures. Sometimes the two sides will switch colors, or even rotate 90 8. This phenomenon resembles an illusory conjunction of location and color (a feature-binding failure). If the two sides are made equal in luminance, the border disap-pears and the colors ﬂow together to form mixture colors of reddish-green, inviolation of Hering's law of color opponency. Sometimes these colors are uniform, other times they form a smooth gradient ranging from reddish green to greenish red. Similarly, stabilizing an equiluminant blue-yellow bipartite ﬁeld results inperception of bluish-yellow forbidden colors. These phenomena can be modeled by a breakdown of competitive interactions between some color-labeled cortical units [13]. 1 4 8and 24 8are degrees of visual angle for which the stabilized stimulus subtended horizontally and vertically, respectively. Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 88 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
4 Castelo-Branco, M. et al. (2000) Neural synchrony correlates with surface segregation rules. Nature 405, 685-689 5 Singer, W. (2001) Consciousness and the binding problem. Ann. N. Y. Acad. Sci. 929, 123-146 6 Singer, W. (2003) Synchronization, binding and expectancy. In The Handbook of Brain Theory and Neural Networks (Arbib, M. A., ed. ), pp. 1136-1143, MIT Press 7 Robertson, L. C. (2003) Binding, spatial attention and perceptual awareness. Nat. Rev. Neurosci. 4, 93-102 8 Humphreys, G. W. (2003) Conscious visual representations built from multiple binding processes: evidence from neurophysiology. Prog. Brain Res. 142, 243-255 9 Ursino, M. et al. (2003) Binding and segmentation of multiple objects through neural oscillators inhibited by contour information. Biol. Cybern. 89, 56-70 10 Bertrand, O. and Tallon-Baudry, C. (2000) Oscillatory gamma activity in humans: a possible role for object representation. Int. J. Psychophysiol. 38, 21 1-223 11 Muller, M. M. et al. (2000) Modulation of induced gamma band activity in the human EEG by attention and visual information processing. Int. J. Psychophysiol. 38, 283-299 12 Gruber, T. et al. (2002) Modulation of induced gamma band responses in a perceptual learning task in the human EEG. J. Cogn. Neurosci. 14, 732-744 13 Billock, V. A. et al. (2001) Perception of forbidden colors in retinally stabilized equiluminant images: an indication of softwired cortical color opponency? J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 18, 2398-2403 14 Crane, H. D. (1994) The Purkinje image eyetracker, image stabiliz-ation, and related forms of stimulus manipulation. In Visual Science and Engineering (Kelly, D. H., ed. ), pp. 15-89, Dekker 15 Grossberg, S. and Wyse, L. (1992) Figure-ground separation of connected scenic ﬁgures: boundaries, ﬁlling-in and opponent proces-sing. In Neural Networks for Vision and Image Processing (Carpenter, G. A. and Grossberg, S., eds), pp. 161-194, MIT Press 16 Levine, D. S. (2000) Introduction to Neural and Cognitive Modeling, Erlbaum 17 Terman, D. and Wang, D. L. (1995) Global competition and local cooperation in a network of neural oscillators. Physica D. 81, 148-176 18 Marr, D. et al. (1978) Analysis of a cooperative stereo algorithm. Biol. Cybern. 28, 223-239 19 Ditchburn, R. W. (1973) Eye movements and Visual Perception, Clarendon 20 Ingling, C. R. Jr and Martinez-Uriegas, E. (1985) The spatiotemporal properties of the r-g X-cell channel. Vision Res. 25, 33-38 21 Merigan, W. H. and Maunsell, J. H. R. (1993) How parallel are the primate visual pathways? Annu. Rev. Neurosci. 16, 369-402 22 Ingling, C. R. Jr and Grigsby, S. S. (1990) Perceptual correlates of magnocellular and parvocellular channels: seeing form and depthin afterimages. Vision Res. 30, 823-828 23 Benardete, E. A. et al. (1992) Contrast gain control in the primate retina: P cells are not X-like, some M-cells are. Vis. Neurosci. 8, 483-486 24 Coppola, D. and Purves, D. (1996) The extraordinarily rapid disappearance of entopic images. Proc. Natl. Acad. Sci. U. S. A. 93, 8001-800425 Weintraub, D. J. (1964) Successive contrast involving luminance and purity alterations of the Ganzfeld. J. Exp. Psychol. 68, 555-562 26 Kelly, D. H. (1981) Disappearance of stabilized chromatic gratings. Science 214, 1257-1258 27 Cohen, H. B. (1961) The effect of contralateral visual stimulation on visibility with stabilized retinal images. Can. J. Psychol. 15, 212-219 28 Pritchard, R. M. et al. (1960) Visual perception approached by the method of stabilized images. Can. J. Psychol. 14, 67-77 29 Pritchard, R. M. (1961) Stabilized images on the retina. Sci. Am. 204, 72-78 30 Davies, P. (1973) The role of central processes in the perception of visual after-images. Br. J. Psychol. 64, 325-338 31 Evans, C. R. (1965) Some studies of pattern perception using a stabilized retinal image. Br. J. Psychol. 56, 121-133 32 Evans, C. R. (1967) Further studies of pattern perception and a stabilized retinal image. Br. J. Psychol. 58, 315-327 33 Cardu, B. et al. (1971) The inﬂuence of peripheral and central factors on the way that stabilized images disappeared. Vision Res. 11, 1337-1343 34 Evans, C. R. and Wells, A. M. (1967) Fragmentation phenomena associated with binocular stabilization. Br. J. Physiol. Opt. 24, 45-50 35 Sacks, O. (1999) Migraine, Vintage Books 36 Hess, R. F. et al. (1990) The puzzle of amblyopia. In Vision: Coding and Efﬁciency (Blakemore, C., ed. ), pp. 267-280, Cambridge University Press 37 Roelfsema, P. R. et al. (1994) Reduced synchronization in the visual cortex of cats with strabismic amblyopia. Eur. J. Neurosci. 6, 1645-1655 38 Lehmann, D. et al. (1965) Changes in patterns of the human electroencephalogram during ﬂuctuations of perception of stabilizedretinal images. Electroencephalogr. Clin. Neurophysiol. 19, 336-343 39 Keesey, U. T. and Nichols, D. J. (1967) Fluctuations in target visibility as related to the occurrence of the alpha component of theelectroencephalogram. Vision Res. 7, 959-977 40 Gregory, R. L. (1977) Vision with isoluminant colour contrast. Perception 6, 113-119 41 Livingstone, M. S. and Hubel, D. H. (1987) Psychophysical evidence for separate channels for perception of form, color, movement and depth. J. Neurosci. 7, 3416-3468 42 Cavanagh, P. (1991) Vision at equiluminance. In Limits of Vision (Kulikowski, J. J. et al., eds), pp. 234-250, CRC Press 43 Mullen, K. T. and Kingdom, F. A. A. (1991) Colour contrast in form perception. In The Perception of Colour (Gouras, P., ed. ), pp. 198-217, CRC Press 44 Buck, S. L. et al. (1977) Initial distinctness and subsequent fading of minimally distinct borders. J. Opt. Soc. Am. 67, 1126-1128 45 Finkel, L. H. and Edelman, G. M. (1989) Integration of distributed cortical systems by reentry: a computer simulation of interactivefunctionally segregated visual areas. J. Neurosci. 9, 3188-3208 46 Schillen, T. B. and Ko ¨nig, P. (1994) Binding by temporal structure in multiple feature domains of an oscillatory neuronal network. Biol. Cybern. 70, 397-405 47 Poggio, T. et al. (1988) Parallel integration in visual modules. Science 242, 436-440 48 Niebur, E. et al. (1993) An oscillation-based model for the neuronal basis of attention. Vision Res. 33, 2789-2802Opinion TRENDS in Neurosciences Vol. 27 No. 2 February 2004 89 www. sciencedirect. com	What do catastrophic visual binding failures look like -- Vincent A Billock Brian H Tsou -- Trends in Neurosciences 2 27 pages 84-89 2004 feb -- 10_1016_j_tins_2003_12_003 -- df63d349ed603638608e0a965
A Role for Cortical Crosstalk in the Binding Problem: Stimulus-driven Correlations that Link Color, Form, and Motion Vincent A. Billock1and Brian H. Tsou2 Abstract &The putative independence of cortical mechanisms for color, form, and motion raises the binding problem—how isneural activity coordinated to create unified and correctly segmented percepts? Binding could be guided by stimulus-driven correlations between mechanisms, but the nature ofthese correlations is largely unexplored and no one has(intentionally) studied effects on binding if this joint informa-tion is compromised. Here, we develop a theoretical frame-work which: (1) describes crosstalk-generated correlationsbetween cortical mechanisms for color, achromatic form, and motion, which arise from retinogeniculate encoding; (2) showshow these correlations can facilitate synchronization, segmen-tation, and binding; (3) provides a basis for understanding perceptual oddities and binding failures that occur forequiluminant and stabilized images. These ideas can be testedby measuring both perceptual events and neural activity while achromatic border contrast or stabilized image velocity is manipulated. & INTRODUCTION How does the brain coordinate the activity of visual mechanisms to create unified percepts? There are twoparticularly important aspects to this binding problem(see Treisman, 1996 for a review of all aspects)—thebinding of different parts of the same image and thebinding of different image features (e. g., form, color, and motion). Part-and feature-binding are closely related because each feature class contributes to the segmenta-tion of images from backgrounds (Regan, 2000). Thisarticle is not a theory of binding, nor is it a theory ofmotion, color, and luminance defined form—on whichvast literatures already exist. Rather, we create a frame-work in which binding and its perceptual attributes maybe better understood. Towards this end, we tackle some key questions that are too seldom addressed in binding studies: (1) Just how independent are putatively inde-pendent cortical mechanisms for color, achromatic form,and motion, and what is the nature of any crosstalkbetween them? (2) How could crosstalk between thesemechanisms contribute to binding? (3) If ordinary visionis a resounding binding success, what would a catastroph-ic part-binding failure look like (feature-binding failures are called illusory conjunctions; Treisman, 1996), and under what conditions would binding failures occur?Background on Parallel Perceptual Mechanisms and Binding Theory Ample physiological and psychophysical evidence sug-gest that human vision is mediated by multiple mecha-nisms, each responding to selective combinations ofimage attributes (color, spatial frequency, orientation,motion, etc. ; for reviews see Regan, 2000; Zeki, 1993). This parallel processing—although crucial to under-standing visual detection and appearance—raises a bind-ing problem: How are unified and correctly segmented percepts created from the activity of putatively indepen-dent, spatially separated c ortical mechanisms? One theory is that binding stems from coordination of mech-anisms responding to an image (von der Malsburg, 1981,1995). Electrophysiological studies show coordinated g-band (generally 30-90 Hz) oscillations between cells inthe same and different orientation columns of the same cortical area, in different cortical areas, and in different hemispheres (for review and debates, see Gray, 1999;Shadlen & Movshon, 1999; Singer, 1999; Singer & Gray,1995). Recently, a series of compelling studies have tiedspecific g-band EEG activity in humans to specific per-cepts (Gruber, Mu ¨ller, & Keil, 2002; Tallon-Baudry & Bertrand, 1999). This impressive corpus of work onbinding and neural synchronization compels us to ad-dress a key question: What information governs binding? In principle, a sufficien tly potent coupling could synchronize any two neurons' activity, but indiscrimi-1General Dynamics,2U. S. Air Force Research Laboratory D2004 Massachusetts Institute of Technology Journal of Cognitive Neuroscience 16:6, pp. 1036-1048	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
nant perceptual binding is undesirable. There are two key kinds of information available to bind sensorymechanisms: location coding and correlated activity. Some studies suggest that binding exploits locationcoding; for example, two cells in different cortical areas,each responding to a different image feature (e. g., colorand orientation), are likely to synchronize if both areactive and both are coded for the same retinotopic location (for review, see Finkel & Edelman, 1989). This is plausible because of the substantial retinotopicallyspecific connections between cortical areas. We do notdenigrate the role of retinotopic mapping in binding,but because large receptive fields in the extrastriatecortex degrade localization, it is useful to have anindependent supplemental source of information forbinding. Moreover, supplementing location coding should speed and improve binding (time is at a premi-um when segmentation/binding takes only 50-200 msec;von der Malsburg, 1999). We suggest that a type ofstimulus-specific common neural input may provideinformation useful for binding. THEORETICAL BUILDING BLOCKS A Role for Stimulus-induced Correlations in Oscillatory Binding A tenet of binding theory is that binding is driven by (and makes manifest) correlations between mechanismsresponding to common image features. In principle, aninitial weak correlation between oscillatory mechanisms is amplified into mutual synchronization (von der Mals-burg, 1981). However, what guarantees that synchroni-zation will be correctly linked to real stimulus features?To avoid binding everything, competitive networkscould select the strongest correlations to be synchro-nized. Correlated activity in neural mechanisms is in-duced by causal relations between temporal events (e. g.,by perception-action interactions) and by common neu-ral inputs (Feng, 2000; von der Malsburg, 1999). One particularly promising source of correlated activity—crosstalk—is both stimulus-driven and induced bycommon neural inputs. Crosstalk is the unintendedportion of a selectively filtered signal. For example, Pcells carry both hue and luminance information (De Valois & De Valois, 1988). A cortical cell that filters the P-cell input to extract luminance may also extract a portion of the chromatic signal as well (Billock, 1995). This crosstalk—a nuisance in visual parallel processingtheory—has been neglected in binding theory. Theclosest exception is Horn, Sagi, and Usher's (1991)segmentation/binding network, which uses correlatednoise to synchronize neural oscillators responding todifferent visual modalities. Consider two such systems(E 1,E2) coupled through a common set of inhibitory (I) neurons. d E1=dt¼/C0E1þ Ft ða E1/C0b I/C0/C18E/C0c R1þ INPUT 1þ Q1;2Þ d R1=dt¼ð1=d/C01ÞR1þ E1 d E2=dt¼/C0E2þ Ft ða E2/C0b I/C0/C18E/C0c R2þ INPUT 2þ Q1;2Þ d R2=dt¼ð1=d/C01ÞR2þ E2 d I=dt¼/C0Iþ Fðf E1þg E2/C0h I/C0/C18IÞ Here E1,E2are firing rates driven by different visual features (INPUT 1, INPUT 2), which in this article could be achromatic contrast, color, or motion. R1,R2are dy-namic thresholds; a,b,c,d,g,u,tare constants; and Ft(x) = 1/(1+ e/C0x/t). Key attributes of Equation 1 are that E1and E2oscillate (fire) in response to their sensory inputs and can synchronize those oscillations. The Q1,2 term is an input common to both systems which speeds and strengthens synchronization—a result likely genericto coupled systems. In Horn et al. 's (1991) model, it re-presents correlated noise. Because Horn et al. considerbinding of color and form, they speculate that somenoisy ''early mixed representation of shape and color information exists in the input layer. '' Another model uses a cell class tuned along two stimulus dimensionsand couples it to a cell class that has a common stimulusdimension (Roelfsema, Engel, Ko ¨nig, & Singer, 1996; e. g., oriented cells for chromatic edges coupled to similarlyoriented cells for achromatic contrast). The correlatednoise and common dimension approaches are moresimilar than they seem. An ''early mixed representation'' is an apt description of retinogeniculate multiplexing. Attempts to create cortical cells tuned along (and labeledfor) one stimulus dimension inevitably induce stimulus-specific crosstalk from other stimulus dimensions (e. g.,Billock, 1995). Below, we quantify this crosstalk for twospecific sensory interactions. Fragmentation and Fading—Characteristic Failure Modes of Segmentation Networks Generally, within a sensory modality, segmentation in-volves cooperation between mechanisms that agree andcompetition between mechanisms that disagree. Thesemethods are generic and apply whether an image is to be segmented on the basis of achromatic form, color, motion, depth, and so forth (Levine, 2000; Wang &Terman, 1997; Grossberg & Wyse, 1992; von der Mals-berg & Buhmann, 1992). Competitive networks definethe location of edges and borders. Cooperative net-works fill in areas between borders and reinforce theresponse of retinotopically neighboring cells signaling(1) Billock and Tsou 1037	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
an edge. Such segmentation systems have characteristic failure modes: If cooperation runs amok, then all mech-anisms agree and the image is grayed-out, rather thansegmented. If such a failure occurs in an already seg-mented image, the image would appear to melt or fade. If competition is too strong, or cooperation too weak,then the image may fragment. For example, Terman and Wang's (1995) neural network (using nearest neighbor connectivity) tends to fragment images. Similarly, stere-opsis segmentation networks fragment images if thresh-olds for keeping cells active and/or entraining adjacentinactive cells are set too high (Marr, Palm, & Poggio,1978). Below, we examine situations (equiluminance andretinal stabilization) where percepts fade, melt, and/orfragment and argue these phenomena are understand-able in a segmentation/part-binding framework. These fading/fragmentation problems may be alleviated if mul-tiple segmentation networks share information aboutlocation of borders. Reinforcement between parallelnetworks—called ''cross-modal construction'' (Finkel& Edelman, 1989)—has the potential to reinforce andrefine estimates of image features (Roelfsema et al.,1996; Schillen & Ko ¨nig, 1994; Horn et al., 1991; Poggio, Gamble, & Little, 1988). Cross-modal construction is intuitive: Realistic segmentation networks have re-sponse thresholds; weak cooperative interactions be-tween segmentation networks responding to differentfeatures of the same image (e. g., chromatic and achro-matic form) allow the more active network to give theother network's units a boost above threshold, but donot entrain units not driven by sensory input, if theirthresholds are chosen appropriately (Horn et al., 1991). We argue that cross-modal construction depends on a stimulus-linked variant of the correlated noise that Horn used for synchronization. ORIGIN AND NATURE OF CROSSTALK-BASED CORRELATIONS BETWEEN ACHROMATIC AND CHROMATIC CORTICAL FORM MECHANISMS Retinogeniculate Origins of Correlations between Chromatic and Achromatic Mechanisms There is much evidence for chromatic/achromatic inter-actions in human vision (Mullen & Kingdom, 1991). Some interactions arise early in the visual pathway. About 80% of LGN cells have Type I receptive fields(center of different spectral sensitivity than its spatiallyopponent surround; Wiesel & Hubel, 1966); these P cellsrespond to both chromatic and achromatic stimuli (De Valois & Pease, 1971). P cells account for both the threshold chromatic and achromatic spatio-temporal contrast sensitivity functions (except for the achromaticlow spatial, high temporal frequency corner; Merigan &Maunsell, 1993; Kelly, 1983). Although the P cell's mixedsignal seems ambiguous, Ingling and Martinez's (1983,1985) algebraic identity factors the P cell's sensitivity into psychophysically meaningful terms. The expansion forr+g/C0cells (similar identities obtain for other cell types and for mixed cone surrounds; Billock, 1996) is: Prþg/C0¼RSe Te/C0GSi Ti ¼ð Rþ GÞfð Se/C0Si Þð Teþ Si Þþð Seþ Si Þð Te/C0Ti Þg=4 (Achromatic, Spatio-temporally bandpass) þð R/C0GÞfð Se/C0Si Þð Te/C0Ti Þþð Seþ Si Þð Teþ Ti Þg=4 (Chromatic, Spatio-temporally lowpass Þ Here, Rand Gare absorptions in L-and M-cones, Se,Si are center (excitatory) and surround (inhibitory) spatial weighting functions, and Te,Tiare center and surround temporal impulse response functions. Se,Si,Te,Tiare all lowpass functions of spatial/temporal frequency, so their sums are lowpass and differences are bandpass. A variation of Equation 2 models both the chromaticand achromatic contrast sensitivity functions (Burbeck& Kelly, 1980). For our current purposes, this formula-tion is unnecessarily complex; we develop some basicideas using a simplification and then return to thefull version when necessary. Temporarily neglectingtime gives Prþg/C0¼RCenter /C0GSurround ¼ð Rþ GÞð Se/C0Si Þ=2 (achromatic, bandpass tuning) þð R/C0GÞð Seþ Si Þ=2 (chromatic, lowpass tuning)ð3Þ For simplicity, we restrict space to one dimension ( x) and define the stimulus in terms of L and M coneabsorptions: Z(x)={ R(x),G(x)}. The response of the P cell to the stimulus is Prþg/C0ðx Þ/C10Zðx Þ¼0:5f Rðx Þþ Gðx Þg /C10 f Seðx Þ/C0Siðx Þg (Achromatic response) þ0:5f Rðx Þ/C0Gðx Þg /C10 f Seðx Þþ Siðx Þg (Chromatic response) where the convolution of two functions A(x)/C10B(x)=R A(t)B(x/C0t)dtand tis a dummy variable of in-tegration. Equations 3 and 4 reveal a subtle encoding of chromatic and achromatic information that is deci-pherable by cortical decoders. To see how, let theexcitatory center ( S e) and inhibitory surround ( Si) spatial weighting functions of this Type I cell be representedby Gaussians. The achromatic term is a difference of Gaussians that closely approximates a second spatialderivative of a Gaussian. The chromatic term is the sum of these Gaussians and is fit by a Gaussian with a space constant ( s) about 1. 83 times that of the achromatic term. If some simple assumptions (Billock, 1995) hold,(2) (4) 1038 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
then the spatial tuning of the P cell is the Fourier transform ( F[P]) of Equation 3. F1/2Prþg/C0/C138¼0:5ð Rþ GÞa2f2 s Hsðfs Þ(Achromatic tuning) þð R/C0GÞH1:83sðfs Þ(Chromatic tuning) ð5Þ where Hs(fs) is exp[ /C02(psfs)2] (the Fourier transform of the center Gaussian), a=2p, and fsis spatial frequency. The achromatic term is a spatial bandpassfilter; the chromatic term is lowpass, multiplexing thechromatic and achromatic information into different frequency ranges. If the multiplexing filters had no overlap, two well-chosen cortical filters could perfectlyseparate (demultiplex) the chromatic and achromaticinformation (Kingdom & Mullen, 1995; Billock, 1991,1995). For P cells, the encoding filters overlap, inducingcrosstalk in the cortical decoders, especially at moderatespatial frequencies where the overlap is greatest. Consider the extraction of achromatic information from P cells by the simplest spatial bandpass filter—spatial differentiation (Billock, 1995). Taking the nth local derivative of the P-cell array is equivalent to differ-entiating Equations 3 or 5, with respect to a spatialdimension; in neural terms it corresponds to buildingcortical dual-opponent cells (Thorell, De Valois, &Albrecht, 1984) by lateral inhibition between afferentgeniculate cells (for details, see Billock, 1995). In the spatial domain, such derivative operators look like simple cells; 1-D differentiation produces oriented cellsand the order of differentiation determines the numberof alternating excitatory and inhibitory lobes; Equation 6describes a cell that has n+ 3 receptive field lobes when probed with achromatic spots, but n+ 1 lobes for equiluminant hue probes. The spatial frequency re-sponse ( A(f s)) of such an achromatic mechanism to a stimulus (defined in terms of L-and M-cone absorp-tions) Z(fs)={ R(fs),G(fs)} is Aðfs Þ¼Zðfs Þð F1/2Dn Pðfs Þ/C138Þ ¼0. 5[R(fs)+G(fs)]anþ2fnþ2 s HS(fs) (achromatic response) þ1/2Rðfs Þ/C0Gðfs Þ/C138anfn s H1:83sðfs Þ (chromatic crosstalk)ð6Þ This bandpass filtering model correctly predicts the spatial tuning of cortical cells sensitive to both achro-matic and chromatic contrast (but presumably labeledonly for achromatic contrast; Billock, 1995). Increasingnshifts the tuning to higher spatial frequencies and narrower bandwidths. The typical cortical cell has anachromatic bandwidth of 1. 4 octaves (De Valois & De Valois, 1988) matching the achromatic term of Equation 6 if n= 2 (i. e., matched filtering for lumi-nance signals in P cells). Similarly, color signals can beextracted by matched lowpass filtering (second-order integration), resulting in achromatic crosstalk at all butthe lowest spatial frequencies. Cðfs Þ¼Zðfs Þð F1/2D/C02Pðfs Þ/C138Þ ¼0:51/2Rðfs Þþ Gðfs Þ/C138H1:83sðfs Þ (luminance crosstalk) þ[R(fs)--G(fs)]H3:35s(fs) (chromatic response) An engineer would find this odd—this kind of cross-talk is generally undesired and avoided in communica-tions systems. Yet, as discussed below, crosstalk is useful information for binding. Estimation of Stimulus-induced Correlations between Spatial and Chromatic Mechanisms The discussion of crosstalk above was deliberately over-simplified to illustrate general principles. To estimatecrosstalk-based correlations, we generalize Equations 6and 7, to include the temporal response. Selective two-dimensional ( x,t) spatio-temporal matched filtering of Equation 2 yields Aðfs;ft Þ¼[R(fs,ft)+G(fs,ft)][Se(fs)Te(ft)--Si(fs)Ti(ft)]2 f Achromatic response g þ1/2Rðfs;ft Þ/C0Gðfs;ft Þ/C1381/2Seðfs ÞTeðft Þ þ Siðfs ÞTiðft Þ/C1381/2Seðfs ÞTeðft Þ/C0Siðfs ÞTiðft Þ/C138 f Chromatic Crosstalk g Cðfx;ft Þ¼[R(fs,ft)--G(fs,ft)][Se(fs)Te(ft)+Si(fs)Ti(ft)]2 f Chromatic Response g þ1/2Rðfs;ft Þþ Gðfs;ft Þ/C1381/2Seðfs ÞTeðft Þ þ Siðfs ÞTiðft Þ/C1381/2Seðfs ÞTeðft Þ/C0Siðfs ÞTiðft Þ/C138 f Achromatic Crosstalk g Although not as elegant as Equations 6 and 7, Equa-tions 8 and 9 are computable if the Se,Si,Te,Tifunctions can be estimated (see Kelly, 1989; Burbeck & Kelly,1980; for methods and estimated functions). We could use Equations 8 and 9 as INPUT 1, INPUT 2in Equation 1 and ignore Q1,2; the correlated input is implicit. Howev-er, it is enlightening to use only the achromatic andchromatic response terms (the terms in bold type) as INPUT 1, INPUT 2and to use Equations 8 and 9 (with the crosstalk terms) to estimate the stimulus-driven correla-tions ( Q 1,2in Equation 1) between spatial mechanisms that extract information from P cells about achromatic form and other mechanisms that extract information about color or chromatic form. We define spectralcorrelation as the integrated overlap of the Fourierspectra of two functions (an analog of Signal Detection(7) (8) (9) Billock and Tsou 1039	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
Theory's cross-ambiguity function). The spectral corre-lation between A(fs,ft) and C(fs,ft)i s Q1;2¼Aðfs;ft Þ2Cðfs;ft Þ ¼Z Aða;b ÞCðfsþa;ftþb Þdadb ð10Þ where a,bare dummy variables of integration. For an equiluminant stimulus, the achromatic signal to theachromatic mechanism is zero and the correlatedcrosstalk consists of the correlation between the chromatic signal in the chromatic pathway [the bold term in the C(f s) equation] and the chromatic crosstalk extracted by the achromatic pathway [the plain typeterm in the A(f s) equation]. Thus, at equiluminance, the visual system attempts to bind something to nothing,based on the correlation between something andbandpass filtered noise. This situation corresponds tosome very odd perceptual effects. Effects of Equiluminance on Perception At equiluminance perception of form, depth and mo-tion are degraded (Cavanagh, 1991; Livingstone & Hubel, 1987; Gregory, 1977). Equiluminance can alsodisrupt segmentation and binding. Segmentation bybinocular disparity is severely degraded at equilumi-nance for random dot stereograms (Lu & Fender,1972). Surfaces are not linked together if their featuresare defined only by color; an image made of equilu-minant colors appears as patches of those colors, not as a unified whole (Cavanagh, 1991; Livingstone & Hubel, 1987). Moreover, perception of equiluminousforms can be unstable. Gregory (1977) found equilu-minant images ''looked uns table in contrast and 'jazzy'. '' This instability is sometimes attributed to aninability of ocular accommodation to use color infor-mation, but this is contradicted by Kotulak, Morse, and Billock (1995). Liebmann (1927) found that there is a critical luminance zone within which ''everything flows... glimmers... everything is soft, jelly-like, colloidal. Often... parts which belong together in the normal figure now have nothing to do with each other. (It is) aworld without firm things, without solidity'' (translatedin Cavanagh, 1991). Studies using luminance minimizedborders show these effects are particularly severe forborders defined by S-cone (tritan) modulation; the borders collapse and the color fields blend into a continuous color gradient (Buck, Frome, & Boynton,1977). No Evidence for Explanations Based on Lack of Parvo Inputs to Central Pathways Livingstone and Hubel (1987) ascribe the detrimental effects of equiluminance to a lack of parvo inputs tomotion, form, and depth channels, but evidence con-tradicts this (Merigan & M aunsell, 1993; Ingling & Grigsby, 1990; Schiller, Logothesis, & Charles, 1990). Moreover, there are chromatic mechanisms sensitive toform, motion, and depth (Regan, 2000; Cavanagh,1991; Mullen & Kingdom, 1991). Indeed, paradoxically,some chromatic phenomena are adversely affected byequiluminance, including color discrimination, color contingent aftereffects, and color rivalries (for review, see Mullen & Kingdom, 1991; Livingstone & Hubel,1987), suggesting a more subtle origin for the effectsof equiluminance. Several possibilities are consideredbelow. Evidence Contradicting Explanations Based on Low Chromatic Acuity Although the low acuity of chromatic mechanisms af-fects some visual phenomena (Cavanagh, 1991), it does not explain the poor and unstable contrast of some equiluminous borders. Border sharpness is not strictly ahigh spatial frequency phenomenon; a blurred edgelooks sharp if the missing harmonics of its Fourier seriesare below detection threshold (Campbell, Hopwell, &Johnstone, 1978). Similarly, reducing acuity by dimmingillumination has little effect on contrast over a largerange and edges modulated at 15 Hz appear sharp, even though acuity is reduced by a factor of 2. 5 (Livingstone & Hubel, 1987). Finally, as Mullen and Kingdom (1991)put it, ''it would be surprising if the lower borderdistinctness rated for S cone mechanisms compared to M-L cone ones was due to their differences in acuitysince a greater difference in acuity occurs betweenluminance and M-L chromatic mechanisms with no lossof border distinctness. '' Luminance as a Master Signal? Gregory (1977) posits that luminance is a master signal necessary for demarcating borders. This fits luminancecaptures color phenomena but does not explain whytritan equiluminous borders suffer excessively relative toequiluminous borders that stimulate the red/green sys-tem. Nor can the vulnerability of tritanopic borders bedue to a lack of S-cone driven color contrast mecha-nisms; double opponent b-y cells (Livingstone & Hubel, 1984), are well suited for transducing S-cone modulatedchromatic contrast and multiple S-cone driven spatialfrequency channels are found psychophysically (Human-ski & Wilson, 1993). A Binding-level Explanation? A possible explanation for the insalience of tritanopic borders stems from the origins of the r/C0g,r+g (luminance) and b-y signals. The major input of r/C0g 1040 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
signal to the cortex comes from spatially opponent P cells that multiplex luminance and color signals, induc-ing crosstalk between cortical luminance and r/C0gcolor mechanisms. Conversely, it appears that a majorsource of retinogeniculate y-b signal is from cells withspatially nonopponent Type II receptive fields (de Monasterio & Gouras, 1975; Wiesel & Hubel, 1966). These cells carry no multiplexed luminance signal and therefore do not induce correlations between cortical luminance and color detectors. The paucity of S-coneinputs to luminance (for review, see Cavanagh, 1991)removes another source of correlations between y-band luminance mechanisms. If correlations betweenmechanisms responding to borders are exploited forbinding, perhaps the y-b system, deprived of correla-tions to luminance borders, evolved only a weak input to the border contrast system. We suggest that binding allows achromatic mechanisms to reinforce r/C0gmech-anisms involved in segmentation and border formation(the luminance-driven form mechanism may need lessreinforcement from the chromatic system because—asdiscussed later—it receives reinforcement from motionmechanisms for unstabilized stimuli). CORRELATED CORTICAL SPATIAL/TEMPORAL MECHANISMS Why Form and Motion (or Space and Time) Are Not Independent There is evidence for separate mechanisms mediating detection of spatial and temporal variation (for review,see Zeki, 1993). There is also evidence for interactions between spatially and temporally tuned mechanisms (Burt, 1987; Breitmeyer & Ganz, 1976). Such interac-tions—as unavoidable as chromatic/achromatic interac-tions—are due to the physical entanglement of spatialand temporal information, and to spatio-temporal mul-tiplexing in retinogeniculate neurons. The physicalentanglement of motion and form is obvious; thevelocity (deg/sec) of a moving grating is V=f t (cycles/sec)/ fs(cycles/deg), where fs,ftare spatial and temporal frequency (each spatial and temporal fre-quency in the stimulus's Fourier representation canbe so treated). The neural entanglement of spatialand temporal information is more complicated, butcan be analyzed analogously to chromatic/achromaticinteractions. Consider the psychophysical achromaticspatio-temporal contrast sensitivity function. In princi-ple, if the function is separable (decomposable into the product of spatial and temporal functions), then inde-pendent spatial and temporal information can be ex-tracted by a homomorphic filter (a log transformfollowed by a matched filter). Unfortunately, contrastsensitivity is inseparable. However, Burbeck and Kelly(1980) showed that an Ingling-Martinez identity couldachieve a limited separati on, decomposing contrast sensitivity into the sum of two separable surfaces. Fora stimulus Z(f x,ft) the frequency response (assuming equal integrated excitatory/inhibitory sensitivity) is Xðfx;ft Þ¼0:5Zðfx;ft Þ1/2Seðfx Þ/C0Siðfx Þ/C1381/2Teðft Þþ Tiðft Þ/C138 Sustained (spatially tuned) response þ0:5Zðfx;ft Þ1/2Seðfx Þþ Siðfx Þ/C1381/2Teðft Þ/C0Tiðft Þ/C138 Transient (temporally tuned) response ð11Þ where Se,Te,Si,Tiare the excitatory and inhibitory spatial and temporal response functions defined in Equation 2. In Equation 11, the terms are labeledsustained and transient. The sustained term hasbandpass tuning for spatial frequencies and lowpasstuning for temporal frequencies. The transient term has lowpass tuning for spatial frequency and bandpass tuning for temporal frequency. In the psychophysicalliterature, the terms sustained and transient usually referto separate retinogeniculate spatial and temporalprocessing pathways. Yet, Burbeck and Kelly's analysisshows that both kinds of responses are embedded in allretinogeniculate cells with center/surround receptivefields, and are well modeled by Equation 11 for cells that obey superposition (P cells and X-like M cells). More-over, although the sustained and transient signals can beseparated algebraically, there is no plausible spatio-temporal filter that enables a strict spatial and temporalseparation by physiological means; the spectral contentof the terms overlap and attempts to extract one signalwill extract a small crosstalk signal as well. Quantifying Stimulus-induced Spatial and Temporal Mechanism Correlations Here, we estimate the stimulus-driven correlation be-tween temporally and spatially tuned pathways, createdby cortical filtering of the spatio-temporal signals carriedby LGN afferents. To extract the maximum signal fromthe spatially tuned (sustained) component of an array of active X-cells, we apply a filter matched to the sustained component, yielding a frequency response SPðfx;ft Þ¼0. 5Z( fx,ft)[Se(fx)--Si(fx)]2[Te(ft)+Ti(ft)]2 (Spatially tuned term) þ0:5Zðfx;ft Þ1/2Seðfx Þ2/C0Siðfx Þ2/C1381/2Teðft Þ2/C0Tiðft Þ2/C138 (Common information term) Similarly, a matched filter can be used to attempt to extract the temporally tuned (transient) component ofthe X-cell signals, yielding a frequency response TEðfx;ft Þ¼0:5Zðfx;ft Þ1/2Seðfx Þ2/C0Siðfx Þ2/C1381/2Teðft Þ2/C0Tiðft Þ2/C138 (common information term) þ0. 5Z(fx,ft)[Se(fx)+Si(fx)]2[Te(ft)--Ti(ft)]2 (temporally tuned term)(12) (13) Billock and Tsou 1041	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
Note that the matched filters are only partially suc-cessful in extracting signals tuned along one stimulusdimension. In both cases, the undesired portions of thesignal—tuned along both space and time—are identical,and so spectral correlation between the mechanisms isdominated by the crosstalk. As before, we use the boldterms of Equations 12 and 13 as input signals INPUT 1, INPUT 2in Equation 1 and estimate correlated crosstalk (Q1,2in Equation 1) by computing the spectral correla-tion between Equations 12 and 13 (neglecting unknownsources of correlated noise; e. g., Horn et al., 1991) Q 1;2¼SPðfs;ft Þ2TEðfs;ft Þ ¼ZZ SPða;b ÞTEðfsþa;ftþb Þdadb ð14Þ where aand bare dummy variables of integration. These spatio-temporal interactions present two possi-bilities for crosstalk-mediated segmentation: (1) Mutualfeedback between correlated form and motion mecha-nisms both responding to moving edges (the cross-modal construction model discussed above for form/color interactions); (2) Motion-correlated segmentationwithin the form pathway; that is, stimulus motion givesrise to two responses in the form pathway: the sustained form signal (the first term in Equation 12) and the trans-ient crosstalk signal (the second term in Equation 12). Because these crosstalk signals arise only in formdetectors stimulated by the target's motion, they are apotential cue for binding by correlated motion. A Binding Failure—Perception of Stabilized Images During normal vision images are in constant motion; microtremors of the eye make volitional stabilizationalmost impossible. However, images can be stabilized mechanically or by producing afterimages on the retina. These stabilized images can fade away rather quickly, aneffect often attributed to the temporal response prop-erties of retinogeniculate neurons. We argue that thisexplanation is grossly inadequate and that a failure of cor-tical segmentation mechanisms is indicated. However,the cortex can only operate on what afferent mecha-nisms send it, so next we analyze what early and cortical mechanisms contribute to stabilized image perception. What Early Mechanisms Do to Perception of Stabilized Images Some stabilized image percepts are consistent with retinogeniculate cell properties. Low contrast, low spa-tial frequency stimuli (favoring transduction by transientmagno [M] cells) usually fade faster than high contrast,high spatial frequency stimuli (favoring transduction bysustained parvo [P] cells; Ingling & Grigsby, 1990). Moreover, nonbleaching chromatic afterimages elevateboth achromatic and chromatic thresholds, suggesting a common P cell-like pathway (Kelly & Martinez-Uriegas,1993). There are some sustained M cells that respond tostatic images for a few seconds, but their temporalresponse becomes transient for high contrast stimuli(Benardete, Kaplan, & Knight, 1992), leaving only P cellsto mediate perception of afterimages. Moreover, King-Smith, Rosten, and Alvarez (1980) describe a subject who (on psychophysical grounds) appears to be missing the parvo system; this subject was also unable to per-ceive afterimages. Hence, transduction by P cells is anecessary condition for perception of most stabilizedimages. However, retinogeniculate properties cannotaccount for at least seven lines of evidence that centralmechanisms are responsible for some binding-failure-like oddities of stabilized images. Three Properties of Stabilized Image Perception Paradoxical to P-and M-Cell Properties (1) Anomalously Rapid Fading of Images that Should Tap High Acuity Sustained Channels Recall that Equation 2 shows that P cells have a sustained response for high spatial frequencies. Near the fovea,these cells have midget receptive fields capable oftransducing spatial frequencies up to 60 c/deg (conesampling limit). Additionally, these P cells (unlike M cells) are sensitive to high contrasts (Benardete et al., 1992). So, it is odd that the stabilized pattern of highcontrast shadows cast by blood vessels on the retinafades faster than other stabilized images (Coppola &Purves, 1996). Moreover, the closer the blood vessels areto the fovea, the faster they fade (as fast as 80 msec forthe highest spatial frequency components), in contra-diction to the response characteristics and retinal distri-bution of P cells. (2) Anomalously Rapid Fading for Stimuli that Tap Chromatic Sustained Mechanisms P cells have a chromatic response that is lowpass in both space and time (Equation 2 shows a bandpass chromaticcomponent as well, but this merely adds to the responseat moderate spatial and temporal frequencies without driving down the response to low frequencies). Al-though Equation 2 must hold if P cells obey linearsuperposition, it does not model psychophysical chro-matic contrast sensitivity for very low temporal frequen-cies (<0. 2 Hz), where the temporal CSF slope isconsistent with a first-order temporal differentiation(Kelly, 1981); this extra temporal derivative is probablya cortical process acting on P-cell inputs. A perceptual manifestation of this is the dramatic elevation of stabi-lized chromatic grating detection thresholds (a factor ofat least 45 greater than for chromatic unstabilized gra-tings and much higher than the elevation for achromatic 1042 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
gratings; Kelly, 1983). To reemphasize, unlike achromat-ic information, which is also transmitted by magno units,color is carried by P cells with sustained temporalproperties. If retinogeniculate mechanisms are solelyresponsible for fading of stabilized images, one wouldexpect color to be affected less by stabilization relative tostabilized luminance signals, not more. (3) Fading of Dynamic Images is Paradoxical to a Peripheral Mechanism Explanation Empty figures defined by a twinkling random dot back-ground fade if fixated steadily (Spillman & Kurtenbach,1992). Dynamic image fading cannot be due to transientperipheral mechanisms; fading is actually faster fordynamic stimuli than for static noise. This suggests a central mechanism that requires coherent modulation to extract kinetic edges. Four Lines of Evidence that Central Mechanisms Mediate Stabilized Image Phenomena (1) Effects of Binocular and Other Sensory Interactions(a) Stabilized monocular image fading can be reversed by similarly patterned stimulation of the other eye(Cohen, 1961), especially if placed on correspondingpoints of the stabilized and unstabilized retinae (effec-tiveness drops off monotonically with retinal disparity). In dichoptic presentation, modulating the background eye slows fading of the image in the stabilized eye(Gerling & Spillmann, 1987). (b) Conversely, fading ofa stabilized image makes unstabilized stimuli in theother eye less visible (Krauskopf & Riggs, 1959). (c)Even after a stabilized image has disappeared in one eye,it can be combined with an unstabilized duplicatemoving image in the other eye, to yield a sensation of motion in depth (Crane, 1994). (d) Some stabilized images induce an abrupt absolute blindness (Billock,Gleason, & Tsou, 2001; Ditchburn, 1973). Both blind-ness and recovery are binocularly simultaneous, imply-ing a central mechanism. (e) Visibility of stabilizedimages can be maintained by listening to auditorystimuli; the effect decays to baseline in about 15 min ifthe stimulation is not varied, suggesting a role for attention (Ditchburn, 1973). (2) Fragmentation Davies (1973), Evans (1965), and Pritchard, Heron, and Hebb (1960) find complex stabilized images dis/reap-pear, not as wholes but in fragmented forms (forillustrations of fragmentation percepts, see Billock & Tsou, 2004). This image fragmentation is not explained by the properties of peripheral neurons (e. g., fragmen-tation can eliminate contours that are present only in abinocular image, but not in the separate retinal images;Evans & Wells, 1967). Moreover, image fragments that wax and wane together obey Gestalt-like rules (Evans,1965, 1967): (a) Short lines appear and disappear as aunit. Fragmentation probability increases with linelength; for foveal vision fragmentation is likely for linelengths of roughly 45 arc min (an order of magnitudelarger than foveal retinogeniculate receptive fields),suggesting a range for the cortical mechanisms in-volved. (b) Disappearance and reappearance of parallel lines are correlated. (c) Random patterns are moreunstable than meaningful ones. (d) Angular patternsare more fragmented and perceptually unstable thanrounded patterns. (e) Fragmentation and fading in oneregion of a field is strongly affected by activity inneighboring regions. (f) Complex patterns are morelikely to fragment than simple ones. Simulations show that Gestalt-like grouping rules are emergent properties of cooperative segmentation and binding networks(Horn & Opher, 2000; Wang & Terman, 1997; Sporns,Tononi, & Edelman, 1991). Interestingly, some modelsof synchronized segmentation tend to fragment imagesif the spatial properties of the model do not pool over alarge enough set of units to discount noisy stimulusinputs (Terman & Wang, 1995). Similar fragmentation is reported by some subjects with migraine (mosaic vi-sion; Sacks, 1995) and amblyopia (especially for highspatial frequency stimuli; Hess, Field, & Watt, 1990); alink between amblyopia and neural synchronization issuggested by Roelfsema, Ko ¨nig, Engel, Sireteanu, and Singer (1994) finding that cortical cells are normal inspatial frequency selectivity, but cells driven by theamblyopic eye do not synchronize well, especially for high spatial frequency stimuli. (3) Filling-in of Stabilized Images If a broken figure is stabilized and fades—then regener-ates—the reappearance of the figure is often marked bycompletion of the break (Ditchburn, 1973). Moreover,Cardu, Gilbert, and Stabel (1971) find that objects made up of dashed lines often (45% of the time) exhibit completion just before fading. This effect resemblesthe completion of images broken by scotomas and iscompatible with binding algorithms that incorporatecooperative interactions. (4) Stabilization Effects on EEG Correlated to Perceptual Phenomena Although no stabilization experiment has looked for synchronization of g-band activity, several studies find that the relative level of a-rhythm activity increases when stabilized images disappear or when perceptualblanks occur during Ganzfeld viewing (Keesey & Nich-ols, 1967, 1969; Lehmann, Beeler, & Fender, 1967,Lehmann, Beeler, & Fender, 1965; Evans & Smith, 1964). Following stabilization, power in the EEG a-band Billock and Tsou 1043	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
(9-10 Hz) rises about 0. 7-1. 0 sec before target disap-pearance and higher frequencies are suppressed untilabout 1 sec prior to reappearance of target structures(Keesey & Nichols, 1967; Lehmann et al., 1965). Overview of Stabilized Image Studies: Only a Central Explanation Will Do The textbook account of stabilized image perception is that fading is due to the transient response propertiesof some retinogeniculate units. That analysis cannotaccount for counterintuitively rapid fading of stimulithat favor highly sustained parvocellular mechanisms. Nor can it account for any of the explicitly corticaleffects discussed above. Of these, image fading andfragmentation are consistent with known failure modes of segmentation networks that require a balance of competitive and cooperative interactions; fading canstem from relative weakness of competitive interac-tions; fragmentation can stem from weakness of coop-erative interactions. The Gestalt properties shown instabilized image studies correspond to emergent prop-erties of cooperative/competitive segmentation net-works (Horn & Opher, 2000; Wang & Terman, 1997; Sporns et al., 1991). As early as 1960, Pritchard et al. argued that fragmentation and partial reappearance offaded stabilized images is evidence for the dynamicformation of Hebbian cell assemblies—essentially thesame argument advanced in modern binding theory. Here, we argue that binding of units responding tononstabilized images is facilitated by the common in-formation in each form and motion mechanism induced by eye movements perturbing the retinal image, allow-ing motion and form-based segmentation networks toreinforce each other. DISCUSSION Summary of Reasoning At this point, it is useful to summarize our reasoning and the evidence that supports it so that we can lay out its strengths and weaknesses, and point out where futurework needs to be done: 1. Feng (2000) and Horn et al. (1991) showed that a preexisting correlation between mechanisms facilitates synchronizing those mechanisms' activity, a form offeature binding. In Horn et al. 's and Feng's work, theseinitial correlations stem from unspecified noise. What weadd here is an explicit physiological source for suchsignals (crosstalk induced by cortical demultiplexing of LGN signals), which results in stimulus-driven correla-tions between cortical mechanisms for achromatic form and color, and between achromatic form and motion. 2. Several investigators point out that feature-binding between mechanisms responding to common parts ofan image can improve part-binding/segmentation (Schil-len & Ko ¨nig, 1994; Finkel & Edelman, 1989; Poggio et al., 1988), a process called crossmodal reinforcement. 3. Other investigators point out that some part-binding (segmentation) models can fail in interestingways, for example, fragmentation of images (Terman &Wang, 1995; Marr et al., 1978). What we add to this is adiscussion of various ways a segmentation networkcould fail (depending on whether cooperative or com-petitive interactions were inadequate). 4. We point out that some odd percepts (e. g., fragmentation of stabilized images and melting ofequiluminant images) closely resemble these expectedfailure modes of segmentation models. We suggestthat these part-binding failures may be due to loss ofcrossmodal reinforcement (Point 2) that under normalcircumstances is available and relatively easy to bind because of stimulus-driven correlations (Point 1). Modeling—What Remains To Be Done Although each point in the above analysis is supported by experiments or mathematical models, it would bedesirable to supplement this analysis with a formalintegrated simulation and with a set of experimentalpredictions to be tested. We provide some such predic-tions in the sections below. A formal integrated simula-tion lies beyond the scope of this article, but we can lay out what would be required to implement one. For simplicity, we discuss form and color binding here—the steps required for form and motion binding arequite similar. (a) Start with a segmentation model, like Terman and Wang's (1995) that fails in one of theinteresting ways described in Point 4, and manipulatethe balance of competition and cooperation in themodel until it so fails. Two copies of this segmentation network are created, although they need not be sym-metrical (only failure in the color segmentation system isbeing studied in this example). (b) Create two sets ofinput mechanisms to drive the networks in (a)—an arrayof color units obeying Equation 9 and an array ofachromatic form units obeying Equation 8. To parame-terize these equations, the S e,Te,Si, and Tifunctions describing the spatial and temporal excitatory and in-hibitory responses of P cells need to be specified using psychophysical or physiological data. Suitable functions(derived psychophysically) can be found in Kelly (1989)and Burbeck and Kelly (1980). Under ordinary (non-equiluminant) conditions, there should be a correlationbetween the color inputs described by Equation 9 andthe achromatic units described by Equation 8. Thiscorrelation is implicit in the crosstalk terms of these equations, but can be made explicit using Equation 10. Now run the segmentation networks so that they syn-chronize for normal stimuli (stimuli that contain bothhue and luminance information). (c) Set the stimuli tobe equiluminant. The response of the achromatic form-1044 Journal of Cognitive Neuroscience Volume 16, Number 6	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
driven network should become disorganized (because it is being driven by noise), and the chromatic networkshould desynchronize from it. Segmentation mistakes inthe unreinforced chromatic network should now occur. Combining Equiluminance and Stabilization— Effects and Predictions Many of the binding failures discussed above seem connected. Equiluminant border melting has obvioussimilarities to border fading in image stabilization. Sim-ilarly, the ''jazziness'' of equiluminant borders may be afast time-scale analog to the fragmentation and dis/reappearance of some stabilized images. The comple-tion of gaps in reappearances of stabilized imagesresembles filling-in phenomena seen for some equilumi-nous images. In general, nulling the signal to a reinforc-ing channel should be detrimental to the channel itreinforced: The reinforcement is lost and lost informa-tion is replaced with mere noise. Significantly, for sub-jects who have one of the color, motion, and achromaticform systems disrupted, interrupting another systemresults in a more dramatic effect. For example, Sacksand Wasserman (1987) find that for an achromatopsic observer, boundaries between areas dissolved during periods of steady fixation (see below for analogousfindings in color normals). Similarly, stimuli that stimu-late only the b-y pathway (tritanopic stimuli) are partic-ularly susceptible to dissolution of contours (Cavanagh,1991). If the visual system requires two cooperatingsystems to achieve segmentation, then we would predictthat combining equiluminance and stabilization should be especially devastating. In fact, steadily fixated colored stimuli do fade more readily when equiluminant to theirsurroundings (Livingstone & Hubel, 1987; Buck et al.,1977; Krauskopf, 1967). Still more dramatic effects areobtained by combining retinal stabilization and equilu-minance. Billock et al. (2001) and Crane and Piantanida(1983) report that if a stabilized red/green or blue/yellowbipartite field is viewed through an unstabilized aper-ture, the colors flow and mix across the faded border. Some observers perceive uniform reddish greens oryellowish blues, in violation of color opponency. Otherobservers report formation of fine color textures orunstable islands of one color continuously forming anddissolving in a sea of the other color. Billock et al. foundthis spurious segmentation occurs only when there is aluminance difference between the two stabilized colored fields. Moreover, while melting together of adjacent equiluminant fields is normally seen only for tritanopicpairs, Billock et al. found that the phenomenon occursfor any stabilized color pair. (Related and extreme casesof binding failures are found in some neurologicaldisorders; Friedman-Hill, Robertson, & Treisman, 1995;Sacks, 1995; Zeki, 1993; Critchley, 1965). Given our reasoning and these preliminary results, a test of our reinforcement hypothesis would be tomeasure both perceptual segmentation and electro-physiological measures of binding while equiluminanceand stabilization are manipulated. The psychophysics isstraightforward. For experiments on equiluminance, animage is made of at least two colors, and radianceratios of the colors are varied through the range inwhich the equiluminance ratio must lie. Whether lossof neural binding signatures occurred at equiluminance can be determined by measuring figure and back-ground luminosities using standard psychophysicaltechniques (Cavanagh, 1991; Wyszecki & Stiles, 1982). Specific information should be gathered with respect toperceptual state (e. g., border instability, border melting,surface delinking). Similarly, image stabilization isstraightforward, albeit technically challenging and ex-pensive (Crane, 1994); useful results can often be had with afterimages or steady fixation (Ditchburn, 1973). Electrophysiological states should be correlated withthe specific perceptual state elicited (e. g., fragmenta-tion, fading, etc. ). If a stabilization system is available,t h ed e g r e eo fs t a b i l i z a t i o nc a nb em a n i p u l a t e db ymoving the image independently of eye movementsor by manipulating the gain of the eye-movement $ image-movement feedback loop (Crane, 1994). Finding what neural signatures to use (and how to test for them noninvasively in responding humans) is more challeng-ing. Since g-band activity is implicated in binding, the loss of higher-frequency EEG power during perceptuallosses in image stabilization is suggestive (Keesey &Nichols, 1967; Lehmann et al., 1965). There are twolimitations with this approach: Some sources of g-band power may be unrelated to binding and some binding failures may represent incorrect bindings and therefore not be identifiable by EEG. Recent experiments offerhelp on both counts, by studying gestalt perception(pop-out) of fragmented images hidden in camouflage. This fragmentation-to-order shift is complementary tothe order-to-fragmentation effects we wish to study. Subjects had g-band activity whether they saw a coher-ent image pop-out or not, but for coherent percepts the outputs of widely spaced electrodes became more correlated (as would be expected for synchronization;Gruber et al., 2002) and a different kind of g-band activity ''the induced gamma response'' was recorded. On every physically identical trial, there was a g-band response phase-locked to the stimulus, but only ontrials where the gestalt percept was obtained was thereanother EEG signature, consisting of g-band bursts in variable phase to the stimulus (Gruber et al., 2002; Tallon-Baudry & Bertrand, 1999; Tallon-Baudry, Ber-trand, Delpuech, & Pernier, 1996). We therefore predictthat EEG signatures like ''induced gamma'' will bereduced during binding failures induced by retinalstabilization (especially fragmentation) and equilumi-nance (e. g., loss of Gestalt symmetry in Glass patterns;Cavanagh, 1991). Moreover, we posit that these signa-tures should be further reduced by combining equilumi-Billock and Tsou 1045	A Role for Cortical Crosstalk in the Binding Problem_ -- Billock Vincent A_ Tsou Brian H_ -- Journal of Cognitive Neuroscience 6 16 pages -- 10_1162_0898929041502742 -- ab6ea96496b8ff8a46e6cd28b8e6456
nance with stabilization, and should be monitored when color boundaries collapse and forbidden colorsare perceived (Billock et al., 2001). An easier variation ofthis experiment could be done with steady fixation on aminimized border (Buck et al., 1977); under these con-ditions, some color boundaries collapse, creating uniformcolor mixtures. Acknowledgments We thank Randolph Blake, Angela Brown, Patrick Cavanagh, Viktor Jirsa, Daniel Levine, Lynn Olzak, Wolf Singer, De Liang Wang, and Scott Watamaniuk for critical readings of the manuscript. Special thanks to J. A. Scott Kelso for suggesting this problem. Reprint requests should be sent to V. A. Billock, General Dynamics, Suite 200, 5200 Springfield Pike, Dayton, OH 45431,USA, or via e-mail: Vince. Billock@wpafb. af. mil. REFERENCES Benardete, E. A., Kaplan, E., & Knight, B. W. (1992). Contrast gain control in the primate retina: P cells are not X-like, some M-cells are. Visual Neuroscience, 8, 483-486. Billock, V. A. (1996). 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RESEARCH Pattern forming mechanisms of color vision Zily Burstein1, David D. Reid1, Peter J. Thomas2, and Jack D. Cowan3 1Department of Physics, University of Chicago, Chicago, IL, USA 2Department of Mathematics, Applied Mathematics, and Statistics; Department of Biology; Department of Cognitive Science, Case Western Reserve University, Cleveland, OH, USA 3Department of Mathematics, University of Chicago, Chicago, IL, USA Keywords: Color vision, V1, Pattern formation, Turing mechanism ABSTRACT While our understanding of the way single neurons process chromatic stimuli in the early visual pathway has advanced significantly in recent years, we do not yet know how these cellsinteract to form stable representations of hue. Drawing on physiological studies, we offer adynamical model of how the primary visual cortex tunes for color, hinged on intracortical interactions and emergent network effects. After detailing the evolution of network activity through analytical and numerical approaches, we discuss the effects of the model 's cortical parameters on the selectivity of the tuning curves. In particular, we explore the role of the model 's thresholding nonlinearity in enhancing hue selectivity by expanding the region of stability, allowing for the precise encoding of chromatic stimuli in early vision. Finally, in theabsence of a stimulus, the model is capable of explaining hallucinatory color perception via a Turing-like mechanism of biological pattern formation. AUTHOR SUMMARY We present a model of color processing in which intracortical neuronal dynamics within thevisual cortex serve as the substrate for hue perception. Our analytical and numerical treatments of the emergent behavior seek to characterize the population dynamics underlying chromaticprocessing within the visual cortex, as well the roles of the various cortical parameters in determining the selectivity of the steady-state network response. We show that the system is self-organizing, capable of encoding stable representations of hue regardless of the stimulusstrength, and generating spontaneous color hallucinations in the absence of any input. INTRODUCTION Our experience of color begins in the early visual pathway, where, from the moment lightstrikes the retina, cone-specific neuronal responses set off the mechanisms by which the pho-tons'chromatic information is converted to the hues we ultimately see. While color vision scientists agree that the single-cell processing of chromatic stimuli occurs along the two inde-pendent cone-opponent L-M and S-(L+M) pathways ( Conway, Eskew, Martin, & Stockman, 2018 ;Kaiser & Boynton, 1996 ), there is yet no consensus as to how and where the divergent signals are synthesized to encode hue. To complicate matters, cone-opponency, observed in electrophysiological recordings of single neurons ( Shapley & Hawken, 2011 ), is often con-founded with hue-opponency, a phenomenon first theorized by Ewald Hering in the 19than open access journal Citation: Burstein, Z., Reid, D. D., Thomas, P. J., & Cowan, J. D. (2023). Pattern forming mechanisms of color vision. Network Neuroscience,7(2), 679-711. https://doi. org/10. 1162/netn_a _00294 DOI: https://doi. org/10. 1162/netn_a_00294 Received: 19 August 2022 Accepted: 17 November 2022 Competing Interests: The authors have declared that no competing interests exist. Corresponding Author: Zily Burstein ceburst@gmail. com Handling Editor: Gustavo Deco Copyright: © 2023 Massachusetts Institute of Technology Published under a Creative Commons Attribution 4. 0 International (CC BY 4. 0) license The MIT Press Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
century and later mapped out in clinical studies by Jameson and Hurvich ( De Valois, Cottaris, Elfar, Mahon, & Wilson, 2000 ;Jameson & Hurvich, 1955 ;Shevell & Martin, 2017 ). Best depicted in the Derrington-Krauskopf-Lennie (DKL) stimulus space (Figure 1 ), cone-opponency predicts that neurons tuned to either the L-Mo r S-(L+M) pathway will not respond to light whose wavelengths isolate the other ( Derrington, Krauskopf, & Lennie, 1984 ). It is tempting to equate these null responses to the four unique hues of color-opponent theory, in which unique blue, for example, is observed when the “redness ”and“greenness ”of a per-ceived color exactly cancel. But the wavelengths of the unique hues specified by perceptualstudies ( Jameson & Hurvich, 1955 ) only roughly match the wavelengths isolating either cone-opponent pathway ( Wool et al., 2015 ;Wuerger, Atkinson, & Cropper, 2005 ;Xiao, 2014 ), and, more fundamentally, we do not yet understand the mechanisms behind the processing that theanalogy implies ( Mollon & Jordan, 1997 ;Stoughton & Conway, 2008 ;Valberg, 2001 ). That is, how do we get from the single neurons 'chromatic responses to our perception of color? The necessary processing has often been attributed to higher-level brain function ( De Valois & De Valois, 1993 ;Lennie, Krauskopf, & Sclar, 1990 ;M. Li, Liu, Juusola, & Tang, 2014 ; Mehrani, Mouraviev, & Tsotsos, 2020 ;Zaidi & Conway, 2019 ) or yet unidentified higher order mechanisms ( Valberg, 2001 ;Wuerger et al., 2005 ). A central question of color vision research is whether these mechanisms rely on paralle l or modular processing to encode stimulus chromaticity ( Conway, 2009 ;Garg, Li, Rashid, & Callaway, 2019 ;Liu et al., 2020 ;Nauhaus, Nielsen, Disney, & Callaway, 2012 ;Schluppeck & Engel, 2002 ;Shapley & Hawken, 2011 ). If signaling about chromaticity is transmitted w ith information about other visual features, such as brightness, orientation, and spatial frequency, how do these features get teased apart? If not, where is the purported color center of the brain ( Conway, Moeller, & Tsao, 2007 ; Gegenfurtner, 2003 )? Several authors have addressed these questions through combinatorial models that param-eterize the weights of the L, M, and S cones contributing to successive stages of processing(De Valois & De Valois, 1993 ;Gegenfurtner & Ennis, 2015 ;Judd, 1949 ;Mehrani et al., 2020 ; Stockman & Brainard, 2010 ). Though differing in their assumptions of modularity, the theories share a mechanistic framework for the transitio n of single-cell receptive field properties Figure 1. The DKL space maps chromatic stimuli onto a circle with two “cardinal ”axes represent-ing the L-Ma n d S-(L+M) pathways. The excitatory or inhi bitory effect of a stimulus on cone-opponent cells tuned to either pathway can be thought of as a projection of its location in DKLspace onto the relevant axis. Stimuli isolating the two pathways correspond roughly to wavelengthsassociated with the red, green, blue, and yellow unique hues of color-opponent theory, leading tothe common, but mistaken, conflation of chromatic stimulus and color perception. Stimulus space: A geometrical construct in whichchromatic stimuli are represented bythe relative cone responses they yield. Color-opponent theory: Theory of color appearance thatpostulates that the four unique hues — red, green, blue, and yellow —are perceived antagonistically. That is,there is no such thing as a reddishgreen or a bluish yellow. Cone-opponency: Receptive field property of neuronsin the early visual pathway, by whichchromatic stimuli are processedthrough the comparison of therelative L, M, and S cone responses. Network Neuroscience 680Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
(Brown, 2014 ). Starting with cells in the retina and lateral geniculate nucleus (LGN) known to be tuned broadly to the cone-opponent axes, these proposed mechanisms build up to cells invarious cortical areas more narrowly tuned to divergent (and debated) chromatic directions in DKL space. While parsimonious, this formalism comes at the cost of tuning the cone weights arbitrarily, disregarding specific properties of real neurons 'receptive fields ( Eskew, 2009 ; Kaiser & Boynton, 1996 ;Stockman & Brainard, 2010 ). Furthermore, the linear combinatorial mechanism is not, on its own, able to account for the variety of color cells observed in the visual cortex ( Garg et al., 2019 ;Johnson, Hawken, & Shapley, 2001 ;Shapley & Hawken, 2011 ). In addition to the forward flow of chromatic information through the successive stages ofprocessing, the encoding of color reflects t he neuronal dynamics within each. Modelers agree that the next forays into a mechanistic theory of color vision should consider these intracortical circuits, but di sagree about where such interactions first become important (De Valois & De Valois, 1993 ;Hanazawa, Komatsu, & Murakami, 2000 ;Liu et al., 2020 ; Wachtler, Sejnowski, & Albright, 2003 ). Electrophysiological studies of macaque visual cortex have shed some light on this ques-tion, showing that the processing of individual hues previously associated with higher level mechanisms has its origins in the primary visual cortex (V1) ( Garg et al., 2019 ;Gegenfurtner, 2003 ;Hanazawa et al., 2000 ;Li et al., 2022 ;Wachtler et al., 2003 ;Xiao, 2014 ;Xiao, Casti, Xiao, & Kaplan, 2007 ). These experiments have identified the emergence of neurons in V1 tuned to the gamut of hues in DKL space, as well as to the role of processing nonlinearitiesin determining their tuning curves ( De Valois et al., 2000 ;Hanazawa et al., 2000 ;Lennie et al., 1990 ;Wachtler et al., 2003 ). Puzzlingly, these cells mainly inhabit the so-called CO “blobs, ” patchy regions rich in cytochrome oxidase that display a sensitivity to stimuli modulating either of the cone-opponent axes rather than the full set of hues ( Landisman & Ts 'o, 2002b ; Li et al., 2022 ;Livingstone & Hubel, 1984 ;Salzmann, Bartels, Logothetis, & Schüz, 2012 ). Some have speculated that this colocaliza tion stems from a mixing of cell populations encoding the two cardinal pathways (Li et al., 2022 ;Xiao, 2014 ) while others indicate a distinct population of hue-sensitive neurons in the “interblob ”regions, more conclusively associated with orientation tuning ( Garg et al., 2019 ;Landisman & Ts 'o, 2002a ). As a whole, however, these studies point to the need for a p opulation theory of chromatic processing remarkably early in the visual pathway. In this article, we present a model of color pr ocessing in which intracortical neuronal dynamics within V1 serve as the substrate for hue perception. Drawing on the canonical Wilson-Cowan neural field equations and the ring model of orientation tuning, we show that this population approach allows us to account for cells responsive to the full rangeof DKL directions without the need to fine-tune input parameters ( Ben-Yishai, Bar-Or, & Sompolinsky, 1995 ;Burstein, 2022 ;Hansel & Sompolinksy, 1998 ;Wilson & Cowan, 1972,1973 ). The threshholding we employ bears in mind the input-response nonlinearities of previous combinatorial models, but zooms out of the single-cell, feedforward interpreta-tion of input as the stimulus-driven LGN afferents to individual neurons. Rather, we model input as the total synaptic current into a population of cells, taking into account both thecone-opponent LGN afferents as well as the hue-dependent connectivity between distinct neuronal populations. The resulting demarcation between the cone-opponent and the hue-selective mechanisms in the same population of cells points to the importance of V1 in the transition from chromatic stimulus to color perception. To characterize this role, we study the effects of the model 's con-nectivity parameters and processing nonlinearities on the narrowness and stability of the hue tuning curves. In the final part of the paper, we show that the model is able to explain color Cardinal pathways: In DKL space, the two orthogonalaxes representing stimuli isolating the L-M and S-(L+M) cone-opponent pathways. Wilson-Cowan neural field equations: Coupled set of partial differentialequations describing the networkdynamics of excitatory and inhibitoryneural populations. Network Neuroscience 681Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
responses in the absence of LGN input, evoking color hallucinations via a Turing-like mech-anism of spontaneous pattern formation in DKL space. MODEL In light of the patchy distribution of color-sensitive cells reported in Landisman and Ts 'o (2002b), Li et al. (2022),Livingstone and Hubel (1984), and Salzmann et al. (2012), we model the color map of V1 as a set of neuronal networks, each encoding the chromaticity of its corresponding region of the visual field. This organization brings to mind the hypercolumnar structure of ori-entation preference within V1 ( Hubel & Wiesel, 1974 ), which, on the basis of its feature-based connectivity properties, allows for the representation of network activity as a function of a local-ized feature space. Here, we assume a mean hue-dependent activity a(θ,t) where θrepresents a direction in the DKL stimulus space, a strictly physiological conception of “hue”from the hues categorizing color perception, as explained above. In drawing this distinction, and in agreementwith Wool et al. (2015) and Li et al. (2022), we give no special status to V1 cells tuned to the DKL directions associated with the unique hues of color-opponent theory, while simultaneously emphasizing the cone-opponent nature of feedforward afferents from the LGN. The resulting activity a(θ,t) of a network of hue-preferring cells, expressed as a firing rate in units of spikes/second, is dominated by the membrane properties of its constituent cells, whose potential variations occur on the order of the membrane time constant τ 0, taken to be 10 msec (Ben-Yishai et al., 1995 ;Carandini & Ringach, 1997 ;Izhikevich, 2010 ). In the vein of previous neural mean field models of feature detection ( Bressloff & Cowan, 2002,2003b ;Bressloff, Cowan, Golubitsky, Thomas, & Wiener, 2001 ;Dayan, Abbott, & Labahn, 2001 ;Ermentrout, 1998 ;Gutkin, Pinto, & Ermentrout, 2003 ), and in close analogy to the ring model of orien-tation tuning ( Ben-Yishai et al., 1995 ;Hansel & Sompolinksy, 1998 ), we let a(θ,t) evolve according to the single-population firing-rate formulation of the Wilson-Cowan equations: τ0daθ;tð Þ dt¼-aθ;tð Þ þ ghθ;tð Þ1/2/C138 ; (1) where h(θ,t), the synaptic input, takes into account both excitatory and inhibitory afferents into a population of cells preferring hue θ, and g(h) is an activation function, as described below. To analyze the relationships between feedforward and recurrent processing and to distin-guish between their respective effects on a(θ,t), we write h(θ,t) as a sum of the stimulus-driven synaptic input from the LGN and the intracortical input resulting from the hue-dependentnetwork connectivity within V1: hθ;tð Þ ¼ h ctxθ;tð Þ þ hlgnθð Þ: (2) We express the input as the average effect of the net synaptic current on the membrane potential of a cell, following the conventions of Ermentrout (1998) and Carandini and Ringach (1997). Thus, h(θ,t) has units of m V and can take on both positive and negative values, chosen here so that a(θ,t) typically ranges from 0 to 60 spikes/sec, consistent with electrophysiological experiments penetrating individual color-responsive cells ( Conway, 2001 ;Johnson et al., 2001 ;Landisman & Ts 'o, 2002a ;Wachtler et al., 2003 ). The input is converted to a firing rate according to the nonlinear activation function ghð Þ ¼ β⋅h-T ð Þ ⋅Hh-Tð Þ ; (3) where H(x) is the Heaviside step function defined as H(x) = 1 for x> 0 and zero for x≤0. Note that in the context of machine learning, this form of activation is also known as the rectified Spontaneous pattern formation: A system 's ability to self-generate new symmetries in the absence ofexternal input. Activation function: A function mapping the afferentinput into a population of neurons(expressed as a current or membranepotential) to the population 's firing rate or probability of firing. Network Neuroscience 682Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
linear unit function, or Re LU for short. By constraining the network activity to levels below 60 spikes/sec, we ignore the effects of neuronal saturation commonly implemented in modelsofg(h)(Ben-Yishai et al., 1995 ;Ermentrout, 1998 ). Here, Tis the threshold potential of a neuron, below which the synaptic input has no effect on the mean firing rate of the network. Interestingly, as a processing feature, this thresholding nonlinearity has been speculated toaccount for the chromatic responses of individual neurons in V1 ( Hanazawa et al., 2000 ). The amplification of these responses, and thus the mean network response, is modulated by β, the neural gain measured in spikes · sec-1/m V. We assume that βis determined by far-ranging internal and external influences, from attentional mechanisms to hallucinogenicinput ( Ferguson & Cardin, 2020 ;Michaiel, Parker, & Niell, 2019 ). Feedforward Input To parameterize the input, prior work has relied on the direct relationship between cortical feature preferences and properties of the visual stimulus ( Ben-Yishai et al., 1995 ;Bressloff & Cowan, 2003b ). Cells in the cortex labeled, for instance, by their spatial frequency preferences can be mapped directly onto a visual space parameterized by the same variable. Thus, theactivity of each neuronal population is no longer labeled purely by its position on the corticalsheet, but also by its preferred stimulus in an analogous feature space. The corresponding network topology may be modeled on the cortical histology, such as the orientation map of Bosking, Zhang, Schofield, and Fitzpatrick (1997) or spatial frequency maps addressed in Bressloff and Cowan (2002),Bressloff and Cowan (2003a), and Bressloff and Cowan (2003b). Conversely, it may be based entirely on functional considerations, as, for instance, in the orientation tuning model of Sompolinksy et al. ( Ben-Yishai et al., 1995 ; Hansel & Sompolinksy, 1998 ), also known as the “ring model, ”which posits a topology based on the experimentally motivated assumption that populations with similar orientation prefer-ences are maximally connected ( Ben-Yishai et al., 1995 ) and on the argument that the impor-tant features of such a connectivity are captured by its first-order Fourier components ( Hansel & Sompolinksy, 1998 ). Our model deviates in this regard by emphasizing that the stimulus 's chromatic information is first discretized along the two cone-opponent pathways. We incorporate this aspect of early processing by projecting the stimulus 's DKL space position θ/C22onto the two cardinal axes: l¼cosθ/C22 s¼sinθ/C22:(4) The magnitudes of landsare thus taken to represent the normalized strengths of the L-M and S-(L+M) cone-opponent signals respectively. The feedforward input is then given by h lgn¼clcosθþssinθ ð Þ ; (5) where cis the signal strength, or contrast, expressed as the mean postsynaptic coarse mem-brane potential (in units of m V) of the target hue population generated by the presynaptic LGN neurons ( Carandini & Ferster, 2000 ). Formulated in this way, the input captures the colocali-zation of cone-opponency and hue selectivity in the activity of V1 cells as observed in Li et al. (2022) and Xiao et al. (2007). The hue tuning networks, parameterized by θ, are not only responsive to the individual cone-opponent stimulus signals, land s, but also implement the combinatorial mechanisms by which they are first mixed ( De Valois et al., 2000 ). Substitut-ing the expressions for land sinto Equation 5, we obtain hlgn¼ccosθ-θ/C22ð Þ : (6) Network Neuroscience 683Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
With this form, we point out the similarity of our combinatorial scheme to that of Mehrani et al. (2020), in which the input from cone-opponent V2 cells into hue tuning V4 cells is weighted as a function of the difference in their preferred hue angles. Most evidently, we differ from this model by first combining the cone-opponent signals in V1 rather than V4, in accordance with the above-mentioned studies. But beyond pointing to V1 as the origin of mixing, these exper-iments indicate that the combinatorial feedforward scheme is not sufficient to account forthe variability of neuronal hue preferences. Li et al. (2022) showed, for instance, that the con-tribution of signals isolating the S-(L+M) pathway is too small to explain the shifting of hue preferences away from the L-M axis by purely combinatorial means. As put forward by Shapley and Hawken (2011),Wachtler et al. (2003),a n d Lehky and Sejnowski (1999),a more complete understanding of neuronal hue encoding within V1 requires us to consider the nonlinear population dynamics therein. Recurrent Interactions We begin by characterizing the connectivity of the target hue tuning populations with a trans-lation invariant cortical connectivity function w(\|x-x0\|), such that the interactions between neurons in a single CO blob (length scale ∼0. 5 mm) depend only on the cortical distance between them ( Bullmore & Sporns, 2012 ;Salzmann et al., 2012 ). The network 's connectivity comprises the interactions of both its excitatory and inhibitory populations, wx-x0jjð Þ ¼ wexcþwinh; (7) each of which we model as a sum of an isotropic and distance-dependent term: wexc¼E0þ E1cos x-x0jjð Þ winh¼-I0-I1cos x-x0jjð Þ :(8) We set E0≥E1> 0 and I0≥I1> 0 so that wexcandwinhare purely excitatory and inhibitory, respectively, in accordance with Dale 's law ( Ben-Yishai et al., 1995 ;Dayan et al., 2001 ). Next, we map the weighting function onto hue space, drawing from the hue tuning micro-architecture revealed by the imaging studies of Liu et al. (2020) and Xiao et al. (2007). These studies point to a linear relationship between distance and hue angle difference, which min-imizes the wiring length of cells tuned to similar hues ( Liu et al., 2020 ). The hue-preferring cells inhabit the so-called “color regions, ”defined as such for their activation by red-green grating stimuli ( Liu et al., 2020 ). These regions predominantly overlap with the V1 CO blobs (Landisman & Ts 'o, 2002b ;Li et al., 2022 ) and are responsive to the full range of hues, much like the patchy distribution of orientation maps within the V1 hypercolumns. Thus, in a similarmanner to the local feature processing models of Bressloff and Cowan (2003b) and Ben-Yishai et al. (1995), we model the CO blob as a single color-processing unit consisting of Nneurons labeled by the continuous hue preference variable θ2[-π,π](Bressloff & Cowan, 2003b ). Figure 2 shows the distribution of hue-responsive neurons within a typical color region (Figure 2A ) as well as a more coarse-grained demarcation of peak activity within several of these regions ( Figure 2B ). To describe the spatial organization of their hue preference data, Xiao et al. (2007) and Liu et al. (2020) applied a linear fit to the cortical distance between two cell populations as a function of the difference in their preferred hue stimuli Δθ≡\|θ-θ 0\| apart in DKL space. Note, this implies a discontinuity between θ= 0 and θ=2π, allowing for the 2 πperiodicity of the hue preference label. Liu et al. (2020) report that the linear fit was able to capture the micro-organization of 42% of their tested hue maps, and a regression per-formed by Xiao et al. (2007) on an individual hue map gave a squared correlation coefficient of R2= 0. 91. Network Neuroscience 684Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
In agreement with these findings, we let \| x-x0\|=\|θ-θ0\|, absorbing the regression param-eters into the connectivity strength values E0,E1,I0,a n d I1in Equation 8. Substituting this change of variables and setting J0=E0-I0,J1=E1-I1(measured in m V/spikes · sec-1) gives wθ-θ0ð Þ ¼ J0þ J1cosθ-θ0ð Þ : (9) As detailed in Figure 3, for J1> 0, this functional form captures the local excitation and lateral inhibition connectivity ansatz typically assu med in neural field models as an analogy to diffusion-driven pattern formation ( Amari, 1977 ;Bressloff, 2003 ;Hoyle, 2006 ;Kim, Rouault, Druckmann, & Jayaraman, 2017 ;Turing, 1952 ). Notably, neurons in close proximity in both cortical and hue space maximally excite each other, and those separated by Δθ=πmaximally inhibit each other, evoking the hue-opponency of perception on a cellular level. We empha-size, however, that this choice of metric is guided by our physiological definition of hue and does not associate a perceived color difference to measurements in hue space. Here, it is also important to distinguish between the connectivity function and the center-surround receptive fields of single-and double-opponent color cells ( Shapley & Hawken, 2011 ). While the structures of both can be approximated by the same functional form, the resemblance is superficial: the former characterizes the interactions between different neuro-nal populations, and the latter is a property of single cells, often adapted for computer vision algorithms ( Somers, Nelson, & Sur, 1995 ;Turner, Schwartz, & Rieke, 2018 ). Finally, we weigh the influence of the presynaptic cells by convolving the connectivity function with the network activity, arriving at the cortical input to the target hue population at time t: h ctxθ;tð Þ ¼Zπ-πwθ-θ0ð Þ aθ0;tð Þ dθ0: (10) The recurrent input is thus a continuous function in θ, derived from the population-level inter-actions. As put forward by the above-mentioned imaging studies, these interactions are colo-calized with the cone-opponent feedforward input, h lgn, within the same CO blob regions of Figure 2. (A) Hue map of individual hue-selective cells obtained by 2-photon calcium imaging of neuronal responsiveness to seven test hues. Scale bar: 200 μm. (B) Regions of peak response to test hues (solid contours). The dashed white lines demarcate the color-preferring regions, colocalized with the CO blobs. Scale bar: 0. 5 mm. Modified with permission from Liu et al. (2020). Network Neuroscience 685Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
V1. Collectively, our formulation of h(θ,t) implements the mixing rules posited by these experiments, without requiring us to arbitrarily fine-tune the relative weights of the afferentsignals. RESULTS Evolution of Network Activity We start by observing that by virtue of the invariance of w(θ-θ0) under translations of θ, the convolution operator Tww*f(θ)=Rπ-πw(θ-θ0)f(θ0)dθ0is diagonalizable by the Fourier eigenfunction basis ^eμθð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ(11) with μ2andêμnormalized to integrate to 1 on [-π,π]. To calculate the eigenvalues λμof the corresponding linear transformations, Zπ-πwθ-θ0ð Þ1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ0dθ0¼λμ1ﬃﬃﬃﬃﬃﬃ 2πp eiμθ; (12) Figure 3. Cortical connectivity functions. (A) A difference of two Gaussians, one characterizing the excitatory interactions (here with σE= 40°) and the other the inhibitory interactions ( σI= 90°). This is the connectivity typically assumed in mean field models of cortical processing. (B)The difference of cosines formulation (Equation 9), with J 0=-2 and J1= 3, captures the local exci-tation and lateral inhibition assumed in panel A. Network Neuroscience 686Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
we make the change of variables θ-θ0=ϕ, so that the left-hand side of Equation 12 can be rewritten as-Zθ-π θþπwϕð Þe-iμϕ 1ﬃﬃﬃﬃﬃﬃ 2πp eiμθdϕ¼Zπ-πwϕð Þe-iμϕ 1ﬃﬃﬃﬃﬃﬃ 2πp eiμθdϕ: (13) The eigenvalues are thus: λμ¼Zπ-πwϕð Þe-iμϕdϕ: (14) Next, we assume a(θ,t) is separable in tand θand bounded on [-π,π] so that we may expand it in the eigenbasis of the convolution operator as: aθ;tð Þ ¼X μcμtð Þ^eμθð Þ: (15) Substituting the expansion into Equation 10, we have for Equation 2 hθ;tð Þ ¼X μcμtð ÞZπ-πwθ-θ0ð Þ ^eμθ0ð Þdθ0/C20/C21 þhlgnθð Þ; (16) where w(θ-θ0) is our choice for the connectivity function (Equation 9) and hlgn(θ) is defined as in Equation 6. Evaluating the integrals, we obtain hθ;tð Þ ¼ λ-1c-1tð Þ^e-1θð Þ þ λ0c0tð Þ^e0θð Þ þ λ1c1tð Þ^e1θð Þ þ ccosθ-θ/C22ð Þ ; (17) with λ0=2πJ0and λ1=λ-1=πJ1. Note here that only the zeroth and first-order complex Fourier components remain. Substituting the expansion Equation 15 and the explicit form of the activation function Equation 3 into Equation 1 yields: τ0X∞ μ¼-∞dcμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞cμtð Þ^eμθð Þ þ βhθ;tð Þ-T ð Þ H hθ;tð Þ-T ð Þ : (18) In the absence of the nonlinearity, each of the eigenmodes êμ(θ) would evolve indepen-dently of the others, and a complete analysis of the time-dependent system would seek to solve a set of equations for cμ(t) (see Methods :Linear Solution ). However, in our setup, the thresholding introduces a coupling of these coefficients, as the critical hue angles, δ1and δ2, at which the input is cut off and is determined by the combined cμ(t) at each point in time. While an analytical solution to this system is in most cases intractable, it is nonetheless infor-mative to break down the rate equation to a coupled system of equations for the evolution ofthe coefficients c μ(t). Taking the inner product of Equation 18 with êνand using < êν\|êμ>=δμν, we obtain: τ0dcνtð Þ dt¼-cνtð Þ þ ^eνβh-T ð Þ H h-T ð Þ ¼-cνtð Þ þ βZδ2tð Þ δ1tð Þhϕ;tð Þ ^e/C3 νϕð Þdϕ(19) where the Heaviside restricts the domain of the inner product to [ δ1(t),δ2(t)]. The time depen-dence of the cutoff angles reflects the evolution of this curve, which requires that the thresh-olding be carried out continuously throughout the duration of the dynamics. Network Neuroscience 687Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
To determine δ1and δ2, we reformulate the Heaviside as a function of θ. Given that the input h(θ,t) is a real-valued function, c02ℝand c1=c/C3-1. For mathematical convenience, we then rewrite Equation 17 in terms of c0, Re( c-1)≡c R-1, and Im( c-1)≡c I-1as hθ;tð Þ ¼λ0c0tð Þﬃﬃﬃﬃﬃﬃ 2πp þclþﬃﬃﬃ 2 πr λ-1c R-1tð Þ ! cosθð Þ þ csþﬃﬃﬃ 2 πr λ-1c I-1tð Þ ! sinθð Þ: (20) Setting q R¼clþﬃﬃﬃ 2 πr λ-1c R-1tð Þ q I¼csþﬃﬃﬃ 2 πr λ-1c I-1tð Þ q0¼λ0c0tð Þﬃﬃﬃﬃﬃﬃ 2πp(21) the input takes the form hθ;tð Þ ¼ q0tð Þ þ chtð Þcosθþγtð Þ 1/2/C138 (22) where tan( γ)=-q I q Rand ch(t)=ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q2 Rþq2 Ip. The Heaviside can then be expressed as Hh-T1/2/C138 ¼ H q0þchcosθþγ ð Þ-T 1/2/C138 ¼H cosθþγ ð Þ-α 1/2/C138(23) where α≡T-q0 ch, and the time arguments are suppressed for simplicity. In this formulation, the Heaviside sets the limits of integration in Equation 19 as the angles θ=δ1,δ2where αintersects with cos( θ+γ), as shown in Figure 4. Figure 4. The limits of integration δ1and δ2in Equation 19 are the angles corresponding to the intersection of α(in gray) and cos( θ+γ) (in black). Here, c=1, β= 1, and T=-1. θ/C22=π/8. J0and J1 are as in Figure 3. Network Neuroscience 688Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
With this reformulation, the system of equations for the evolution of the coupled cν (Equation 19) takes the more explicit form: τ0dcνtð Þ dt¼-cνtð Þ þ βZδ2 δ1q0tð Þ þ chtð Þcosϕþγtð Þ 1/2/C138 1/2/C138 ^e/C3 νϕð Þdϕ: (24) Note that, for all cν, the integrand of Equation 24 is a function of q0(t),ch(t), and γ(t) and therefore, implicitly, only of the coefficients c0(t),c-1(t), and c1(t). Thus, the dynamics are determined in full by the evolution of c\|ν\|≤1(t): τ0dc0tð Þ dt¼-c0tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 dϕ τ0dc1tð Þ dt¼-c1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 e-iθϕð Þdϕ τ0dc-1tð Þ dt¼-c-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Zδ2 δ1hϕ;tð Þ1/2/C138 eiθϕð Þdϕ;(25) with h(ϕ,t) as in Equation 22. Separating Equation 25 into its real and imaginary parts, and noting that a real-valued activity profile a(θ,t) requires c02ℝand c1=-c/C3-1, reduces the system to a set of equations forc0(t),c R-1(t), and c I-1(t). Evaluating the integrals, we obtain: τ0dc0tð Þ dt¼-c0tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp chsinδ2þγ ð Þ-sinδ1þγ ð Þ 1/2/C138 þ T-q0 ð Þ δ1-δ2 ð Þ fg τ0dc R-1tð Þ dt¼-c R-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp/C26ch 2cosγδ 2-δ1 ð Þ þ cosγþδ1þδ2 ð Þ sinδ2-δ1 ð Þ 1/2/C138 þ T-q0 ð Þ sinδ1-sinδ2 ð Þ/C27 τ0dc I-1tð Þ dt¼-c I-1tð Þ þβﬃﬃﬃﬃﬃﬃ 2πp/C26ch 2sinγδ 1-δ2 ð Þ þ sinγþδ1þδ2 ð Þ sinδ2-δ1 ð Þ 1/2/C138 þ T-q0 ð Þ cosδ2-cosδ1 ð Þ/C27 ; ð26Þ where the time arguments of q0,ch,γ,δ1, and δ2are suppressed for clarity. Written in this form, the system provides a representation of the time evolution of a(θ,t)i n terms of the coupled evolution of the constants c\|ν\|≤1. It is important to note that these equa-tions are nonlinear due to the implicit Heaviside in our determination of δ1(t) and δ2(t). While our reformulation of the right-hand side of Equation 24 allows for the explicit representation ofthe coupling of c νvia the nonlinearity, it is also this coupling that proves the analytical solution of the trajectories intractable. Thus, to describe the behavior of the time-dependent solution, we turn next to a numerical analysis of the system 's phase portrait —that is, to an exploration of the features and stability of the system 's emergent steady states. Steady-State Solution We approach the solution to Equation 1 with a Forward Euler method, propagating the activity from a random array of spontaneous initial values between 0 and 0. 2 spikes/sec to its steady-Network Neuroscience 689Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
state value. Within each time step (typically chosen to be 1 msec), we coarse-grain the net-work into n= 501 populations with hue preferences separated evenly across the DKL angle domain [-π,π]. The choice of an odd nallows us to numerically integrate Equation 10 using the Composite Simpson 's Rule, whereupon we rectify { h(θ,t)-T} and evaluate the right-hand side of Equation 1. Below, we use the term tuning curve only in reference to the emergent steady-state activity profiles. Figure 5 shows an example of a hue tuning curve obtained with this method. Note that the peak of the tuning curve is located at the LGN hue input angle /C18/C22, which is equivalent to the steady-state value of-γin Equation 26 (see Methods :Evolution of Peak Angle ). Furthermore, the steady-state solution requiresda∞θð Þ dt= 0 so that Equation 1 becomes a∞θð Þ ¼ gh∞θð Þ1/2/C138 : (27) Thus, the shape of the activity profile at the steady state is equivalent to the net cortical input, cut off by gatδ1≡θ⋆ c1andδ2≡θ⋆ c2. Here, θ⋆ c1andθ⋆ c2are the critical cutoff angles for the steady-state activity profile, beyond which a∞(θ) would take on negative values. We emphasize that the values of the cortical parameters J0,J1,c,T, and βare bounded by the physiological properties of V1. Varying these parameters in the subsequent analysis is therefore an investigation of their relative effects on hue processing, and we are not fine-tuning their weights to obtain specific hue tuning curves. Here, we explore a range of values for the cortical and stimulus parameters under the con-straint that the network activity remains between 0 and 60 spikes/sec, as motivated above. We further restrict J1> 0 and J0< 0 to elicit the local excitation and global inhibition connectivity ansatz of previous neural field models. Our main aim is to graphically characterize the relative effects of the parameters on the width, Δc=θ⋆ c2-θ⋆ c1, and peak height, a∞(θ/C22), of the network tuning curves. Together, these two properties reflect the network selectivity and emergent sig-nal strength, respectively. Note that these effects are robust to small additive white noise and may also be gleaned from the net input, expressed as in Equation 20 and evaluated at the steady-state values of the coefficients. Figure 5. Steady-state activity profile for a neuronal network encoding stimulus θ/C22=π/8. Parameters are as in Figure 4. Network Neuroscience 690Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
It is also important to note here the difference between a network tuning curve and a single-neuron tuning curve. The former is a coarse-grained representation of the CO blob response,with the horizontal axis representing the gamut of hue preferences within a single network. A relatively large tuning width would therefore indicate considerable responses from a wide range of hue tuning cells and poor network selectivity. The single-neuron tuning curve, onthe other hand, is an electrophysiological recording of an individual cell 's response to a set of hue stimuli, with the horizontal axis representing the range of stimulus hue angles used in the experiment. The peak location of the single-neuron tuning curve would therefore indicate the hue preference of the individual neuron, while the width would represent its selectivity forthat specific hue. Thus, though the two types of tuning curves are labeled and shaped similarly, the latter is only useful to characterize our network 's constituent neurons and notthe emergent properties of the population as a whole ( Bressloff & Cowan, 2003b ). Roles of the stimulus strength and cortical threshold. We begin by considering the role of the stimulus signal strength con the hue tuning width and peak height. Figure 6 shows typical tuning curves for two values of J1. We find that the stimulus strength has a quickly saturating effect on Δcfor all J1> 0, which is more pronounced at lower values of cas J10. Above saturation, the main contribution of the chromatic signal is to increase the network response, that is, to increase a∞(θ/C22). We also note that at T= 0, the trend reverses, such that increasing chas no effect on the tuning width at T= 0 and a widening effect for T>0. Figure 7 illustrates this reversal with four tuning curves of matched parameters and varying values of T. The coupling of cand Tmust be considered because some neural field models (see Amari, 1977 ;Carandini & Ringach, 1997 ; Dayan et al., 2001 ) take T= 0 for mathematical simplicity. Indeed, we might expect that there is no more physiological significance to choosing a threshold potential of T= 0 m V than any other value, beyond their relative magnitudes to h(θ,t). However, the independence of cand Δcat T= 0 and the significance of the relative signs of cand Telsewhere suggest quite the opposite. The effect of the chromatic input on tuning the network hue selectivity weakens notonly once the anisotropic strength parameter, J 1, is large enough to predominate, but also as T0. The coupling of cand Tis equally significant to the effects of Ton the tuning curve prop-erties. Figure 8 shows that below a certain value, Tprimarily modulates a∞(θ/C22). However, for Figure 6. Effect of con the tuning curve properties. The tuning role of cquickly saturates, while its effect on the network response rate grows without bound. For θ/C22=0, β=1, T=-1, and J0=-1. (A) J1=0. 2. ( B ) J1= 0. 7. Network Neuroscience 691Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
Figure 8. Effect of Ton the tuning curve properties. θ/C22=0, β=1, J0=-3,J1= 2, and c= 10. Figure 7. Effect of con the tuning curve for varying values of Twith β=1, J0=-1,J1= 0. 2, and θ/C22= 0. Note that the small network response rates are due to the low values of cchosen here. (A) T=-5. (B) T=-1. (C) T=0. ( D ) T= 0. 5. Network Neuroscience 692Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
comparable magnitudes of the stimulus strength and threshold, \| c\|∼\|T\|, we see a transition in which Talso begins to sharpen the tuning curve and continues to do so until the threshold surpasses h(θ,t)f o ra l l θ(i. e., for δ⋆ 1=δ⋆ 2= 0). Accordingly, for higher stimulus strengths, the thresholding nonlinearity plays a greater role in modulating the network selectivity at lower and a wider range of Tvalues. Roles of the cortical weights. The anisotropic connectivity strength J1exhibits similar relation-ships to the tuning curve properties to those of c. That is, for T<0, a∞(θ/C22) grows and Δcnarrows with increasing J1(see Figure 9A ). The trend with respect to Δcreverses for T>0( Figure 9B ), whereas the trend with respect to a∞(θ/C22) remains unaffected. These similarities are a mark of the competition between the external input and the cortical parameters in driving the network selectivity and reflect the fact that both parameters modulate the anisotropic terms of the model. This means that the role of J1in driving network selectivity becomes more significant with decreasing stimulus strength (see Figure 10 ). However, a large external input does not suppress the contribution of J1to the overall network activity. That is, increasing J1results in raising a∞(θ/C22), regardless of the strength of the stimulus. Similarly, a Figure 9. Effect of J1on the tuning curve properties for varying values of T. β=1, c= 0. 3, J0=-10, and θ/C22=0. ( A ) T=-1. (B) T= 0. 2. Figure 10. Effect of J1on the tuning curve properties for different stimulus strengths. β=1, T=-5,J0=-9, and θ/C22=0. ( A ) c=1. ( B ) c=3. Network Neuroscience 693Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
relatively large value of J1does not restrict the growth of the network response with increasing stimulus strength. Thus, the anisotropic tuning introduced by the external input and the recur-rent interactions act cooperatively to raise the network 's response to the stimulus hue, and competitively to tune its selectivity. In contrast, J 0acts cooperatively with the external stimulus to sharpen the curves. As shown in Figure 11, the tuning curves narrow with decreasing values of J0, that is, with an increase in the relative strength of global inhibition to global excitation, a trend which is conserved for various stimulus strengths. Furthermore, there is no trend reversal at T= 0. Rather, for much of the parameter space, J0acts with the thresholding to sharpen the tuning curves, as is illustrated in Figure 12. This could be expected from the fact that at each point throughout the dynamics, both Tand J0act isotropically on all hue preferences, lowering or raising the input for all con-tributing neurons. However, this commonality also means that for \| T\|> >\| c\| (where the effect of TonΔcsaturates, as explained above), the thresholding suppresses the role of J0, analogous to the competition between cand J1. Finally, Figures 11 and12also show that increasing the global inhibition acts to reduce the value of a∞(θ/C22) for all cand T. Figure 11. Effect of J0on the tuning curve properties for varying stimulus strengths. β=1, T=-2,J1= 2, and θ/C22=0. ( A ) c=1. ( B ) c=6. Figure 12. Effect of J0on the tuning curve properties for varying values of T. β=1, c=1, J1= 2, and θ/C22=0. ( A ) T=-5. (B) T= 0. 2. Network Neuroscience 694Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
We thus conclude that the emergent hue curves in V1 are both inherited from the LGN and built on the recurrent interactions. The competition between J1andcpoints to a continuum of regimes in which either hlgnorhctxdominates. However, in all regimes, J0works cooperatively with cto narrow the curves, and all the parame ters work together to raise the network response. Likewise, the competition between J0and T(both cortical parameters) is modulated by the value of c, and the location of the peak is always completely determined by the LGN signal, regardless of the relative magnitudes of the cortical and stimulus strength parameters (see Methods :Evolution of Peak Angle ). Comparisons with the orientation tuning ring model. Finally, we seek to compare the emergent properties of the hue tuning model with those of the orientation tuning ring model ( Ben-Yishai et al., 1995 ;Hansel & Sompolinksy, 1998 ). This leads us to separate the analysis into two regions: one corresponding to the analytical regime with J0<1 2πβand J1<1 πβ, and the other to the extended regime with J1≥1 πβand J0constrained as described in the section Stability Analysis. As detailed in Methods :Linear Solution, the former regime defines the ( J0,J1) param-eter space wherein the model permits a closed-from stable solution for cases in which the input into all cells is above threshold. By contrast, the dynamics in the latter, extended regimealways implement thresholding and thus do not permit the linear closed-form solution. For comparison purposes, note that these parameter regimes are analogous to the orientation model 's homogeneous and marginal regimes, respectively, labels which refer to the system 's responses to unoriented stimuli. An important difference between our two models is our choice to assume modularity for the color vision pathway. As described above, there is no consensus as to when and how the var-ious visual features are separated along the visual pathway. That is, we do not yet understand how the brain recognizes the extent to which an activated color-and orientation-preferring neuron is signaling for a stimulus 's color or orientation. And moreover, we do not know at which point of the visual pathway the differentiation becomes important. We have therefore chosen to emphasize the unoriented color selective cells localized in the CO blob regions of V1, though the model is intended to describe the color-processing pathway broadly, for anycolor-preferring neurons regardless of other feature tuning capabilities. Thus, the choice ofmodularity is not to reject the possibility of joint feature processing, but rather to parse out the color mechanism for a separate analysis. Furthermore, it is in keeping with perceptual studies which indicate that the red-green and blue-yellow color-opponent systems are onlyresponsive to color stimuli and not to broadband, white light ( Stockman & Brainard, 2010 ). The difference between our two models thus comes to our choice to consider the purely chro-matic component of the input afferent from the LGN, whereas the orientation model incorpo-rates external inputs with varying degrees of anisotropy, that is, h extθð Þ ¼ c1-/C15þ/C15cos 2 θð Þ 1/2/C138 ;0≤/C15≤1=2 (28) where /C15represents the degree of anisotropy. The differing assumptions underlying the formulation of h(θ,t) have important implications for the subsequent parameter analyses adopted by our two models. In the orientation tuning model, the authors detail the pronounced shift in the relative roles of the cortical and stimulusparameters in narrowing the tuning curve. In this setup, for /C150. 5, an increase in cwidens the tuning curve, whereas for /C150, the tuning curve selectivity is completely determined by the cortical parameters. The latter scenario constrains the value of the analogous anisotropiccortical parameter, J 2, to the marginal regime. Network Neuroscience 695Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
In contrast, our model does not apportion separate regions of the parameter space to exter-nal and recurrent mechanisms. Rather, in both the analytical and extended regimes, the rolesofcand J 1exist on a spectrum, where the effect of each parameter is suppressed by larger values of the other. Of course, this suppression is more stark in the extended regime because it covers larger values of J1. In this sense, our color model draws a similar conclusion to that of the orientation model: when the anisotropic tuning provided by the recurrent interactions islarge, the tuning from the stimulus is negligible, and vice versa. However, we emphasize that the transition is not sharp and that cdoes have an effect on the tuning curve selectivity in the extended regime (see Figure 6B ), as does J 1in the analytical regime. In this regard, the two models are more consistent in their interpretations of J0's contribution to the selectivity of the tuning curves. That is, in the two regimes of each model, the inhibitionacts cooperatively with the thresholding to sharpen the tuning curves. Here again, the orien-tation model makes a distinction between the marginal phase (i. e., /C15= 0 and J 22marginal regime), wherein the tuning curve width is completely determined by the cortical anisotropy, and all other cases, where the isotropic inhibition and stimulus come into play. For thesecases, the authors argue, J 0does not act alone to narrow the curve: though J0may sharpen the tuning curves, it is the anisotropy from the input or cortical interactions which acts as the source of the orientation selectivity. Although our color model 's tuning mechanism, too, requires a source of anisotropy, we have emphasized above that there is no single source of hue selectivity. When J1is small, in either regime, both the stimulus and the uniform inhibition are significant to the hue tuning mechanism. Ultimately, the orientation model sets up a dichotomy between two specific regions of parameter space. In the nonmarginal case, cis the primary player in the tuning mechanism, and in the marginal case, this role belongs to J2. The uniform inhibition is thus given a sec-ondary “sharpening ”role. By contrast, in choosing a fully anisotropic hlgn, our color model does not encompass an analogous marginal phase with an always dominating J1. Rather, even for large J1, the uniform inhibition is at least equally important to the modulation of the tuning width. In fact, as we have shown above, for larger values of c,J0is more effective than J1in narrowing the tuning curves, for both the analytical and extended regimes. We thus stress that the two regimes, though analogous to those of the orientation model, do not constitute a division in the hue processing mechanism. Rather, we define the boundary between the analytical and extended regimes solely by whether or not the linear case exists. It is therefore determined by the values of J0and J1for which the linear solution applies, given that the values of c,T,a n d βkeep the input above threshold throughout the dynamics (Methods :Linear Solution ). We note that for each combination of J0and J1within the analytical regime there exists also a nonlinear case, in which h(θ,t) is cut off by the thresholding non-linearity and, thereby, the linear solution does not apply. Our definition differs from that of theorientation model, which demarcates the boundary between the homogeneous and marginal phases based on the emergent steady-state tuning curves alone. For more on this approach, see the discussion of the broad and narrow profiles in Hansel and Sompolinksy (1998). A sw e will show next, the boundary is integral to the corresponding stability analysis of the steady-state tuning curves. Stability Analysis To analyze the stability of the emergent tuning curves, we turn once more to our separable activity ansatz assumed in the eigenfunction decomposition of Equation 15. This means thatwe are faced again with a nonlinearity induced coupling of the time-dependent coefficients Network Neuroscience 696Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
and, consequently, the analytical intractability of the associated stability analysis. We therefore set up the Jacobian matrix for a numerical analysis of the local stability. We begin by adding a small perturbation of the form δaθ;tð Þ ¼X μDμtð Þ^eμθð Þ (29) and substituting the resulting activity into Equation 1. The eigenmodes then evolve according to the following equation for the coefficients Dμ(see Methods :Linear Stability Analysis ): τ0d Dνtð Þ dt¼-Dνtð Þ þ βZδ⋆ 2 δ⋆ 1δq0tð Þ þ δq Rtð Þcosϕð Þ þ δq Itð Þsinϕð Þ 1/2/C138 ^e/C3 νϕð Þdϕ; (30) where δ⋆ 1andδ⋆ 2are the critical cutoff angles of the steady-state solution, obtained numerically. We observe that the integrand of Equation 30 is a function of D0,DR-1, and DI-1alone, and, as such, the stability of the steady-state tuning curve is completely determined by the stability of these first-order coefficients. Evaluating the integrals for ν= 0 and ν=-1, and noting from Equation 21 that δq0¼ﬃﬃﬃﬃﬃﬃ 2πp J0δc0≡ﬃﬃﬃﬃﬃﬃ 2πp J0D0 δq R¼ﬃﬃﬃﬃﬃﬃ 2πp J1δc R-1≡ﬃﬃﬃﬃﬃﬃ 2πp J1DR-1 δq I¼ﬃﬃﬃﬃﬃﬃ 2πp J1δc I-1≡ﬃﬃﬃﬃﬃﬃ 2πp J1DI-1;(31) we obtain the following system of equations for the evolution of the characteristic coefficients: τ0d D 0 dt¼βJ0δ⋆ 2-δ⋆ 1/C0/C1-1/C2/C3 D0 þβJ1sinδ⋆ 2-sinδ⋆ 1/C0/C1/C2/C3 DR-1 þβJ1cosδ⋆ 1-cosδ⋆ 2/C0/C1/C2/C3 DI-1 τ0d DR-1 dt¼βJ0sinδ⋆ 2-sinδ⋆ 1/C0/C1/C2/C3 D0 þβJ1 42δ⋆ 2-2δ⋆ 1þsin 2 δ⋆ 2/C0/C1-sin 2 δ⋆ 1/C0/C1 /C0/C1-1/C20/C21 DR-1 þβJ1 4cos 2 δ⋆ 1/C0/C1-cos 2 δ⋆ 2/C0/C1 /C0/C1/C20/C21 DI-1 τ0d DI-1 dt¼βJ0cosδ⋆ 1-cosδ⋆ 2/C0/C1/C2/C3 D0 þβJ1 4cos 2 δ⋆ 1/C0/C1-cos 2 δ⋆ 2/C0/C1 /C0/C1/C20/C21 DR-1 þβJ1 42δ⋆ 2-2δ⋆ 1þsin 2 δ⋆ 1/C0/C1-sin 2 δ⋆ 2/C0/C1 /C0/C1-1/C20/C21 DI-1(32) Network Neuroscience 697Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
The entries of the corresponding Jacobian matrix consist of the bracketed prefactors, and may equally be obtained from the general system of equations for the global network dynamics, asfollows: J¼∂f1 ∂c0∂f1 ∂c R-1∂f1 ∂c I-1 ∂f2 ∂c0∂f2 ∂c R-1∂f2 ∂c I-1 ∂f3 ∂c0∂f3 ∂c R-1∂f3 ∂c I-12 66666666643 7777777775 c⋆ 0;c R⋆-1;c I⋆-1(33) where f1,f2,a n d f3are the right-hand sides of the equations in Equation 26 and the first-order partial derivatives are evaluated at the steady-state values of c0,c R-1, and c I-1. The stability of the tuning curve is then determined by the eigenvalues of J. We note that the existence of a steady state is a function of the cortical strengths J0and J1. By fixing the values of β,c,θ/C22, and T, we are left with a two-parameter family of differential equations, allowing us to analyze this de pendence numerically in the associated ( J0,J1) parameter space. Carrying out a parameter sweep across thi s space, we find that the system features a bifurcation curve, below which the model permits steady-state solutions and above which no equilibrium exists. To determine stability within the former region, we compute Jat the emergent steady-state tuning curves of various points in the parameter space. Solving the associated characteristic equations, we observe that the eigenvalues are always real and negative, and thus conclude that all emergent steady-state tuning curves are stable. Figure 13 shows the bifurcation diagrams for two families of equations, distinguished by their values for T. Most notable is the extended regime, which permits stable steady-state solu-tions beyond the boundary set by the linear case ( Methods :Linear Solution ). As this parameter regime is not accessible to the linear solution, the tuning curves in this regime are necessarily a product of the thresholding nonlinearity and are thus always cut off below \| θ\|=π. The thresh-olding nonlinearity therefore not only expands the region of stability, but also ensures that thetuning curves emerging within the extended regime are selective for hue. As we have seen, thisexpansion is pivotal when the external input is weak and the anisotropic cortical strength plays the larger role in narrowing the tuning curves. Furthermore, regardless of input strength, it allows for a larger overall network response, as the peak activity, a ∞(θ/C22), grows with increasing J1. Finally, as we will see in the following section, in the absence of any stimulus (i. e., for c= 0), the extended regime features the spontaneous generation of stable tuning curves and may thus serve as the bedrock for color hallucinations. However, looking back at Figure 13, perhaps most striking is the horizontal portion of the bifurcation curve at J0=1 2πβfor J1<1 πβ, which sets the same stability conditions on J0and J1as in the linear case. This is despite the fact that many of the points in the analytical regimes of the two featured families correspond to solutions that implement thresholding, thus signifying thatthe analytical regime is not an exclusively linear one. The key to understanding the shape of this region lies in noticing that the bifurcation dia-gram does not change for varying values of c,T, and θ/C22, as shown in Figure 13 for the two values Bifurcation curve: A curve in parameter space defininga transition in the system 's dynamics and stability. Network Neuroscience 698Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
of T. The stability conditions on J0and J1are thus uniquely determined by βalone. Further-more, for the general diagram (i. e., with βfixed and c,Tunfixed), each point of the analytical regime permits linear solutions, in addition to the ones that implement thresholding. Accord-ingly, the uniqueness of the bifurcation diagram implies that at each point of the analytical ( J0, J1) subspace, the stability of the latter, nonlinear solutions is equivalent to that of the linear solutions. This means that the boundary at J0=1 2πβset by the linear case ( Methods :Linear Solution ) applies to the full, nonlinear model as well. A Turing Mechanism for Color Hallucinations Biological Turing patterns. Underpinning our hue tuning model is the mathematics of reaction-diffusion systems, for which, in particular, Alan Turing 's treatment of biological pattern forma-tion offers many valuable insights ( Turing, 1952 ). The general Turing mechanism assumes a system of two interacting chemicals, whose local reaction and long-range diffusion propertiesgovern the dynamics of their relative concentrations. In the original framework these chemi-cals are termed “morphogens ”to elicit their form-producing capabilities within a developing embryo, whose anatomical structure emerges as a result of their underlying concentrationdynamics. This, for instance, may be attributed to the morphogens 'catalysis of organ forma-tion in different parts of the developing organism. Most analogous to our model is the formulation which distributes the morphogens across a continuous ring of tissue, parameterized by the cellular position θ. Assuming that the system Reaction-diffusion system: Mathematical model of two or moreinteracting substances consisting oflocal dynamics (via the reactionterms) and spatial dynamics (via thediffusion terms). Figure 13. Bifurcation diagram for β= 1 and c= 1 for two values of threshold (shown in the leg-end). The gray and white regions correspond to the analytical and extended regimes, respectively. The black dashed line is the bifurcation curve, above which the tuning curves grow without bound. The overlaid symbols correspond to points tested in a parameter sweep over the extended regime. Notably, the parameter sweep produces the same bifurcation curve for both values of T. Here, we must note that for critical values of T, for which the input is not large enough to generate activity, the model permits the trivial a(θ) = 0 steady-state solution in both the analytical and extended regimes. This solution, however, is unstable to perturbations large enough to make the input cross the thresh-old. For more on bifurcation theory in the context of neural fields, see Gross (2021) and Hesse and Gross (2014). See also Hansel and Sompolinksy (1998) for an analogous “phase diagram ”analysis of the orientation ring model. Network Neuroscience 699Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
never deviates far from the underlying homogeneous steady state, the two dynamical state equations for their concentrations, Xand Y, take the linear form d Xθ;tð Þ dt¼a Xθ;tð Þ þ b Yθ;tð Þ þ DX∇2Xθ;tð Þ d Yθ;tð Þ dt¼c Xθ;tð Þ þ d Yθ;tð Þ þ DY∇2Yθ;tð Þ ;(34) where a,b,c, and drepresent the chemical reaction rates, and DXand DYare the diffusion rates of Xand Y, respectively. Here, we set a,c> 0, so that increasing the concentration of X activates the production of both Xand Y, and b,d< 0 so that Yhas an inhibitory effect on the production of both chemicals ( Hoyle, 2006 ). In the absence of diffusion (i. e., with DX=DY= 0), the system has a homogeneous steady-state solution, ( X,Y) = 0, whose stability is determined by a Jacobian composed of the reaction rates,ab cd/C20/C21, and hence by the system 's local chemical properties alone ( Hoyle, 2006 ). Note that at this point the system is circularly symmetric with respect to interchanging any two cells on the ring. Assuming the existence of a stable steady-state solution, and the corresponding require-ments on the rate parameters a-d, we next set the diffusive terms DX,DY> 0, taking the sep-arable ansatz for the general solution: X¼X∞ μ¼-∞Aμeλμteiμθ Y¼X∞ μ¼-∞Bμeλμteiμθ:(35) Furthermore, we set DX<DYto generate the local excitation and lateral inhibition of the mor-phogen concentrations ( Hoyle, 2006 ;Murray, 2003 ), evoking the connectivity function ansatz Equation 9. The underlying steady state then remains stable if the real parts of the eigenvalues λμ, obtained from the modified Jacobian, are negative. With the reaction rates fixed from the stability conditions above, these eigenvalues are functions of the diffusion parameters alone. Thus, the conditions for stability may be thought of in terms of a bifurcation diagram in the ( D X, DY) phase space, comparable to Figure 13. From here, a set of additional conditions may be placed on DXand DYso that the system undergoes a Turing bifurcation, wherein at least one λμbecomes positive and the homoge-neous steady state loses its stability. With the addition of a small random perturbation, the instability results in the growth of the corresponding eigenmodes eiμθ, such that, over time, Equation 35 is dominated by the eigenmodes with largest λμ. These represent stationary waves whose wavelengths are set by the circumference of the ring (i. e., by the spatial properties of the medium) and whose growth is bounded by the higher order terms that had been initiallyignored in the near-equilibrium formulation ( Murray, 2003 ;Scholz, 2009 ;Turing, 1952 ). The underlying circular symmetry is thus broken and a spatial pattern is formed. In his seminal paper, Turing extrapolated this mechanism to explain various biological phe-nomena, such as the development of petals on a flower, spotted color patterns, and the growth Network Neuroscience 700Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
of an embryo along various directions from an original spherical state. A hallmark of each of these examples is that there is no input into the system, so the emergent patterns reflect amechanism of spontaneous symmetry breaking, onset by a perturbation of “some influences unspecified ”(Turing, 1952 ). In light of this, we ask, can the visual cortex self-generate the perception of hue? Spontaneous symmetry breaking and color hallucinations. To assess our model 's ability to self-organize in the absence of visual input, we set c= 0 and seek to establish the presence of a Turing mechanism marked by the following three features: 1. A system comprised of local excitation and long-range inhibition. 2. Spontaneous symmetry breaking in the absence of input within a region of a parameter space defined by the relevant bifurcation parameter(s). 3. The emergence of patterns that are bounded by the system 's nonlinearities. As noted above, requiring DX<DYin Equation 34 sets up the diffusion-driven activator-inhibitor dynamics governing the evolution of the morphogen concentration across the ring of cells. With these assumptions, Turing 's reaction-diffusion equations bear a strong resem-blance to our one-population generalization of the excitatory and inhibitory color cell dynam-ics in the absence of LGN input: τ0daθ;tð Þ dt¼-aθ;tð Þ þ g Zπ-πJ0þ J1cosθ-θ0ð Þ ð Þ aθ0;tð Þ dθ0/C20/C21 ; (36) where the local excitation and long-range inhibition are incorporated in the anisotropic inter-action term J1cos(θ-θ0), and the reaction terms a X(θ,t),b Y(θ,t),c X(θ,t), and d Y(θ,t) find their neural analogue in the term-a(θ,t). Importantly, the notions of “local ”and“long-range ” here describe interactions in the DKL space, and not in the physical cortical space correlate to Turing 's ring of tissue. Accordingly, we treat J1as the Turing bifurcation parameter and look for spontaneous color tuning beyond a bifurcation point J1=JT 1. Additionally, we observe that the onset of pattern formation is determined by a critical value of T, so that the relevant parameter space for our exploration is ( J1,T)(Figure 14 ). This analysis is summarized in Figure 15. Figure 14. The onset of color hallucinations in the ( J1,T) parameter space. The model generates spontaneous hue tuning curves beyond J1=JT 1≡1 πβand below T=0. Network Neuroscience 701Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
Figure 15. Spontaneous pattern formation in the absence of input ( c=0 ). β=1. ( A-B):J0=-2,J1= 0. 1 (A) T= 0 (B) For T< 0, the homo-geneous steady-state value increases. Here, T=-10. (C-D): Pattern formation in the extended regime for J0=-2,J1= 0. 4. (C) No hue tuning curve emerges for T≥0. Here, T=0. ( D ) T=-10. A hue tuning curve is generated in the absence of external input. (E-F):T=-10,J0=-7,J1= 6. The emergent tuning curve is more selective for larger values of J1. For each run, the activity is peaked about a different angle, set by the random initial conditions. The peak value and tuning width are consistent between trials. Network Neuroscience 702Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
We observe that within the analytical regime, the system generates a stable homogeneous steady-state solution a∞(θ) = const ≥0 for all values of the parameters β,T,J0,a n d J1(Figure 15A-B). As such, from the closed-form linear steady-state solution ( Methods :Linear Solution ), we obtain a∞θð Þ ¼-βT 1-2πβJ0for T≤0 0 for T>0:8 < :(37) We further observe that beyond J1=1 πβ, a stable homogeneous steady-state solution remains at a∞(θ) = 0 for T≥0(Figure 15C ). However, at T= 0, this radial symmetry is broken, and the cortex generates spontaneous tuning curves with peak locations determined by the random initial conditions ( Figure 15D-F). Thus, the system bifurcates when J1=1 πβand T= 0, permitting the onset of color hallucinations in a region defined by these two values (Figure 14 ). Note that the unimodal tuning curves predict stationary, single-hued phosphenes. Extension to other CO blob networks would therefore indicate a hallucination comprised ofmultiple phosphenes of varied hues, each determined by the local cortical activity at hallu-cinatory onset. Bearing these predictions in mind, we point to a recent functional MRI study of blind patients experiencing visual hallucinations ( Hahamy, Wilf, Rosin, Behrmann, & Malach, 2021 ). The study attributes these visions to the activation of the neural networks underlying normal vision, precipitated by the hyper-excitability of the cortex to spontaneous resting-stateactivity fluctuations when it is deprived of external input. This is suggestive of the required lowering of neuronal threshold at the onset of color hallucinations predicted here. Notably, a reduction in membrane potential threshold h a sa l s ob e e na t t r i b u t e dt ot h ea c t i o no f hallucinogens ( De V os, Mason, & Kuypers, 2021 ;Varley, Carhart-Harris, Roseman, Menon, & Stamatakis, 2020 ). Finally, we note that the stability of the emergent tuning curves is determined by the bifur-cation diagram of Figure 13. This means that, in addition to expanding the region of stability in the presence of chromatic stimuli, the model 's nonlinearity allows for stable, spontaneous color hallucinations in their absence. Having thus established a Turing-like mechanism for our model 's self-organization, we end with an analogy to Turing 's original diffusion-driven formulation. In his concluding example, Turing applies the mechanism to explain the growth of an embryo along various axes of itsoriginal spherical state. This growth is driven by diffusion, directed by the “disturbing influ-ences, ”shaped by the system 's chemical and physical properties, and bounded by the sys-tem's nonlinearities. It is all too clear to see the parallels with our hue tuning model, wherein a hallucination is driven by the anisotropy of the cortical interactions, its hue determined by the initial conditions, its selectivity shaped by the cortical parameters, and its stability ensured by the thresholding nonlinearity. DISCUSSION This paper presents a neural field model of color vision processing that reconceptualizes thelink between chromatic stimuli and our perception of hue. It does so guided by the premisethat the visual cortex initiates the mixing of the cardinal L-Ma n d S-(L+M) pathways and Network Neuroscience 703Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
thereby transforms the discrete cone-opponent signals to a continuous representation of chromatic information. Such mixing mechanisms have been implemented by previous com-binatorial models of color processing, though through a largely feedforward approach or at the level of the single neuron. Our theory bears in mind the mixing mechanism, but reframes the stage-wise combina-torial scheme to one based on the nonlinear population dynamics within the visual cortex. Accordingly, we propose a hue-based cortical connectivity, built upon the cortical hue map micro-architecture revealed by recent optic al imaging studies of V1. By considering the intracortical network interactions, we have accounted for V1 cells responsive to the gamut of DKL directions without the need to fine-tune the cortical parameters. We do so without restricting to a particular category of V1 neuron, as both single-opponent and double-opponent, and altogether novel types of cells, have been suggested as the primary messen-gers of chromatic information. Rather, we zoom out from the individual neuron 's receptive field to model the aggregate, population-level properties and, in particular, the stable repre-sentation of hue. We thereby offer that chromatic processing in the visual cortex is, in itsessence, a self-organizing system of neuronal-activity pattern formation, capable of encoding chromatic information in the presence of visual stimuli and generating information in their absence. Further, in assuming modularity for chromatic processing, we have not ruled out a mech-anism for joint feature processing. Our choice to focus on the unoriented color cells within the CO blob regions allowed us to parse out the chromatic pathway for an independentanalysis, but should not be interpreted as a claim about its functional independence. We leave open the question of the functional and anatomical separation of the various visual streams. Equally unsettled is the question of how much S cone input contributes to the mixing of the cone-opponent channels, with some studie s showing a relatively weak S cone input into the neurons of V1, compared to its L and M cone counterparts ( Li et al., 2022 ;Xiao, 2014 ). The variations across these experiments may stem, in part, from differences in optical imaging and electrode penetration techniques, including the particulars of the chromatic stimulus used (Li et al., 2022 ;Liu et al., 2020 ;Polimeni, Granquist-Fraser, Wood, & Schwartz, 2005 ; Salzmann et al., 2012 ). On the whole, however, single-cell recordings have identified two main types of color-responsive regions: color patches that contain neurons tuned exclusively to stimuli modulating either of the cone-opponent pathways, and patches with neurons exhi-biting a mixed sensitivity to a combination of the two ( Landisman & Ts 'o, 2002a ;Li et al., 2022 ;Livingstone & Hubel, 1984 ). Further experiments on the connectivity between these regions, and among the single-and double-opponent color cell populations of which theyconsist, may point to added micro-architectures for the hue maps, along the lines of the geo-metric orientation models of Bressloff and Cowan (2003a) and Bressloff et al. (2001). Finally, we emphasize that the mechanism we offer departs from previous combinatorial color models that predict hue sensation at the final stage of processing ( De Valois & De Valois, 1993 ;Mehrani et al., 2020 ), as well as neural field models that conflate cone-and color-opponency in their interpretations ( Faugeras, Song, & Veltz, 2022 ;Smirnova, Chizhkova, & Chizhov, 2015 ;Song, Faugeras, & Veltz, 2019 ). The emergent hue tuning curves we have characterized are a network property reflective of the physiological neuronal responses, and should not be confounded with our perception of hue. A photon of wavelength 700 nm strik-ing a retina is no more “red”than any other particle —color is a perceptual phenomenon not yet represented in these first stages of vision. By recognizing that the hue tuning mechanism of Network Neuroscience 704Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
the visual cortex is an early stop in the visual pathway, we point to the need for further field theory approaches to our understanding of color perception. METHODS Linear Solution We assume in the linear case that the net input h(θ,t) is above threshold throughout the dynamics such that the activity profile is never cut off and H(h(θ,t)-T)=1∀θ2{-π,π}. Equation 18 therefore takes the linear form: τ0X∞ μ¼-∞dcμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞cμtð Þ^eμθð Þ þ βhθ;tð Þ-T ð Þ (38) Taking the inner product of Equation 38 with êνon the full domain ≡{-π,π}, we obtain the system of equations for all the coefficients cν: τ0dc0tð Þ dt¼2πβJ0-1 ð Þ c0tð Þ-ﬃﬃﬃﬃﬃﬃ 2πp βT τ0dc1tð Þ dt¼πβJ1-1 ð Þ c1tð Þ þﬃﬃﬃπ 2r βcl-isð Þ τ0dc-1tð Þ dt¼πβJ1-1 ð Þ c-1tð Þ þﬃﬃﬃπ 2r βclþis ð Þ τ0dcνtð Þ dt¼-cνtð Þ∀νjj>1(39) We may thus solve for each of the coefficients independently, yielding equations for the evolution of each. Substitution into the activity expansion of Equation 15 then gives the closed-form solution for the evolution of the activity: aθ;tð Þ ¼ K0e-1-2πβJ0 ð Þ t=τ0-ﬃﬃﬃﬃﬃﬃ 2πp βT 1-2πβJ0/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp þ K-1e-1-πβJ1 ð Þ t=τ0þﬃﬃﬃπ 2rcβlþis ð Þ 1-πβJ1/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp e-iθ þ K1e-1-πβJ1 ð Þ t=τ0þﬃﬃﬃπ 2rcβl-isð Þ 1-πβJ1/C26/C271ﬃﬃﬃﬃﬃﬃ 2πp eiθþ Kν1ﬃﬃﬃﬃﬃﬃ 2πp e-t/C26/C27 eiνθjνjj>1 (40) where the constants Kνare determined by the Fourier coefficients cν(0) of the initial activity a(θ, 0). For J0<1 2πβand J1<1 πβ, the solution approaches the globally asymptotic stable steady-state tuning curve a∞θð Þ ¼-βT 1-2πβJ0þcβcosθ-θ/C22 ð Þ 1-πβJ1: (41) We call the corresponding ( J0,J1) parameter space the analytical regime. Globally asymptotic stable steady state: As y s t e m 's equilibrium solution which attracts all other solutions regardless ofinitial conditions. Network Neuroscience 705Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
Evolution of Peak Angle We first assume that upon receiving a stimulus θ/C22at time t= 0, the network has a random spon-taneous firing rate a(θ, 0). Using Equation 15, with c02ℝandcμ=c/C3-μ, we expand the activity profile in terms of the initial values of the corresponding coefficients cμ(0): aθ;0ð Þ ¼X μcμ0ð Þ^eμθð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp c00ð Þ þ X μ≥12c R-μ0ð Þcosμθð Þ þ 2c I-μ0ð Þsinμθð Þ/C16/C17() ¼1ﬃﬃﬃﬃﬃﬃ 2πp c00ð Þ þ X μ≥12rμ0ð Þcosμθ-ϕμ0ð Þ/C0/C1/C2/C3()(42) with tan( ϕμ)=c I-μ c R-μand r2 μ=(c I-μ)2+(c R-μ)2such that ϕμ(0) are completely determined by the initial conditions. Thus, at t= 0 the activity profile is composed of an infinite sum of cosine functions, each peaked about a corresponding disparate angle ϕμ, and therefore has no dis-cernible peak. To characterize the evolution of the network activity from these initial condi-tions to its hue tuning profile at t∞, we seek to obtain the steady-state values of ϕμand the corresponding tuning curve peak inductively as follows. Let us first take μ= 1. As seen in Figure 4, we note that δ1(t) and δ2(t) are symmetric about γ(t) such that δ2+γ=2π-(δ1+γ). Using this symmetry, we factor out cos( γ) and sin( γ), respectively, in the equations for c R-1and c I-1in Equation 26: τ0dc R-1 dt¼-c R-1þβﬃﬃﬃﬃﬃﬃ 2πp F1cosγ τ0dc I-1 dt¼-c I-1-βﬃﬃﬃﬃﬃﬃ 2πp F1sinγ(43) with F1¼ch 2δ2-δ1 ð Þ þ sinδ2-δ1 ð Þ 1/2/C138 þ 2T-q0 ð Þ sinγþδ1 ð Þ (44) and time arguments suppressed. We let F⋆and γ⋆denote the steady-state values of Fand γ, respectively, allowing for the following expressions for the steady-state values of c R-1and c I-1: c R⋆-1¼βﬃﬃﬃﬃﬃﬃ 2πp F⋆ 1cosγ⋆ c I⋆-1¼-βﬃﬃﬃﬃﬃﬃ 2πp F⋆ 1sinγ⋆:(45) Thus, we have tanϕ⋆ 1/C0/C1 ¼c I⋆-1 c R⋆-1¼-tanγ⋆ð Þ : (46) Similar calculations for the steady-state values of the higher order coefficients yield the general equations τ0dc R-μtð Þ dt¼-c R-μtð Þ þβﬃﬃﬃﬃﬃﬃ 2πp Fμtð Þcosγtð Þð Þ τ0dc I-μtð Þ dt¼-c I-μtð Þ-βﬃﬃﬃﬃﬃﬃ 2πp Fμtð Þsinγtð Þð Þ :(47) Network Neuroscience 706Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
As before, we note that the evolution of cμ(t), and therefore of Fμ(t),∀μ2ℤdepends only on the first-order coefficients c\|μ\|≤1(t). Therefore, the steady-state values of the higher order coefficients c R⋆-μ¼βﬃﬃﬃﬃﬃﬃ 2πp F⋆ μcosμγ⋆ð Þ c I⋆-μ¼-βﬃﬃﬃﬃﬃﬃ 2πp F⋆ μsinμγ⋆ð Þ(48) and the corresponding ϕμ, that is, tanϕ⋆ μ/C16/C17 ¼c I⋆-μ c R⋆-μ¼-tanμγ⋆ð Þ ; (49) are fully determined by the solution to Equation 26. Substitution of Equation 48 into Equation 15 then gives: a∞θð Þ ¼1ﬃﬃﬃﬃﬃﬃ 2πp c⋆ 0þβ πX μ≥1F⋆ μcosμγ⋆ð Þ cosμθð Þ-F⋆ μsinμγ⋆ð Þ sinμθð Þ/C16/C17 ¼1ﬃﬃﬃﬃﬃﬃ 2πp c⋆ 0þβ πX μ≥1F⋆ μcosμθþγ⋆ð Þð Þ ;(50) so that θ=-γ⋆represents the peak angle of the steady-state profile a∞(θ). Further, from Equation 22, we have tanγ⋆ð Þ ¼-q⋆ I q⋆ R ¼-csinθ/C22-β πλ-1F⋆sinγ⋆ ccosθ/C22þβ πλ-1F⋆cosγ⋆(51) which requires γ⋆¼-θ/C22: (52) That is, the steady-state peak-γ⋆is equivalent to the LGN hue input θ/C22. Linear Stability Analysis This section presents the mathematical details for obtaining Equation 30. Adding a small perturbation δa(θ,t) to the steady-state tuning curve and substituting the resulting network activity aθ;tð Þ ¼ a∞θð Þ þ δaθ;tð Þ (53) into Equation 1, we obtain: τ0dδaθ;tð Þ dt¼-a∞θð Þ þ δaθ;tð Þ ð Þ þ βh∞θð Þ þ δhθ;tð Þ-T ð Þ H h∞θð Þ þ δhθ;tð Þ-T ð Þ ;(54) Network Neuroscience 707Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
where δh(θ,t) is a perturbation to the input due to δa(θ,t). Taylor expanding the right-hand side of Equation 54 in h(θ,t)≡h∞(θ)+δh(θ,t) about h(θ,t)=h∞(θ) then yields τ0dδaθ;tð Þ dt¼-a∞θð Þþδaθ;tð Þ ð Þ þβh∞θð Þ-T ð Þ H h∞θð Þ-T ð Þ þ δhθ;tð Þ H h∞θð Þ-T ð Þ þ Oδh2/C0/C1 /C8/C9 : (55) For small perturbations, the higher order terms in δh(θ,t) are negligible, and, using a∞(θ)= β(h∞(θ)-T)H(h∞(θ)-T), we rewrite Equation 55 as τ0dδaθ;tð Þ dt¼-δaθ;tð Þ þ βδhθ;tð Þ H h∞θð Þ-T ð Þ : (56) Next, expanding δa(θ,t) as in Equation 29, we obtain τ0X∞ μ¼-∞d Dμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞Dμtð Þ^eμθð Þ þ βδhθ;tð Þ H h∞θð Þ-T ð Þ ; (57) wherein we express δh(θ,t) in terms of Equation 21 to yield: τ0X∞ μ¼-∞d Dμtð Þ dt^eμθð Þ ¼-X∞ μ¼-∞Dμtð Þ^eμθð Þ þβδq0tð Þ þ δq Rtð Þcosθð Þ þ δq Itð Þsinθð Þ 1/2/C138 H h∞θð Þ-T ð Þ : (58) Finally, taking the inner product of Equation 58 with êν(θ), and reformulating the thresh-olding nonlinearity in terms of the critical cutoff angles δ1and δ2as in section Evolution of Network Activity, we arrive at τ0d Dνtð Þ dt¼-Dνtð Þ þ βZδ⋆ 2 δ⋆ 1δq0tð Þ þ δq Rtð Þcosϕð Þ þ δq Itð Þsinϕð Þ 1/2/C138 ^e/C3 νϕð Þdϕ; (59) where δ⋆ 1andδ⋆2are the steady-state values of the cutoff angles. ACKNOWLEDGMENTS The authors acknowledge the fruitful and stimulating discussions with Wim van Drongelen and Graham Smith, and research support from the Oberlin College libraries. AUTHOR CONTRIBUTIONS Zily Burstein: Conceptualization; Data curation; Formal analysis; Investigation; Methodology;Project administration; Resources; Software; Supervision; Validation; Visualization; Writing-original draft; Writing-review & editing. David D. Reid: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Software; Supervision;Validation; Visualization; Writing-review & editing. Peter J. Thomas: Conceptualization; Methodology; Resources; Writing-review & editing. Jack D. Cow an: Conceptualization; Methodology; Project administration; Resources; Writing-review & editing. FUNDING INFORMATION Peter J. Thomas, Directorate for Mathematical and Physical Sciences ( https://dx. doi. org/10. 13039/100000086 ), Award ID: DMS-2052109. Peter J. Thomas, Office of Research Infrastruc-ture Programs, National Institutes of Health ( https://dx. doi. org/10. 13039/100016958 ), Award ID: R01 NS118606. Network Neuroscience 708Pattern forming mechanisms of color vision Downloaded from http://direct. mit. edu/netn/article-pdf/7/2/679/2118501/netn_a_00294. pdf by guest on 09 August 2024	netn_a_00294.pdf
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Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms Vincent A. Billock and Brian H. Tsou U. S. Air Force Research Laboratory An extraordinary variety of experimental (e. g., flicker, magnetic fields) and clinical (epilepsy, migraine) conditions give rise to a surprisingly common set of elementary hallucinations, includingspots, geometric patterns, and jagged lines, some of which also have color, depth, motion, andtexture. Many of these simple hallucinations fall into a small number of perceptual geometries—the Klu¨ver forms—that (via a nonlinear mapping from retina to cortex) correspond to even simpler sets of oriented stripes of cortical activity (and their superpositions). Other simple hallucinations(phosphenes and fortification auras) are linked to the Klu ¨ver forms and to pattern-forming cortical mechanisms by their spatial and temporal scales. The Klu ¨ver cortical activity patterns are examples of self-organized pattern formation that arise from nonlinear dynamic interactions between excit-atory and inhibitory cortical neurons; with reasonable modifications, this model accounts for a widerange of hallucinated patterns. The Klu ¨ver cortical activity patterns are a subset of autonomous spatiotemporal cortical patterns, some of which have been studied with functional imaging tech-niques. Understanding the interaction of these intrinsic patterns with stimulus-driven corticalactivity is an important problem in neuroscience. In line with this, hallucinatory pattern formationinteracts with physical stimuli, and many conditions that induce hallucinations show interestinginteractions with one another. Both types of interactions are predictable from neural and psycho-physical principles such as localized processing, excitatory-inhibitory neural circuits, lateral inhi-bition, simultaneous and sequential contrast, saccadic suppression, and perceptual opponency. Elementary hallucinations arise from familiar mechanisms stimulated in unusual ways. Keywords: intrinsic neural activity, Klu ¨ver form constant, migraine fortification aura, phosphene, spatiotemporal pattern formation What are these Geometrical Spectra? and how, and in what depart-ment of the bodily or mental economy do they originate?... there isa kaleidoscopic power in the sensorium to form regular patterns by thesymmetrical combination of causal elements.... —The astronomer Sir John F. W. Herschel (1867) on hallucinationsseen during migraine and during surgery “under the blessed influ-ence of chloroform. ” I had two days spoiled by a psychological experiment with mescal, an intoxicant used by some of our Southwestern Indians in their religiousceremonies; a sort of cactus bud, of which the U. S. Government haddistributed a supply to certain medical men, including Weir Mitchell, who sent me some to try. He himself had been in “fairyland. ” It givesthe most glorious visions of color.... I took one bud three days ago, was violently sick for 24 hours, and had no other symptoms.... I will take the visions on trust! —William James (1920) in a June 11, 1896, letter to his brother Henry. For many, mention of visual hallucinations brings to mind complex visual imagery: the stuff of waking dreams. This ismisleading; simple forms and patterns are more typical than Freud-ian drama. Elementary hallucinations—generated by a host ofclinical and laboratory-induced states—range from simple spots tothe flashing serrated arcs that accompany migraines to the kalei-doscopic patterns induced by some drugs. Until recently, this sheervariety made hallucinations seem like a collection of disparatecuriosities ill-posed for serious study. However, there are at leastfour reasons why this view is mistaken. First, there is much moreorder in these conditions than is generally appreciated. There aremany links and interactions between hallucinatory conditions. Moreover, even very different-seeming elementary hallucinationstake place on common spatial and temporal scales that correspondto identifiable neural mechanisms. Indeed, some different-seeminghallucinations can have the same pattern of cortical activity, dif-fering only by cortical position and orientation. The strong con-nections between elementary visual hallucinations and particularneural systems led Frances Wilkinson (2004) to call hallucinations This article was published Online First March 26, 2012. Vincent A. Billock, National Research Council, U. S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio; Brian H. Tsou, U. S. Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio. This research was funded in part by a National Research Council/Air Force Office of Scientific Research Senior Research Award to Vincent A. Billock. We thank Scott Kelso and Mingzhou Ding for introducing us to this topic;Oliver Sacks for suggesting that we look at relationships between patternsformed under migraine and image stabilization; Angela Brown, Lynn Olzak,Keith White, and Ewen King-Smith for helpful discussions on migraine,functional imaging limitations, and retinal sampling; and Bard Ermentrout and Whitman Richards for comments on an earlier version of the manuscript. Correspondence concerning this article should be addressed to Vincent A. Billock, NRC, Room 210, Building 248, Wright-Patterson Air Force Base, OH 45433. E-mail: vincent. billock. ctr@wpafb. af. mil Psychological Bulletin In the public domain 2012, Vol. 138, No. 4, 744-774 DOI: 10. 1037/a0027580 744	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
“windows on the visual brain. ” Second, elementary hallucinations illustrate an important general principle in cognitive science. It isnow well recognized that complex systems, such as the humanbrain, have collective properties that are not inherent in the indi-vidual neural elements; it has become common to ascribe somecognitive behaviors as emergent properties of the collective sys-tem. However, such attributions are seldom well worked out andare most often reasoned by analogy to better studied complexphysical systems. Elementary hallucinations—and especially geo-metric hallucinations—provide particularly well-worked-out ex-amples of a neural emergent behavior (self-organized spatiotem-poral pattern formation) that corresponds to specific percepts. Third, a rather perverse aspect of the study of the human brain asa complex system is that the brain typically behaves in a decep-tively stable and veridical fashion—much like a linear system; ifwe want to understand the brain as a nonlinear dynamic complexsystem we need to push it outside of its normal pseudolinearregime. 1Perceptual catastrophe then becomes a tool of systems identification. Finally, recent advances in neuroscience have given new importance to understanding the relationship betweenstimulus-driven and autonomous neural pattern formation. A con-vergence of research in complexity theory, neural modeling, visualpsychophysics, and functional brain imaging dramatically ad-vances understanding of perceptual pattern formation and high-lights interactions between intrinsic neural activity and physicalstimulation. Hallucinatory Conditions and the Percepts They Induce An Overview of Elementary Hallucinatory Visual Percepts This review focuses on elementary hallucinations and excludes complex imagery associated with dreams, dementia, delirium, andthe latter phases of drug intoxication and sensory deprivation(useful reviews of complex hallucinations include Aleman &Larøi, 2008; Collerton, Perry, & Mc Keith, 2005; Horowitz,1978). 2We also exclude the fascinating disturbances of shape, size, motion, and color that accompany fever and neurological disorders (e. g., the Alice in Wonderland syndrome that can heraldmononucleosis). An amazing variety of external and internal con-ditions lead to elementary hallucinations (see Figure 1 and Table1). A partial list includes migraine, epilepsy, hypoglycemia, theearly stages of some drug intoxications, ocular pressure, retinaland cortical electric stimulation, transcranial magnetic stimulation(TMS), and photopic stimulation, especially by uniform flickeringlights. The ubiquity, ease of generation, and geometric nature ofthese hallucinations have been a clue to neural modeling and agoad to perceptual experimentation. There are three basic types ofelementary hallucination: phosphenes, geometric forms, and forti-fications. The simplest illusory forms are phosphenes—usuallysmall spots of light—often induced by focal stimulation of the eyeor brain (e. g., by ocular pressure, electricity, and, more exotically,cosmic rays; Sannita, Narici, & Picozza, 2006). Phosphenes varyin size with position in the visual field and have a variety ofshapes, colors, and movements. The ability to generate phospheneshas been an opportunity to those who would create visual pros-theses for the blind and a nuisance for those who would keep the induced percepts simple. Often, phosphenes are induced in greatnumbers (polyopia) and are organized by some geometric principle(see, e. g., Figure 1E); they may appear to be hexagonally packedor to be texture elements on some greater geometric figure. Themost common geometric figures—the Klu ¨ver forms—are lattices (often hexagonal), polar webs, fan shapes, pinwheels, spirals,concentric circles, and some three-dimensional analogues (tunnels,funnels, etc. ). Fortifications are bright flashing serrated arc-shapedpatterns often seen in migraine; occasionally, similar zigzag forms,like a piece of a fortification arc, are seen in epileptic seizure,hypoglycemia, and other conditions (Klu ¨ver, 1966; Purkinje, 1819/1823). When hallucinations accompany migraine or epi-lepsy, they are referred to clinically as auras. Migraine and Fortification Illusions Migraine hallucinations range from Klu ¨ver's geometric forms to complex multisensory hallucinations (which resemble temporal-lobe epilepsy). Even within an observer, more than one hallucina-tion can occur; for example, Oliver Sacks and John Herschelreported both Klu ¨ver geometries and fortification patterns, some-times during the same attack (Sacks, 1995a). In subjects experi-encing more than one percept, phosphenes are generally seenbefore fortifications, which are seen before Klu ¨ver patterns (Sacks, 1995b); geometric patterns, like lattices, are also sometimesglimpsed between fortification features. However, as fortificationauras are considered characteristic of migraine, we focus on for-tifications here. Fortifications are bright line segments set atroughly 60° angles to one another along a vertical arc (see Figure1H). In migraine, many patients report that the line segmentsflicker at about 10 Hz, but most patients have no experience inaccurately naming flicker rates. Crotogino, Feindel, and Wilkinson(2001) flicker-matched migraine scintillations to a physical stan-dard in 11 observers: The average matched scintillation rate was17. 8 Hz, and individual subjects made consistent flicker matchesacross attacks. The segments are usually bright white, but colorscan sometimes be seen between them (Sacks, 1995a) or on theirtips, like match-ends; red, yellow, and blue, in that order, are mostcommon (Richards, 1971). The jagged structure resembles an oldstyle of fortification with many angled sides to deprive primitivewall-breaching artillery of a clean nonglancing shot. The illusiongenerally starts near the fovea in one visual hemifield and movesslowly toward the periphery. Near the fovea, the angled featuresare small, and the arc is nearly circular. The angles between theserrations are between 45° and 60°. As the arc moves toward theperiphery, its individual angular features enlarge, and the arcstraightens into a nearly vertical shape that appears to move morequickly through the visual field. Dahlem and Mu ¨ller's (2003) 1The required push may not be very large. There is some evidence that the cortex is critically poised, in the sense that relatively small changes can trigger avalanches of neuronal activity (Plenz & Thiagarajan, 2007; Wer-ner, 2007). This is in accord with the relatively modest amount of stimu-lation needed to induce some hallucinations. 2Some investigators report that complex hallucinations can develop from simpler ones (e. g., a geometric pattern of radiating lines becomes the legs of a spider upon introspection; Horowitz, 1978; Hughlings Jackson,1958). 745 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
model describes the overall changes in arc shape and apparent speed as it propagates across cortex; this behavior is generic forweakly excitable media and can be mimicked by reaction-diffusionmodels (Dahlem & Hadjikhani, 2009). A temporary scotoma(blind region) is left in the wake of the fortification arc's move-ment. Mapping the serrated arcs to cortical coordinates reveals thateach serration covers about 1 mm of cortex and that the arc movesat a constant and rather stately speed of 2-3 mm/min on cortex(Gru¨sser, 1995; Lashley, 1941; Richards, 1971; Wilkinson, Fein-del, & Grivell, 1999), requiring 20-25 min to traverse one entireside of striate cortex. Richards (1971) suggested that the angularity of fortifications was consistent with activation of hexagonally packed orientationhypercolumns as a trigger wave swept through them (see Figure 2). The packing that Richards predicted is strikingly similar to thecortical iso-orientation pinwheel organization ultimately revealedby neuroanatomy (Bonhoeffer & Grinvald, 1991; Swindale, Mat-subara, & Cynader, 1987). For other developments in fortificationmodels, see Dahlem, Engelman, Löwel, and Mu ¨ller (2000); Reggia and Montgomery (1996); and Schwartz (1980). When the psycho-physics of migraine are compared to the topographical mappingqualities of visual cortices, the likely cortical loci of migrainepercepts are Areas V1, V3a, and V8. Jagged arcs are consistentwith the orientation processing in V1, but there is no reason toassume that other visual areas cannot be activated during migraine,and if activated, there is no reason to assume that this activitycould not affect V1 via feedback. (Indeed, based on studies ofcortical spreading depression, the condition could spread over theentire occipital lobe of the affected hemisphere but could havedifficulty crossing prominent fissures between cortical areas, likethe parieto-occipital sulcus. ) Sacks (1995a) reported a range ofphenomena consistent with the activation of many sensory areas. Interestingly, Hadjikhani et al. (2001) had a subject with an un-usual exercise-induced aura—a drifting crescent-shaped cloud of TV-like noise—shown by functional magnetic resonance imaging(f MRI) to originate in V3a. (This percept resembled the twinklingtextures induced adjacent to a centrally viewed patch of TV noise;Tyler & Hardage, 1998. ) Hadjikhani et al. suggested that classicfortification illusions may arise in V1 and color effects in V8. Functional imaging also shows cortical thickening abnormalities inareas V3a and MT of the brains of migraineurs, which is interest-ing because MT is important in motion perception and migraineursare especially susceptible to visual motion-induced sickness(Granziera, Da Silva, Snyder, Tuch, & Hadjikhani, 2006). The slow movement of the fortification arcs suggests a diffusive-triggering process. The closest physiological analogue tothe spread of a migraine fortification arc (and its accompanyingscotoma) is a wave of cortical spreading depression, triggered inanimal preparations by an infusion of potassium. The depressionaspect is a matter of temporal scale: Initially, the spreading waveof extracellular potassium renders affected neurons briefly hyper-excitable, but as potassium concentration increases, the neuronsbecome so depolarized that further action is suppressed for alonger period. In humans, Wilkinson (2004) suggested that a“wavefront of neural excitation operating on intrinsic corticalnetworks is presumed to underlie the positive hallucinations andthe subsequent neuronal depression, the scotoma” (p. 308). Had-jikhani et al. (2001) found eight aspects of f MRI imageryduring migraine corresponding to known aspects of cortical Figure 1. Some characteristic elementary visual hallucinations. A-D: These LSD flashbacks painted by Oster (1970) come in circular, radial and spiralgeometries, three of the most common percepts cataloged by Klu ¨ver (1966) for many hallucinatory conditions. E: A proliferation of identical phosphenes (poly-opia) induced by THC and arranged in a spiral geometry (Siegel & Jarvik, 1975). F-G: Some more complicated lattice-like patterns produced by THC intoxication(Siegel & Jarvik, 1975) and by binocular pressure on the eyes (Tyler, 1978). H:Superposition of fortification patterns produced by migraine; actual patterns flashand move across retina (Richards, 1971). Panels A-D from “Phosphenes,” by G. Oster, 1970, Scientific American, 222 (2), p. 82. Reprinted with permission. Copy-right 1970 Scientific American, a division of Nature America, Inc. All rights reserved. Panels E-F from “Drug-Induced Hallucinations in Animals and Man,”by R. K. Siegel and M. E. Jarvik, in R. K. Siegel and L. J. West (Eds. ),Hallucinations (pp. 117 & unnumbered page [Color Plate 6] following p. 146), 1975, New York, NY: John Wiley & Sons. Copyright 1975 by John Wiley &Sons. Reprinted with permission. Panel G from “Some New Entopic Phenomena,”by C. W. Tyler, 1978, Vision Research, 18, p. 1637. Copyright 1978, with permission from Elsevier. Panel H from “The Fortification Illusions of Migraines,”by W. Richards, 1971, Scientific American, 224 (5), p. 90. Copyright 1971 by W. Richards. Reprinted with permission. 746 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
spreading depression. One discrepancy is that the symptoms of migraine normally do not spread as far as the cortical spreadingwave, suggesting that the strength of the wave falls below anactivation threshold in the unaffected region (Dahlem & Had-jikhani, 2009). Since the first neural effect of cortical spreading depression is thought to be transient excitation, it is interesting that a variety ofevidence suggests that the brains of migraineurs with aura arechronically hyperexcitable between attacks (Aurora & Wilkinson,2007). For example, thresholds for phosphene generation by TMSare reduced in subjects who experience migraine aura (Aurora,Ahmad, Welch, Bhardhwaj, & Ramadan, 1998; Fierro et al., 2003;Mulleners, Chronicle, Palmer, Koehler, & Vredeveld, 2001; Mul-leners, Chronicle, Vredeveld, & Koehler, 2002). Psychophysicalevidence consistent with hyperexcitability or hypo-inhibition hasbeen adduced by Chadaide et al., 2007; Chronicle, Wilkins, and Coleston (1995); Mulleners, Aurora, et al. (2001); Palmer, Chron-icle, Rolan, and Mulleners (2000); and Wilkinson, Karanovic, and Wilson (2008). Epilepsy In epilepsy, like migraine, many unusual events are possible, depending on the cortical locus of the seizure. Compared to mi-graine, epileptic hallucinations are brief, with typical durations ofseveral seconds rather than many minutes (a strong diagnosticdiscriminator between migraine aura and petit mal occipital epi-lepsy; Panayiotopoulos, 1999). This is consistent with electrophys-iological findings that epileptiform activity is orders of magnitudefaster than the 3 mm/min cortical spread of migraine (Chervin,Pierce, & Connors, 1988; Penfield & Rasmussen, 1950). Thepercentage of epilepsy cases with visual symptoms (about 11% in Penfield & Kristiansen, 1951) seems surprisingly small (given theshare of cortex involved in some ways with vision); symptomsrange from simple phosphenes to the complicated scenic halluci-nations of temporal lobe epilepsy (Fried, Spenser, & Spenser,1995; Penfield & Kristiansen, 1951; Wilkinson, 2004; Williamson, Thadani, Darcey, Spenser, & Mattson, 1992). Phosphenes have avariety of simple shapes (stars, streaks, spots, annuli, wedges, andoccasionally a zigzag like a piece of a fortification illusion) andmay be strongly colored. In occipital epilepsy, circular phosphenesand annular forms (e. g., doughnut phosphenes) seem particularlyprevalent (Panayiotopoulos, 1999). Single-wedge and double-wedge (butterfly or bow-tie) phosphenes are also seen. Wilkin-son's (2004) review found that “components may be stationary andlong-lasting, but more commonly flicker, pulsate, twinkle or move. Several patterns of motion (translational, rotary, expansion, con-traction and random) have been described” (p. 311). If a subjecthas a characteristic epileptic aura, it can often be triggered by focalelectrical stimulation during neurosurgery (Penfield & Rasmussen,1950). Occipital epilepsy and its associated hallucinations can becontrolled by medicines that dampen neuronal excitability (e. g.,carbamazepine; Panayiotopoulos, 1999). Ocular-Pressure-Induced Forms It has been known since antiquity that illusory images are induced by applying pressure to the eyeballs; a light spot (phos-phene) appears opposite the point of pressure (if localized) and canbe seen in the dark. Some believed that ocular pressure createdinternal light, but Descartes and Newton correctly anticipated thatpressure mechanically stimulates the retina (Gru ¨sser & Hagner, 1990; Wade & Brozek, 2001). Phosphenes can sometimes be seenduring eye movements and accommodation in the dark, or whenocular pressure is abruptly raised (e. g., sometimes by coughing;Gru¨sser & Landis, 1991). The immediate retinal effect of ocular pressure is to activate retinal on-center ganglion cells (after a delayof 0. 2 s) and inhibit off-center ganglion cells, resulting in percep-tion of a light increment (Gru ¨sser, Gru ¨sser-Cornhels, Kusel, & Przybyzewski, 1989; Gru ¨sser, Hagner, & Przybyswewski, 1989). Gru¨sser and colleagues (Gru ¨sser, Gru ¨sser-Cornhels, et al., 1989; Gru¨sser, Hagner, & Przybyswewski, 1989) theorized that defor-Table 1 Conditions That Trigger Hallucinations and the Nature of the Resulting Hallucinations Condition Geometric pattern Fortification Phosphene Complex Drugs O, C, D Sa S O,C,D Photopic O, C, D Va S, C Vb Ocular pressure O, C, D OMigraine S, C O, C, D S, C SEpilepsy S, C S a S, C, D S Hypoglycemia S Sa Transcranial magnetic stimulationc O, C, D O Electrical (cortex)d S, P O, D Se Electrical (retina)f CO Charles Bonnet syndrome O O O, CSensory deprivation O O S Note. O/H11005often; S /H11005sometimes; V /H11005very seldom; C /H11005colored; D /H11005dynamic; P /H11005pinwheel (usually small and rotating). a Sometimes a serrated form resembling a fortification illusion occurs but is usually stationary and usually does not flash. b If only one eye is stimulated by flicker while the other eye is kept dark, binocular rivalry between the eyes leads to switching between the geometric hallucinations and dark-phase hallucinations, which are swirling amorphous structures (Smythies, 1959a). c Noninvasive focal electrical stimulation of neural tissue, by induction, using a temporally alternating, spatially focused magnetic field. d Usually done with implanted electrodes as part of visual prosthesis studies, or during surgery. e Depends on which cortical area is stimulated. f Usually done with current applied to entire orb. 747 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
mation stretches the horizontal cell layer, depolarizing on-center bipolar cells and hyperpolarizing off-center cells. Continued deepocular pressure (for more than about 40 s) results in a temporaryblindness via ischemia (like applying a choke hold directly to theretina)—a useful technique often employed by psychophysicists. In the time between phosphene induction and eventual blindnesslies an interesting perceptual phenomenon: Sustained simultaneousbinocular pressure evokes complicated Klu ¨ver-like percepts, like the lattice spiral in Figure 1G (Tyler, 1978). The requirement forbinocular stimulation is intriguing; the instructive exception is thatfor some amblyopic subjects, stimulation of the good eye is suf-ficient to induce geometric forms (Gru ¨sser & Landis, 1991. ) Most pressure percepts are not consistent with the entopic (intrinsicanatomical) retinal structures posited by Purkinje (1819/1823), andtheir commonality with other geometric hallucinations suggeststhat pressure-induced excitation drives a cortical pattern-formingmechanism. Electrical Stimulation of Retina After Galvani's dramatic demonstrations of electrically in-duced muscle contraction and Volta's invention of the battery,it was not long until electric current was applied to the head andespecially the eye. As for so many visual effects, Jan Purkinje(1819/1823) led the way. Light sensations were generally re-ported at onset and offset of current. Some studies placed one electrode on the roof of the mouth and the other on the fore-head; others applied current directly to the eye, sometimesimmersing the eyes in salt water and passing the current thoughthe solution. Vivid colors, phosphenes, and patterns were pro-duced in these experiments. At high frequencies, large coloredcrescent, horseshoe, smoke-ring and ameba-shaped phosphenesappear, which persist to stimulation rates of 100 Hz in the darkand up to 210 Hz in light-adapted eyes (Wolff, Delacour,Carpenter, & Brindley 1968); the retina can follow higher rateswith electricity than light, in part because the photochemicalbottleneck in the photoreceptors is bypassed. Wolff et al. (1968)found that geometric patterns seen for 10-40 Hz sinusoidalelectrical stimulation were equivalent to those seen for flicker-ing light. There is an understated heroic quality to many ofthese early experiments, which mostly used the investigators assubjects. Brindley (1955) noted that his experiment, whichapplied the electrodes directly to the conjunctiva of the eye,required topical anesthesia; despite this, the experiment wasdifficult because the current through his anesthetized eye spreadto other nerves and made his teeth ache severely. Alternately, electric retinal stimulation can produce effects unique to retinal dynamics. Carpenter (1973) electrically stimu-lated the eye with 100 Hz AC current and found that if a darkobject passes through the visual field, a series of thin light illusorycontours form in the passing object's wake. These contours obeyfour rules: (a) The illusory contours disappear promptly when thecurrent is turned off. (b) One contour forms for each cycle of thealternating current completed while the moving object transitsthe visual space. (c) Half of the contours move with the wake; theother half move against it. (d) If two contours collide, they neitherpass through one another nor mutually annihilate, but rather, theymerge to form loop-like structures. Drover and Ermentrout (2006)modeled the formation of such contours by populations of gap-coupled neurons responding to every other cycle of the drivingfrequency: Some contours are generated by even-numbered stim-ulus cycles and others generated by odd-cycles, each with a dif-ferent motion bias as they form traveling waves. These wakecontours are reminiscent of Charpentier's bands (Mc Dougall,1904): a series of three to four afterimages produced over about 50ms when a bright bar is moved in the dark. This periodicitycorresponds to a sampling rate of at least 60 Hz. Drover and Ermentrout's half-sampling mechanism for wake contours, if ap-plied to the 120 Hz oscillations found in the electroretinogram(ERG; King-Smith, Loffing, & Jones, 1986), might account for Charpentier's bands as well. Electric Stimulation of Cortex (Including TMS) High voltages applied to the scalp near visual cortex produce diffuse cortical stimulation, creating textured phosphenes sev-eral degrees across (Merton & Morton, 1980). Knoll, Kugler,Eichmeier, and Höffer (1962) used temporal electrodes drivenby 5-100 Hz electric current to induce geometric patterns in 46observers similar to patterns seen in flickering lights and indirect electric cortical stimulation experiments (Penfield &Rasmussen, 1950). Many of Knoll et al. 's subjects saw thehallucinatory patterns better when light-adapted, a fact notedrepeatedly for other hallucinatory inducers. Eichmeier and Nie-Figure 2. An interesting interpretation of a migraine attack like that depicted in Figure 1H (a superposition of several migraine fortificationpatterns seen by one subject during the course of an attack). Each verticalarc in Figure 1H represents one snapshot of the illusion. Here, Richards(1971) has connected features from successive fortification patterns, re-vealing a hexagonal-like pattern of underlying activity that closely resem-bles the hexagonal packing of cortical orientation hypercolumns revealedby neuroanatomy. From “The Fortification Illusions of Migraines,” by W. Richards, 1971, Scientific American, 224 (5), p. 93. Reprinted with permis-sion. Copyright 1971 Scientific American, a division of Nature America, Inc. All rights reserved. 748 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
dermaier (1976) found that it was easier to electrically stimulate geometric hallucinations at high altitudes, presumably as a sideeffect of hypoxia. This is interesting because hypoxia alsofacilitates triggering the cortical spreading depression phenom-enon linked to migraine (Dahlem & Mu ¨ller, 2004; Grafstein, 1963; Lea ˜o, 1963). Some transcranial electric stimuli may also inadvertently excite the eye (Motokawa, 1970; Paulus, 2010);the persistence of hallucinations after temporarily pressure-blinding the eyes is a useful check. Direct electric stimulation of the human cortex (V1-V3) was used by Penfield and Rasmussen (1950) while locating surgi-cally excisable epileptic foci. Stimulation of V1 usually leads tosmall colored phosphenes. Stimulation outside V1 often leadsto moving colorless phosphenes. Pulse trains are more effectivethan single stimuli or direct current, and phosphene thresholdsare smallest in cortical layers 4-6 (Bak et al., 1990). Pulsetrains longer than about 15 s become ineffective (Dobelle,Mladejovsky, Evans, Roberts, & Girvin, 1976; Dobelle, Quest,Autunes, Roberts, & Girvin, 1979). Electrode separations of 1mm or less result in a single phosphene (Bak et al., 1990),consistent with estimates of human cortical column size. Sur-face electrodes tend to induce flicker, while deeper electrodestend to induce steady deeply colored circular phosphenes. In-creasing stimulus strength raises the phosphene's brightnesswhile reducing its size (Bak et al., 1990, and Evans, Gordon,Abramov, Mladejovsky, & Dobelle, 1979, found a logarithmicrelation for phosphene brightness). Brindley and Lewin (1968)used arrays of phosphene-inducing electrodes while developinga neural prosthetic for the blind. Maps of phosphene size as afunction of electrode position can be used to estimate corticalmagnification: Phosphenes generated in central vision are punc-tuate, while phosphenes in peripheral vision may be severaldegrees wide (Dobelle et al., 1976, 1979). TMS is a special case, intermediate to transcranial electric stimulation and direct cortical stimulation using small elec-trodes. An alternating magnetic field—applied across theskull—induces electric currents in the cortex. Unlike transcra-nial electric stimulation, TMS stimuli are generally painless. Special antennas localize stimulation to several centimeters ofcortical area, which, although not as precise as those induced bydirect cortical electric stimulation, is more convenient; based onother induced hallucinations, the increased spatial extent ofstimulation is likely to bias the system to geometric hallucina-tions rather than simple phosphenes. In Marg and Rudiak(1994) and Kammer, Puls, Erb, and Grodd (2005), the mostcommon percepts were wedges and butterfly patterns (doublewedges); grids and hexagonal lattice patterns were also re-ported. Single pulses of TMS do not produce illusory features ininexperienced observers (Kammer et al., 2005), but short trainsof stimulation (two to five pulses) reliably produce hallucina-tions in all observers (Boroojerdi et al., 2002; Ray, Meador,Epstein, Loring, & Day, 1998). The position and shape of thephosphene in Krammer et al. 's experiment did not vary muchwith the skull position of the coil over occipital cortex; toexplain this invariance, Kammer et al. speculated that their TMS activated cortex indirectly, via the optic radiations to V1and back-projecting fibers from V2 and V3 onto V1. Flicker-Induced Hallucinations Flicker-induced hallucinations were discovered by Purkinje (1819/1823) and Brewster (1834). Purkinje evoked colored pat-terns while waving his spread fingers between a bright light andhis shut eyes. Brewster (the inventor of the kaleidoscope) sawillusory colored patterns while rushing past a sunlit fence with hiseyes shut—a result enjoyably replicable for passengers on fastdrives through sunlit forests (Billock & Tsou, 2010). Both casesillustrate the optimal stimulus for flicker-induced hallucina-tions—a strong source of spatially homogeneous temporal modu-lation (closed eyes are relatively uniform diffusers). Later studiesby Fechner (1838) and others used rotating black-and-white sectordisks: At a sufficiently high rotation rate, the individual rotatingsectors fuse into a uniform flickering gray. Many observers seehexagonal pastel-colored patterns on the disk under these circum-stances, but other patterns are reported as well. Most modernstudies of flicker-induced hallucinations use stroboscopic illumi-nation of a uniform surface (a ganzfeld) or the flickering emptyscreen of a CRT. In clinical studies, illusory forms and colors area frequently reported side effect of flicker ERG/electroencephalo-gram (EEG) tests. W. G. Walter (1956) stimulated thousands ofpatients with stroboscopic-illuminated ganzfelds, all of whom re-ported illusory colors, movements, and patterns; the optimumfrequency for evoking the effects varied greatly between observ-ers, and the most common patterns were moving concentric ringsand spoke patterns. Mundy-Castle (1953) made the same claimsfor his 1,000 subjects. Many studies find that flicker rates of about4-25 Hz elicit illusory color and forms (Brown & Gebhard, 1948;Freedman & Marks, 1965; Mundy-Castle, 1953; Remole, 1971,1973; Smythies, 1959a, 1959b, 1960; V. J. Walter & Walter,1949). In our experience, hallucinations do occur at low flicker rates but are more salient at moderate rates (ca. 10-20 Hz). The form ofthe hallucinated pattern may be frequency dependent (e. g.,Allefeld, Pu ¨tz, Kastner, & Wackermann, 2011; Becker & Elliot, 2006; Young, Cole, Gamble, & Rayner, 1975). There seems to belittle consistency between studies, possibly due to differences instimulation techniques. The most ambitious attempt to determinefrequency dependency is Allefeld et al. 's (2011) strobed-ganzfeldstudy. They plotted frequency-of-seeing distributions for about3,000 occurrences of 17 different spatial percepts and found thatthey occur mostly in the 5-26 Hz range. Several of their 17perceptual patterns correspond to Klu ¨ver's forms (and Ermentrout-Cowan cortical stripe formation). Both spirals and concentriccircles (ripples) have perceived frequency distributions that peak atabout 15-18 Hz. Three separate categories correspond to radialhallucinations (wheel, sun, star) and have peaks between about 7and 18 Hz. Hexagonal (honeycomb) and rectilinear (raster) latticeshave peaks near 15 and 30 Hz, respectively. Phosphene-like illu-sions (spot, organic) are more common at 25-30 Hz. Color in thesehallucinations is most common at 11-18 Hz (much higher than for Fechner-Benham illusory colors, which are most vivid for flickerrates near 6 Hz). Remole (1971, 1973) measured luminance contrast thresh-olds for flicker-induced illusory geometric patterns as a func-tion of temporal frequency; some of these tuning functionsresemble resonance curves. For binocular vision, his subjectshad lower thresholds for flash rates of 10-18 Hz. For monoc-749 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
ular vision, thresholds were higher, and the function's minimum was (for two thirds of observers) shifted to higher frequencies. Slightly lower thresholds for binocular vision are expected oninformation theoretic grounds (by a factor of √2), which con-trasts with the near absolute requirement of binocular pressurefor ocular pressure hallucinations. Remole's results lie mostlybetween these extremes: At some frequencies, binocular thresh-olds are an order of magnitude lower than for monocular vision,while at other frequencies, the binocular advantage drasticallynarrows. Remole also examined wavelength effects: Flickeringyellows and reds required less luminance than blues and greensto induce hallucinations, and the threshold differences for redversus green and blue versus yellow were highest at about 14Hz; Remole believed this implicates color opponency in hallu-cinatory pattern formation. Reports vary on the qualia of col-ored hallucinations, with some subjects finding the colors un-earthly (W. G. Walter, 1956) and either too vivid or not vividat all (we have found pastel colors in our experiments, likethose induced in Fechner-Benham subjective color; Billock &Tsou, 2006). As W. G. Walter (1956) pointed out, on some levelthe colors are real enough: Brady (1954) conditioned subjects togive a galvanic skin response to red lights and found thatflicker-induced red led to the conditioned response. Although W. G. Walter (1956) made much of the strobo-scope's very short pulse width, Smythies (1960) tested variouswaveforms and light-to-dark pulse widths and concluded thatthe waveform's exact shape has little effect, if temporal alter-ation of light and dark is vigorous. In a foreshadowing ofexperiments using visual noise (Billock & Tsou, 2007; Mac-Kay, 1965), Smythies (1960) noted that random flicker alsoworks well. Smythies reported that sudden changes in theflicker frequency while hallucinating can cause the hallucinatedfeatures to become finer/more numerous with higher temporalfrequency and coarser/less numerous with lower temporal fre-quency. Smythies (1960) also tried stimulating the two eyeswith out-of-phase flicker and noted that the perceptual effectswere similar to doubling the frequency of stimulation. Given Remole's (1973) results, it would be interesting to know whateffect this has on the pattern-formation thresholds. When onlyone eye is stimulated, hallucinations vary with the state ofbinocular rivalry. Light-phase hallucinations (the geometricpercepts addressed throughout this article) occur during rivalryperiods when the stimulated eye is dominant, while dark-phasepercepts (swirling amorphous patterns, boiling, ameba-likeblobs, etc. ) occur when the patched eye is dominant (Brown &Gebhard, 1948; Smythies, 1959a). Smythies (1959a) arguedthat the patterns are visualizations of the noise in the patchedeye. This is supported by one study of spatiotemporal fractalnoise, which gave rise to percepts of similar character (Billock,Cunningham, Havig, & Tsou, 2001). They also resemble someof the smoke-like contours reported for some apparent-motion-induced hallucinations (Billock & Tsou, 2007) and for somenoise stimuli (Fiorentini & Mac Kay, 1965; Mac Kay, 1965). Other Photopic-Induced Hallucinations Motion-induced hallucination. There are at least two other kinds of photopically induced patterns: motion-induced hallucina-tions and Mac Kay effects. Among motion-induced illusions are theillusory and supernumerary spokes reported by Tynan and Sekular (1975); Holcombe, Macknik, Intrilgator, Seiffert, and Tse (1999);and Purves, Paydarfar, and Andrews (1996). A particularly spec-tacular motion-induced pattern-forming effect was discovered by Mayzner (1975), while manipulating the marquee light illusion. Mayzner linked flashing lights into a closed circular or squarecircuit (see, e. g., Figure 3); the flashing icons appear to moveabout the circuit, taking corners like ducks in a shooting gallery. When four or five equally spaced illusory-moving lights areviewed in the dark, the imaginary space defined by the movementfills with illusory colors and rotating shapes. The two studies toaddress this phenomenon only differ in the nature of the color. Mayzner described a succession of vibrant colors filling the space. Billock and Tsou (2007) found smoky swirls of more pastel colors,often green, purple, and gray, interlaced into a pulled-taffy-liketexture. This texture evolves into a storm-like pattern, with grayarms reaching out toward the icons. The arms straighten out into a Figure 3. An artistic rendition of a Mayzner illusion induced by illusory motion. Illusory motion is induced by turning icons on and off in turn (themarquee light illusion). Here, a square is defined by 20 such icons. If everyfourth icon is lit and then extinguished, followed by the icons clockwise tothem, and so on, the percept is of five icons moving around an imaginarysquare. Active icons are depicted in white, newly extinguished icons (stillfaintly visible due to visual persistence) are depicted in gray, and othericons are outlined in dashes. When viewed in the dark, the center of thesquare fills with a smoky or taffy-like colored texture that resembles astorm pattern. During viewing, radial arms appear in the storm pattern andreach out toward the moving icons, forming a five-bladed hallucinatorypropeller. However, the number of blades may be as great as the numberof visible icons (both the active icons and the ones visible because of visualpersistence). From “Neural Interactions Between Flicker-Induced Self-Organized Visual Hallucinations and Physical Stimuli,” by V. A. Billockand B. H. Tsou, 2007, PNAS: Proceedings of the National Academy of Sciences, USA, 104, p. 8493. Copyright 2007 National Academy of Sci-ences, USA. Reprinted with permission. 750 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
propeller shape rotating in time with the illusory movements of the flashing icons (see Figure 3). Mac Kay effects. Donald Mac Kay (1957a, 1957b, 1965, 1978) described a series of phenomena that have become known as Mac Kay effects, without much separate consideration of theirdiverse nature. Most of these effects involve some kind of oppo-nency between certain spatial forms. If a pattern of concentriccircles is steadily fixed for many seconds and then replaced with ablank field, a faint noisy fan-like shape can be seen as an after-image. Similarly, if the same pattern is viewed in flickering lightin an otherwise dark room, a fan-shaped pattern will be seensuperimposed on the concentric circles. Mac Kay treated fan shapesand concentric circles as complementary (opponent forms) insequential contrast with one another and interpreted the effects offlickering light as two phases of perception superimposed inone another: The physical concentric circles are seen during theflashes, and their complementary afterimage is seen during thedark phase. Similar superimposed patterns can be seen whileviewing closely spaced patterns under steady room light. Here, aflicker-like temporal stimulation is supplied by eye movementsacross the closely spaced features. This interpretation is reinforcedby the absence of this third form of the Mac Kay's hallucinatoryeffect under retinal image stabilization. Another variation of the Mac Kay effect was seen when the physical patterns were printedon transparencies and illuminated by dynamic spatial noise (e. g.,TV static) from behind. Mac Kay reported two forms of motion forthis stimulation: one that seems to flow along the transparencies'contours and one that seems to flow at right angles to thosecontours. The latter is like a Mac Kay opponent effect, but actualillusory contours are weak or absent. The strengths of the twoeffects seem to vary with the channel the TV noise is taken from,suggesting that the statistics of the noise matter. Billock and Tsou(2007) were able to strengthen the Mac Kay noise effect usingfractal noise; when viewed through concentric circles, hallucina-tory rotating fan shapes are seen; when viewed through fan-shapepatterns, pulsating hallucinatory circles are seen. Finally, Mac Kay(1965) and Fiorentini and Mac Kay (1965) noted that noise canaggregate into moving shapes under certain conditions: If framesof spatial noise are interspaced with frames of uniform luminance,the noise pixels seem to aggregate into maggot-like wrigglingforms. The effect is strongest in binocular vision and does notoccur if the noise is viewed by one eye and the blank frames by theother eye. Drug-Induced Hallucinations Hallucinations seen in the early stage of drug intoxications range from simple phosphenes to patterns as complicated as intri-cate Persian rugs. Klu ¨ver (1966) compared reports on drug-based hallucinations and found that most hallucinations fit into four basiccategories: (a) lattices (rhomboidal and hexagonal), (b) cobwebs,(c) radial and concentric forms (fan shapes and bull's-eye-likepatterns) whose three-dimensional versions resemble tunnels andfunnels, and (d) spirals. Figures 1A-1G show a variety of these Klu¨ver geometries. Klu ¨ver also noted that these same forms can be seen under many other conditions. Drug hallucinations are oftenreported to be very bright (similar in this respect to migrainefortification patterns) and vividly colored (in contrast to the pastelcolors usually seen in flicker-induced hallucinations). Paintingsand drawings of these vivid images show that the same forms are seen by independent observers across different cultures (Siegel &Jarvik, 1975). These drawings also provide evidence of a charac-teristic spatial scale (roughly 2 mm on cortex) for drug hallucina-tions, estimated by comparing the visual cortex's dimensions to thenumber of illusory periodic features. Hallucinogenic drugs gener-ally modulate one of four neurotransmitter-based mechanisms:acetylcholine (e. g., scopolamine), dopamine (amphetamine, co-caine), glutamate (ketamine, PCP) or serotonin (LSD, mescaline,psilocybin). For a useful recent review of hallucinogens and neu-rotransmitters, see Aleman and Larøi (2008). Not surprisingly,clinical conditions that affect these neurotransmitters can alsoinduce hallucinations; for example, about 25% of patients with Parkinson's disease (a dopamine defect; Wolters & Berendse,2001) report hallucinations, as do 60% of patients with Lewy bodydementia (an acetylcholine defect; Aleman & Larøi, 2008). As inepilepsy, the locations of neural defects and lesions matters: Lewybody dementia patients who have visual hallucinations tend toform more Lewy bodies in visually specialized areas of temporallobe than patients who are hallucination free (Harding, Broe, &Halliday, 2002). Sensory Deprivation and Charles Bonnet Syndrome In the 1950s, Donald Hebb was interested in the effects of conditions rumored to have been employed in brainwashing andcoerced confessions (Heron, 1961). Students in his lab discoveredthat systematically depriving the brain of differentiated sensoryinputs could lead to hallucination (Bexton, Heron, & Scott, 1954;Heron, Doane, & Scott, 1956). For example Heron et al. 's (1956)subjects wore translucent goggles to provide spatially unstructuredstimulation, had audition masked by ventilation fan noise, andwore cardboard sleeves to limit tactile stimulation. All 14 subjectsreported visual hallucinations—at first phosphenes and geometricforms but later some more complex hallucinations as well (thesame evolution reported for hallucinogenic drugs). Interestingly,studies in which the subjects were in darkness reported fewerhallucinations, and more time was often required before halluci-natory onset (cf. Vernon, 1963; Zubek, 1969), but other differ-ences between studies may have influenced the incidence of hal-lucination (Zuckerman & Cohen, 1964). Sometimes, these sensorydeprivation effects occurred as a side effect: About 39% of direc-tors of medically protective environments like laminar flow units(for treatment and prevention of infection) reported hallucinationsin their patients—the fourth most common side effect in this study(Kellerman, Rigler, & Siegel, 1977). Nature provides a clinicalversion of a tightly controlled sensory deprivation experiment—Charles Bonnet syndrome (CBS)—in which subjects who havevisual losses experience both simple and complex visual halluci-nations (ffytche, 2005; Wilkinson, 2004). In the literature, thecomplex illusions get more emphasis, but actually, the simplehallucinations seem to predominate. For example in Lepore's(1990) 59 patients with CBS, 63% had only elementary halluci-nations, and 27% had both elementary and complex hallucinations. Of the simple percepts, grids and lattices are common, especiallyfor losses of central vision (e. g., macular degeneration). Functionalimaging (f MRI) during these elementary hallucinations showsincreased activity in occipital cortex (ffytche, Howard, Brammer,Woodruff, & Williams, 1998); patients who hallucinate in color751 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
also show activity in the posterior fusiform gyrus, which is known to contain concentrations of color sensitive neurons. Bexton et al. (1954) recognized from the beginning that the hallucinations re-ported for sensory deprivation were probably related to CBS(although they did not use the term); they cited similar cases in thenondemented elderly with conditions like cataract. ffytche (2008)remarked that the percepts reported by his CBS patients resembleflicker-induced hallucinations. Geometric Hallucinations and Their Potential Neural Correlates Of the three kinds of elementary hallucinations, the geometric hallucinations described by Klu ¨ver (1966) are of special interest because they are ubiquitous in conditions that lead to any kind ofelementary hallucination and because these shapes connect di-rectly to certain kinds of neural pattern-forming mechanisms. Chief among these Klu ¨ver forms are lattice and cobweb-like structures, fan shapes, concentric circles, spirals, and related three-dimensional structures like tunnels and funnels. Siegel and Jarvik(1975) proposed expanding Klu ¨ver's categories to include charac-teristic motions, but for our purposes, the actual forms are a goodstarting point; these Klu ¨ver form constants are fodder for neural theorists, and their characteristic motions and other dynamicsemerge naturally from the theoretical treatment of the forms. Ermentrout and Cowan (1979) noted that spiral, fan-shaped, andcircular Klu ¨ver hallucinations (see Figure 1) are all perceptual correlates of stripe patterns on visual cortex (see Figures 4 and 5). This stems from the nonlinear neural mapping of retina to cortex(see the Appendix): Parallel stripes of cortical activity are gener-ated by viewing physical concentric circles, fan shapes, or spirals(these cortical stripes are seen with functional imaging; see, e. g.,Figure 5). Conversely, if stripes of activity autonomously form onvisual cortex, then their orientation on cortex determines the shapeof the resulting hallucination (Ermentrout & Cowan, 1979). Scroll-ing of the cortical stripes yields rotation for fans/spirals andinward/outward movement for concentric patterns. Rotation of thestripes yields morphing between the percepts; a radial form cantwist into a spiral, which tightens until it is a set of concentriccircles. Kaleidoscopic changes between Klu ¨ver-like patterns are seen under many conditions; Sacks (1995b) estimated that mor-phological changes can occur about 10 times per second duringmigraine. More than one set of stripe patterns can be generated ata time. Superposition of simple cortical activity patterns modelsmore complicated lattice and cobweb hallucinations (Ermentrout& Cowan, 1979). Competition between coevolving cortical stripepatterns may be another route to changes in perceived form. Related stripe-like patterns arise in nature: the parallel cylindri-cal rolls of rising hot and falling cold fluid formed during fluidconvection, the intricate patterns on seashells, the spot-and-stripecamouflage adorning many animal skins (see Figures 6, 7, and 8;Bestehorn & Haken, 1991; Ermentrout, Campbell, & Oster, 1986;Kondo & Miura, 2010; Meinhardt, 2003; Murray, 1988; Turing,1952). Nothing dictates the point-by-point behavior of thesesystems—self-organized patterns arise from nonlinear dynamicinteractions of many neighboring units. To explain autonomouscortical stripe formation, Ermentrout and Cowan (1979) created anexcitatory-inhibitory neuronal network; under some conditions, ifexcitation is uniformly increased above a critical level (either byexternal stimuli or internal conditions), then this neuronal network generates parallel stripes of cortical activity. The orientation of thiscortical pattern (and its perceptual correlate) is unpredictable andcan be unstable. Basic Models of Self-Organized Hallucinatory Neural Pattern Formation Most models of cortical pattern formation build on the Ermentrout-Cowan model, a member of the class of Wilson-Cowan models used in a wide range of nonlinear dynamic neuraland perceptual problems (Ermentrout & Cowan, 1979; Wilson,1999; Wilson & Cowan, 1973). Like other spontaneous pattern-forming systems, there are two structural requirements: an asym-metry between two interacting mechanisms and a diffusion-like Figure 4. Mapping of retinal geometric patterns to cortical stripe pat-terns. Physical geometry imaged on retina (left) is mapped nonlinearly ontocortex (right), resulting in stripe patterns of neural activation on cortex. Conversely, if oriented stripes of neural activity form on cortex, theyshould evoke the corresponding hallucinatory percept on the left: the Ermentrout and Cowan (1979) thesis. Some more complicated percepts(e. g., polar webs, hexagonal lattices) can result from superposition ofdifferent cortical stripe patterns. From “Neural Interactions Between Flicker-Induced Self-Organized Visual Hallucinations and Physical Stim-uli,” by V. A. Billock and B. H. Tsou, 2007, PNAS: Proceedings of the National Academy of Sciences, USA, 104, p. 8491. Copyright 2007 Na-tional Academy of Sciences, USA. Reprinted with permission. 752 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
mechanism for spreading their influences. Both can manifest in subtle ways. For the cortex, the conceptually simplest model is aset of coupled integrodifferential equations, each pair of whichrepresents dynamic neural interactions in one particular corticallocation. 3 /H11509E /H11509t/H11005/H11002E/H11001SE/H20853a W EE/H11569E/H11002b W IE/H11569I/H11001Sensory Input /H20854, /H11509I /H11509t/H11005/H11002I/H11001SI/H20853c W EI/H11569E/H11002d W II/H11569I/H11001Sensory Input /H20854. (1) Here, SEand SIare sigmoidal neural response nonlinearities, and the W/H11569Eand W/H11569Iterms are spatial convolutions of neural activity with excitatory ( E) and inhibitory ( I) spatial weighting functions; for example, WEE(x, y)/H11569E(x, y)/H11005/H20848 /H20848 WEE(/H92701,/H92702)E(x/H11002 /H92701,y/H11002/H92702)d/H92701d/H92702(/H92701and/H92702are dummy integration variables). The neural interactions are asymmetric: Neighboring excitatory cells excite each other and inhibitory cells alike, while inhibitory cellsinhibit both other inhibitory cells and excitatory cells (relatedsubtle asymmetries lie at the heart of every pattern-forming sys-tem). The Gaussian-like neural-weighting functions ( W i,j) describe the influence that neighboring neurons have on particular neurons as a function of distance; the resulting interplay between excitationand inhibition mimics the spatial derivative operator found in otherpattern-forming systems. The range of cortical excitation—about 1mm—is roughly the width of a human cortical orientation oculardominance column (a fundamental unit of cortical organization),while inhibition can span several columns. This excitatory rangematches the spatial scale seen in the migraine fortification hallu-cination—a broad flattened vertical arc made up of bright flicker-ing line segments set at roughly 60° angles to each other—whenadjusted for retinotopic location, each segment is approximately1. 2 mm long on cortex (Richards, 1971). Similarly, the hypercol-umn spacing in human cortex (about 2 mm; Horton, 1996; Horton& Hedley-White, 1984) roughly fits the maximum frequency ofhallucinatory features (for a fan geometry) with about 15 features(and blank spaces between features) per cortical hemifield (inaccord with the rough number of cortical hypercolumns that couldbe organized along the circa 60-mm extent of each hemisphere's V1; Bressloff et al., 2001). (Each 2-mm-wide hypercolumn con-tains two 1-mm orientation ocular dominance columns, one foreach eye; differential activation between the two ocular dominancecolumns could account for the sensation of depth seen in many Klu¨ver hallucinations. ) Biased flicker-induced hallucinations (see Figure 6) are a conspicuous exception to the 1-to 2-mm corticalscale, with a spatial scale similar to the scale of the biasingstimulus; this may be helpful when studying hallucinatory patternformation using functional imaging (Billock & Tsou, 2007). Relation of the Ermentrout-Cowan Model to Some Familiar Pattern-Forming Systems When first encountering self-organizing models as complex as Equation 1, there is a temptation to accept autonomous pattern for-3Because neural density is high, the neural interactions can be handled in a mathematically continuous fashion. This is sometimes called a neural field model, in analogy to the idealized fields used in physics (Amari, 1977; Coombes, 2005). Some models (Baker & Cowan, 2009; Bressloff, Cowan, Golubitsky, Thomas, &Wiener, 2001; Cowan, 1985; Henke et al., 2009; Wiener, 1994) use a singleequation that explicitly deals with excitatory neurons but builds in interactionsequivalent to the effects of inhibitory neurons. This is mathematically elegant andcomputationally advantageous, but for our neural and pedagogical purposes, it isuseful to employ a structure like Equation 1. Figure 5. A physical demonstration of cortical stripes and their perceptual correlates. The nonlinear mapping from retina to primary visual cortex (V1) introduced in Figure 2 can be seen using functional imaging of humancortex (in this case, functional magnetic resonance imaging) while viewing physical patterns. A-B: Radialpatterns on retina (s1, s2) evoke horizontal stripes on V1. C-D: Concentric patterns on retina (s3, s4) evokevertical stripes on V1. The dynamic reversing checkerboard patterns shown to retina are used to force both on-and off-cells (with both sustained and transient temporal properties) to respond nearly continuously, yielding astrong blood oxygen level-dependent response. From “Functional Analysis of Primary Visual Cortex (V1) in Humans,” by R. B. H. Tootell et al., 1998, PNAS: Proceedings of the National Academy of Sciences, USA, 95, p. 813. Copyright 1998 National Academy of Sciences, USA. Reprinted with permission. 753 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
mation in the spirit of “... and then a miracle occurs...,” like Aphrodite sprung from the sea foam. To better understand self-organized pattern formation in neural systems, it is useful to examineother pattern-forming systems. Early work on hallucinatory modelingused fluid dynamics as an analogy: A pan of liquid, heated frombelow and cooled from above, can form patterns in the convectingfluid. As the bottom-heated fluid rises and the top-cooled fluid falls,currents are set up, at first at random, across the container. Reinforc-ing currents combine constructively, while opposing currents cancel. If a high enough temperature difference is maintained, the system canevolve into a set of mutually reinforcing rolling concentric cylindersof convecting fluid, which (if viewed from above) resemble thecortical stripes that Ermentrout and Cowan (1979) invoked to explainhallucination patterns. Moreover, three such interacting roll patternssuperimposed at 60° angles create a hexagonal cell pattern (oftenfound in atmospheric phenomena; warm air rises in the center of eachcell, and cold air descends on the edges) like the hexagonal texturesseen in hallucinations. In simple fluid dynamics, rolls are favoredwhen surface tension is eliminated, while hexagonal cells form ifsurface tension at the top of the container is substantial. In complexitytheory, a variable whose change can force a sudden qualitative changein pattern formation is called a control parameter. Here, the temper-ature difference between top and bottom is the main control param-eter, determining if patterns can form, and surface tension is a sec-ondary control parameter, with influence over the type of patternsformed. The constant input of heat from the bottom is dissipated to thesurface by convection and requires constant renewal. Such patternsare called dissipative structures; this term has also been applied tohallucinations like those induced by flicker (Stwerka, 1993) and ratherappropriately too since the hallucinations cease promptly when theoutside forcing (the flicker) is removed. The effects of control param-eters are also analogous in the two systems: In flicker-induced hallu-cinations, hexagonal forms are more often seen for weak stimulation,and fan shapes, concentric circles, and spirals (all of which stem fromcortical stripe formation) are more often seen during stronger stimu-lation. Rule, Stoffregen, and Ermentrout's (2011) model predicts asimilar effect for flicker rate. However, although fluid-dynamic pat-tern formation resembles hallucinatory pattern formation, mathemat-ically there are closer systems. Here, we visit two other systems—reaction-diffusion and population dynamics—that mathematicallyresemble neural pattern formation to seek additional insights into thebehavior of Equation 1. Turing (1952) introduced reaction-diffusion systems. Such sys-tems consist of two diffusing agents; one agent can activate amarker of some kind (like a skin or fur pigment), while the otheragent inhibits the activation of the marker. They take the form Figure 6. Some reaction-diffusion simulations of a spotted coat growing on an animal skin of fixed shape and variable size ( /H9253). Murray (1989) found that in this case, the pattern was dependent on a particular spatial scale and suggested that this is why textured coats are common in medium-sized animals but uncommon in mice andelephants. From Mathematical Biology (p. 445), by J. D. Murray, 1989, Berlin, Germany: Springer-Verlag. Copyright 1989 by Springer-Verlag. With kind permission from Springer Science /H11001Business Media. 754 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
Figure 7. Reaction-diffusion (RD) simulations of pattern formation compared to actual biological patterns. A: Some stable states that RD systems can generate. B: Some two-dimensional simulations produced by a simple Turing model. C top: Actual and simulated shell patterns produced by Meinhardt's (2003) RD model. C bottom:Actual and simulated fish skin patterns produced by Sanderson, Kirby, Johnson, and Yang's (2006) RD modelon fish skins. From “Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Forma-tion,” by S. Kondo and T. Miura, 2010, Science, 329, p. 1618. Copyright 2010 by the American Association for the Advancement of Science. Reprinted with permission from AAAS. 755 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
/H11509A//H11509t/H11005F/H20849A,I/H20850/H11001DAƒ2A;/H11509I//H11509t/H11005G/H20849A,I/H20850/H11001DIƒ2I, (2) where DAand DIare diffusion rates for the activator ( A) and inhibitor ( I) chemicals, respectively, and ƒ2is the second derivative (Laplacian) operator /H115092x//H11509x2/H11001/H115092y//H11509y2, which is used to make the diffusion flowdown the Aor Iconcentration gradient. F( A,I) and G( A,I) are generally quadratic functions of chemical concentration but are specific to thesystem being modeled; for example, in Gierer and Meinhardt (1972), F/H20849A,I/H20850/H11005k 1/H11002k2A/H11001k3A2/I;G/H20849A,I/H20850/H11005k4A2/H11002k5I. (3) Turing was interested in the spot-and-stripe patterns of animal coats, such as leopards and zebras, and showed that a necessary condition for thediffusion rate constants is that D A/HS11005DI. Because of the asymmetry between substance A's and substance I's actions ( Iinhibits A's action, but not vice versa), the diffusion can drive an instability. This was a revolu-tionary development—diffusion had previously been thought an agent ofstability—but the combination of the two Turing asymmetries creates the Turing instability. Murray (1988, 1989) made some compelling simula-tions of animal coats using Turing-like reaction-diffusion models (see,e. g., Figure 6). Shoji, Iwasa, Mochizuki, and Kondo, (2002) and Kondoand Miura (2010) extended Turing models to complicated patterns foundon fish skins and other surfaces (see Figure 7). The wave of corticalspreading depression—believed to be the physiological basis of mi-graine—is an example of a reaction-diffusion process in an excitablemedium. It is therefore not surprising that several investigators havecreated reaction-diffusion models of the spread of the migraine fortifica-tion arc and its accompanying scotoma. Reggia and Montgomery's(1996) reaction-diffusion model acting on orientation specific neuronscan result in a moving arc with irregular jagged lines that come and go,similar to the flickering serrations of fortification illusions. Dahlem and Chronicle (2004) presented an improved reaction-diffusion model thatclosely matches patient's migraine fortification patterns. Methods devel-oped for studying traveling waves in reaction-diffusion systems haveproven useful in studying similar waves produced by Ermentrout-Cowan-like models. Perhaps closer to Ermentrout-Cowan models are population-based models that resemble the simple reaction-diffusion model. As a bridge between models, Murray (1989) gifts us with the Parable of the Sweaty Grasshoppers. Imagine a host of grasshop-pers in a field of dry grass. The grasshoppers have a mutation thatmakes them sweat excessively if they get hot. If the dry grasscatches fire, nearby grasshoppers fly a distance away and startsweating so much that their surroundings become too wet to ignitewhen the flame-front arrives. If the sparks from the initial flameignite many random areas, causing many random grasshoppermigrations-cum-sweating-attacks, the result is a scattered patternof burnt and unburnt areas, not unlike a Dalmatian's spotted coat. The key is that the grasshoppers must move faster than the fire(D I/H11022DAin Equation 2), buying time to sweat a firebreak. Now consider a slightly different system: The unfortunate grass-hoppers avoid the fire this time but attract the attention of a group ofhungry praying mantises. The interaction between the two popula-tions is asymmetrical in several ways. The mantises inhibit the grass-hoppers' numbers Gdirectly by consuming them and indirectly by reducing the number of breeders available. The grasshoppers excitethe mantis population Mby contributing nutrients for the mantis to use for reproduction. By itself, this can lead to a temporal pattern—aboom-and-bust cycle, where the mantis numbers soar until the grass-hopper numbers plummet and the mantises starve, allowing the grass-hopper numbers to recover. However, there is another possibility. Thegrasshoppers diffuse away from the mantises' territory faster than themantises can breed pursuers while continuing to cover their existingterritory. Segel and Jackson (1972) expressed a model of a similarsystem (sans insect drama) as Figure 8. Biasing a classic pattern-forming system—convection patterns formed in a pan of fluid heated from below and cooled from above. Convec-tion organizes into parallel cylinders of rising hot and falling cool fluid, whichlook like Ermentrout and Cowan's (1979) cortical stripes when viewed fromabove. The orientation of the stripes forms randomly unless biased by anoutside force. The top row of the simulations shows fluid injections (whitestripes) delivered at the beginning of the simulation (first studied by Bestehorn& Haken, 1991). In Columns a and b, by 200 time steps, the biasing geometryhas imposed its orientation on the entire emergent pattern. In Column c, twoinjections are made, one slightly stronger than the other, which compete untilthe stronger injection eventually dominates the entire pattern. From Principles of Brain Functioning (pp. 37 & 241), by H. Haken, 1996, Berlin, Germany: Springer-Verlag. Copyright 1996 by Springer-Verlag. With kind permissionfrom Springer Science /H11001Business Media. 756 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
/H11509G//H11509t /H11005 a G/H11002 b GM /H11001 c G2/H11001 DGƒ2G prey population growth births violent deaths G-cooperation diffusion, /H11509M//H11509t /H11005 e M/H11001 f GM /H11002 g S2/H11001 DMƒ2M predator population growth births feeding M-competition diffusion. (4) For some parameterizations, this kind of model leads to patchy and even periodic distributions of predator and prey (Segel &Levin, 1976), much like the structured coloring of the Turinganimal skins. Some similar models can lead to waves of pursuitand evasion (which is an interesting way to think about theadvancing migraine wave front, with its leading excitatory fortifi-cations and following inhibitory scotoma). By now, several of theconnections between the Ermentrout-Cowan model and thepredator-prey model should be clear, and it is not surprising thatthe Wilson-Cowan model (Cowan, 1968; Wilson & Cowan, 1972,1973) which Ermentrout and Cowan's work was based on, can beinterpreted as a sophisticated kind of population dynamicsmodel—one in which the populations become active or quietinstead of flourishing or extinct. In particular, note the terms in Equation 4 that contain the cand gcoefficients; without these terms, the model does not produce spatial patterns. The cterm says that the grasshoppers cooperate with one another; the /H11002gterm says that the predators fight with one another. Taken together with theother terms, this asymmetry mirrors the asymmetry of the Ermen-trout and Cowan equations, where excitatory cells excite bothexcitatory and inhibitory cells, while inhibitory cells inhibit theirfellow inhibitory cells as well as excitatory cells. Yet the carefulreader should wonder, Where in the Ermentrout-Cowan model arethe diffusion-like (second spatial derivative) terms that are crucialto pattern formation in both reaction-diffusion systems and popu-lation dynamics? 4The answer is that they are hidden in the very interactions of the two excitatory and two inhibitory terms in Equation 1. Because the neural excitatory and inhibitory weightingfunctions are Gaussian-like, with different variances, their effec-tive interaction kernels resemble differences of Gaussians andespecially second derivatives of Gaussians. These effective inter-action kernels dictate the periodicity and standard scale of neuralpattern formation (see Figure 9). Because a second derivative of a Gaussian is a linear operation, one can rearrange terms to isolate anaked second derivative operator, which is the Laplacian operatorused in the diffusion terms of reaction-diffusion equations andpredator-prey models (for the relevant math in a wonderfullydifferent context, see Marr & Hildreth, 1980). 5 Pattern Formation in the Ermentrout-Cowan Model If the spatial ranges of cortical excitation and inhibition differ and the level of spatially uniform excitation is raised sufficientlyhigh, the neuronal network in Equation 1 gives rise to periodicparallel stripes of neural activity and inactivity that can form at anyorientation (Ermentrout & Cowan, 1979). The patterning is due tothe local excitation-lateral inhibition connectivity built into theexcitatory cells in Equation 1. The pattern begins as a small localfiring rate increase but grows due to local excitation, resulting inan inhibited firing rate in the flanking regions. The inhibited flankscannot themselves inhibit their flanks, allowing excitation to growin the flanks of the flanks and so on. Thus, stripes of activation andinhibition form a periodic grating-like pattern whose scale is determined by the ranges of the local excitation and inhibition. There are two routes to raising the overall level of spatiallyuniform excitation. The first—used to model drug hallucinationsand epilepsy/migraine aura—involves breaking the balance be-tween the neural strengths of excitation and inhibition. Ermentroutand Cowan (1979) induced internal excitation by increasing aand cin Equation 1. However, the near symmetry of the equations suggests that a similar result could be obtained by decreasinginhibition ( bandd), as some have considered for percepts induced by migraine and epilepsy (Dahlem & Chronicle, 2004; Tass, 1995,1997). Tass (1997) studied pattern formation for every combina-tion of increasing or decreasing the four excitatory and inhibitoryparameters in Equation 1. Some parameterizations allow only onehallucinatory state (dictated by random initial conditions) toemerge; other parameterizations allow as many as four hallucina-tory states to alternate, leading to perceptual multistability. Weakercoupling between the excitatory and inhibitory elements of the network (the bandcterms of Equation 1) favors simple and stable geometric hallucinations, while stronger coupling leads to oscilla-tion between competing states (e. g., perception of fan shapesalternating with perception of spirals and concentric circles; Tass,1997). The other route is via a spatially uniform input (the Input term in Equation 1). Because visual neurons have little response to staticspatially uniform stimuli, many relatively uniform stimuli thatproduce hallucinatory pattern formation are temporally modulated(e. g., flicker, pulsed TMS, AC electric stimulation). This couldhelp account for the temporal instability of many induced hallu-cinations—each temporal modulation is both a stimulus to patternformation and a perturbation, disturbing the previously elicitedstate—but other factors likely influence the temporal scales ofhallucinations. Rule et al. (2011) experimented with variations of Equation 1 (which autonomously produces produce 13 Hz dampedoscillations). At low forcing frequencies (e. g., below 13 Hz), thesystem responds at the forcing frequency and tends to generatestable hexagonal gratings, like those that Fechner (1838) reportedon rotating black-and-white tops. At high forcing frequencies, thesystem breaks into two populations, each of which responds toevery other cycle of the stimulus, and tends to generate stripepatterns. This subsampling (called frequency demultiplication byneural theorists) was also a feature of Drover and Ermentrout's(2006) retinal-based model of electrically induced wake contour 4Here, diffusion-like indicates spread of neural activity, not movement of substances. 5The exact form of the spatial derivative term is probably not crucial; Cowan's (1985) model shows a fourth derivative-of-a-Gaussian-like inter-action kernel for the entire system (see his Figure 12. 7, p. 236), but inspatial frequency terms, this is much like a second derivative tuned to anarrower range of higher spatial frequencies. 757 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
hallucinations. Because each location in perceptual space is rep-resented by a cell that responds on every other cycle, this createsstanding wave patterns on cortex, with the two populations re-sponding both temporally and spatially out of phase with oneanother. The overall effect should resemble the directionally mul-tistable rotating fan shapes and pulsating bull's-eye patterns re-ported by Billock and Tsou (2007). Rule et al. found that spatialscale of flicker-induced hallucinations in their model depends onthe temporal forcing frequency. An interesting implication of themodeling is that the production of stripes rather than hexagons athigh temporal frequencies is guaranteed if the resting level ofcortical activity is sufficiently high that the average neuron will benear an inflection point in its sigmoidal input-output function. This is the hallucinatory analogue of Werner's (2007) and Plenzand Thiagarajan's (2007) contention that the nervous system iscritically poised. The result may also cast light on some otherinteractions between intrinsic neural rhythms and hallucinatoryactivity. In addition to the standing wave phenomena described by Rule et al. (2011), there are also motion instabilities in neural fieldmodels (like Ermentrout and Cowan's) that can result in travelingwaves (for reviews, see Coombes, 2005; Ermentrout & Kleinfeld,2001; Ermentrout & Terman, 2010). These waves are similar tothose generated by reaction-diffusion equations but travel synap-tically, rather than by volume transmission. Recently, it has be-come possible to study these waves in slices of living surgicallyremoved cortical tissue (e. g., Golomb & Amitai, 1997). One keytheoretical result, which fits nicely with Rule et al. 's findings, is that the speed of the traveling wave is dependent on the threshold(the lower inflection point) of the sigmoidal firing function (e. g.,S Ein Equation 1; Coombes, 2005); this has been manipulated experimentally by applying electric fields across the tissue slice (Richardson, Schiff, & Gluckman, 2005). The similarities between the optimal flicker rate for Ermentrout-Cowan simple stripe-type (see, e. g., Figure 4) hallucinations (ca. 10-20 Hz; Becker & Elliot, 2006; Billock & Tsou, 2007), thepsychophysically matched flicker rate of migraine fortificationaura (18 Hz; Crotogino et al., 2001), the roughly 10 Hz rate forkaleidoscopic changes of Klu ¨ver-like hallucinations sometimes experienced during migraine (Sacks, 1995b), and the content of thealpha and beta bands of EEG (8-12 and 13-20 Hz, respectively)are intriguing, yet poorly understood. Certainly, the frequencydependency and temporal scaling of hallucinations may be relatedto resonance between the external modulation and intrinsic neuralactivity (Ermentrout & Terman, 2010; Herrmann, 2001; Rule etal., 2011). It was recognized early on that despite reducing thebrightness of flicker, inducing hallucinations with flicker throughclosed eyes is surprisingly effective, perhaps because eye closureincreases the background alpha-band EEG signal, which the flickermay interact with (V. J. Walter & Walter, 1949). This is supportedby Shevelev et al. 's (2000) finding that when flicker is synchro-nized to the observer's alpha rhythm, pattern formation takes2-5 s, compared to 10-15 s for nonsynchronized trials. Similarly,Kanai, Chaieb, Antal, Walsh, and Paulus (2008) found that phos-phenes are best excited by 10-12 Hz current applied to cortexwhen the observer is in the dark and by 16-20 Hz current when theobserver is in the light, in accord with synergistic interactions withunderlying EEG cortical activity measured in darkness (alphaband) and light (beta band). In addition to providing evidence for interactions between in-trinsic neural activity and outside stimulation in hallucinatorypattern formation, such studies may shed light on discrepanciesbetween studies in frequency dependencies of flicker-induced hal-lucinations. Mundy-Castle (1953) pointed out that in addition tothe broad range of driving frequencies, the cortex responds with aneven wider range of harmonics and subharmonics, making itdifficult to identify the mechanism of action. Some subjects havelarger evoked responses at the first harmonic than at the funda-mental driving frequency, allowing them to respond to rather lowstimulus frequencies (V. J. Walter & Walter, 1949). Seeminglyminor stimulation differences could have a strong effect by mod-ifying the intrinsic activity. Some studies use eyelids as diffusersfor a bright flickering source, while other studies use open eyes,with rotating tops, strobe-illuminated ganzfelds (empty texturelessfields), or flickering monitors as sources. Wackerman, Pu ¨tz, and Allefeld (2008) pointed out that although alpha-band intrinsicactivity is present in both eyes-closed and eyes-open-ganzfeldviewing, the measured content of the alpha band is different for thetwo cases, with the alpha-band peak shifting to higher frequenciesfor ganzfeld viewing (similar to the viewing conditions of Kanai etal. 's, 2008, study). Becker, Gramann, Muller, and Elliott (2009)found a similar frequency shift and an additional increase ingamma-band (ca. 40 Hz) activity accompanying the developmentof the illusory percept. Figure 9. Second derivative-like spatial interaction kernels embedded in the Ermentrout-Cowan model (see Equation 1 in the text; Murray, 1989). These kernels are a systems-level mathematical analogue to the line-spreadfunctions measured in single cell electrophysiology. Here, W a(the solid line) is the interaction built into the top equation of Equation 1; for example, it represents the a WEEE/H11002b WIEIinteraction in the E (excitation) growth equation. Wi(the dashed function) is the c WEIE/H11002 d WIIIinteraction in the I(inhibition) growth equation. Thus, inhibition is the mirror image of excitation, and every stripe of excitatory activity generated by the pattern is flanked by stripes of inhibitory activity (andvice versa). The aand iindices that Murray (1989) used mirror the activator-inhibitor terms of reaction-diffusion theory and were used toaddress anatomical stripe development on striate cortex. Each three-lobedfunction is like a scale-specific version of the Laplacian (second derivative)terms in Equations 2 and 4 in the text. Alternately, one can think of thesefunctions as Laplacians that have operated on Gaussians. Laplacian-of-Gaussian operators are ubiquitous in vision and neuroscience, especially inthe neuroscience of lateral inhibition (where they go by names like Mexican-hat functions, difference of Gaussians, and difference of offset Gaussians). From Mathematical Biology (p. 491), by J. D. Murray, 1989, Berlin, Germany: Springer-Verlag. Copyright 1989 by Springer-Verlag. With kind permission from Springer Science /H11001Business Media. 758 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
Intriguing Synergies Between Hallucinatory Conditions The interplay between increasing excitation via the coupling constants (e. g., a, c) and via the Input term in Equation 1 suggests that condition pairings that drive both should interact synergisti-cally. Five lines of evidence support this: (a) Subjects given asubhallucinatory dose of mescaline immediately perceivemescaline-quality geometric hallucinations (more intense and or-nate than those normally seen in flicker) when exposed to flick-ering light (Smythies, 1960). Similarly, Knoll, Kuger, Höffer, and Lawder (1963) reported similar hallucinatory interactions betweenelectrical cortical stimulation and several hallucinogenic drugs. (b) As discussed above, when flicker is synchronized to the subject'salpha EEG rhythm, pattern formation is 3 to 5 times faster than forunsynchronized trials (Shevelev et al., 2000). (c) Oster (1970)reported that LSD flashbacks (see Figures 1A-1D) are easilyevoked by ocular pressure, and Klu ¨ver (1966) reported that in the early prehallucinatory stage of intoxication, ocular pressure speedsdrug hallucinations. Similarly, Siegel's (1992) patient triggered LSD flashbacks (resembling Figure 1F) with flickering neonlights, but only after heavy stimulant (nicotine and caffeine) use. Inkeeping with this, about 83% of heavy amphetamine abusers reporthallucinations as a side effect (Paulseth & Klawans, 1985). Anumber of studies are consistent with a net excitatory effect ofhallucinogenic drugs on the visual system (Winters, 1975). Forexample, LSD increases visual evoked responses at both the lateralgeniculate nucleus (LGN) and cortex (Purpura, 1956a, 1956b). Interestingly, diphenylhydantoin, a drug used to treat epilepsy bydampening neural excitability (usually by deactivating some so-dium channels in neural membranes), is also effective in suppress-ing LSD flashbacks (Thurlow & Girvin, 1971), and subjects with LSD flashbacks show an epileptic-like increase in the coherence of EEG in the absence of stimulation (Abraham & Duffy, 2001). (d)Migraineurs, based on many lines of evidence, appear to be chron-ically hyperexcitable (Aurora & Wilkinson, 2007) and, most rel-evantly here, between attacks have reduced thresholds for TMS-induced phosphenes (Aurora et al., 1998). The same reduction inphosphene-induction thresholds has been documented in Ecstasyusers who hallucinate (Oliveri & Calvo, 2003). (e) Flicker triggershallucinatory auras in some migraineurs and epileptics. Like ordi-nary flicker-induced and ocular-pressure-induced hallucinations,photo-induced epileptic auras are easier to obtain for two eyes thanone (Hess, Harding, & Drasdo, 1974). Moreover, in animals dosedwith GABA-antagonists (which block inhibition), a migraine-likecortical wave is triggered by flicker (van Harreveld & Stamm,1955). Many of these interactions make sense as a synergy be-tween excitability (internal parameters set by neural wiring anddrug dosage) and the Input term (an external variable tapped by flicker, ocular pressure, and TMS) of Equation 1; both mecha-nisms increase excitation in a spatially homogeneous way. Con-sider a pedestal of internal activation to which external stimulationis added: If the combined effects of steady internal activation andtransient external stimulation exceed a critical level, then patternformation occurs. In the case of drug flashbacks, previous hallu-cinatory experiences may influence the long-term value of Equa-tion 1's a-dparameters via Hebbian learning. Experimental Evidence of Spontaneous Pattern Formation in Visual Cortex Many hallucinations are explained by models based on Ermen-trout and Cowan (1979). Yet the Ermentrout-Cowan model'sgreater significance may be that it suggests that autonomous neuralactivity could be perceptually meaningful. This has become animportant issue in sensory neuroscience (Arieli, Sterkin, Grinvald,& Aertsen, 1996; Kenet, Bibitchkov, Tsodyks, Grinvald, & Arieli,2003; Ringach, 2003, 2009). It has long been known that evenunstimulated visual cortex forms neural patterns; once thought tobe mere noise, this activity is spatiotemporally structured and mayinteract in perceptually significant ways with spatiotemporal ac-tivity induced by images. For example, Kenet et al. (2003) usedvoltage-sensitive dyes to make high-resolution video recordings ofneural activity in the visual cortex of anesthetized and unstimu-lated cats; this autonomous cortical activity is highly structuredand matches cortical activity induced in the same animals byviewing oriented grating patterns (see Figure 10; Kenet et al., 2003; Ringach, 2003). These autonomously formed patterns arereminiscent of the Ermentrout-Cowan patterns; Goldberg, Rokni,and Sompolinsky (2004) treated both the Kenet et al. and Ermentrout-Cowan patterns as subsets of attractors in a dynamicphase space of neural states visited by the underlying nonlineardynamic system, an interpretation reinforced by recent work onneural attractor dynamics (Ghosh, Rho, Mc Intosh, Kötter, & Jirsa,2008; Luczak, Bartho, & Harris, 2009; Ringach, 2009). Thisdescription is supported by Kenet et al. 's (2003) results: Thespontaneous neural activity continually and unpredictablyswitched between states describing different grating orientations,reminiscent of the unpredictability of the Ermentrout-Cowanmodel and the instability of many induced hallucinations. Ofcourse, Kenet's patterns (if present in unanesthetized humans) aresubliminal, since we do not perceive illusory oriented gratingswherever we look; Ringach (2009) suggested that sensory depri-vation hallucinations (including clinical conditions like CBS) maystem from these patterns becoming visible. However, even sub-threshold neural patterns could alter perception by interacting withstimulated activity. Opponent Interactions Between Hallucinations and Stimulus-Generated Activity Several lines of evidence suggest that cortical pattern formation interacts with stimulus-generated cortical activity to influence per-ception. Wilkinson (2004) proposed that the ictal blindness thatfollows visual epilepsy may be the result of high autonomousactivity in visual cortex that never settles into an organized pattern,interfering with stimulus-driven visual activity without giving riseto percepts of its own. More prosaically, Fiser, Chiu, and Weliky(2004) showed that as cortex ages, the cortical modulation ofsensory input by the underlying activity changes as well. Billockand Tsou (2007, 2010) attacked the problem from the other direc-tion, using small physical stimuli to bias the otherwise randomhallucinations evoked by flickering light. Previous studies of self-organized pattern formation in nonlinear dynamic systems foundthat pattern-forming systems could be biased by introducing afeature that was aligned in the desired orientation. For example, in Figure 8, a fluid-dynamic system forms oriented cylinders of rising759 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
hot and falling cold fluid, which from above appear like the stripes in the Ermentrout-Cowan model. As in the Ermentrout-Cowanmodel, there is no preferred orientation to these stripes. However,even a single line of injected fluid can dictate the final orientationof the entire system (Bestehorn & Haken, 1991). Billock and Tsou(2007) adapted this idea to flicker-induced hallucinations, by plac-ing small fan-shaped or bull's-eye patterns in the midst of auniform flickering field. They expected the hallucinatory patternsto grow from the biasing stimulus, like a geometric seed dictating crystal growth in a supersaturated solution. Instead, they found aninteresting perceptual opponency between orthogonal hallucina-tory geometries: Flickering around a small physical bull's-eyepattern induces a hallucinatory rotating fan shape in the flicker,whereas flickering around a small fan shape induces hallucinatoryconcentric circular forms that can wobble and pulsate (see Figure11). Some patterns are more visible than others: Induced fan-Figure 10. Spontaneous cortical pattern formation in cats. Top: Single-frame images of voltage-dependent dyes on the visual cortex evoked by oriented physical stimuli (Ringach, 2003). Bottom: Similarity of sponta-neous states arising in sensory-deprived cats to the activity evoked by oriented stimuli (Kenet, Bibitchkov,Tsodyks, Grinvald, & Arieli, 2003). a: Cortical activity evoked by presentation of vertical gratings, averagedover 160 frames. b: Single frame of spontaneous activity resembling the activity in Panel a. c: Single frame ofimaged neural activity evoked by vertical grating, shown for comparison. Top panel from “Neuroscience: Statesof Mind,” by D. L. Ringach, 2003, Nature, 425, p. 913. Reprinted by permission from Macmillan Publishers Ltd., copyright 2003. Bottom panels from “Spontaneously Emerging Cortical Representations of Visual Attributes,” by T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, 2003, Nature, 425, p. 954. Reprinted by permission from Macmillan Publishers Ltd., copyright 2003. 760 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
shaped hallucinations are more salient than induced circular pat-terns, which in turn are more salient than induced spirals. Thispattern preference resembles Kenet et al. 's (2003) finding thatsome spontaneous cortical patterns recur much more often thanothers. In general, biased hallucinations and their interactions with visual stimuli obey the same rules as perception of ordi-nary stimuli; these hallucinations are spatiotemporally localizedphenomena, limited in space to the flickering field and rigidlylocked to the presence of the physical stimulus (Billock & Tsou,2007). There is evidence for temporal opponency with these same orthogonal patterns: Mac Kay (1957a, 1957b, 1965, 1978) saworthogonal hallucinations (during the off phase of flickered illu-mination) superimposed on flickered physical geometric patterns(a sequential contrast phenomenon); as discussed earlier, he sawthe same illusions as brief afterimages following prolonged view-ing of a geometric pattern and in noise streaming behind transpar-encies of these geometries. These Mac Kay effect afterimage op-ponencies are influenced by the orientation, motion, and spatialscale of the adapting features (Georgeson, 1976, 1985). A similaropponency is seen in ambiguous textures with both circular andradial correlations; for example, after adapting to a radial geome-try, a previously ambiguous pattern of random dots (Glass pat-terns) appears circular (Clifford & Weston, 2005). Two (not mu-tually exclusive) theories on functional interactions of thesegeometries have been advanced. Some researchers have noted thatgeometries like the fan shapes and concentric circles used by Mac Kay (1957a, 1957b) are the basis functions of Lie grouptheories of spatial vision (Caelli, 1977; Dodwell, 1991). Addition-ally, some masking phenomena found by Vidyasagar, Buza ´s, Kis-va´rday, and Eysel (1999) suggest that geometric opponency re-duces the persistence of some visual percepts (Wede & Francis,2006). Since a variety of drugs and clinical conditions prolongvisual persistence, it would be interesting if any of them alsoimpact geometric opponency. A possible fourth line of evidence for interactions between spontaneous pattern formation and physical stimuli stems fromsome particularly odd illusory conjunctions. Treisman (1996)treated illusory conjunctions as feature misbindings; for example,in peripheral vision, a red square and a green triangle may bemisperceived as a green square and a red triangle. One experi-ment—on perception of forbidden colors—used vertical adjacentred/green equiluminous stripes (Billock, Gleason, & Tsou, 2001;Billock & Tsou, 2010). When stabilized on the eye, the borderbetween the stripes usually collapses, and observers perceive theentire field as a novel reddish green. However, sometimes theperception of the stripes becomes multistable, and the colorsappear to change sides. Although this can be understood as anordinary illusory conjunction, Billock, Gleason, and Tsou (2001)had one subject who saw the vertical color stripes rearrange intohorizontal stripes. In this particular experiment, the tops and bot-toms of the vertical stripes were rounded (see Billock & Tsou,2004, their Figure 3), so there was no horizontal structure in theactual image for the color to misbind to. However, if humansgenerate Kenet-type cortical orientation patterns, odd occasionalmisbindings to the orientation of the Kenet patterns would beexpected. If the Kenet patterns are spatially opponent for orthog-onal geometries, misbindings would be expected in the orthogonalorientation to the original stimulus. Advanced Models of Hallucinatory Pattern Formation The spatial weighting functions ( Wijin Equation 1) used by Ermentrout and Cowan (1979) and Tass (1995, 1997) are spatially symmetric, and pattern formation arises from local reinforcementby neighboring members of the neural network; allowable patternsgenerated by the network include any doubly periodic pattern thattiles the visual cortex. Some neural network models allow aniso-tropic connectivity on cortex (Cowan, 1997; Bressloff et al., 2001,2002; Golubitsky, Shiau, & Török, 2003; Jirsa & Kelso, 2000;Qubbaj & Jirsa, 2007; Wiener, 1994), which creates new possibil-ities for modeling pattern perception. These networks could beuseful in modeling spatial opponency between radial and concen-tric patterns (Billock & Tsou, 2007). One can imagine a set of Ermentrout-Cowan-like networks, each biased to form stripes inhorizontal, vertical, or oblique angles along V1, with competitionbetween networks for the right to fire. Such a system couldproduce spatial opponency at a distance, if the vertical and hori-zontal stripe-producing networks had mutual geometrically antag-onistic long-range connections preferentially arranged along par-ticular directions on V1. Anisotropic reaction-diffusion models Figure 11. Biasing hallucinations. If a physical geometry (bold figures) is placed within a flickering field, the hallucination induced by the flicker(gray figures) is orthogonal to the physical stimulus; flickering around aphysical set of concentric circles induces hallucinatory rotating fan shapes;flickering around physical fan shapes induces wobbling or pulsating con-centric circular patterns. If the entire field flickers in phase, the hallucina-tion extends through the physical stimulus. If the flicker does not extendthrough the physical form or if the form is flickered out of phase, then thehallucination is confined to the empty region beyond the physical biasingfield. From “Neural Interactions Between Flicker-Induced Self-Organized Visual Hallucinations and Physical Stimuli,” by V. A. Billock and B. H. Tsou, 2007, PNAS: Proceedings of the National Academy of Sciences, USA, 104, p. 8492. Copyright 2007 National Academy of Sciences, USA. Reprinted with permission. 761 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
(mathematically similar to Ermentrout-Cowan neural networks) produce similar stripe-orientation opponencies; for example, inpattern formation on fish skins, if diffusion is easier in one direc-tion than another and the inhibitor species has greater range thanthe activator, then stripes form parallel to the direction of diffusionuntil the activator's range is exceeded and then switch orientationto the perpendicular direction. Similar results occur if the activatorand inhibitor species have different preferred directions of diffu-sion (Shoji et al., 2002). An anisotropic model should also facil-itate modeling the serrated appearance of migraine fortificationsand the finding that fans and concentric circles are easier toproduce and more salient than spirals (Billock & Tsou, 2007),whereas isotropic models show no such preference. Bressloff et al. 's (2001, 2002) neural pattern-formation model illustrates the power of stimulus-specific networks for understand-ing complicated hybrid hallucinations, like textures superimposedon a geometric pattern (see, e. g., Figures 1F and 12, which showa finer lattice texturing a coarser spiral structure). Cortical area V1is functionally subdivided into an array of hypercolumns, whichcontain neurons with various preferences in feature orientation. In Bressloff et al. 's models, the neurons within a column interactwithout respect to orientation preference, but connections betweencortical columns are between neurons of common orientationpreference. These models behave almost as if there is a separate Ermentrout-Cowan-like pattern-forming network for each orientedpopulation that competes for the right to fire; the patterns that arisecan appear textured if the winning neurons are perceptually labeledfor orientation (see also Golubitsky et al., 2003). 6The orientation of the resulting cortical stripe patterns determines the overall geometry of the illusion, while the orientation preferences of theactivated neurons determine the texture attached to the geometry. Moreover, the resulting individual stripes of activity need not havethe same orientation labels; richly complicated percepts, like her-ringbone textures, can result. Bressloff et al. 's work drasticallyexpands the range of hallucinatory percepts that can be modeledand points the way to still more powerful treatments; one canimagine Bressloff et al.-like networks with neurons perceptuallylabeled for contrast, depth, color, and motion. A vast array ofperceptual phenomena could be modeled by such networks. Bakerand Cowan (2009) created a vector-based model that expands thenumber of perceptual attributes that such models can pin to spon-taneous hallucinatory pattern formation. In another interestingvariation, Golubitsky et al. (2003) introduced a weak level ofanistrophy into the model by making connection strength vary withorientation preference (but any stimulus preference should apply). Curiously, this also creates time-varying pattern formation, withrotating and pulsating hallucinations, and tunnel-like percepts withstructure seeming to emerge from or disappear into the opening ofthe tunnel. Stochastic Resonance in Neural/Perceptual Pattern Formation Stochastic resonance—a qualitative change in a system's behav-ior when noise is added—comes in three forms: (a) paradoxicalincrease in signal-to-noise ratio, (b) induced multistability, and (c)autonomous spatiotemporal pattern formation. The first two phe-nomena are often enhanced by using random fractal (1/f n) noise in place of the usual white noise. However, pattern formation via6This explanation is an oversimplification that does not do complete justice to Bressloff et al. 's (2001, 2002) approach. For example, their model not only forms orientation-specific textures but also sharpens theorientation specificity of the network, relative to the specificity of theindividual cortical neurons. Figure 12. Neural connectivity affects perceptual pattern formation. Top: In Bressloff, Cowan, Golubitsky, Thomas, and Wiener's (2001, 2002)models, these connections can be specific to sensory neural preferences,like feature orientation. Here, local connections ( /H110211 mm) within a cortical orientation hypercolumn are indiscriminate, while connections betweenhypercolumns are dependent on orientation selectivity. Bottom: Bressloffet al. 's model leads to textured hallucinations that resemble some complexreported hallucinations, in this case Siegel and Jarvik's (1975) THC-induced hallucination (see Figure 1F). From “Geometric Visual Halluci-nations, Euclidean Symmetry and the Functional Architecture of Striate Cortex,” by P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, &M. C. Wiener, 2001, Philosophical Transactions of the Royal Society of London: Series B. Biological Sciences, 356, pp. 306 & 325. Copyright 2001 by the Royal Society. Reprinted with permission. 762 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
stochastic resonance—first predicted for mathematical systems (Ditzinger, Ning, & Hu, 1994)—is not yet well documented inneural systems; there are only a few relevant models and onerelated psychophysical study. For example, in a simulation of200/H11003200 coupled neural oscillators, 1/f noise was far more efficient than white noise in inducing spatiotemporal pattern for-mation (Busch, Garcia-Ojalvo, & Kaiser, 2003; Busch & Kaiser,2003). Similarly, a noise input to Tass's (1995) model inducessimulated concentric circles that pulsate and spirals and fans thatrotate (which corresponds to scrolling of the cortical stripe pat-terns). These are the same dynamics that emerged in Golubitsky etal. 's (2003) model, but via a very different mechanism. Tass'smodel illuminates a fascinating experimental finding: Viewingdynamic fractal noise through a transparency of concentric circlesgenerates a hallucinatory rotating fan shape, while viewing iden-tical noise through a transparency of radial blades generates pul-sating circles (Billock & Tsou, 2007). The noise's statistics matter;spatially fractal noise requires much less contrast to generate thehallucinations than does spatially flat (white) noise. The temporalnoise statistics affect the hallucinatory rotation or pulsation rate. Aspects of Perceptual Pattern Formation That Require Additional Work Despite these advances, gaps remain in the study of cortical and perceptual pattern formation. Many of these gaps could be filled byimproved electrophysiological mapping/functional imaging tech-niques, but the technical problems are formidable. The moreprosaic problem involves resolution of neural events. In this re-gard, the best spatiotemporal resolution is obtained with voltage-specific dyes on exposed primate cortex (Kenet et al., 2003)—forexample, see Figure 10—but the method is unsuited for humanstudies. Conversely, imaging studies do map migraine triggerwaves in V1 and other activity in V3a (Hadjikhani et al., 2001), butnot at the resolution required to infer the percept. Biased halluci-nations (see Figure 11)—which produce stable, coarse-featuredpatterns—may be helpful here (Billock & Tsou, 2007). However,a more serious problem stems from the difficulties imposed byneural coding. Coding of Hallucinatory Features: Neural Substrates and Functional Imaging One difficulty with functional imaging is the way that the imaging signal relates to the neural activity underlying the hallu-cination. For example, in the Ermentrout-Cowan models that ex-plicitly represent the excitatory and inhibitory cell populations, thestationary patterns consist of stripes of high neural activity inwhich excitatory activity dominates, flanked by stripes of lowneural activity; such stripes—if spatiotemporally stable—shouldbe imageable. However, for scrolling patterns, which in our expe-rience are more common (e. g., rotating fans and spirals, pulsatingcircles), the inhibitory activity lags the excitatory activity, filling inthe cortical grating patterns with neural activity. Inhibitory cellsare at least as energetically demanding as excitatory cells, and so,their activity will blur the f MRI blood oxygen level-dependentsignal (and just about any kind of imaging that is not neurotrans-mitter specific), limiting detection of the coarse structure of thehallucination. Even worse for imaging are Rule et al. 's (2011)standing wave patterns, where the stripes of excitation and inhi-bition exchange positions every 50-100 ms. 7Imaging the fine structure is more difficult still; Bressloff et al. 's (2001, 2002) work suggests that while Ermentrout-Cowan cortical stripe formationdetermines the overall geometry of hallucinations, the specificneurons activated in those stripes determine what that geometrylooks like: its texture, depth, contrast, color, and more. It isinteresting then that many experimental conditions induce pastel orgrey hallucinatory features, suggesting small asymmetries in theactivation of complementary neural populations. In the early visualsystem, perceptually opponent mechanisms often balance one an-other; for example, activated off-response cells signal featuresdarker than their background, while activation of on-responseneurons should trigger perception of a bright feature. This codingof images by complementary neural populations imposes limita-tions on imaging those responses. Consider, for example, a light-and-dark hallucination generated by patterns of on-and off-cortical cell activity, respectively; if both yield the same imageableactivity on V1, then the structure of the percept would not beresolved using current functional imaging approaches. 8 However, there are exceptions to the population balance rule. Brilliant colors are often reported in drug-induced hallucinations. Similarly, migraine fortifications are reported as bright, oftenblindingly so. Of course, this may be a reporting bias—one speaks of the bright fortification lines and not of the dark spaces betweenthem. Indeed, some drawings of migraine fortifications depict theindividual fortifications as more like grating patches than linesegments (e. g., Dahlem & Chronicle, 2004, p. 358, their Figure 5),suggesting that both on-and off-neurons can be activated duringmigraine, but out of phase with one another (a spatiotemporalphase shift could also account for the perceptual scintillations seenin migraine fortifications). This is essentially the same spatiotem-poral phase shift considered above for scrolling of the Ermentrout-Cowan patterns, and it imposes the same filling-in of imageableactivity problem. Similar problems will limit ability to discernhallucinatory texture, color, motion, and depth until advances inimaging overcome these limitations. The nuances of hallucinatory perception just discussed also highlight a related problem. Although we now understand whypercepts like spirals, fan shapes, and concentric circles form andcan readily imagine why such forms would take on attributes likecolor and depth, we have relatively little understanding aboutdifferences between subjects or conditions in inducing color anddepth. For example, in flicker-induced hallucinations, some ob-servers describe the illusory colors as vivid and even unearthly,while other observers (including, alas, ourselves) see the colors aspastels. Small differences in experimental design also seem to havelarge effects: Most biased hallucinations are achromatic Klu ¨ver forms, although subjective colors may appear in the inducing fields(Billock & Tsou, 2007). 7A standing wave (whose psychophysical correlate is a contrast-reversing grating) is mathematically equivalent to two traveling waves, propagating through one another in opposite directions (Kelly, 1985). 8Tootell et al. 's (1998) retinocortical mapping cleverly avoided this problem by using an image (see Figure 3) whose voids were designed to elicit minimal neural response, while the image features (made of contrast-reversing checks) were designed to create maximal imaging signals. 763 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
Other Complications: Alternate Pattern-Formation Mechanisms and Neural Loci Complicating this picture are alternative mechanisms that could lead to hallucinatory geometric patterns. For example, considerrotating spirals and pinwheels: In the Ermentrout-Cowan model,these correspond to a pattern of oblique stripes of neural activityon cortex (angle of orientation on cortex determines spiral tight-ness), scrolling perpendicularly to the orientation of the stripes. Alternately, some systems of reactive media produce spreadingscroll-like waves that in two dimensions are actual spirals. Fohlmeister, Gerster, Ritz, and van Hemman (1995) modeled an Ermentrout-Cowan-like system that produces both stripe patterns and actual scrolling spiral waves on cortex, whose perceptualcorrelates could be hard to distinguish under some conditions. Ermentrout and Terman (2010) created a coupled neural oscillatormodel that produces rotating spiral waves on simulated cortex; anetwork of excitatory cells with time delays can behave similarly(Sacks, 1995b). Models incorporating synaptic-like time delayscan also yield temporal oscillations (ca. 10 Hz) that resemble theillusory flicker seen in CBS and visual epilepsy (Henke, Robinson,Drysdale, & Loxley, 2009). Similarly, Ermentrout (1984) and Ruleet al. (2011) noted that the Ermentrout and Cowan (1979) modelcan producing oscillations of 13-20 Hz, similar to migraine scin-tillations. Another complication is that some classes of visualhallucinations may be influenced by activity in areas other thanstriate cortex (V1). For example, some cells in cortical area V4 aresensitive to concentric circles and fan shapes (Gallant, Connor,Rakshit, Lewis, & Van Essen, 1996); V3a (and possibly V8) isactive in some migraines (Hadjikhani et al., 2001). Significantly,ffytche (2008) found increases in activity throughout visual andfrontal-parietal cortex during flicker-induced hallucinations; activ-ity in visual cortex was centered on V4. Activity in V1 was notparticularly elevated compared to nonhallucinating flicker con-trols, but the Ermentrout-Cowan model does not require a netincrease in V1 activity, but rather an increase in spatial structuringof excitatory and inhibitory activity, which is not currently image-able (as discussed above). If other cortices are involved in auton-omous spatiotemporal pattern formation, they may further compli-cate matters by feeding back their activity to V1. It would beextremely interesting to find out if perception of circular andfan-shaped hallucinations could be altered by deactivating V4temporarily (e. g., by TMS) or permanently by surgery in a patientwith pathological V4 lesions. An analogous experiment wouldmonitor migraine percepts before and after V3a and V8 deactiva-tion. It would also be desirable to monitor activity in V1 layer 6neurons. Although flicker is a fine stimulus for the LGN, ffytschefound an interesting loss of activity in LGN during flicker-inducedhallucinations, relative to flicker trials without hallucinations. Al-though LGN is often thought of as a relay station from retina tocortex, there are an order of magnitude more neurons contacting LGN from layer 6 of V1 than there are LGN cells projecting to V1. This massive feedback pathway is believed to selectively modu-late/regulate LGN sensitivity and may be activated by pattern-forming activity in V1. Because the layer 6 feedback is spatiallyspecific, it may also spatially shape the input to V1 pattern-forming mechanisms. Pattern-Formation Mechanisms in Phosphene and Fortification Percepts Much effort has gone into elucidating the self-organized pattern-formation mechanisms that underlie geometric hallucinations. Wehave already seen several connections between geometric halluci-nations, phosphenes, and fortification percepts, especially in theirspatial and temporal scales and in the ways that the same condi-tions can elicit combinations of the three percepts. Although not asmuch effort has gone into understanding the nonlinear dynamics ofphosphenes and fortifications auras, it is worthwhile to examinesome exceptions. Single spot-like phosphenes—most common forocular pressure and direct cortical electric stimulation—are oftenbest understood as focal activation of one or more neighboringneurons, within a single cortical column. They can however becreated by pattern-forming mechanisms, like Sacks's (1995b)time-delay excitatory neural network, which also models Klu ¨ver patterns. Multiple phosphenes are an interesting problem. Ermentrout-Cowan models can produce regular arrays of phos-phenes—the polyopia phenomena often noted for some clinicalconditions and often seen for drug and flicker-induced hallucina-tions. Reaction-diffusion-like mechanisms—which can model mi-graine fortification arcs (Dahlem & Chronicle, 2004)—would alsobe able to model phosphene polyopia and stripe formation invision (which resembles the spot-and-stripe formation on animalcoats that motivated reaction-diffusion theory). For example, whennonequiluminant color borders are retinally stabilized, the bor-ders can collapse, and the colors can seem to slowly diffuseacross the former boundaries, like an image made of meltingwax. As the colors flow and mix, they can form patterns, likered streaks on a green background or blue glitter on a yellowbackground (Billock, Gleason, & Tsou, 2001; Billock & Tsou,2010), much like the spotted and striped animal coats that areformed by reaction-diffusion systems (Murray, 1989; Turing,1952). Although, at first blush, diffusion seems like an oddmechanism to invoke, filling-in mechanisms are ubiquitous invision and often mimic a high-velocity diffusion-like process(see Paradiso & Nakayama, 1991, who found a fast corticalspread speed of at least 150 mm/s by using temporal masking asa probe). In this regard, the slowest color diffusion seen by Billock, Gleason, and Tsou (2001) would lie between the nor-mal very fast filling-in process and the very slow movement ofmigraine activity. Of course, given the conceptual similarity ofthe Ermentrout-Cowan model to a reaction-diffusion system, aunified model that tackles the sequential development of pho-sphenes, fortifications, and Klu ¨ver geometries within a single migraine attack should be feasible. Along these lines, it isworthwhile to consider cases that blur the distinction betweenphosphenes and Klu ¨ver patterns. The Blurry Distinction Between Phosphenes and Klu¨ver Forms As discussed earlier, phosphenes show the same kind of nonlinear scaling with position in the visual field as geometrichallucinations and migraine fortifications. When phosphenesoccur in large numbers (polyopia), they are often arranged inthe same geometries as Klu ¨ver patterns. Yet this does not exhaust the potential links between these phenomena, which764 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
sometimes appear to be more a continuum of states than a set of discrete percepts. For example, in TMS of cortex, single-anddouble-wedge (bow-tie/butterfly) phosphenes are common. Kammer et al. (2005) found that single wedges are seen at lowerstimulus intensities and that butterfly wedges extend into bothvisual fields at higher intensities. These also occur in epilepsy(Babb, Halgren, Wilson, Engel, & Crandall, 1981), but a morecommon form for occipital epilepsy is a doughnut-shaped pho-sphene. This favoritism for certain forms recalls Kenet et al. 's(2003) finding that some spontaneously formed cortical activitypatterns occur more frequently than others. It is also interestingto note that these particular phosphenes would correspond tosingle vertical or horizontal stripes of cortical activation (adouble wedge results if a horizontal stripe forms symmetricallyon both occipital cortices). The length and position on cortex ofthese stripes would determine the size and location of thephosphene. Since isolated stripes could occur early in thepattern-formation process and could wane before seeding a fullperiodic pattern, this could provide a route to understandingsuch phosphenes. Alternately, some pattern-forming systemscan form single stripes as a final state (known as a bump inneural field theory; see Kishimoto & Amari, 1979; see also Coombes, 2005; Ermentrout & Terman, 2010). Similarly, thesmall rotating pinwheels and spoked stars seen in some elec-trical stimulations of cortex (Dobelle & Mladejovsky, 1974;Penfield & Rasmussen, 1950) could be small Klu ¨ver patterns created by Ermentrout-Cowan mechanisms but could also rep-resent a small wave front of excitation, activating in sequenceorientation-specific cell populations as the wave moves througha single orientation column (Schwartz, 1980), similar to amechanism sometimes evoked to explain migraine scintillation. Eye Movements, Edge-Sharpening Mechanisms, and Hallucination In ordinary vision, the visual system compensates for eye move-ments in various ways, perceptually stabilizing the moving world,deblurring image motion for some eye movements, and blockingvisual perception during fast saccades and blinks (Volkmann,1986). Similar phenomena can be found in visual hallucinations,but the overall picture is incomplete because these phenomena aremostly the subject of observer commentary rather than systematicstudy. Clearly, dream imagery is compatible with rapid eye move-ments, as is eidetic imagery, and neither seems to shift with eyemovements (Gru ¨sser & Landis, 1991). However, migraine hallu-cinations and experimentally induced phosphenes all shift in thedirection of gaze for saccadic and pursuit eye movements (Gru ¨sser, 1986). The differences may lie in whether the cortical area inwhich the hallucination arises is directly involved in corollary-discharge-based eye movement compensation. Evidence for addi-tional postprocessing exists. For example, in CBS, clinical visualloss leads to sensory deprivation hallucinations that can sometimesbe abolished by making saccadic (but not pursuit) eye movements(Kölmel, 1985), especially for saccades toward the normal portionof the visual field (Vaphiades, Celesia, & Brigell, 1996). This joinsfindings that saccades disrupt not only active vision but visualphosphenes (Riggs, Merton, & Morton, 1974) and visual cognitiondriven by mental imagery (Brockmole, Carlson, & Irwin, 2002). Curiously, when Charles Bonnet hallucinations originate in nowblind areas of peripheral visual field, they seem sharper than is normally seen in periphery. Similarly, Richards (1971) noted thatconverging eye movements can make migraine hallucinationsseem sharper than otherwise. In both cases, the anomalous sharp-ening is similar to the neural deblurring that occurs for someviewing conditions (e. g., Hammett & Bex, 1996). Pattern Formation in Other Sensory Cortices: What Would Cortical Stripes Sound Like? The cytoarchitecture of visual cortex is replicated in other sensory cortices; pattern formation is likely to occur in othersensory cortices and should be experienced in characteristicways as auditory, somatosensory, or other hallucinations. Forexample, an unfortunate somatosensory hallucination that canoccur in migraine is formication, a sensation of insects movingon or under the skin. This might be explained as the analogueof a visual fortification arc; a scintillating fortification wavefront moving through somatosensory cortex could result in asensation of spreading irregular movement across a section ofskin. Similarly, Klu ¨ver (1966) described a curious synesthesia experienced by a mescaline-dosed subject who reported notonly seeing hexagonal textures (which he called fretwork) butalso that he himself seemed to be made of the same pattern. Another subject could feel a moving cobweb sensation on histongue. A third subject reported that his legs and feet becamerotating spirals, identical to the rotating spiral he was seeing inhis visual field, and that he could feel them as rotating spirals;“one has the sensation of somatic and optic unity” (p. 71). Several of these reports came from physicians and show usefulinsights but might carry little weight if not for similar reports ofgeometric percepts in somatosensory synesthesia without intox-ication (Cytowic, 1989). Some of Klu ¨ver's reports are sugges-tive of sensory binding between visual and somatosensorycortex. There is precedent for this: Tactile two-point discrimi-nation on a limb improves if the limb is viewed under magni-fication, even if the magnified view is uninformative, suggest-ing that visual magnification forces some kind of slavedmagnified remapping of somatosensory cortex (Jackson, 2001;Kennett, Taylor-Clarke, & Haggard, 2001). Moreover, entrain-ment of pattern formation between cortices does not requiremassive coupling; in simulations intended to probe the conti-nuity of visual hallucinations between the two visual hemi-spheres, Rule et al. (2011) found that only a thin band ofsimulated connecting neurons was enough to cause identicalstripe formation in the two cortices. This begs the question, Ifpattern formation occurs in nonvisual cortex (and given thestructural similarities between sensory cortices, it is likely),would the orientation of the stripes be as perceptually mean-ingful as they are in visual cortex? To answer this may requirethat the work that illuminated visual hallucinations in visualcortex be recapitulated for the other senses and their associatedcortices. Such an effort would have three main thrusts: First,like Klu ¨ver, we need to synthesize reports of nonvisual hallu-cinations in synesthesia, migraine, epileptic seizures, and otherinducing conditions, seeking recurrent archetyped nonvisualhallucinations that are analogues of Klu ¨ver's visual form con-stants. Some of these may correspond to stripe patterns onspecific sensory cortices, as Klu ¨ver's visual forms did. Second,765 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
some theoreticians use the mapping from geometric patterns on retina to stripes on V1 to argue that these mappings evolved toallow the cortex to exploit certain regularities and invariancesin image processing (Caelli, 1977; Dodwell, 1991; Schwartz,1977, 1980); if other sensory cortices exploit similar ap-proaches, it should be possible to derive what forms would beof special significance to the information processing takingplace in these sensory cortices. Third, the insights from the firsttwo lines of research could be used to design functional imagingstudies of nonvisual cortices, using nonvisual stimuli to revealcortical stripe formation, as Tootell, Switkes, Silverman, and Hamilton (1988; Tootell et al., 1998) did with visual cortex. This then would provide feedback to the researchers engaged inthe other two lines of research. Conclusions It is in this sense, finally, that migraine is enthralling; for it shows us, in the form of a hallucinatory display, not only an elemental activityof the cerebral cortex, but an entire self-organizing system, a universalbehavior, at work. It shows us not only the secrets of neuronalorganization, but the creative heart of Nature itself. —Oliver Sacks (1995b) The variety of hallucinatory patterns generated by self-organized cortical pattern formation is dauntingly broad, and ourunderstanding of these phenomena is still incomplete, but thatshould not keep us from celebrating the remarkable multidisci-plinary convergence described above. Until recently, hallucina-tions were treated as disparate curiosities of vision, ill-conditionedfor serious study. Today, all evidence suggests that elementaryhallucinations are grounded in nonlinear dynamic neuronal net-works and that hallucinations provide a window into the innerworking of those networks. Geometric hallucinations are linked tophosphenes and migraine fortifications by their common spatialand temporal scales, by various theoretical commonalities, byinducers that lead to various combinations of these hallucinations,and by interactions between the many conditions that induce them. Further advances are expected from developments in functionalimaging and sophisticated mathematical models. These studiesshould illuminate ordinary visual processing as well; the interac-tions of geometric hallucinations with one another and with phys-ical stimuli obey many simple rules found in ordinary stimulus-driven vision and operate on familiar spatial and temporal neuralscales. Specifically, the behavior of hallucinatory percepts sug-gests that neural pattern formation in visual cortex is governed bythe same cortical properties of localized processing, lateral inhi-bition, simultaneous and sequential contrast, saccadic interference,and perceptual opponency that govern ordinary vision. 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