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Several factors contribute to this: Receptive fieldcenters in foveal retina are smaller than in the periphery, fovealoptics are better than peripheral optics, retinal ganglion cell densityis higher near fovea, and visual cortex devotes more neurons tofoveal inputs to peripheral ones. These factors fit neatly together;the cortex devotes its resources in accordance to the availability ofretinal information. Thus, the cortex magnifies the foveal repre-sentation and progressively minifies the peripheral representation. Originally, cortical magnification C Mwas thought of as a ratio of neural image size on cortex to optical image size on retina. An application of this is M-scaling: the physical magnification that must be applied to a stimulus at a peripheral location in the retinafor a particular perceptual task to be completed that could becompleted with an unmagnified stimulus presented to the fovea. For example, contrast sensitivity is relatively constant if spatialfrequency is expressed in cycles/millimeter on visual cortex. Thecortical magnification factor can also be inferred from the scalingof fortification serrations in migraine (Gru ¨sser, 1995). In general, cortical magnification C Mat a distance r(in degrees of visual angle) from the fovea along a constant retinal direction /H9258, is fit by a simple monotonically decreasing function CM/H11005k//H20849r/H11001a/H20850. (A1) Because CMcan be defined as a ratio of small changes of cortical and retinal extent, it resembles a differential. If we naively inte-grate a function of this form—as Fechner did for Weber's law (see Billock & Tsou, 2011, on Fechner's law)—a logarithmic functionis obtained. However, since both the retina and cortex are two-dimensional sheets and isotropy is not guaranteed, the situation isa little more complex (Schwartz, 1994). In 1977, two independentinvestigators noted that the cortical magnification factor can beintegrated in such a way as to yield a sensible mapping from thepolar ( r,/H9258) retina to the more rectilinear ( x,y) cortex (Cowan, 1977; Schwartz, 1977). x/H11005/H9252ln/H20851/H20849√/H20849εr/H20850/H11001/H20849w o2/H11001εr2/H208501/ 2/H20850/2wo/H20852;y/H11005/H9252r/H9258//H20849wo2/H11001εr2/H208501/ 2, (A2)where ris radial distance (in degrees of visual angle) on retina from fovea, /H9258is the angular direction from the fovea, wois the mean diameter of foveal receptive fields, εis the rate of increase of that diameter with r, dis the density of receptive field overlap, and/H9252/H11005 (4d//H9266ε)1/2. Equation A2 is the version that was employed by Ermentrout and Cowan (1979) and Ermentrout (1984); varia-tions can be found in other studies, but for our purposes, thedifferences are of no consequence. For distances greater than onedegree of visual angle from the fovea (e. g., for all but a tiny portionof the ca. 135° vertical /H11003150° horizontal degree visual field), this function is well approximated as x/H11005/H9252ln(r√ε/w o);y/H11005/H9252 /H9258. This mapping is sometimes called a complex logarithmic function (be-cause the same form can be used to represent the logarithm of acomplex number like ln[ r,i/H9258]; Cowan, 1977; Schwartz, 1977). This conceptual advance did not get the attention it deserved until Tootell, Silverman, Switkes, and De Valois (1982) found a directand astonishingly convincing way to image the mapping in pri-mates. They created a polar grid visual stimulus consisting ofradial lines intersecting logarithmically spaced concentric circles. Markings on the grid continually reversed in contrast, assuring thatthey would activate both on-and off-cells and both sustained-andtransient-cells for maximum neural response. The stimuli were shownfor 30 min to one eye of a macaque who had been dosed withradioactive 14C-labeled 2-deoxy-D-glucose. Increased neural activity led to increased uptake of the radioactive glucose; after metabolism, the radioactive tag accumulated in the active neurons. The animal wasthen sacrificed, and its occipital cortex was extracted and pressedagainst a photographic plate, which was exposed by emitted radiation(see Figure A1). The delineation between stimulated and inactivecortex is remarkably crisp. Although modern imaging techniquesallow the same mapping to be studied in intact humans (see Figure 5),this remains the most convincing single direct demonstration of anyneural inference that we are aware of. A1 Although the nonlinear retinocortical mapping has important implications for how the cortex may be organized to processspatial information (Schwartz, 1977, 1980), for our purposes itsmost important implication is that three very different kinds of A1Schwartz (1994) recounted that when Tootell first showed Figure A1 at a conference (the Association for Research in Vision and Ophthalmology annual meeting, ca. 1980), the audience spontaneously stood and applauded. (Appendix continues )772 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
Figure A1. The clearest and most compelling demonstration of the nonlinear mapping from retinal to cortical coordinates. Equation A2 is a complex logarithmic mapping from retinal polar coordinates to cortical Cartesiancoordinates, which formed the basis for Ermentrout and Cowan's (1979) hypothesis. The polar web pattern onthe top (A) was used to stimulate the visual system of a macaque treated with radioactively labeled glucose. Aftersufficient time to accumulate the tracer in the most active cells (for details, see the Appendix), the radiograph(B) of the animal's cortex was made. The radial and concentric features of the polar web stimulation on retinaare transformed to the nearly rectilinear grid shown on V1. Almost two decades later, it became possible for Tootell et al. (1998) to recreate this demonstration noninvasively in living humans using functional magneticresonance imaging (see Figure 5), but the clarity of this early demonstration has not yet been equaled. From“Deoxyglucose Analysis of Retinotopic Organization in Primate Striate Cortex,” by R. B. H. Tootell, M. S. Silverman, E. Switkes, & R. L. De Valois, 1982, Science, 218, p. 902. Copyright 1982 by the American Association for the Advancement of Science. Reprinted with permission from AAAS. (Appendix continues )773 HALLUCINATIONS AND CORTICAL PATTERN FORMATION	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
retinal stimulus patterns would result in identical cortical activity patterns: fan shapes, concentric circles, and spirals imaged on retinaall map to cortex as parallel stripes of cortical activity. Some corol-laries of this are the following: (a) If parallel stripes of cortical activityform spontaneously, their percept is governed by orientation of thestripes on cortex (Ermentrout & Cowan, 1979). (b) Perceptuallyopponent shapes, like fan shapes and concentric circles, are associatedwith orthogonal patterns of cortical activity (as are clockwise andcounterclockwise spirals). (c) Scrolling stripe patterns on cortex leadsto rotating fan shapes and spirals and to expanding/contracting con-centric circles. (d) Rotating stripes of cortical activity morphs per-cepts—fan shapes twist into spirals, which progressively tighten until becoming concentric circles; multistability between hallucinatory pat-terns could result. Much hallucinatory diversity can thus be explainedby the interaction of cortical stripe formation with the nonlinearretinocortical mapping. Received May 13, 2011 Revision received October 12, 2011 Accepted November 16, 2011 /H18546774 BILLOCK AND TSOU	Elementary Visual Hallucinations and Their Relationships to Neural Pattern-Forming Mechanisms.pdf
r THE ROYAL ‹d Ü SOCIETY Geometric visual hallucinations, Euclidean symmetry and the functional architecture ofstriatecortex Paul C. Bressloff', jack D. Cowan", Martin Golubitsky, Peter J. Thomas' and Matthew C. Wiener' 'Department of Mathematics, University of Utah,Salt Lake City,UT84112,USA "Department of Mathematics, University of Chicago,Chicago IL60637,USA 'Department of Mathematics, k'nitersit of How. Eton Houston, TX77204-3476,USA 'Computational Xeurobiolo Laboratory, Salt Institutefor Biological Studies,POBoxd5800,San Diego,CA92186-5800, USA ''Laboratory of Menrops Jcholo, National Institutes of Health,Bethesda, MD20892,USA This paper isconcerned witha striking visual experience: that ofseeinggeometric visualhallucinations. Hallucinatory images were classified by Klüver into four groups called form constants comprising (igratings, lattices, fretworks, filigrees, honeycombs andchequer-boards, (ii)cobwebs, (iii)tunnels, funnels, alleys, cones and vessels, and (iv)spirals. Thispaperdescribes a mathematical investigation of their origin based on the assumption that the patterns ofconnection between retinaandstriatecortex (henceforth referred toas V1}—theretinocortical map—and ofneuronal circuitsin VI,bothlocaland lateral, determine their geometry. Inthefirstpartofthepaperweshowthatformconstants, whenviewedin V1coordinates, essentially correspond tocombinations ofplanewaves,thewavelenqths ofwhichareintegral multiples ofthewidth ofa human Hubel—Wiesel hypercolumn, ca. 1. 33-2mm. \Venextintroduce a mathematical description of the large-scalc dynamics of V1intcrmsofthecontinuum limitofa latticc ofintcrconncctcd hypcr-columns, each ofwhichitselfcomprises a number ofinterconnected iso-orientation columns. Wethen show that thepatterns ofinterconnection in V1exhibitaveryinteresting symmetry, i. e. theyareinvariant under the action of the planar Euclidean group E(2}—thegroupofrigidnotions intheplane— rotations, reflections and translations. \Vhat isnovelisthatthelateralconnectivity of V1issuchthata new group action isneededtorepresent itsproperties: byvirtueofitsanisotropy itisinvariant with respect to certain shifts and twists oftheplane. Itisthisshift—twist invariance thatgenerates new rcprcscntations of E(2). Assuming thatthestrcngth oflateralconncctions iswcakcomparcd withthatof local connections, we next calculate the eigenvalues and eigenfunctions ofthecortical dynamics, using Rayleigh—Schrödinger perturbation theory. The result isthatintheabsence oflateralconnections, the eigenfunctions are degenerate, comprising both even and odd combinations ofsinusoids in‹ö,thecortical label fororientation preference, andplanewavesinr,thecortical position coordinate. 'Switching-on' the lateral interactions breaks the degeneracy and either even orelseoddeigenfunctions areselected. Thèse results can beshowntofollowdirectly fromthe Euclidean symmetry wehaveimposed. Inthesecondpartofthepaperwcstudythenatureofvarious evenandoddcombinations ofeigen-functions orplanforms, thesymmetries ofwhicharesuchthattheyremaininvariant undertheparticular action of E(2)wehaveimposed. Thèsesymmetries correspond tocertainsubgroups of E(2),theso-called axial subgroups. Axial subgroups are important inthattheequivariant branching lemmaindicates that when a symmetrical dynamical system becomes unstable, new solutions emerge which have symmetries corresponding totheaxialsubgroups oftheunderlying symmetry group. Thisisprecisely thecasestudied inthispaper. Thuswestudythevarious planforms thatemergewhenourmodel V1dynamics become unstableunderthepresumed actionofhallucinoqens orflickering lights. Weshowthattheplanforms correspond totheaxialsubgroups of E(2),undertheshift—twist action. \Vethencompote whatsuch planforms would look likeinthevisualfield,givenanextension oftheretinocor ticalmaptoincludeits action on local edges and contours. What ismostinteresting isthat,givenourinterpretation ofthe correspondance between Vl planforms and perceived patterns, the setofplanforms generates represent-ativcs ofalltheformconstants. Itisalsonotcworthy thattheplanforms dcrivcd fromourcontinuum model naturally divide V1 into what are called linear regions, inwhichthepattern hasa nearconstant orientation, reminiscent oftheiso-orientation patchesconstructed viaopticalimaginé. Theboundaries of such regions form fractures whose points ofintersection correspond tothewell-known 'pinwheels'. * Autlior forrorrespondence (eoivanu math. urhi‹-ago. edu). Phil. Trans. R. Soc. Land. 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300 P. C. Bressloff andothers Gromr/rir riisualhallucinatians To complete the study we then investigate the stability oftheplanforms, usingmethods ofnonlinear stability analysis, including Liapunov—Schmidt reduction and Poincaré—Lindstedt perturbation theory. We finda close correspondence between stable planforms and form constants. The results are sensitive to the detailed specification ofthelateralconnectivity andsuggestaninteresting possibility, thatthecortical mechanisms by which geometric visual hallucinations are generated, ifsitedmainlyin V1,areclosely related tothoseinvolved intheprocessing ofedgesandcontours. Keywords: hallucinations; visual imagery; flicker phosphenes; neural modelling; horizontal connections; contours '... thehallucination is... nota staticprocessbuta dynamic process, the instability of which reflects an instability in itsconditions oforigin' (Klüver (1966), p. 95,ina comment on Mourgue (1932). 1. INTRODUCTION Geometric visual hallucinations are seen inmanysitua-tions, for example, after being exposed to flickering lights (Purkinje 1918; Helmholtz 1924; Smythies 1960 after the administration ofcertainanaesthetics (\Vinters 1975}, on waking up or falling asleep (Dybowski 1939a, following deep binocular pressure on one's eyeballs (Tyler 1978a, and shortly after the ingesting ofdrugssuch as LSDandmarijuana (Oster1970;Siegel1977}. Patterns that may be hallucinatory are found preserved inpetro-glyphs (Patterson 1992) and incavepaintings (Clottes& Lewis-\Villiams 199S}. Therearemanyreports ofsuch experiences (Knauer & Maloney 1913, pp. 429-430a: 'Immediately before my open eycs area vastnumberof rings, apparently made ofextremely finesteelwire,all constantly rotating inthedirection ofthehandsofa clock;thesecirclesareconcentrically arranged, theinner-most being infinitely small, almost pointlikc, the outermost bcing abouta mctcr anda halfindiameter. The spaces between the wires seem brighter than the wires themselves. Now thewiresshinelikedimsilverin parts. Nowa beautiful lightviolettinthasdeveloped in them. As I watch, the center seems torecedeintothe depth oftheroom,leavingtheperiphery stationary, till thewholeassumestheformofa deep tunnel ofwirerings. The light,whichwasirregularly distributed amongthe circles,hasrecededwiththecenterintotheapexofthe funnel. Thecenterisgradually returning, andpassingthe position when alltheringsareinthesameverticalplane, continues toadvance, tilla cone forms with itsapex toward me... Thewiresarenowflattening intobands orribbons,witha suggestion oftransverse striation, and colored a gorgeous ultramarine blue, which passes in places intoanintenseseagreen. Thesebandsmoverhyth-mically, ina wavy upward direction, suggesting a slow endless procession ofsmallmosaics, ascending thewallin singlefiles. Thewholepicturehassuddenly receded, the center much more than thesides,andnowina moment, high above me, isa dome ofthemostbeautiful mosiacs,. Thedomehasabsolutely nodiscernible pattern. But circles are now developing upon it;the circles are becoming sharp and elongated... nowtheyarerhombics now oblongs; and now allsortsofcuriousanglesare forming; and mathematical figuresarechasingeachother wildly acrosstheroof.. Klüver (1966) organized the many reported images into four classes, which he called form constants: (I gratings, lattices, fretworks, filigrees, honeycombs and chequer-Phil. Trans. R. Soc. Lond. B (2001s boards; II cobwebs; (III tunnels and funnels, alleys, cones, vessels; and(IVAspirale. Someexamples ofclass I form constants arc shown infigure1,whilecxamplcs of theotherclassesareshowninfigures2-4. Such images are seen both by blind subjects and in sealed dark rooms (Krill eta/. 1963}. Various reports (Klüver 1966)indicate thatalthough theyaredifficult to localise inspace,andactually movewiththeeyes,their positions relative toeachotherremainstablewithrespect tosuchmovements. Thissuggests thattheyaregenerated not intheeyes,butsomewhere inthebrain. Oneclueon their location inthebrainisprovided byrecentstudiesof visual imagery (Miyashita 1995s. Although controversial, the evidence seems tosuggestthatareas V1and V2,the striate and extra-striate visual cortices, are involved in visual imagery, particularly iftheimagerequires detailed inspection (Kosslyn 1994). More precisely, ithasbeen suggested that(Ishai& Sagi 1995, p. 1773s '[the]topological representation [provided by VI]might subserve visual imagery when thesubjectisscrutinizing attentively local features ofobjectsthatarcstoredin memory'. Thus visual imagery isseenastheresultofaninteraction between mechanisms subserving the retrieval of visual memories and those involving focal attention. In this respect itisinteresting thatthereseemstobecompetition between the seeing ofvisualimagery andhallucinations (Knauer & Maloney 1913,p. 433): '. aftera picture had been placed ona background and then removed “Itriedtoseethepicturewithopeneyes. Innor:asewas I successful; only [hallucinatory] visionary phenomena covered theground”' Competition between hallucinatory images and after-images was alsoreported Klüver19G6,p. 35: 'Insomeinstances, the[hallucinatory] visionspreyentcd the appearance ofafter-images entirely; [howexer] in most cases a sharply outlined normal after-image appeared fora while... whilethevisionary phenomena were stationary, theafter-images movedwiththeeyes'. As pointed out tousbyoneofthereferees, thefused image ofa pair ofrandom dotstereograms alsoseemsto be stationary with respect toeyemovements. Ithasalso been argued that because hallucinatory images are seen ascontinuous acrossthemidline, theymustbelocatedat higher levels in the visual pathway than V1 or V2 (R. Shapley, personal communication. ) In this respect there isevidence thatcallosal connections alongthe VU V2 border can act tomaintain continuity oftheimages across thevertical meridian (Hubel& Wiesel 1967a. 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Figure 1. (a)'Phosphene' produced bydeepbinocular pressure ontheeyeballs. Redrawn from Tyler(1978). (fi)Honeycomb hallucination generated bymarijuana. Redrawn from Clottes& Lewis-\\'illiams (1998. images. Inourviewsuchimagesaregenerated in VIand stabilized with respect toeyemovements bymechanisms present in V2andelsewhere. Itislikelythattheactionof“ such mechanisms israpidly fedbackto V1 (Lee etal. 1998}. Itnowfollows, because allobservers reportseeing Klüver's form constants or variations, that those properties common toallsuchhallucinations shouldyield information about the architecture of V1. Wetherefore investigate that architecture, i. e. thepatterns ofconnec-tion between neurons in the retina and those in Vl, together with intracortical Vl connections, on the hypothesis that such patterns determine, in large part, the geometry of hallucinatory form constants, and wc defer untila later study, the investigation ofmechanisms that contribute totheircontinuity acrossthemidline and totheirstabilité inthevisualfield. (b) Fh Phxn«n wsfnocowfcœfnæp The firststepistocalculate whatvisualhallucinations look like, not in the standard polar coordinates ofthe visual field,butinthecoordinates of Vl. Itiswellestab-lished that there isa topographic map ofthevisualfield in V1,theretinotopic representation, andthatthecentral region ofthevisualfieldhasa much bigger representation in VIthanitdoesinthevisualfield(Sereno ctal. 1995). The reason forthisispartlythatthereisa non-uniform Phil. Trans. R. Soc. Land. Y›('200l) Geametric riiua/Àa//ucina/ion,i P. C. Bresslolf andothers 301 (b) Figure ?. (a)Funneland(é)spiralhallucinations generated by LSD. Redrawn from Oster(1970. Figure 3. (n)Funneland(b)spiraltunnelhallucinations generated by USD. 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302 P. C. Bressloff andothers Geametric nrenalhallucinatians Figure 4. Cobwebpctroqlyph. Redrawn from Patterson (1992). distribution of retinal ganglion cells, each of which connects to V1viathelateralgeniculate nucleus (LGN}. This allows calculation ofthedetailsofthemap(Cowan 1977). Let pg be the packing density ofretinalganglion cellsperunitareaofthevisualfield,p the corresponding density per unit surface area ofcellsin V1,and[R RI retinal or equivalently, visual field coordinates. Then pzrpdryd Bgisthenumber ofganglion cellaxonsina retinal element ofarea Rdryd R By hypothesis these axons connect topographically tocellsinanelement of V1 surface area dxdy,i. e. topdzdycortical cells. (V1is assumed to be locally flatwith Cartesian coordinates. ) Empirical evidence indicates that p is approximately constant (Hubel & Wiesel 1974a,5},whereas PRdeclines from the origin ofthevisualfield,i. e. thefovea,withan inverse square law (Dras do 1977) 3c/2 (c) (b) 1 where z 0 A nd t are constants. Estimates ofm 0 — 0. 087 andc = 0. 051 inappropriate unitscanbeobtained from published data (Drasdo 1977}. From the inverse square law one can calculate the Jacobian ofthemapandhence V1 coordinates (x,y) asfunctions ofvisualfieldorretinal coordinates (R RI The resulting coordinate transforma-tion takes theform O t x = —In1+—R where o and fiare constants in appropriate units. Figure 5 shows themap. The transformation has two important limiting cases: (i)nearthefovea,ci R<mt, itreducesto and (ii, sufficiently farawayfromthefovea,^R mt, it becomes x = —ln Cr R Case (i)isjusta scaled version oftheidentitymap,andcase (ii)isa scaled version ofthecomplex logarithm aswasfirst recognized by Schwartz (1977). To see this, let 3g-xg+iyg=rpexp[i8p], bethecomplex representation l“igureñ. 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(b) Figure 6. Actionoftheretinocortical maponthefunnelform constant. (a)Imageinthevisualfield;(é)VI map oftheimage. ofa retinal point (xg,yg = (rq,8R, then x +ip = In(Rexp[i BR]/=In R-I-i8RThusx = In R = R (c)Forrmconstantsasspontaneous cortical patterns Given that the retinocor ticalmapisgenerated bythe complex logarithm (except near the fovea), itiseasyto calculate the action ofthetransformation oncircles,rays, and logarithmic spirals in the visual field. Circles of constant r R inthevisualfieldbecome vertical linesin V1, whereas rays of constant R become horizontal lines. Interestingly, logarithmic spirals become oblique lines in V1: the equation of such a spiral isjust 8q =a Inrg whence = nx under the action of‹R 2. Thusform constants comprising circles, rays and logarithmic spirals inthevisualfieldcorrespond tostripesofneuralactivity atvarious anglesin V1. Figures6 and 7 show the map action on the funnel and spiral form constants shown in figure 2. A possible mechanism forthespontaneous formation of stripes ofneuralactivityundertheactionofhallucinogens was originally proposed by Ermentrout & Cowan (1979a. They studied interacting populations of excitatory and inhibitory neurons distributed within a two-dimensional (2D cortical sheet. Modelling the evolution ofthesystem intermsofa setof Wilson—Cowan equations (Wilson& Cowan 1972, 1973) they showed how spatially periodic activity patterns such as stripes can bifurcate from a homogeneous low-activity state via a Turing-like instability (Turing 1952). The model also supports the formation of other periodic patternssuchashexagons and squares—under the retinocortical map these Phil. Trans. R. Soc. Land. Y›('200I) Geometric r/iu‹z/hallucinatians P. C. Bressloff andothers 303 Figure 7. Actionoftherctinocortical maponthespiralform constant. (u)Imageinthevisualfield;b V1 map oftheimage. generate more complex hallucinations inthevisualfield such as chequer-boards. Similar results are found ina reduced single-population model provided that the inter-actions are characterized by a mixture of short-range excitation and long-range inhibition (the so-called 'Mexican hat distribution'). (d) Owient«tiontxn*ngin F2 The Ermentrout—Cowan theory ofvisualhallucinations isover-simplified inthesensethat VIisrepresented asifit were justa cortical retina. However, VI cells do much more than merely signalling position inthevisualfield: most cortical cellssignalthelocalorientation ofa contrast edge orbar—they aretunedtoa particular local orienta-tion (Hubel & \Viesel 1974a}. The absence oforientation representation in the Ermentrout—Cowan model means thata number oftheformconstants cannotbegenerated by the model, including lattice tunnels (figure 42),honey-combs and certain chequer-boards (figure1, and cobwebs (figure 4). These hallucinations, except the chequer-boards, are more accurately characterized aslatticesof locally orientated contours oredgesratherthanintermsof contrasting regions oflightanddark. In recent years, much information has accumulated about the distribution oforientation selective cellsin V1, and about their pattern ofinterconnection (Gilbert 1992}. Figure S shows a typical arrangement of such cells, obtained viamicroelectrodes implanted incat Vl. 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304 P. C. Bressloff andothers Geametric visualhallucinatians Figure S. (n Orientation tuned cellsin V1. Notethe constancy olorientation prel'erence ateachcorticallocation (electrode tracks1 and3, and therotationoforientation preference ascorticallocationchanges(electrode track 2). (b)Receptive fieldsfortracks1 and 3. (c)Expansion ofthe receptive fieldsoftrack2 toshowtherotationoforientation preference. Redrawn from Gilbert(199?. preference reappears, i. e. thedistribution isa-periodic in the orientation preference angle. The second panel shows the receptive fieldsofthecells,andhowtheychangewith V1 location. The third panel shows more clearly the rotation ofsuchfieldswithtranslation across V1. How are orientation tuned cells distributed and inter-connected? Recent work on optical imaging has made it possible toseehowthecellsareactually distributed in VI (Blasdel 1992), and a variety of stains and labels has made itpossible to see how they are interconnected (G. G. Blasdel and L. Sincich, personal communication), (Eysel 1999; Bosking rfal. 1997. Figures 9 and 10show such data. Thus, figure 9a shows that the distribution of orientation preferences isindeedroughly-periodic, in that approximately every 0. 5mm(inthemacaque} there isaniso-orientation patchofa given preference, and figure 10showsthatthereseemtobeatleasttwolength-scales: (i)local—cells lessthan0. 5 mm apart tend to make connections with most of their neighbours ina roughly isotropic fashion, asseeninfigure9é,and (ii)lateral—cells makecontacts onlyevery0. 5mmor so along their axons with cells in similar iso-orientation patches. In addition, figure 10 shows that the long axons which support such connections, known as intrinsic lateral or horizontal connections, and found mainly inlayers IIand Phil. Trans. R. Soc. Land. B (2001a l“igure9 (a)Distribution oforientation preferences in macac{ue V1 obtained viaopticalimaging. Redrawn from Blasdel(1992. (h)Connections madebyaninhibitory interneuron incat V1. Redrawn from Eyscl(1999). III of V1,andtosomeextentinlayer V(Rockland & Lund 1983a, tend tobeorientated alongthedirection of their cell'spreference (Gilbert 1992;Bosking ctal. 1997, i. e. theyrunparallel tothevisuotopic axisoftheircell's orientation preference. These horizontal connections arise almost exclusively from excitatory neurons (Levitt & Lund 1997; Gilbert & \Viesel 1983), although 20% terminate on inhibitory cellsandcanthushavesignificant inhibitory effects (Mc Guire ctal. 1991}. There issomeanatomical andpsychophysical evidence (Horton 1996;Tyler1982) that human VI has several times the surface area ofmacaque V1witha hypercolumn spacing ofca. 1. 33-2mm. Intherestofthispaperwe work with this length-scale to extend the Ermentrout— Cowan theory ofvisualhallucinations toincludeorienta-tion selective cells. A preliminary account of this was described in\Viener (1994)and Cowan(1997}. 2. A MODEL OF V1 WITH ANISOTROPIC LATERAL CONNECTIONS (a) Z'liertiofef The state ofa population ofcellscomprising aniso-orientation patch atcortical position r CR 2 attime is/C112/C114/C101/C102/C101/C114/C101/C110/C99/C101 /C114/C101/C97/C112/C112/C101/C97/C114/C115/C44 /C105/C46/C101/C46 /C116/C104/C101 /C100/C105/C115/C116/C114/C105/C98/C117/C116/C105/C111/C110 /C105/C115 /C25/C45/C112/C101/C114/C105/C111/C100/C105/C99 /C105/C110 /C116/C104/C101 /C111/C114/C105/C101/C110/C116/C97/C116/C105/C111/C110 /C112/C114/C101/C102/C101/C114/C101/C110/C99/C101 /C97/C110/C103/C108/C101/C46 /C84/C104/C101 /C115/C101/C99/C111/C110/C100 /C112/C97/C110/C101/C108 /C115/C104/C111/C119/C115 /C116/C104/C101 /C114/C101/C99/C101/C112/C116/C105/C118/C101 /C162/C101/C108/C100/C115 /C111/C102 /C116/C104/C101 /C99/C101/C108/C108/C115/C44 /C97/C110/C100 /C104/C111/C119 /C116/C104/C101/C121/C99/C104/C97/C110/C103/C101 /C119/C105/C116/C104/C86/C49 /C108/C111/C99/C97/C116/C105/C111/C110/C46 /C84/C104/C101 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Figure10. Lateralconnections madebycellsin(a)owl monkey and b)treeshrew V1. Aradioactiv e tracerisusedto show thelocations ofallterminating axonsfromcellsina central injection site,superimposed onanorientation map obtained keyopticalimaging. Redrawn from G. G. Blasdeland L. Sincich(personal communication) and Boskingctal. (1997). characterized by the real-valued activity variable apr,e,/), where ee [0, istheorientation preference of the patch. Vl istreatedasan(unbounded continuous 2D sheet ofnervous tissue. Forthesakeofanalytical tract-ability, we make the additional simplifying assumption that d›and r are independent variables—all possible orientations are represented at every position. A more accurate model would need to incorporate details concerning the distribution oforientation patches inthe cortical plane (asillustrated infigure9a). Itisknown,for example, thata region ofhuman V1ca. 2. 67mm'onits surface and extending throughout itsdepthcontains at least two setsofalliso-orientation patches intherange 0 e <r, one for each eye. Such a slab was called a hypercolumn by Hubel & \Viesel (1974a). Ifhuman V1as awhole (in one hemisphere) hasa surface area ofca. 3500 mm" (Horton 1996},thisgivesapproximately 1300 such hypercolumns. So one interpretation ofourmodel Phil. Trans. R. Soc. Land. Y›('200I) Geametric visualhallucination. ‹ P. C. Bressloff andothers 3Oñ would be that itisa continuum version ofa lattice of hypercolumns. However, a potential difficulty with this interpretation isthattheeffective wavelength ofmanyof the patterns underlying visual hallucinations isofthe order oftwicethehypercolumn spacing (see,forexample, figure 2, suggesting that lattice effects might be impor-tant. A counter-argument for the validity of the continuum model (besides mathematical convenience) is tonotethattheseparation oftwopointsinthevisual field—visual acuity— (ata given retinal eccentricity of rp, corresponds to hypercolumn spacing (Hubel & Wiesel 1974a), and sotoeachlocation inthevisualfield there corresponds toa representation in Vlofthatloca-tionwithfiniteresolution andallpossible orientations. The activity variable a z,m,t)evolvesaccording toa generalization ofthe Wilson—Cowan equations (\Vilson& Cowan 1972, 1973} that takes into account the additional internal degree of freedom arising from orientation preference: xw[u(r',d›',t)]d 'da'-I-b(r,d›,/), (l where o and are decay and coupling coefficients, b(r,e,t)isanexternal input,in(r,e\|r',e')istheweight of connections between neurons at r tuned to e and neurons atr'tunedtod›',and [¿]isthesmoothnonlinear function forconstants and §. Y\'ithout lossofgenerality wemay subtract from w[¿]a constant equal to[1-I-e"]*'toobtain the (mathematically) important property that w[0] = 0, which implies that for zero external inputs the homoge-neous state a(r,e,t ——0 for allr,e,I isa solution to equation (1). Fromthediscussion in§1 (d),wetakethe pattern of connections in(r,‹D\|r',e') to satisfy the following properties (seefigure I). There existsa mixture oflocalconnections withina hypcrcolumn and (anisotropic) lateral connections between hypercolumns; the latter only connect elements with the same orientation preference. Thus inthecontinuum model,inisdecomposed as with..— =../e/ (ii)Lateral connections between hypercolumns only join neurons that liealongthedirection oftheir (common) orientation preference d›. Thus in the continuum model (4) where rt (1,0)andfi istherotation matrix/C99/C104/C97/C114/C97/C99/C116/C101/C114/C105/C122/C101/C100 /C98/C121/C116/C104/C101 /C114/C101/C97/C108/C45/C118/C97/C108/C117/C101/C100 /C97/C99/C116/C105/C118/C105/C116/C121/C118/C97/C114/C105/C97/C98/C108/C101 /C97/C40/C114/C44/C30/C44/C116/C41/C44 /C119/C104/C101/C114/C101 /C30/C50/C137/C48/C44/C25/C41 /C105/C115 /C116/C104/C101 /C111/C114/C105/C101/C110/C116/C97/C116/C105/C111/C110 /C112/C114/C101/C102/C101/C114/C101/C110/C99/C101 /C111/C102 /C116/C104/C101 /C112/C97/C116/C99/C104/C46/C86/C49/C105/C115 /C116/C114/C101/C97/C116/C101/C100 /C97/C115 /C97/C110 /C40/C117/C110/C98/C111/C117/C110/C100/C101/C100/C41 /C99/C111/C110/C116/C105/C110/C117/C111/C117/C115 /C50/C68 /C115/C104/C101/C101/C116 /C111/C102 /C110/C101/C114/C118/C111/C117/C115 /C116/C105/C115/C115/C117/C101/C46 /C70/C111/C114 /C116/C104/C101 /C115/C97/C107/C101 /C111/C102 /C97/C110/C97/C108/C121/C116/C105/C99/C97/C108 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306 P. C. Bressloff andothers Gromr/rir nisualhallucinatians :, “ lateralconnections hypercolumn Figure II. Illustration ofthelocalconnections withina hypercol umn and theanisotropic lateralconnedtionsbetween hypercol umns. cos8 —sin8 i sin8 cos8 The weighting function g(i determines how the strength of'lateral connections varies with thedistance ofseparation. Wetakeg(i}tobeoftheparticular form g(/ = [2x\j] '”exp-2 2\-A+,[2z ] with(„ < t„and A„p<l,whichrepresent acombi-nation of short-range excitation and long-range inhibition. This isanexample ofthe Mexican hat distribution. (Note that one can view the short-range excitatory connections asarisingfrompatchy local connections within a hypercolumn. ) Itispossible toconsider moregeneralchoicesofweight distribution inthat (i}allow for some spread in the distribution of lateral connections (sec figure 12, and (inincorporate spatially extended isotropic localinter-actions. An example ofsuchadistribution isgivenbythe following generalization ofequations (3 and (4): with A„(— = t„(d›}=0 for\|‹5\| >et„ and At„ (\|r\|) =0 for r>{9. Moreover, equation (5}ismodi-fiedaccording to with r#=(cos(8), sin(8 and §(— 8)=§(8. The para-meters ed and determine the angular spread oflateral connections with respect to orientation preference and Phil. Trans. R. Soc. Land. B (2001a Figure12. Example ofanangularspreadintheanisotropic lateralconnections between hypcrcolumns withrespectto both space (8t) and orientation preference (d›t. space, respectively, whereas {t determines the (spatial) range oftheisotropic localconnections. (b) Euclidean symmetry Suppose that the weight distribution in satisfies equations (7)and(8. We show that inisinvariant under the action ofthe Euclidean group E(2 ofrigidmotions in the plane, and discuss some of the important conse-quences ofsucha symmetry. (i)Euclidean groupaction The Euclidean group iscomposed ofthe(semi-direct) product of O(2, the group ofplanarrotations andreflec-tions, with R', the group of planar translations. The action ofthe Euclidean groupon R 2 x S'isgenerated by forall C O(2)+R 2 and the action on in P\|P')is where xisthereflection / f— 2 and fi#isa rotation by 8. The corresponding group action on a function a:R 2 x S'—+Rwhere P ——(r,‹o)isgivenby (10} The particular form of the action of rotations in equations (9} reflects a crucial feature of the lateral connections, namely that they tend tobeorientated along thedirection oftheircell'spreference (seefigure11). Thus, ifwejustrotate Vl,thenthecellsthatarenowconnected atlongrangewillnotbeconnected inthedirection of their preference. This difficulty can be overcome by permuting the local cells in each hypercolumn so that cells that are connected at long range are again connected inthedirection oftheirpreference. Thus,in the continuum model, the action ofrotation of V1by8 corresponds torotation ofrby8 while simultaneously sending e toe+ 8. Thisisillustrated infigure13. The action ofreflections isjustified ina similar fashion. (ii)/nrar/an/ weightdistributionin thenowprovethatinasgivenbyequations (7)and(8)is invariant under the action ofthe Euclidean groupdefined by equations (9}. 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twopoints P,Q'=R 2 X [0,x) 0. 5 —05 —1 —0. 5 0 0. 5 l 0. 5-Geametric visualhallucinatians P. C. Bressloff andothers 307 rotationby8 =x/6 —0. 5-' / {r8) —I —0'. 5 0 0'. 5 l satisfying equations (3—(5)isalso Euclidean invariant. Translation invariance ofinisobvious, i. e. Invariance with respect toa rotation by8 follows from Finally, invariance under a reflection xaboutthex-axis holds because We have used the identity u R_ ——R nandtheconditions (iii Implications of Euclidean symmetry Consider the action of on equation (1)forñ(r,/)=0: R"x S' R' x S' ———aa>°'P,I + fi in(P\|P”)°[a(«*'P”, t ]d P”, R'x S' since d[>°' P]——Ad Pandinis Euclidean invariant. Ifwe rewrite equation (l)asanoperator equation, namely, da d/ then itfollowsthat>F[a] F[ma]. Thus F commutes with CE(2 and F issaidtobergu/rarinn/ withrespecttothe Phil. Trans. R. Soc. Land. Y›('200I) Figure 13. Actionofa rotation by symmetry group E(2) (Golubitsky et‹z/. 1955). Theequiv-ariance oftheoperator F with respect totheactionof E(2 has major implications for the nature ofsolutions bifurcating trom the homogeneous resting state. Let be a bifurcation parameter. \Ve show in§4 that neara point forwhichthesteadystatea(r,e,y}=0 becomes unstable, there must existsmoothsolutions totheequilibrium equa-tion G[a i-,m,y)] 0 that are identified by their symmetry (Golubitsky etal. 19SS). \Vefindsolutions that are doubly periodic with respect toa rhombic, square or hexagonal lattice by using the remnants of'Euclidean symmetry on these lattices. Theseremnants arethe(semi-directs products I ofthetorus T'oftranslations modulo the lattice with the dihedral groups 2, and D„ the holohedries of the lattice. Thus, when a i-,d›,fi =0 becomes unstable, new solutions emerge from the instability with symmetries that are broken compared with f. Sufficiently close tothebifurcation pointthese patterns are characterized by (finite}linearcombinations of eigenfunctions of the linear operator L ——DG obtained by linearizing equation (IQaboutthehomoge-neous statea = 0. Thesecigcnfunctions arcderivedin§3. For the sake ofmathematical convenience, werestrict our analysis inthispapertothesimpler weightdistribu-tion given by equations (3 and (4) with satisfying either equation (5 or(SQ. Themostimportant property of inisitsinvariance undertheextended Euclidean group action (9, which isitselfa natural consequence ofthe anisotropic pattern oflateralconnections. Substitution of equation (3) into equation (1)gives (for zero external inputs) + fi zt„(r—r',ew[a(r',e,t ]de', R' (1I where we have introduced an additional coupling para-meter fithatcharacterizes therelative strength oflateral interactions. Equation (11aisofconvolution type,inthat the weighting functions are homogeneous intheirrespec-tivedomains. However, theweighting function mt,t(r,e) isanisotropic, asitdepends on‹o. 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308 P. C. Bressloff andothers Geametric visualhallucinatians analyse the full model described by equation (11),itis useful to consider two limiting cases, namely the ring model oforientation tuningandthe Ermentrout—Cowan model (Ermentrout & Cowan 1979). (i The r/ngmodelofoi'ien Ia!ion tuning The firstlimiting caseistoneglectlateralconnections completely by setting fi=0 inequation (11). Eachpointr inthecortexisthenindependently described bytheso-called ring model of orientation tuning (Hansel & Sompolinsky 1997; Mundel r/al. 1997;Ermentrout 1998; Bresslolf r/a/. 2000a): 0 (12} Linearizing this equation about the homogeneous state a r,m,I) 0 and considering perturbations oftheform a(r,m,I)=e"'a(r,‹o)yieldstheeigenvalue equation Introducing the Fourier series expansion a r,m) c„,(r} e"'”'-I-c. c. generates thefollowing discrete dispersion relation fortheeigenvalue 7. : where w = dv[c]/dc evaluated atc =0 and (13 Note that because zt„ (d›}isarealandevenfunction of&, Let W¿ = max{W„ n C Z“} and suppose that § is unique with W >0 and § 1. It then follows from equation (13) that the homogeneous statea r,‹D =0 is stable forsufficiently smally,butbecomes unstable when increases beyond the critical value y,=o/wt W¿dueto excitation of linear eigenmodes of the form a z,‹b) (re"°-I-(r)e* 2 '*°, where (r isanarbitrary complex function ofr. Itcanbeshownthatthesaturating nonlinearities ofthesystemstabilize thegrowing pattern ofactivity (Ermentrout 1998;Bressloff r/a/. 2000a}. In terms of polar coordinates c(r = /(r) e"°"' we have a r,e)-(r)cos(2§[‹D—‹o(r)]. Thus ateachpointrin the plane the maximum (linear) response occurs atthe orientations e(r} +ta/§,1 0,1,...,§ —1 when § 0. Of particular relevance from abiological perspective are the cases§ =0 and§ = 1. Inthefirstcasethereisa bulk instability inwhichthenewsteadystateshowsno orientation preference. Any tuning isgenerated inthe genicocortical map. We call this the 'Hubel—\Viesel' mode (Hubel & Wiesel 1974a}. In the second case the response isunimodal withrespecttoe. Theoccurrence of a sharply tuned response peaked atsomeangled›(r)ina local region of V1corresponds tothepresence ofa local contour there, the orientation ofwhichisdetermined by the inverse ofthedoubleretinocortical mapdescribed in §5(a). Anexample oftypicaltuningcurvesisshownin Phil. Trans. R. Soc. Land. B (2001a 0 /2 orientation & I Figure 14. Sharporientation tuningcurvesfora Mexican hat weight kernel with „ =20°, t„=6(J°and A„ = 1. The tuning curve ismarginally stablesothatthepeakactivitya at earthpointinthe‹corticalplaneisarbitrary. Thear:tivityis truncated atw=0 inlinewiththechoiceofw[0]=0. figure 14,whichisobtained bytakingin„(e) tobea difference of Gaussians overthedomain [—a/2,n/2]: (13) with „ < t„and A„p<1. The location ofthecentre‹D(r)ofeachtuningcurveis arbitrary, which reflects the rotational equivariance of equation (12a under the modified group action 8: (r,d›) (r,e+t9). Moreover, in the absence of lateral interactions the tuned response isuncorrelated across different points in Vl. Inthispaperweshowhow the presence o1anisotropic lateralconnections leadsto periodic patterns ofactivity across V1inwhichthepeaks ofthetuningcurveatdifferent locations arecorrelated. (ii)The Ermentrout— Cocanmodel The other limiting case istoneglect theorientation label completely. Equation (11) then reduces toa one-population version ofthemodelstudiedby Ermentrout & Cowan (1979a: Inthismodelthereisnoreasontodistinguish anydirec-tion in V1,soweassumethatin,t(r—r' —› mt„(\|r—r'\|, i. e. mt„depends onlyonthemagnitude ofr—r'. Itcan be shown that the resulting system isequivariant with respect tothestandard actionofthe Euclidean groupin theplane. 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which upon Fourier transforming generates a dispersion relation fortheeigenvalue 2 asa function ofg =k\|,i. e. where W(q} = fi 1,t(A i S the Fourier transform of m 1 „(\|r\|. Note that2 isreal. Ifwechoosezt,t(\|r\| tobein the form ofa Mexican hat function, then itissimpleto establish that 7. passesthrough zeroata critical para-meter value v,signalling thegrowthofspatially periodic patterns with wavenumber q„ where W(q, = max,{ W(q ). Close to the bifurcation point these patterns can be represented as linear combinations of plane waves a(r) c,exp('ifr, r), with \|k, ——q,. Asshownby Ermentrout & Cowan (1979 and Cowan (1982}, the underlying Euclidean symmetry oftheweighting function, together withtherestriction to doubly periodic functions, then determines the allowable combinations ofplanewavescomprising steady-state solu-tions. Inparticular, stripe,chequer-board andhexagonal patterns ofactivity canforminthe V1mapot“thevisual field. In this paper we generalize the treatment by Ermentrout & Cowan (1979) toincorporate theeffectsof orientation preference—and show how plane waves of cortical activity modulate the distribution of tuning curves across thenetwork andleadtocontoured patterns. 3. LINEAR STABILITY ANALYSIS The first step in the analysis of pattern-forming instabilities inthefullcortical modelistolinearize equa-tion (Il about the homogeneous solution o(r,e}=0 and tosolvetheresulting eigenvalue problem. Inparticular, we wish tofindconditions underwhichthehomogeneous solution becomes marginally stable due tothevanishing ofoneofthe(degenerate) eigenvalues, andtoidentify the marginally stable modes. This will require performing a perturbation expansion with respect tothesmallpara-meter ficharacterizing therelative strength oftheaniso-tropic lateral connections. (a)£inroriz«Gox We linearize equation (11)aboutthehomogeneous state and introduce solutions oftheforma(r,d›,/) e''ar,m). This generates the eigenvalue equation R' Because oftranslation symmetry, theeigenvalue equation (17)canbewrittenintheform a e,e)=u(e—‹,r)e'*"+ c. c. with A g(cos p,sinp)and Phil. Trans. R. Soc. Land. Y›('200I) (18a Geometric uiiua/hallucinations P. C. Bressloff andothers 309 Here „ k,d›)isthe Fouriertransform ofin„(r,e). Assume that zt„ satisfies equations (4)and(5}sothat the totalweightdistribution zis Euclidean invariant. The resulting symmetry ofthesystemthenrestricts thestruc-ture ofthesolutions oftheeigenvalue equation (19): (i)2 and u(‹O)onlydependonthemagnitude q =\|k\| of thewavevectork. Thatis,thereisaninfinitedegen-eracy due torotational invariance. (ii)Foreach A the associated subspace ofeigenfunctions decomposes intotwoinvariant subspaces (21/ corresponding to even and odd functions, respec-tively and (22/ As noted in greater generality by Bosch Vivancos etal. (199a),thisisaconsequence ofreflection invar-iance, as we now indicate. That is,let t denote reflections about the wavevector k sothatupk ——k. Then upa r,m a u5z,2p—‹D)=u(‹,o—d›}e""+c. c. Since nt isa reflection, any space that itactson decomposes into two subspaces, one on which itacts astheidentity/ and one onwhichitactsas—I. Results (i)and(ii)canalsobederived directly from equation (19). Forexpanding the-periodic function u(e) asa Fourier serieswithrespecttoe (23/ and setting mt„(r,‹5)= (figr) leads to the matrix eigenvalue equation with W, given byequation (14)and 0 R' Itisclearfromequation (24a that item (i)holds. 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310 P. C. Bressloff andothers Geametric rzina/hallucination. ‹ Q <<1 Figure 15. Spliflingofdegenerate eigenvalues dueto anisotropic lateralconnections between hypcrcolumns. (b) Eigenfurtctions and eigext'nfues The calculation oftheeigenvalues andeigenfunctions ofthelinearized equation (17},andhencethederivation ofconditions forthemarginal sta Joility ofthehomoge-neous state, has been reduced totheproblem ofsolving the matrix equation (24),whichwerewriteinthemore convenient form (26} We exploit the experimental observation that the intrinsic lateral connections appear tobeweakrelativetothelocal connections, i. e. fi Wff W. Equation (26s can then be solved by expanding asa power series in fiandusing Rayleigh—Schrödinger perturbation theory. (i)Ca. se0 ——0 Inthelimiting caseofzerolateralinteractions equation (26) reduces toequation (13). Following thediscussion of the ring model in§2(c},let W=max{W„,ne Z“} >0 and suppose that§ =1 (unimodal orientation tuning curves}. Thehomogeneous statea(r,e)=0 isthenstable forsufficiently smally,butbecomes marginally stableat the critical point y,=a/wt Wt due tothevanishing of the eigenvalue 7.. In this case there are both even and odd marginally stable modes cos(2e and sin(2e}. Ifwenowswitchonthelateralconnections, thenthere isa q-dependent splittingofthedegenerate eigenvalue 7. that also separates out odd and even solutions. Denoting the characteristic size ofsucha splitting by h7=G(fi}, we impose the condition that h74/ywt AW,where This ensures that the perturbation does not excite states associated with other eigenvalues of the unperturbed problem (seefigure H). Wecanthenrestrictourselves to calculating perturbative corrections to the degenerate eigenvalue 2t and itsassociated eigenfunctions. Therefore, we introduce thepowerseriesexpansions and Phil. Trans. R. Soc. Lond. B (2001s (27) (28) where J,„ isthe Kronecker deltafunction. \Vesubstitute these expansions into the matrix eigenvalue equation (26} and systematically solve the resulting hierarchy of equations to successive orders in fiusing (degenerate} perturbation theory. This analysis, which iscarriedoutin Appendix A(a), leads tothefollowing results: (i)2 = Jt foreven(+}andodd(—)solutions wheretof'9(fi 2 Hy +COS(2ö) + * ( COSÿ€Œ with (29} (30} (31} (32) (c)ñfnrqiiinf stn6ifi@ Suppose that 6it(q) has a unique maximum at q = gt 0 and let q,=q+ if i'ip(qg G (q and q,=q if G (q >G+(q+. Under such circumstances, the homogeneous state a(r,e)=0 will become margin-ally stable atthecriticalpointy,=a/wt Gt(q,)andthe marginally stable modes willbeoftheform 0 0 where fi,=q,(cosp„sin‹,c, and n(‹DQ=ut(d› forq,-qy. The infinite degeneracy arising from rotation invariance means that allmodeslyingonthecirclek ——g become marginally stable atthecriticalpoint. However, thiscan be reduced toa finitesetofmodesbyrestricting solutions tobedoublyperiodic functions. Thetypesofdoublyperi-odic solutions that can bifurcate from the homogeneous statewillbedetermined in§4. Asa specific example illustrating marginal stability let (r}begivenbyequation (5). 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Figure16. (a)Plotofmarginal stabilitycurvespt(q)fory(. ‹ given bythedifference o1Gaussians (equation (6) with t„=1,I;„ =3,A„ ——I and fi=0. 4Wt. Also set o/wt W=1. Thecriticalwavenumber forspontaneous pattern formation isq,. "1“hemarginally stableeigenmodes are odd functions ofd›. (é)Plotofcriticalwavenumber q¿for marginal stabilityofeven( )andodd(—) patterns asa function ofthestrength ofinhibitory coupling A„. Ifthe inhibition istooweakthenthereisa bulk instability with respect tothespatialdomain. with J,(x the Bessel function of integer order n,we derive the result Next we substitute equation (G)into(34}andusestandard properties of Besselfunctions toobtain 2 4 —A „ex I„ I„ 4 (35) where ( isa modified Bessel function ofintegerordern. The rcsulting marginal stability curvcs =yt(g)= a/w Gt(q)areplottedtofirstorderinfiinfigure16a. The existence ofa non-zero critical wavenumber q,=q at y,= (q, isevident, indicating that the marginally stable eigenmodes are odd functions ofe. Theinclusion of higher-order terms inß does not alter thisbasicresult,at least for small fi. Ifwetakethefundamental unitof length tobera. 400{im,thenthewavelength ofa pattern is2s(0. 400s g,mm, ra. 2. 66 mm at the critical wave-number 2, 1 (seefigure IGE}. Phil. Trane. R. Soc. Land. Y›('200l) Geometric Visualhallucinatians P. C. Bressloff andothers 311 Figure17. Sameasfigure16 except that I(q)satisfies equation (36)with8p= /3ratherthanequation (34a. It can beseenthatthemarginally stableeigenmodes arenoweven functions of‹5. An interesting question concerns under what circum-stances can even patterns be excited by a primary instability rather than odd, intheregimeofweaklateral interactions. One example occurs when there isa suffi-cient spread in the distribution of lateral connections along the lines shown infigure12. Inparticular, suppose that (r)isgivenbyequation (S)with (8}-1 for8w<8# and zero otherwise. Equation (34}thenbecomes Sie 2 " 0) “ (36/ To firstorderinfithesizeofthegapbetween theodd and even eigenmodes at the critical point q,isdeter-mined by2W 2 (q ( S ec equation 29). Itfollowsthatif >/4 then <2 reverses sign, suggesting that even rather than odd eigenmodes become marginally stable first. Thisisconfirmed bythemarginal stability curves shown infigure17. (i)Choosingtheburcationparameter Itisworthcommenting atthisstageonthechoiceof bifurcation parameter y. One way toinducea primary instability of the homogeneous state istoincrease the global coupling parameter inequation (29auntilthe critical point y, isreached. However, itisclearfrom equation (29) that an equivalent way toinducesuchan instability istokeep fixed and increase the slope of the neural output function w. Thelattercouldbeachieved by keeping a non-zero uniform input b(r,e,I)=h9in equation (1)sothatthehomogeneous stateisnon-zero, a z d›,t)——a =f-0 with w = w'(at. Then variation ofthe input ht and consequently wt, corresponds to changing the effective neural threshold and hence the level of network excitability. Indeed, thisisthought tobeoneof the possible effects of hallucinogens. In summary, the mathematically convenient choice of asthebifurcation parameter can be reinterpreted in terms ofbiologically meaningful parameter variations. Itisalsopossible that hallucinogens act indirectly on the relative levels ofinhi-bition and thiscouldalsoplaya role indetermining the type of patterns that emerge—a particular example is discussed below. /C119/C105/C116/C104/C74/C110/C40/C120/C41 /C116/C104/C101 /C66/C101/C115/C115/C101/C108 /C102/C117/C110/C99/C116/C105/C111/C110 /C111/C102 /C105/C110/C116/C101/C103/C101/C114 /C111/C114/C100/C101/C114 /C110/C44/C119 /C101 /C100/C101/C114/C105/C118/C101 /C116/C104/C101 /C114/C101/C115/C117/C108/C116 /C94/C87/C110/C40/C113/C41/C136/C40/C0/C49/C41/C110/C90/C49 /C48/C103/C40/C115/C41/C74/C50/C110/C40/C115/C113/C41/C100/C115/C46 /C40/C51/C52/C41 /C78/C101/C120/C116 /C119/C101 /C115/C117/C98/C115/C116/C105/C116/C117/C116/C101 /C101/C113/C117/C97/C116/C105/C111/C110 /C40/C54/C41 /C105/C110/C116/C111 /C40/C51/C52/C41 /C97/C110/C100/C117/C115/C101 /C115/C116/C97/C110/C100/C97/C114/C100 /C112/C114/C111/C112/C101/C114/C116/C105/C101/C115 /C111/C102 /C66/C101/C115/C115/C101/C108 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/C105/C110/C104/C105/C98/C105/C116/C105/C111/C110 /C105/C115 /C116/C111/C111 /C119/C101/C97/C107 /C116/C104/C101/C110 /C116/C104/C101/C114/C101 /C105/C115 /C97 /C98/C117/C108/C107 /C105/C110/C115/C116/C97/C98/C105/C108/C105/C116/C121/C119/C105/C116/C104 /C114/C101/C115/C112/C101/C99/C116 /C116/C111 /C116/C104/C101 /C115/C112/C97/C116/C105/C97/C108 /C100/C111/C109/C97/C105/C110/C46µ qqc 1 2 3 4 50. 940. 960. 98 0. 92 evenodd 0. 90 /C70/C105/C103/C117/C114/C101 /C49/C55/C46 /C83/C97/C109/C101 /C97/C115 /C162/C103/C117/C114/C101 /C49/C54 /C101/C120/C99/C101/C112/C116 /C116/C104/C97/C116 /C94/C87/C40/C113/C41/C115 /C97 /C116 /C105 /C115 /C162 /C101 /C115 /C101/C113/C117/C97/C116/C105/C111/C110 /C40/C51/C54/C41 /C119/C105/C116/C104 /C18/C48/C136/C25/C47/C51 /C114/C97/C116/C104/C101/C114 /C116/C104/C97/C110 /C101/C113/C117/C97/C116/C105/C111/C110 /C40/C51/C52/C41/C46 /C73/C116 /C99/C97/C110 /C98/C101 /C115/C101/C101/C110 /C116/C104/C97/C116 /C116/C104/C101 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312 P. C. Bresslolf andothers Geametric Visualhallucination. ‹ 1. 0 098 0. 96 0. 94 0. 92 1 2 3 4 5 Figure 18. Plotofmarginal stabilitycurvep9(9)fora bulk instability with respect toorientation and,(i)givenbythe difference of Uaussians (equation (6 with(„t=1,f;„ =3, A,,=1. 0,ß =0. 4W„ando/w,W=1. Thecriticalivave-number forspontaneous patternformation isq,. (d) The Ermentrout—Conanmodelrevisited The marginally stable eigenmodes (equation (33a identified in the analysis consist of spatially periodic patterns of activity that modulate the distribution of orientation tuning curves across V1. Examples ofthese contoured corticalplanforms willbepresented in§4 and the corresponding hallucination patterns in the visual field (obtained by applying an inverse retinocor ticalmap) will be constructed in§5. Itturnsoutthattheresulting patterns account forsomeofthemorecomplicated form constants where contours are prominent, including cobwebs, honeycombs and lattices (figure 4. However, other form constants such aschequer-boards, funnelsand spirals (figures6 and 7)comprise contrasting regionsof light and dark. One possibility isthatthesehallucinations are a result of higher-level processes filling in the contoured patterns generated in V1. Analternative expla-nation isthatsuchregionsareactually formedin V1itself bya mechanism similar tothatsuggested intheoriginal Ermentrout—Cowan model. This raises the interesting issue astowhether ornotthereissomeparameter regime inwhichthenewmodelcanbehaveinasimilarfashion to the 'cortical retina' of Ermentrout & Cowan (1979), that is, can cortical orientation tuning somehow be switched off? One possible mechanism isthefollowing: suppose that the relative level oflocalinhibition, whichis specified by the parameter fi.. in equation (15), is reduced (e. g. bythepossible (indirect) actionofhalluci-nogens). Then Wq = max{ W„ n C Z*} rather than W, and the marginally stable eigenmodes will consist of spatially periodic patterns that modulate bulk instabiliti-tieswithrespecttoorientation. To make these ideas more explicit, we introduce the perturbation expansions and Phil. Trane. R. See. Lond. B (2001s (37) (38) Substituting these expansions into the matrix eigenvalue equation (26) and solving the resulting equations to successive orders infileadstothefollowing results: and u Q =1 + fi u j cos 2m‹P) +O fi, >( with +I(d 3 } (39) 4. DOUBLY PERIODIC PLANFORMS (40} (41 Substituting equation (40} into equation (33a shows that the marginally stable states are now only weakly depen-dent on the orientation e,andtolowestorderinfisimply correspond to the spatially periodic patterns of the Ermentrout—Cowan model. The length-scale of these patterns isdetermined bythemarginal stability curve yt(q)=o/w G9(q), an example of which isshownin figure l S. The occurrence ofa bulk instability in orientation means that for sufficiently small fitheresulting cortical patterns willbemorelikecontrasting regionsoflightand dark rather than alatticeoforiented contours (see§4. However, ifthestrength oflateralconnections fiwere increased, then the eigenfunctions (40} would develop a significant dependence on the orientation e. Thiscould then provide an alternative mechanism forthegeneration ofevencontoured patterns—recall from§3(c)thatonly odd contoured patterns emerge in the case ofa tuned instability with respect to orientation, unless there is significant angular spread inthelateralconnections. As we found in§3 (c)and§3 (d),rotation symmetry implies that the space ofmarginally stableeigenfunctions of the linearized Wilson—Cowan equation is infinite-dimensional, that is,ifu(‹o)e'“" isa solution then so is u(e —p)e"‹“'". However, translation symmetry suggests that we can restrict the space ofsolutions ofthenonlinear Wilson—Cowan equation (11)tothatofdoublyperiodic functions. This restriction isstandard inmanytreatments ofspontaneous pattern formation, butasyetithasno formal justification. There is, however, a wealth of evidence from experiments on converting fluids and chemical reaction—diffusion systems (\Valgraef 1997), and simulations ofneuralnets(Vonder Malsburg & Cowan 1982a, which indicates that such systems tend togenerate doubly periodic patterns intheplanewhenthehomoge-neous state isdestabilized. Givensucha restriction, the associated space of marginally stable eigenfunctions is then finite-dimensional. A finite set of specific eigen-functions can then be identified ascandidate planforms, in the sense that they approximate time-independent solutions ofequation (11)sufficiently closetothecritical point where the homogeneous state loses stability. Inthis section we construct such planforms. /C40/C100/C41/C84/C104/C101/C69/C114/C109/C101/C110/C116/C114/C111/C117/C116 /C94/C67/C111/C119/C97/C110/C109/C111/C100/C101/C108/C114/C101/C118/C105/C115/C105/C116/C101/C100 /C84/C104/C101 /C109/C97/C114/C103/C105/C110/C97/C108/C108/C121/C115/C116/C97/C98/C108/C101 /C101/C105/C103/C101/C110/C109/C111/C100/C101/C115 /C40/C101/C113/C117/C97/C116/C105/C111/C110 /C40/C51/C51/C41/C41 /C105/C100/C101/C110/C116/C105/C162/C101/C100 /C105/C110 /C116/C104/C101 /C97/C110/C97/C108/C121/C115/C105/C115 /C99/C111/C110/C115/C105/C115/C116 /C111/C102 /C115/C112/C97/C116/C105/C97/C108/C108/C121 /C112/C101/C114/C105/C111/C100/C105/C99 /C112/C97/C116/C116/C101/C114/C110/C115 /C111/C102 /C97/C99/C116/C105/C118/C105/C116/C121/C116/C104/C97/C116 /C109/C111/C100/C117/C108/C97/C116/C101 /C116/C104/C101 /C100/C105/C115/C116/C114/C105/C98/C117/C116/C105/C111/C110 /C111/C102/C111/C114/C105/C101/C110/C116/C97/C116/C105/C111/C110 /C116/C117/C110/C105/C110/C103 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Table l. Generators for theplanarlatticesandtheirdual lattice ft k square (I,0) (O,1) (1,0) hexagonal (1,1/3) (0,2/3 (1,0 rhombic (1,—cotp (0,cosccp) (1,0 (—1, ) (COS §, Si Ii\j (a)Restriction todoublyperiodic solutions Let A bea planar lattice; that is,choosetwolinearly independent vectors ftandf2andlet Note that A isa subgroup ofthegroupofplanartrans-lations. A function /:R'x S I —+ R isdoublyperiodic with respect to A if forevery IC A. Let8 be the angle between the two basis vectors ftand I;. Wecanthendistinguish threetypesof lattice according tothevalueof8:squarelattice(8=r2, rhombic lattice (0<8<r/2,8 r/3) and hexagonal (B=x/3}. Afterrotation, thegenerators oftheplanar lattices aregivenintable1 (forunitlatticespacing). Restriction to double periodicity means that the original Euclidean symmetry group isnowrestricted to the symmetry group ofthelattice,f¿ H¿-l-Z'",where Hp istheholohedry ofthelattice,thesubgroup of O(2 that preserves the lattice, and T” isthetwotorusof planar translations modulo the lattice. Thus, the holo-hedry oftherhombic latticeis D;,theholohedry ofthe square lattice is D4andtheholohedry ofthehexagonal lattice isf Ib. Obscrvethatthecorresponding spaceof marginally stable modes isnowfinite-dimensional—we can only rotate eigenfunctions through a finite set of angles (forexample, multiples of /2forthesquarelattice and multiples of 3 forthehexagonal lattice). Itremains todetermine thespace K¿ ofmarginally stable eigenfunctions and the action off¿onthisspace. In§3 we showed that eigenfunctions either reside in UJ (the even case) or U (the odd cases where the length of' fcisequaltothecriticalwavenumber q,. In particular, the eigenfunctions have the form u(e —p)e'“'"whereu is either an odd oreveneigenfunction. Wenowchoosethe size of the lattice so that e'“* isdoublyperiodic with respect tothatlattice,i. e. A isa dual wave vector forthe lattice. In fact, there are infinitely many choices for the lattice sizethatsatisfythisconstraint—we selecttheone forwhichq,istheshortest lengthofa dual wave vector. The generators for the dual lattices are also given in tablel with q,=1. Theeigenfunctions corresponding to dual wave vectors ofunitlengtharcgivenintable2. It follows that K¿ can be identified with the m-dimensional complex vector space spanned by the vectors i..e C“' with m=2 for square or rhombic lattices and m=3 forhexagonal lattices. Itcanbeshown that these form f¿-irreducible representations. The actions of the group f¿ on K¿ can then be explicitly written down forboththesquareorrhombic andhexa-gonal lattices in both the odd and even cases. These actions aregivenin Appendix A(b). Phil. Trans. R. Soc. Land. Y›('200I) Geametric riiwa/hallucination. ‹ P. C. Bressloff andothers 313 Table 2. Ei Menfunctions corresponding toshortestdualinare nectore lattice sc[uare hexagonal rhombic (b) Planfo i-rue We now use an important result from bifurcation theory inthepresence ofsymmetries, namely, theequi-variant branching lemma (Golubitsky ctal. 1988). Forour particular problem, the equivariant branching lemma implies that generically there existsa (unique} doubly periodic solution bifurcating from the homogeneous state for cach oftheaxialsubgroups off¿undertheaction (9)—asubgroup Z C f¿ isaxialifthedimension ofthe space ofvectorsthatarefixedby Z isequaltounity. The axial subgroups are calculated from the actions presented in Appendix A(b)(see Bressloff rfa/. (2000é}fordetails) and lead totheevenplanforms listedintable3 and the odd planforms listed intable4. Thegeneric planforms can then be generated by combining basic properties of the Euclidean group action (equation (9 on doubly peri-odic functions with solutions of the underlying linear eigenvalue problem. The latter determines both the critical wavenumber q,andthe-periodic function u(d›). In particular, the perturbation analysis of§3(c} and §3(d)showsthat(inthecaseofweaklateralinteractions) u(e) can take one ofthreepossible forms: (i)even contoured planforms (equation (30) with "( cos(2m) (ii)odd contoured planforms (equation (31) with u(d›} sin(2d›, (iii)evennon-contoured planforms (equation (40) with u(e) 1. Each planform isanapproximate steady-state solution a(r,e}ofthecontinuum model(equation (11a defined on the unbounded domain R' x S I. Todetermine howthese solutions generate hallucinations in the visual field, we firstneedtointerpret theplanforms intermsofactivity patterns ina bounded domain of Vl,whichwedenoteby C R. Once this has been achieved, the resulting patterns inthevisualfieldcanbeobtained byapplying theinverseretinocor ticalmapasdescribed in§5(a). The interpretation of non-contoured planforms is relatively straightforward, since tolowestorderinfithe solutions are ‹o-independent and can thus be directly treated asactivity patternsa z)in VIwithr C M. At the simplest level, such patterns can be represented as contrasting regions of light and dark depending on whether a(r)>0 ora x')<0. Theseregionsformsquare, triangular, orrhombic cellsthattile asillustrated in figures 19and20. The case ofcontoured planforms ismoresubtle. Ata given location rin V1wehaveasumoftwoorthree sinusoids with different phases and amplitudes (seetables3 and 4),whichcanbewritten asa(r,e) fi(r)cos[2e —2et(r}]. 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314 P. C. Bresslolf andothers Geametric visualhallucination. ‹ Table 3. Evenplanforms inithu(—‹o)=u(e) (The hexagon solutions (0)and( have thesameisotropysubgroup, buttheyarcnotconjugatcsolutions. lattice square rhombic hexagonal even square even roll even rhombic even roll even hexagon (0) even hexagon (s) even roll Table 4. Odd planforms mi/b u(— d›) —u d›) lattice square rhombic hexagonal namc odd square odd roll odd rhombic odd roll odd hexagon triangle patchwork quilt odd roll planform eiqenfunction planform cigcnfunc tion orientation tuning curve atr(seefigure14). Hencethe contoured solutions generally consist of iso-orientation regions orpatches overwhichct(r isconstant, butthe amplitude A(r) varies. As inthenon-contoured case,these patches are either square, triangular orrhombicinshape. However, we now show each patch toberepresented bya locally orientated contour centred atthepointofmaximal amplitude A (rp„ within thepatch. Theresulting oddand even contoured patternsarcshowninfigures21and22for the square latttice, infigures23and24fortherhombic latttice and infigures25and26forthehexagonal lattice. Note that our particular interpretation ofcontoured plan-forms breaks down inthecaseofanoddtriangle ona hexagonal lattice: the latter comprises hexagonal patches inwhichallorientations arepresent withequalmagni-tudes. In this case we draw a 'star'shapeindicating the presence of multiple orientations ata given point, see figure 26b. Note that this planform contains the well-known 'pinwheels' described by Blasdel(1992). 5. FROMCORTICAL PATTERNS TOVISUAL HALLUCINATIONS In§4 we derived the generic planforms that bifurcate from the homogeneous state and interpreted them in terms ofcortical activity patterns. Inordertocompute what the various planforms look like in visual field coordinatcs, weneedtoapplyaninverserctinocortical map. In the case ofnon-contoured patterns thiscanbe carried out directly using the (single) retinocortical map introduced in§1(b). However, forcontoured planforms it isnecessary tospecifyhowtomaplocalcontours inthe visual field as well as position—this is achieved by considering a so-called double retinocortical map. Another important feature ofthemapping between Vl Phil. Trans. R. Soc. Land. B (2001a and the visual fieldisthattheperiodicity oftheangular retinal coordinate 8g implies that they-coordinate in V1 satisfies cylindrical periodic boundary conditions (see figure 5). Thisboundary condition shouldbecommensu-rate with the square, rhombic or hexagonal lattice associated with thedoublyperiodic planforms. (a) T'he double retinocortical map An important consequence oftheintroduction oforien-tation asa cortical label isthattheretinocortical map described earlier needs to be extended to cover the mapping oflocalcontours inthevisualfield—in effectto treat them asa vector field. Letegbetheorientation of sucha local contour, and eitsimagein V1. Whatisthe appropriate map from OR to‹5thatmust Joeaddedtothe map cp —›¿ described earlier? We note thata linein VI ofconstant slopetand›isa levelcurveoftheequation //, =ycos B—isinb, where (x,y) are Cartesian coordinates in V1. 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(b) Geometric visualhallucinations P. C. Bresslolf andothers 315 {a) {b) Figure 21. Contours ofevenaxialeigenfunctions onthe Figure 19. Non-contoured axial eigenfunctions onthesquare square lattice: (a)square,(fi)roll. lattice: (o)square,(b)roll. {b) Figure 20. Non-contoured axial eigenfunctions onrhombic and hexagonal lattices: (a)rhombic, (#)hexagonal. Phil. bans. R. Soc. Land. B (2001) Figure 22. Contours ofoddaxialeigenfunctions onthesquare lattice: (o)square,(b)roll. /C71/C101/C111/C109/C101/C116/C114/C105/C99/C118/C105/C115/C117/C97/C108/C104/C97/C108/C108/C117/C99/C105/C110/C97/C116/C105/C111/C110/C115 /C80/C46/C67/C46 /C66/C114/C101/C115/C115/C108/C111/C161/C97/C110/C100/C111/C116/C104/C101/C114/C115 /C51/C49/C53 /C80/C104/C105/C108/C46/C84 /C114/C97/C110/C115/C46 /C82/C46 /C83/C111/C99/C46 /C76/C111/C110/C100/C46 /C66/C40 /C50 /C48 /C48 /C49 /C41(b)(a) /C70/C105/C103/C117/C114/C101 /C50/C49/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C101/C118/C101/C110 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C115/C113/C117/C97/C114/C101 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41/C115 /C113 /C117 /C97 /C114 /C101 /C44/C40 /C98/C41/C114 /C111 /C108 /C108 /C46 (b)(a) /C70/C105/C103/C117/C114/C101 /C49/C57/C46 /C78/C111/C110/C45/C99/C111/C110/C116/C111/C117/C114/C101/C100 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C115/C113/C117/C97/C114/C101 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41 /C115/C113/C117/C97/C114/C101/C44 /C40 /C98/C41/C114 /C111 /C108 /C108 /C46 (b)(a) /C70/C105/C103/C117/C114/C101 /C50/C48/C46 /C78/C111/C110/C45/C99/C111/C110/C116/C111/C117/C114/C101/C100 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C114/C104/C111/C109/C98/C105/C99 /C97/C110/C100 /C104/C101/C120/C97/C103/C111/C110/C97/C108 /C108/C97/C116/C116/C105/C99/C101/C115/C58 /C40 /C97/C41/C114 /C104 /C111 /C109 /C98 /C105 /C99 /C44/C40 /C98/C41 /C104/C101/C120/C97/C103/C111/C110/C97/C108/C46(b)(a) /C70/C105/C103/C117/C114/C101 /C50/C50/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C111/C100/C100 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C115/C113/C117/C97/C114/C101 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41 /C115/C113/C117/C97/C114/C101/C44 /C40 /C98/C41/C114 /C111 /C108 /C108 /C46	TB010299-output.pdf
316 P. C. Bressloff andothers Geometric visualhallucinations {b) i“““'“““'“›““”““““““'“"“"“!“““““““"““”““'“'\'““”“““““““"'"“““I”“““I““““““›“I““"'“”“”“"“““““I””““““““““"""“"“"“I“”'”“““I““"“““““:“““”I“"“““““““!“:“"““““““““I“““:“““'““” Figure 23. Contours ofevenaxialeigenfunctions onthe rhombic lattice: a)rhombic, (b)roll. (°) (b) Figure 24. Contours ofoddaxialeigenfunctions onthe rhombic lattice: (o)rhombic, (b)roll. Phil. Irans. R. Soc. Lord. B (2001) (a) Figure 25. Contours ofevenaxialeigenfunctions onthe hexagonal lattice: (o)s-hexagonal, (#)0-hexagonal. Figure 26. Contours ofoddaxialeigenfunctions onthe hexagonal lattice: (a)triangular, (é)0-hexagonal. /C51/C49/C54 /C80/C46/C67/C46 /C66/C114/C101/C115/C115/C108/C111/C161/C97/C110/C100/C111/C116/C104/C101/C114/C115 /C71/C101/C111/C109/C101/C116/C114/C105/C99/C118/C105/C115/C117/C97/C108/C104/C97/C108/C108/C117/C99/C105/C110/C97/C116/C105/C111/C110/C115 /C80/C104/C105/C108/C46/C84 /C114/C97/C110/C115/C46 /C82/C46 /C83/C111/C99/C46 /C76/C111/C110/C100/C46 /C66/C40 /C50 /C48 /C48 /C49 /C41(a) (b) /C70/C105/C103/C117/C114/C101 /C50/C53/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C101/C118/C101/C110 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C104/C101/C120/C97/C103/C111/C110/C97/C108 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41/C25/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C44 /C40 /C98/C41 /C48/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C46 (a) (b) /C70/C105/C103/C117/C114/C101 /C50/C54/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C111/C100/C100 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C104/C101/C120/C97/C103/C111/C110/C97/C108 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41 /C116/C114/C105/C97/C110/C103/C117/C108/C97/C114/C44 /C40 /C98/C41 /C48/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C46(b)(a) /C70/C105/C103/C117/C114/C101 /C50/C51/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C101/C118/C101/C110 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C114/C104/C111/C109/C98/C105/C99 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41/C114 /C104 /C111 /C109 /C98 /C105 /C99 /C44/C40 /C98/C41/C114 /C111 /C108 /C108 /C46 (a) (b) /C70/C105/C103/C117/C114/C101 /C50/C52/C46 /C67/C111/C110/C116/C111/C117/C114/C115 /C111/C102 /C111/C100/C100 /C97/C120/C105/C97/C108 /C101/C105/C103/C101/C110/C102/C117/C110/C99/C116/C105/C111/C110/C115 /C111/C110 /C116/C104/C101 /C114/C104/C111/C109/C98/C105/C99 /C108/C97/C116/C116/C105/C99/C101/C58 /C40 /C97/C41/C114 /C104 /C111 /C109 /C98 /C105 /C99 /C44/C40 /C98/C41/C114 /C111 /C108 /C108 /C46	TB010299-output.pdf
Figure27. Thegeometry oforientation tuning. Itiseasytoshowthatthetangent vectorcorresponding tosucha curve takes theform Thus the retinal vector fieldinduced bya constant vector field in V1twistswiththeretinalangle R and stretches with the retinal radius rp. Itfollowsthatifd›pisthe orientation of“a lineinthevisualfield,then (42) i. e. localorientation in V1isrelative totheangular coor-dinate ofvisualfieldposition. Thegeometry oftheabove setup isshowninfigure27. The resulting double map (zp,eg) \|¿,e)hasvery interesting properties. As previously noted, the map g —› takes circles, rays and logarithmic spirals into vertical, horizontal and oblique lines, respectively. What about theextended map?Because thetangenttoa circleat a given point isperpendicular totheradiusatthatpoint, forcircles, R= R* w J2, sothatd›=r/2. Similarly, for rays, ett=Btt,soe = 0. Forlogarithmic spiralswecan write either 8g aln R g=exp[ R] Inretinalcoor-dinates we findthesomewhat cumbersome formula Rsin R-I-e" R cos R ñr cos8 —u”"sin8 However, this can be rewritten astan(‹9p— R = a,so that in V1coordinates, tane = a. Thusweseethatthe local orientations ofcircles,raysandlogarithmic spirals, measured in relative terms, all lie along the cortical images ofsuchforms. Figure2Sshowsthedetails. (b) Plarifoirnainthevisualfield Inordertogenerate a visual fieldpatternwesplitour model Vl domain into two pieces, each running 72 mm along thex direction and 48mmalongthe direction, representing the right and lefthemifields inthe visual field (seefigure5). Because the coordinate corre-sponds toa change from — /2to /2in72mm,which meets up again smoothly with the representation inthe opposite hemifield, we must only choose scalings and rotations ofourplanforms thatsatisfycylindrical periodic boundary conditions inthe direction. Inthex direction, corresponding tothelogarithm ofradialeccentricity, we neglect the region immediately around the fovea and also Phil. Trans. R. Soc. Land. Y›('200I) Geametrie IArea/he//urina/ioui P. C. Bresslolf andothers 317 the faredgeoftheperiphery, sowehavenoconstraint on thepatterns inthisdirection. Recall that each V1 planform isdoublyperiodic with respect toa spatial lattice generated by two lattice vectors 2 The cylindrical periodicity isthusequivalent to requiring that there be an integral combination of'lattice vectors that spans F = 96mminthe direction with no change inthex direction: 0 96 (43} Iftheacuteangleofthelatticep'isspecified, thenthe wavevectors k,aredetermined bytherequirement 1, i=j The integral combination requirement limits which wave-lengths are permitted for planforms in the cortex. The length-scale fora planform isgivenbythelengthofthe lattice vectors \|f\| = 2 :' : 96 COS ' *2 sin(p') + (45) The commonly reported hallucination patterns usually have 30-40 repetitions ofthepatternarounda circumfer-ence of the visual field, corresponding tolength-scales ranging from 2. 4-3. 2 mm. Therefore, we would expect the critical wavelength 2r JJ, forbifurcations tobeinthis range (see§3(c}). Notethatwhenwerotatetheplanform tomatchthecylindrical boundary conditions werotate k and hence the maximal amplitude orientations ct(r by The resulting non-contoured planforms inthevisualfield obtained by applying the inverse single retinocor tical map to the corresponding V1 planforms are shown in figures 29 and 30. Similarly, the odd and even contoured planforms obtained by applying the double retinocortical map are shown in figures 31 and 32 for the square lattice, in figures 33 and 34 for the rhombic lattice, and in figures 35and36forthehexagonal lattice. One ofthestriking features oftheresulting (contoured visual planforms isthatonlytheevenplanforms appear tobecontour completing anditisthesethatrecover the remaining form constants missing from the original Ermentrout—Cowan model. The reader should compare, for example, the pressure phosphenes shown infigure1 with the planform shown infigure35a,andthecobweb offigure4 with that offigure31a. 6. THESELECTION ANDSTABILITY OFPATTERNS Itremains todetermine whichofthevariousplanforms we have presented above inourmodelareactually stable for biologically relevant parameter sets. Sofarwehave used a mixture ofperturbation theoryandsymmetry to construct the linear eigenmodes (equation (33a that are candidate planforms for pattern forming instabilities. 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318 P. C. Bressloff andothers Geametric aisualhallucinatians (b) tb) (ii) \/ Figure 29. Action ofthesingleinverseretinocortical non-contoured sc[uareplanforms: (c)square,(b)roll. —z/2 —z/2 (b) Figure 2S. Action ofthesingle and double maps onlogarithmic spirals. Dashedlinesshowthe localtangents toa logarithmic spiralcontourinthevisualfield, and theresulting imagein V1. Since circleandraycontours in thevisualfieldarejustspecial casesoflogarithmic spirals,the same resultholdsalsoforsuch contours. (n)Visualfield;(b) striatecortex,(i)singlemap, (ii)doublemap. '“' O ' Figure 30. Action ofthesingleinccrscrctinocortical map determine which of these modes are stabilized by the nonlinearities of the system we use techniques such as Liapunov—Schmidt reduction and Poincaré—Lindstcdt perturbation theory toreducethedynamics toa setof nonlinear equations for the amplitudes c,appearing in equation (33) (Walgraef 1997. These amplitude equa-tions, which effectively describe the dynamics ona finite-dimensional centre manifold, then determine the selec-tion and stability ofpatterns (atleastsufficiently closeto Phil. Trans. R. Soc. Land. B (2001a onnon-contoured rhombicandhexagonal planforms: (a)rhombic, (h)hexagonal. the bifurcation point). The symmetries of the system severely restrict the allowed forms (Golubitsky ctal. 1988); however, the coefficients in this form are inher-ently model dependent and have tobecalculated expli-citly. Inthissectionwedetermine theamplitude equation for our cortical model up to cubic order and use this to/C100/C101/C116/C101/C114/C109/C105/C110/C101 /C119/C104/C105/C99/C104 /C111/C102 /C116/C104/C101/C115/C101 /C109/C111/C100/C101/C115 /C97/C114/C101 /C115/C116/C97/C98/C105/C108/C105/C122/C101/C100 /C98/C121/C116/C104/C101 /C110/C111/C110/C108/C105/C110/C101/C97/C114/C105/C116/C105/C101/C115 /C111/C102 /C116/C104/C101 /C115/C121/C115/C116/C101/C109 /C119/C101 /C117/C115/C101 /C116/C101/C99/C104/C110/C105/C113/C117/C101/C115 /C115/C117/C99/C104 /C97/C115/C76/C105/C97/C112/C117/C110/C111/C118 /C94/C83/C99/C104/C109/C105/C100/C116 /C114/C101/C100/C117/C99/C116/C105/C111/C110 /C97/C110/C100 /C80/C111/C105/C110/C99/C97/C114/C101 /C168/C94/C76/C105/C110/C100/C115/C116/C101/C100/C116 /C112/C101/C114/C116/C117/C114/C98/C97/C116/C105/C111/C110 /C116/C104/C101/C111/C114/C121/C116/C111 /C114/C101/C100/C117/C99/C101 /C116/C104/C101 /C100/C121 /C110/C97/C109/C105/C99/C115 /C116/C111 /C97 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bbbb bb(g) Figure 31. Action ofthedoubleinverseretinocortical mapon even square planforms: (nsquare,(é roll. investigate the selection and stability ofbothoddpatterns satisfying u(— d›)=—u(e and even patterns satisfying u —8) = u(‹O). A more complete discussion ofstability and selection based on symmetrical bifurcation theory, which takes into account the possible effects ofhigher-order contributions to the amplitude equation, will be presented elsewhere (Bressloff ctal. 2000a). Assume that sufficiently close tothebifurcation pointat which the homogeneous state a(r,e)=0 becomes marginally stable, the excited modes grow slowly ata rate O(6' where t"= —. Onecanthenusethemethod of multiple-scales toperform a Poincaré—Lindstedt pertur-bation expansion in £. First, we Taylor expand the nonlinear function w[a]appearing inequation (11a, where w = w'[0], w 2 = w”[0] 2,w 3 = w”'[0] 3!,etc. Then we perform a perturbation expansion of equation (11a with respect toe by writing and introducing a slow time-scale i —e 2 /. Collecting terms with equal powers oft then generates a hierarchy of equations as shown in Appendix A(c}. The G(r Phil. Trans. R. Soc. Land. Y›('200I) Geametric visualhallucinatians P. C. Bressloff andothers 319 (a) Figure 32. Action ofthedoubleinverseretinocortical mapon odd square planforms: (u)square,(SQroll. equation isequivalent totheeigenvalue equation (10} with 0,p y,and\|A\| = g,sothat (t)e""!"u(d›—‹,o —l-c. c., (46) with k ——q,( cosp„sin‹,r. Requiring that the U(e' and G(e' equations in the hierarchy be self-consistent then leads toa solvability condition, which in turn generates evolution equations for the amplitudes c (t) (see Appendix A(c)). (i)Squai'eorrhombiclattice First,consider planforms (equation (46) corresponding toa bimodal structure of the square orrhombic type (N 2}. Thatis,take A = g,(1,0 and k 2 ——g com 8), sin(8), with8 = r/2 forthesquarelatticeand0<8<r/2, B 3 fora rhombic lattice. 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320 P. C. Bressloff andothers Geometric riiwa/hallucinations *1 “ I i'' / I y ) / I \1 [a) Figure 33. Action ofthedoubleinverseretinocortical mapon Figure 35. Action ofthedoubleinverseretinocortical mapon even rhombic planforms: (n)rhombic, (#)roll. even hexagonal planforms: (n)w-hexagonal, (é)0-hexagonal. (b) '\ « \” (b) Figure 34. Action ofthedoubleinverseretinocortical mapon Figure 36. Action ofthedoubleinverseretinocortical mapon odd rhombic planforms: (n)rhombic, (é)f Oll. odd hexagonal planforms: (a)triangular, (é)0-hexagonal. Phil. Trans. R. Soc. Land. B (2001)/C51/C50/C48 /C80/C46/C67/C46 /C66/C114/C101/C115/C115/C108/C111/C161/C97/C110/C100/C111/C116/C104/C101/C114/C115 /C71/C101/C111/C109/C101/C116/C114/C105/C99/C118/C105/C115/C117/C97/C108/C104/C97/C108/C108/C117/C99/C105/C110/C97/C116/C105/C111/C110/C115 /C80/C104/C105/C108/C46/C84 /C114/C97/C110/C115/C46 /C82/C46 /C83/C111/C99/C46 /C76/C111/C110/C100/C46 /C66/C40 /C50 /C48 /C48 /C49 /C41(a) (b) /C70/C105/C103/C117/C114/C101 /C51/C51/C46 /C65/C99/C116/C105/C111/C110 /C111/C102 /C116/C104/C101 /C100/C111/C117/C98/C108/C101 /C105/C110/C118/C101/C114/C115/C101 /C114/C101/C116/C105/C110/C111/C99/C111/C114/C116/C105/C99/C97/C108 /C109/C97/C112 /C111/C110 /C101/C118/C101/C110 /C114/C104/C111/C109/C98/C105/C99 /C112/C108/C97/C110/C102/C111/C114/C109/C115/C58 /C40 /C97/C41/C114 /C104 /C111 /C109 /C98 /C105 /C99 /C44/C40 /C98/C41/C114 /C111 /C108 /C108 /C46 (a) (b) /C70/C105/C103/C117/C114/C101 /C51/C52/C46 /C65/C99/C116/C105/C111/C110 /C111/C102 /C116/C104/C101 /C100/C111/C117/C98/C108/C101 /C105/C110/C118/C101/C114/C115/C101 /C114/C101/C116/C105/C110/C111/C99/C111/C114/C116/C105/C99/C97/C108 /C109/C97/C112 /C111/C110 /C111/C100/C100 /C114/C104/C111/C109/C98/C105/C99 /C112/C108/C97/C110/C102/C111/C114/C109/C115/C58 /C40 /C97/C41/C114 /C104 /C111 /C109 /C98 /C105 /C99 /C44/C40 /C98/C41/C114 /C111 /C108 /C108 /C46(a) (b) /C70/C105/C103/C117/C114/C101 /C51/C53/C46 /C65/C99/C116/C105/C111/C110 /C111/C102 /C116/C104/C101 /C100/C111/C117/C98/C108/C101 /C105/C110/C118/C101/C114/C115/C101 /C114/C101/C116/C105/C110/C111/C99/C111/C114/C116/C105/C99/C97/C108 /C109/C97/C112 /C111/C110 /C101/C118/C101/C110 /C104/C101/C120/C97/C103/C111/C110/C97/C108 /C112/C108/C97/C110/C102/C111/C114/C109/C115/C58 /C40 /C97/C41/C25/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C44 /C40 /C98/C41 /C48/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C46 (b)(a) /C70/C105/C103/C117/C114/C101 /C51/C54/C46 /C65/C99/C116/C105/C111/C110 /C111/C102 /C116/C104/C101 /C100/C111/C117/C98/C108/C101 /C105/C110/C118/C101/C114/C115/C101 /C114/C101/C116/C105/C110/C111/C99/C111/C114/C116/C105/C99/C97/C108 /C109/C97/C112 /C111/C110 /C111/C100/C100 /C104/C101/C120/C97/C103/C111/C110/C97/C108 /C112/C108/C97/C110/C102/C111/C114/C109/C115/C58 /C40 /C97/C41 /C116/C114/C105/C97/C110/C103/C117/C108/C97/C114/C44 /C40 /C98/C41 /C48/C45/C104/C101/C120/C97/C103/C111/C110/C97/C108/C46	TB010299-output.pdf
b2bb forall0p<p<rwith (ii Hexagonal lattice Next consider planforms ona hexagonal lattice with I-3,p = 0,p;=2 3,p = —2r J 3. Thecubicampli-tude equations take theform (30/ where = 1,2,3 mod 3, 3 isgivenbyequation (49a forp = 2w/3,and with (51/ (32) In deriving equation (50) we have assumed that the neurons are operating close to threshold such that Cl(c. The basic structure of equations (47) and (50a is universal inthesensethatitonlydepends ontheunder-lying symmetries ofthesystemandonthetypeofbifurca-tion that itisundergoing. Incontrast, theactualvaluesof the coefficients wq andp are model-dependent and have tobecalculated explicitly. Moreover, thesecoefficients are different for odd and even patterns because they have distinct eigenfunctions n(‹5). Note also that because of symmetry the quadratic term in equation (50) must vanish identically inthecaseofoddpatterns. (iii)Evenc0ntouredplanforms Substituting the perturbation expansion oftheeigen-function (equation (30a for even contoured planforms into equations (52)and(49agives with thecoefficients n Jdefinedbyequation (32}. (iv)Oddcontouredp (33/ Substituting the perturbation expansion oftheeigen-function (equation (31) foroddcontoured planforms into equations (52)and(49)gives,respectively, and 3 (56) with the coefficients u, defined by equation (32). Note that the quadratic term inequation (50)vanishes identi-cally inthecaseofoddpatterns. Phil. Trans. R. Soc. Land. Y›('200I) Geametric visualhallucinatians P. C. Bressloff andothers 321 (v) Even non-contoured planfoi'ms Substituting the perturbation expansion oftheeigen-function (equation (40) for even non-contoured plan-forms into equations (52)and(49}gives,respectively, =+;d' [!:«. ]2°'/3et°,› and r 8 =1 +d ['q ] [1+2cos2] +oo with the coefficients u,','definedbyequation (41}. (b) Even aridoddpatterns onsquareozrhombic ¥Ve now use equation (47) toinvestigate theselection and stability of odd or even patterns on square or rhombic lattices. Assuming that w#>0 and d >0, the following three types ofsteadystatearepossible forarbi-trary phases, 2. (iii)squares or rhombics: « = [ + ]e'”', *2 [ 0+ ] e The non-trivial solutions correspond totheaxialplan-forms listed intables3 and 4. A standard linear stability analysis shows that if2>#>yt„ then rolls are stable whereas the square orrhombic patterns areunstable. The opposite holds if 2wp<up. These stability properties persist when higher-order terms in the amplitude equation areincluded (Bresslolf ctal. 2000a}. Using equations (48a,(54),{56}and(58)with3o\|w›/ = 1,wededucethatfornon-contoured patterns and for(oddoreven)contoured patterns Hence, inthecaseofa square orrhombic latticewehave the following results concer ning patterns bifurcating from the homogeneous state close to the point of marginal stability (inthelimitofweaklateralinteractions) : (i)Fornon-contoured patterns onasquareorrhombic lattice there existstablerollsandunstable squares. (ii)For(evenorodd}contoured patterns ona square lattice there existstablerollsandunstable squares. In the case ofa rhombic lattice ofangle8 J2,rolls are stable ifcos(48a>—l/2whereas 8-rhombics are stable ifcos(48)<—1/2,thatis,ifr J6<8<a/3. Itshouldbenotedthatthisresultdiffersfromthatobtained by Ermentrout & Cowan (1979) inwhichstablesquares were shown tooccurforcertainparameter ranges(seealso Ermentrout (1991}). Weattribute thisdifference tothe anisotropy ofthelateralconnections incorporated intothe current model and theconsequent shift—twist symmetry of the Euclidean group action. 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322 P. C. Bressloff andothers Geometric riiun/hallucination,i Figure37. Plotoftheeveneigenfunction coefficient u*(9of equation (32)asa function ofwavcnumbcr q. Alsoplottedis theé7(fi)contribution totheeveneigenvalue expansion, equation (?9), m+(q)=W9(g)-]-2(q). The peak ofm>(q) determines thecriticalwavenumber q,(tofirstorderinfi). Same parameter valuesasfigure17. Next we use equations (50} and (51) to analyse the stability ofevenplanforms ona hexagonal lattice. On decomposing r,=â,e"'',itisa simple matter toshowthat two of the phases d, are arbitrary while the sum 3 _ d,andtherealamplitudes U,evolveaccording totheequations d/ and d L d 3 sind, (60} with i,j 1,2,3 mod 3. Itimmediately follows from equation (60a that the stable steady-state solution will have aphase =0 ifp>0 and aphase = rifp<0. From equations (48a, (54a and (58) with 3n\|w3\|/w =l we seethat forevennon-contoured patterns, and for even contoured patterns. In the parameter regime where the marginally stable modes are even contoured planforms (such asinfigure17awefindthatu J (q, >0. This isillustrated infigure37. Therefore, 2w 2 g/ 3 >w 0 forboththecontoured andnon-contoured cases. Standard analysis then shows that (to cubic order) there exists a stable hexagonal pattern C,——C tori = 1,2,3 with amplitude (Busse 1962) over theparameter range The maxima ot the resulting hexagonal pattern are located on an equilateral triangular lattice for >0 Phil. Trans. R. Soc. Land. B (2001a / z c-hexagons f Olls Figure 38. Bifurcation diagram showing thevariation ofthe amplitude C with theparameter p forevenhexagonal androll patterns withp>0. Solidanddashedcurvesindicatestable and unstable solutions, respectively. Also shown isa secondary branch ofrer'tangu1ar patterns, RA. Higher-order termsin theamplitude equation areneededtodetermine itsstability. (0-hexagons whereas the maxima are located on an equilateral hexagonal lattice for p <0 (-hexagons). Both classes of hexagonal planform have the same D axial subgroup (up to conjugacy), see table A2 in Appendix A(b. One can also establish that rolls are unstable versus hexagonal structures intherange 2 (62) Hence, inthecaseof“a hexagonal lattice, we have the following result concerning the even patterns bifurcating from the homogeneous stateclosetothepointofmarginal stability (in the limit ofweaklateralinteractions). For even (contoured or non-contoured) patterns on a hexagonal lattice, stable hexagonal patterns are the first to appear (subcritically) beyond the bifurcation point. Subsequently, the stable hexagonal branch exchanges stability with an unstable branch of roll patterns, as shown infigure38. Techniques from symmetrical bifurcation theory can be used toinvestigate theeffectsofhigherordertermsinthe amplitude equation (Buzano & Golubitsky 1983): inthe case ofevenplanforms theresultsareidentical tothose obtained in the analysis of Bénard convection in the absence ofmidplane symmetry. Forexample, onefinds that the exchange ofstability between thehexagons and rolls isduetoa secondary bifurcation that generates rectangular patterns. (d) Odd patterns on nhex:agoiial Suffice Recall that inthecaseofoddpatterns, thequadratic term inequation (50)vanishes identically. Thehomoge-neous state now destabilizes viaa (supercritical) pitchfork bifurcation tothefouraxialplanforms listedintables3 and 4. 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Figure 39. Ploto1theoddeigenlunction coefficient u,(9of equation (32)asa function ofwavenumber q. Alsoplottedis the Ö(fi)contribution totheoddeigenvalue expansion, equation (29, m-(q = W9(q)—We(J). The peak ofm-(q) determines thecriticalwavenumber q,(tofirstorderinß). Same parameter values asfigure II. various solutions, and to identify possible secondary bifurcations. Unfortunately, one cannot carry over previous results obtained from the study ofthe Bcnard convection problem with midplane symmetry, even though the corresponding amplitude equation isidentical instructure atcubicorder Golubitsky r/a/. 1984):higher-order contributions totheamplitude equation willdiffer inthetwoproblems duetotheradically different actions ofthe Euclidean groupandtheresulting differences in the associated axial subgroups. I The effects ofsuchcontri-butions on the bifurcation structure ofodd(andevent cortical patterns will be studied in detail elsewhere (Bresslolf r/a/. 2000#. Here, we simply describe the more limited stability results that can be deduced at cubic order. A basic question concerns which ofthefourodd planforms ona hexagonal lattice (hexagons, triangles, patchwork quilts and rolls}arestable. Itturnsoutthatif 2y 2 y/3> t hen rolls are stable, whereas if2wq/ 3 <y0 then either hexagons or triangles are stable (depending upon higher-order terms). Equations (49) and (56) with 3s1 31/ =I imply that In the parameter regime where the marginally stable modes are odd contoured planforms (such asinfigure16a we find that u (q,. <0, and thus 2>;,/. <wt. This is illustrated infigure39. Hence,inthecaseofa hexagonal lattice we have the following result concerning the odd patterns bifurcating from the homogeneous state close to the point of marginal stability (in the limit of weak lateral interactions). For odd (contoured) patterns ona hexagonal lattice there exist four primary bifurcation branches corresponding tohexagons, triangles, patchwork quilts and rolls. Eitherthehexagons orthetriangles arc stable (depending on higher-order terms through a secondary bifurcation) and all other branches are unstable. This isillustrated infigure40. 7. DISCUSSION This paper describes a new model ofthespontaneous generation ofpatterns in V1(seenasgeometric hallucina-tions). Whereas the earlier work of Ermentrout and Phil. Trane. R. Soc. Land. Y›('200l) Geometric ut. ‹nalhallucinatians P. C. Bressloff andothers 323 R Figure 40. Bifurcation diagram showing thevariation ofthe amplitude C with theparameter p foroddpatternsona hexagonal lattice. Solidanddashedcurvesindicatestableand unstable solutions, respectively. Either hexagons (H) or triangles (T)arestable(depending onhigher-order termsin theamplitude equation) whereaspatchwork quilts(PQ and rolls(R)areunstable. Secondary bifurcations (notshown) may arisefromhigher-order terms(Bresslolf etal. 20005). Cowan started witha general neural network and sought the minimal restrictions necessary toproduce hallucina-tion patterns, the current model incorporates data gath-ered over the past two decades to show that common hallucinatory images can be generated bya biologically plausible architecture inwhichtheconnections between iso-orientation patches in V1arelocallyisotropic, but non-locally anisotropic. As we, and Ermentrout and Cowan before usshow,the Euclidean symmetry ofsuch an architecture, i. c. thesymmctry with respect torigid motions in the plane, plays a key role in determining which patterns ofactivation oftheiso-orientation patches appear when the homogeneous state becomes unstable, presumed tooccur,forexample, shortlyaftertheaction ofhallucinogens onthosebrainstemnucleithatcontrol cortical excitability. There are,however, twoimportant differences between the current work and that ot Ermentrout and Cowanin thewayinwhichthe Euclidean groupisimplemented: The group action isdifferent andnovel,andsothe way inwhichthevarioussubgroups ofthe Euclidean group are generated issignificantly different. In particular, the various planforms corresponding to the subgroups are labelled by orientation preference, aswellasbytheirlocation inthecortical plane. It follows that the eigenfunctions that generate such planforms arc also labelled insucha fashion. This adds an additional complication totheproblem of calculating such eigenfunctions and the eigenvalues towhichtheybelong, fromthelinearized cortical dynamics. 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324 P. C. Bressloff andothers Geometric riiun/hallucination,i eigenvalues and eigenfunctions, and therefore the planforms. Given such eigenfunctions we then make use of Poincaré—Lindstedt perturbation theory to compute the stability ofthevarious planforms that appear when the homogeneous state becomes unstable. (ii)Because we include orientation preference in the formulation, we have toconsider theactionofthe retinocortical map on orientated contours oredges. Ineffectwedothisbytreating thelocaltangents to such contours asa vector field. Aswediscussed, this iscarried outbythetangent mapassociated with the complex logarithm, one consequence ofwhichis that e,the V1labelfororientation preference, isnot exactly equal toorientation preference inthevisual field, 8R. but differs from itbytheangle8p,the polar angle ofreceptive fieldposition. \Vccalledthe map from visual fieldcoordinates {rp,8p,‹D R )to Vl coordinates {x,y,‹D}a double map. Its possible presence in V1issubjecttoexperimental verification. Ifthedoublemapispresent, thenelements tunedto the same relative angle d›shouldbeconnected with greater strength than others; ifonlythesinglemap (rz,8z] {x,y} obtains, then elements tuned tothe same absolute angle eg should be soconnected. If, in fact, the double map ispresent, thenelements tuned to the same angle e should be connected along lines atthatanglein VI. Thiswouldsupport Mitchison and Crick's hypothesis on connectivity in V1(Mitchison & Crick 19S2)andwouldbeconsis-tentwiththeobservations of G. G. Blasdel(personal communication) and Bosking r/a/. (1997}. In this connection, itisofinterestthatfromequation (42)it follows that near the vertical meridian (where most oftheobservations havebeenmade},changes ine closely approximate changes in eg. However, a prediction ofthedoublemapisthatsuchchanges should berelatively largeanddetectable withoptical imaging, near thehorizontal meridian. The main advance over the Ermentrout—Cowan work isthatallthe Klüverformconstants cannowbeobtained as planforms associated with axial subgroups of the Euclidean group in the plane, generated by the new representations wc have discovered. There are several aspects ofthisworkthatrcquirccomment. (i)Theanalysis indicates thatundercertainconditions the planforms are either contoured or else non-contoured, depending on the strength ofinhibition between neighbouring iso-orientation patches. If such inhibition isweak,individual hypercolumns do not exhibit any tendency toamplify anyparticular orientation. Innormalcircumstances, sucha prefer-cncc would have tobcsupplicd byinputsfromthe LGN. Inthiscase,V1canbesaidtooperate inthe Hubel—Wiesel mode (see§2(c)}. Ifthehorizontal interactions are stilleffective, thenplanewavesof cortical activité can emerge, with no label fororien-tation preference. The resulting planforms are called non-contoured, and correspond to a subset ofthe Klüver form constants: tunnels and funnels, and spirals. Conversely, ifthere is strong inhibition between neighbouring iso-orientation patches, even Phil. Trans. R. Soc. Lond. B (2001s (ii} weakly biased inputs toahypercolumn cantriggera sharply tuned response such that, under the combined action ofmanyinteracting hypercolumns, plane waves labelled for orientation preference can emerge. The resultinq planforms correspond to contoured patterns and to the remaining form constants described by Klüver—honeycombs and chequer-boards, and cobwebs. Interestingly, allbut the square planforms are stable, but there do exist hallucinatory images that correspond to square planforms and itispossible thatthèsearejusttransi-tional forms. Another conclusion tobedrawnfromthisanalysis is that the circuits in VI,whicharenormally involved inthedetection oforiented edgesandtheformation and processing ofcontours, arealsoresponsible for the generation ofthehallucinator y form constants. Thus, we introduced in§2(aa V1modelcircuitin which the lateral connectivity isanisotropic and inhibitory. (We noted in§1 (d) that 20% of the (excitatory) lateral connections inlayers IIand III of V1endoninhibitory inter-neurons, sotheoverall action of the lateral connections could become inhibitory, especially athighlevelsofactivité. } Aswe demonstrated in§3(c) the mathematical conse-quences ofthisistheselection ofoddplanforms, but thèse do not form continuous contours (see§5(b}). This isconsistent with the possibilité that such connections are involved in the segmentation of visual images (Li1999. Inordertoselectevenplan-forms, which are contour forming and correspond to seenformconstants, itprovedsuflicient toallowfor deviation away from the visuotopic axis by atleast 45º in the pattern oflateralconnections between iso-orientation patches. Thèseresultsareconsistent with observations that suggest that there are two circuits in V1,onedealing withcontrast edges,in which the relevant lateral connections have the anisotropy found by G. G. Blasdel and L. Sincich (personal communication) and Bosking c/a/. 1997s, and another that might beinvolved withtheprocès-sing oftextures, surfaces andcolourcontrast, and which hasa much more isotropic lateral connec-tivity Livingstone & Hubel 1984}. One can interpret the lessanisotropic patternneededtogenerate even planforms asa composite ofthetwocircuits. There are also two other intriguing possible scenarios that are consistent with our analysis. The firstwasreferred toin§3(d}. Inthecasewhere V1is operating in the Hubel—Wiesel mode, with no intrinsic tuning for orientation, and ifthelateral interactions are not asweakaswehaveassumed in our analysis, then even contoured planforms can form. The second possibility stems from the observa-tion that at low levels of VI activité, lateral interactions are all excitatory (Hirsch & Gilbert 1991, sothatabulkinstability occursifthehomo-geneous state becomes unstable, followed by secondary bifurcations topatterned planforms atthe critical wavelength of2. 4-3. 2 mm, when the level of activity rises and the inhibition isactivated. 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Figure41. Tunnelhallucination generated by LSD. Redrawn from Oster (1970). Hopf bifurcations (Ermentrout & Cowan 1980) that give risetooscillations orpropagating waves. In such cases itispossible forevenplanforms tobe selected by the anistropic connectivity and odd planforms by the isotropic connectivity. Inaddition, sucha scenario isactually observed: manysubjects who have taken LSD and similar hallucinogens report seeing bright white light atthecentreofthe visual field,whichthenexplodes intoa hallucinatory image (Siegel & Jarvik 1975s in ca. 3s, corre-sponding toa propagation velocity in V1 of ca. 2. 4cms* 1,suggestive ofslowlymoving epileptiform activity (Milton etal. 1995;Senseman 1999}. (iii)Oneofthemajoraspectsdescribed inthispaperis the presumed Euclidean symmetry of V1. Many systems exhibit Euclidean symmetry, but what is novel here isthewayinwhichsucha symmetry is generated. Thus, equation (9) shows that the symmetry group isgenerated, inlargepart,bya translation orshift{r,d›)—+{r-l-s,‹D}followed bya rotation ortwist{r,e}—›{Phr,e+8}. Itisthefinal twist e—›e +8 that isnovel,andwhichisrequired to match the observations of O. G. Blasdel and L. Sincich (personal communication) and Bosking etal. (1997}. In this respect itis of considerable interest that Zweck & Williams (2001) have introduced a set ofbasisfunctions withthesame shift—twist symmetry as part of an algorithm to implement contour completion. Their reason for doing so istobindsparsely distributed receptive fields together functionally, to perform Euclidean invariant computations. Itremains toexplicate the precise relationship between the Euclidean invariant circuits we have introduced here, and the Euclidean invariant receptive field models introduced by Zweck and Williams. (iv) Finally, itshould also be emphasized that many variants of the Klüver form constants have been described, some ofwhichcannotbeunderstood in terms ofthesimplemodelwchaveintroduced. For example the tunnel image shown in figure 41 exhibits a reversed retinocortical magnification, and corresponds to images described in Knauer & Maloney (1913}. It ispossible that some of the circuits beyond V1, forexample, thoseinthedorsal Phil. Trans. R. Soc. Lond. B (2001) Geometric u/iualhallucinations P. C. Bressloff andothers 325 (a) Figure 42. (o)Lattice-tunnelhallucination generated by rnarijuana. Reproduced from Sieqel(1977, with permission from Alan D. Eiselin. (b A simulation ofthelattice-tunnel. Figure 43. Complcx hallucination gcncratcd by LSD. Redrawn from Oster(1970). segment ofmedialsuperior temporal cortex(MSTd that process radial motion, are involved in the generation of such images, viaa feedback to V1 (Morrone etal. 1995). Similarly, the lattice—tunnel shown infigure42a ismorecomplicated thananyofthesimpleform constants shown earlier. One intriguing possibility is that such images are generated asa result ofa mismatch between the planform corresponding to one ofthe Klüverformconstants, andtheunderlying structure of V1. Wehave(implicitly) assumed that V1 has patchy connections that endow itwithlattice properties. Itshouldbeclearfromfigures9 and 10 that sucha cortical lattice issomewhat disordered. 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32G P. C. Bresslolf andothers Geametric Visualhallucinatians when planforms are spontaneously generated insuch a lattice. Figure 42# shows a computation of the appearance inthevisualfieldofa hexagonal rollona square lattice, when there isaslightincommensur-ability between thetwo. Asa lastexample weshowinfigure43another hallucinatory image triggered by LSD. Such an image does not fitverywellasaformconstant. However, there issomesecondary structure alongthe main (horizontal) axis oftheitsmajorcomponents. (This isalsotrueofthefunnelandspiralimages shown in figure 2, also triggered by LSD. ) This suggests the possibility that at least two differing length-scales are involved in their generation, but thisisbeyondthescopeofthemodeldescribed inthe current paper. Itisofinterest thatsimilarimages have been rcpor ted following stimulation with flick-ering light (Smythies 1960. The authors wish tothank Dr Alex Dimitrov, Dr Trevor Mundel and Dr Crary Blasdel formanyhelpfuldiscussions. Theauthors alsowishtothanktherefereesfora number ofhelpfulcomments and Alan D. Eiselinforpermission toreproduce hisartworkin figure41. Thisworkwassupported inpartbygrant96-24from the James S. Mc Donnell Foundation to J. D. C. Theresearch of M. G. wassupported inpartby NSFgrant DMS-9704980. M. G. wishes tothankthe Centerfor Biodynamics, Boston University, foritshospitality andsupport. Theresearch of P. C. B. wassup-ported bya grant from the Leverhulme Trust. P. C. B. wishesto thank the Mathematics Department, University of Chicago, for itshospitality andsupport. P. C. B. and J. D. C. alsowishtothank Professor Geo Hrey Hinton FRS and the Gatsby Computational Neurosr iences Unit, University College, London, forhospitality and support. P. J. T. wassupported, inpart,by NIHgrant T-32-MH? 0029. APPENDIX A We summarize here the derivation ofequations (29)— (31}. Thisinvolves solvingthematrixequation (A1 using a standard application ofdegenerate perturbation theory. That is,weintroduce theperturbation expansions (A2} (A3 and substitute these into the eigenvalue equation (26). We then systematically solve the resulting hierarchy of equations tosuccessive ordersinfi. (i O(fi) terms Setting m 1inequation (A1)yieldsthe G(ßequation Combining thiswiththeconjugate equation m —1 we obtain thematrixequation Phil. Trane. R. See. Lond. B (2001s Equation (A4) has solutions oftheform i t›(f/ ( {A4} (A5} where plus and minus denote the even and odd solutions. We have used the result. The G(fi)termsin equation (AI) for which m Al generate the corre-sponding first-orderamplitudes The f'9(fi 2 )contribution toequation (A1}form=1 is Combining with theanalogous equation form=—1 yields thematrixequation where (A8} (A9) Multiplying both sides ofequation (A8} on the leftby (2, and using equation (A4 implies that B q 0. This, together with equation (A7), determines the second-order contribution totheeigenvalue: (A10} Having obtained 7. ' 2 ',wecanthenuseequations (AS)and (A5) toobtaintheresult (All The unknown amplitudes ¿ 1 ri nd A I A re determined by theoverallnormalization ofthesolution. Finally, combining equations (A2), (A. SQ, (A6), and (AIO) generates equation (29). Similarly, combining equa-tions (A3), (A6, (A7},(AII)and(23)yieldsthepairof equations (30) and (31}. (b) Gonst iactionofa:xialsubgroups We sketch how toconstruct theaxialsubgroups from the irreducible representations oftheholohedry Hy corre-sponding totheshortest dualwavevectorsasgivenin table 2. Byrescalings wecanassume thatthecritical wavenumber q 1 and that the doubly periodic func-tions are on alattice A whose dual lattice A* isgenerated by wave vectors oflengthunity. 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bbbb Table Al. Tosusactionon Nt-irreducible representamos lattice torusaction hexagonal (e'""'c, e'”"'c;,e*'”'"+'°'r rhombic (e 2 "'r,e 2 " c2) Table A2. Left, 2 + 2 action on rhomb iclattice centre, D T" action onsquarelattice;riph/, fit-I-T"actionon hexagonal lattice) (Foru ‹b)even,£-\|-I; foru(d›)odd,e —1. action O action fi5 action action of N¿onthesesubspaces anddetermine theaxial subgroups. The action of the torus Z'" on the subspace K¿ is derived asfollows: write8C T” as Using the fact that k,-£q——6,q,theresultofthetrans-lation action isgivenintable A1. The holohedries Hp are D4, Dt;and D2forthesquare, hexagonal and rhombic lattices, respectively. In each case, the generators forthesegroupsarea reflection and a rotation. For the square and hexagonal lattices, the reflection isa, the reflection across the x-axis where r=(x,y). For the rhombic lattice, the reflection is<,. The counterclockwise rotation (,through anglesr/2,r3 and r,istherotation generator forthethreelattices. The action of Hp forthevariouslatticesisgivenintable A2. Finally, foreachofthesixtypesofirreducible represen-tations, we compute the axial subgroups—those isotropy subgroups Z that have one-dimensional fixed-point subspaces fix(Z), ineachirreducible representation. The computations for the square and rhombic lattices arc straightforward because we can use the F“ action in table A1 toassume, afterconjugacy, thatrtandr2arereal and non-negative. The computations on the hexagonal lattice are more complicated (Bressloff ctal. 2000a}. The results, up toconjugacy, arelistedintables A3and A4. [c)Derivation oftheamplitude equation Assume that sufficiently close tothebifurcation pointat which thehomogeneous statea z ‹D)=0becomes margin-Phil. Trans. R. Soc. Land. Y›('200I) Geametric ui. ‹ualhallucinations P. C. Bressloff andothers 327 Table A3. Axial subffroupswhenu(—e)=u(d›) (O(2) isgenerated by[0,8]e Z" and rotation by ({onthe rhombic lattice, }' on the square lattice, and }' on the hexagonal lattices. The points(1,1,I and (—1,—1,—1 have thesameisotropy subgroup (Dt,, butarenotconjugated bya group element; therefore, the associated eigenfunctions generate dift'erent planforms. ) lattice subgroup Z hx Z name square fi J(x,{) (1,1 even square O(2 Z;(x) (1,0) even roll rhombic fi2(ap,{) (1,1) even roll O(2 (1,0) even rhombic hexagonal D,(a,() (1,1,l) even hexagon (0) D5(x,{) (—I,—1,—1 even hexagon () O(2 Z (x) (1,0,0) even roll Table A4. Axial subgroupswhenu —e = u(d› (O(2 isgenerated by[0,8;je Z"androtation byr(I;onthe rhombic lattice,{ 2 on the square lattice, and {“ on the hexagonal lattice. ) lattice subgroup Z fix(X name square D,(u[$, $],{) (1,—1 odd square O(2) Z,({'x[, 0]) (1,0 odd roll rhombic fi2(a,;,[$, $],I;) (1,1 odd rhombic O(2 ‹BZ;({'x[, 0] (1,0 odd roll hexagonal Z5(I;) (1,1,1 odd hexagon fi3(x(,(' (i,i,i) triangle D;(a,(') (0,1,—I) patchwork quilt O(2) Z ({'u[$ (I]) (10 0) odd roll allystable,theexcitedmodesgrowslowlyatarate G(c' where e 2 = —y,. Weusethemethod ofmultiple scalesto derive thecubicamplitude equations (47aand(50). (i)Multiple-scaleanalysis \Ve begin by rewriting equation (HQ in the more compact form da with (A12} R' y,+r'. 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328 P. C. Bresslolf andothers Geametric visualhallucinatians and introducing a slow time-scale z t 2 t. Collecting terms with equal powers ofc then generates a hierarchy ofequations oftheform La = 0, La„ ——b,„ n>1, where and 3 (A14) (A16) (ii)Sol0abilitconditions The firstequation inthehierarchy isequivalent tothe eigenvalue equation (26)with2 = 0, = y,and\|k ——q. Therefore, the relevant classes ofsolution areoftheform (equation (46a: (A17) Following §4 we restrict solutions tothespaceofdoubly periodic functions ona square orrhombic lattice(N=2) ora hexagonal lattice (N = 3). Nextwedefinetheinner product of two arbitrary functions a z,a), 5(r,‹D) according to where D isa fundamental domain of the periodically tiled plane (whose area isnormalized to unity). The linear operator f. isself-adjoint withrespecttothisinner product, i. e. (a\|Lb)=(La\|b). Therefore, defining we have (r\|£o,)-(f. r\|a„)=0 forn = 2,3,.... Since f. a,=é„weobtaina hierarchy ofsolvability conditions From equation (A15 the lowest order solvability condition isy,cr2(r\|z a) = 0. Itturnsoutthatinthepresence of lateral interactions the inner product (rt\|in a') can be non-vanishing (inthecaseofevenpatterns), whichleads toa contradiction when 2 0. Thiscanberemedied by assuming that w2=ew\+ G(r' and considering the modified solvability condition (r\|e* 1 é2+é. ) = 0. This generates theequation (A18) An alternative approach tohandling thenon-vanishing of the inner product (r\|in a1) would be to expand the bifurcation parameter as = y,-I-ey-I-e 2 y +....This Phil. Trans. R. Soc. Land. B (2001a would then give a quadratic (rather than a cubic) amplitude equation describing the growth of unstable hexagonal patterns. In the case of odd patterns (u\|in a' 2 ) 0 and no restriction on w2 is required. However, foreaseofexposition wetreattheoddandeven cases inthesameway. (iii)Amplitude equations In order to explicitly derive the amplitude equations (47) and (50) from the solvability condition (A18), we need toevaluate innerproducts oftheform(r\|z a'). Since u isasolution tothelinearequations (Al4), it follows that (A19} Thus, substituting equation (A17) into the left-hand side ofequation (A18)andusing(A19}showsthat a I = [I-I-E'''] {A20} with I I, ' G(fi). Thefi-dependent factorsappearing on the right-hand side ofequation (A20)areeliminated from the final amplitude equations by an appropriate rescaling ofthetimei anda global rescaling oftheampli-tudesc. Similarly, and (A2l) with I"' and I!' given by equations (57} and (58). Note from equation (A21) that the inner product [u\|a 2 )isonly non-vanishing when =3 planforms since we require {corresponding ENDNOTE tohexagonal 0. Onepossible set of wave vectors is A = J,(cos(p, sin(‹,o with ‹,o=0,‹,o;=2 3,p = —2r J 3. Alsonotethatifu(e}is an odd eigenfunction then itimmediately follows that 0. Finally, we substitute equations (A19, (A20), (A21) and (A22) into (A18) and perform the rescaling ert (w Wt+ I!'r. After an additional rescaling of time we obtain the amplitude equations (47) for N =2 and {50) for N=3. 'Interestingly, there does exist an example from fluid dynamics where the modified Euclidean group action (equation (9) arises (Bosch Vivancos ctal. 1995a. REFERENCES Blasdel, CI. U. 1992 Orientation selectivity, preference, and continuity in monkey striate cortex. J.. Neurosci. 12, 3139-3161. Bosch Vivancos, I.,Chossat, P. & Melbour ne I. 1995New planforms insystemsofpartialdifferential equations with Eur:lidean symmetry. Arch. Ration. Mesh. 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Quantitative theory of driven nonlinear brain dynamics J. A. Robertsa,b,c,⁎, P. A. Robinsona,b a School of Physics, University of Sydney, New South Wales 2006, Australia b Brain Dynamics Center, Sydney Medical School —Western, University of Sydney, Westmead, New South Wales 2145, Australia c Queensland Institute of Medical Research, Brisbane, Queensland 4006, Australia abstract article info Article history: Accepted 21 May 2012Available online 29 May 2012 Keywords:Steady state visual evoked potential Entrainment Neural ﬁeld model Nonlinear dynamics EEGPeriodic stimulus Strong periodic stimuli such as bright ﬂashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity, but the underlying mechanisms for these phenomena are poorly understood at present. The dominant features of these experimentally observed dynamics are reproducedby the dynamics of a quantitative neural ﬁeld model subject to periodic drive. Model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements, exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f α, subharmonic entrainment near 2 fα, and harmonic generation. Further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude, including period doubling, bistable phase-locking, hysteresis, wave mixing, and chaos indicated by positive Lyapunov exponents. Moreover, photosensitive seizures arepredicted for physiologically realistic model parameters yielding bistability between healthy and seizure dy-namics. These results demonstrate the applicability of neural ﬁeld models to the new regime of periodically driven nonlinear dynamics, enabling interpretation of experimental data in terms of speci ﬁc generating mechanisms and providing new tests of the theory. © 2012 Elsevier Inc. All rights reserved. Introduction Periodic ﬂashing light, also termed ﬂicker or photic driving, evokes responses in the electroencephalogram (EEG) that have been studied since the 1930s. Despite being widely used to probe visual function and perception ( Müller et al., 2006; Nunez, 1995; Regan, 1989 ), the un-derlying neurophysiological mechanisms of these responses remain poorly understood. Early studies identi ﬁed a clear nonlinear contribu-tion to the driven cortical dynamics ( van der Tweel and Spekreijse, 1969; van der Tweel and Verduyn Lunel, 1965 ). Nonlinearity in large-scale brain activity has seen considerable recent interest ( Stam, 2005 ), particularly in healthy resting state activity ( Freyer et al., 2009, 2011; Stam et al., 1999 ), epileptic seizures ( Breakspear et al., 2006; Lehnertz and Elger, 1998; Robinson et al., 2002; Wendling et al., 2000 ), and func-tional neuroimaging ( Stephan et al., 2008a ). Notable nonlinear features in the EEG response to ﬂicker include entrainment and frequency mixing ( Herrmann, 2001; Rager and Singer, 1998; Regan, 1989; Townsend et al., 1975 ), both of which involve interactions between stimuli and ongoing activity and are thus of central importance to un-derstanding brain function ( Engel et al., 2001 ). Periodic stimuli can also provoke epileptic seizures ( Parra et al., 2005 ), particularly for driving frequencies near the natural alpha rhythm fα(≈10 Hz in adult humans), the most prominent feature of healthy resting EEG ( Nunez, 1995 ). The mechanisms of these seizures are not well understood, nor is there a clear relation to the brain's re-sponse in cases when the stimuli do not cause seizure. Thus there is a need for modeling to unify these nonlinear phenomena within a sin-gle framework. Studies of the nonlinear mechanisms responsible for ﬂicker-induced EEG signals have primarily taken a signal analysis approach ( Kelly, 1981; Liu et al., 2010; Regan, 1989; van der Tweel and Spekreijse, 1969; van der Tweel and Verduyn Lunel, 1965 ), modeling time series data as arising from abstract ﬁlters. Although successful in describing input-output relationships, such models are dif ﬁcult to generalize to other phenomena; it is preferable to have a single systematic theory. In-deed there is a strong need in neuroscience to move beyond models of time series to models of the underlying brain structures and connectiv-ity, from which the observed dynamics emerge ( Breakspear et al., 2010; Stephan et al., 2008b ). Such modeling is now possible given recent de-velopments in neural ﬁeld theory ( Coombes, 2010; Deco et al., 2008 ), particularly as it relates to the corticothalamic system ( Robinson et al., 1997, 2002, 2004 ). Neural ﬁeld models describe the aggregate activity of populations of neurons and are thus especially appro priate for describing large-scale brain dynamics. Building on early work ( Amari, 1975; Ermentrout and Cowan, 1979; Freeman, 1975; Lopes da Silva et al., 1974; Nunez, 1974;Neuro Image 62 (2012) 1947-1955 ⁎Corresponding author at: Queensland Institute of Medical Research, 300 Herston Road, Herston, Queensland 4006, Australia. Fax: +61 7 3845 3511. E-mail addresses: jamesr@physics. usyd. edu. au (J. A. Roberts), robinson@physics. usyd. edu. au (P. A. Robinson). 1053-8119/$-see front matter © 2012 Elsevier Inc. All rights reserved. doi:10. 1016/j. neuroimage. 2012. 05. 054 Contents lists available at Sci Verse Science Direct Neuro Image journal homepage: www. elsevier. com/locate/ynimg	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
Wilson and Cowan, 1973 ), neural ﬁeld models ( Baker and Cowan, 2009; Bressloff et al., 2002; Coombes, 2005; Hutt and Atay, 2005; Jirsa, 2009; Jirsa and Haken, 1996; Pinotsis and F riston, 2011; Pinotsis et al., 2012; Robinson et al., 1997; Taylor and Baier, 2011; Wright and Liley, 1996 ) and their limiting case of neural mass models ( Cosandier-Rimélé et al., 2008; David and Friston, 2003; David et al., 2006; Goodfellow et al., 2011; Jansen and Rit, 1995; Lopes da Silva et al., 2003; Moran et al.,2009; Wendling et al., 2000 ) have successfully de s c r i b e dab r o a dr a n g e of healthy and pathological brain dynamics at the spatial and temporal scales probed by EEGs and magnetoencephalograms. Neural ﬁeld models have been notably successful in describing spatiotemporal dynamics in vi-sual cortex ( Baker and Cowan, 2009; Bre ssloff et al., 2002; Coombes, 2005; Ermentrout and Cowan, 1979 ), in particular explaining pattern for-mation during hallucinations. Visual evoked responses to individually-ﬂashed stimuli have also been studied using neural mass models ( David et al., 2006; Jansen and Rit, 1995 ), and ( Spiegler et al., 2011 ) very recently applied the ( Jansen and Rit, 1995 ) model to study entrainment and other nonlinear features of photic-driven activity. Both classes of model have explained a range of seizure phenomena, relating this pathological activ-ity to healthy dynamics generated by the same models ( Breakspear et al., 2006; Cosandier-Rimélé et al., 2008; Goodfellow et al., 2011; Kim et al., 2009; Lopes da Silva et al., 2003; Nevado-Holgado et al., 2012; Roberts and Robinson, 2008; Robinson et al., 2002; Taylor and Baier, 2011; Wendling et al., 2000 ). In this paper we explain experimentally-observed nonlinear inter-actions between brain activity and ﬂicker by incorporating periodic driving into a corticothalamic model ( Roberts, 2010; Robinson et al., 1997, 2002 ). The results reproduce multiple features of published data and constrain the physiologically allowable states. We further predict various other nonlinear phenomena whose existence can be tested in future experiments, including period doubling, bistable phase-locking, hysteresis, wave-wave interactions, and chaos, and show how to treat entrainment and seizures in the same framework. This extends neural ﬁeld modeling into a new regime and enables new experimental tests of the theory. We focus on EEG responses to sinusoidally modulated light, which are termed steady-state visual evoked potentials (SSVEPs) ( Nunez, 1995; Regan, 1989 ), but our approach is general and could be applied to any periodic stimulus. The SSVEP is dominated by synchronous corti-cal activity concentrated at the drive frequency f Dand its harmonics. Ex-periments have measured the response over a wide range of f Din cats (Rager and Singer, 1998 ) and humans ( Herrmann, 2001 ), showing clear spectral peaks, as seen in Fig. 1 (a) for spatially uniform square wave modulated stimuli. Here, as in the original ﬁgure in Herrmann (2001), the spectrum for each f Dis multiplied by fto enhance peaks at higher frequencies, and the peaks appear as discrete dots due to being sampled on a coarse grid. These peaks involve both responses at f D (which can be linear or nonlinear) and inherently nonlinear effects. The clearest nonlinear effect is the generation of N:1 harmonics (Nresponse oscillations phase-locked to each drive oscillation). Since the square wave input signal used here contains only odd har-monics, the even harmonics in the output (in particular the prominent 2:1 harmonic) must be generated nonlinearly; power at the odd har-monic frequencies likely contains both linear and nonlinear compo-nents. The position of this nonlinearity in the visual pathway is not yet well established, and could involve effects in retinal, thalamic, and/or cortical neuronal populations. Suppression of background alpha activity over a range of f Dis another signiﬁcant nonlinear effect seen in Fig. 1 (a), where the dominant re-sponse tracks f Dorf D/2, while the dominant frequency is unchanged out-side this range. (These oscillations are network-level collective modes, whose frequencies are not individual neural ﬁring rates in general. ) This is an example of entrainment, characterized by a reduction of ongoing ac-tivity in favor of activity phase-locked to the drive, at frequencies harmon-ically related to the drive frequency. Fig. 1 (a) thus shows entrainment to f D(1:1 phase locking) and subharmonic entrainment to f D/2 (1:2 locking). Quantitative modeling, such as that used in the present work, will be critical in analyzing the data to distinguish between the above possibili-ties. In this context, it is particularly important to use established models, rather than introducing a new one ad hoc for each new experiment. Established models enable new phenomena to be incorporated into wider, uni ﬁed frameworks, thereby relating them to other measures and situations, and permitting paramet ers to be constrained a priori, rath-er than being treated as free. The wide use of the Hodgkin-Huxley model is perhaps the best known example of this approach in neuroscience. This paper is organized as follows. The Methods section outlines the nonlinear corticothalamic model. In the Results section the periodically driven dynamics of this model system are shown to reproduce the main features of the experimental SSVEP spectra in Fig. 1 ( a )a sw e l la s predicting rich nonlinear dynamics whose existence and properties can be explored in future experiments. Methods The Corticothalamic model section presents the nonlinear cor-ticothalamic model, the dynamics of which will be analyzed using the time series methods described in the Time series analysis section. Fig. 1. Spectral response for f D=1-50 Hz. Responses to the Nth harmonic of f Dare la-beled N:1; subharmonic entrainment is labeled 1:2. Darker shading denotes higher power, and spectra have been multiplied by fto enhance power at high frequencies as in the original Fig. 5 of Herrmann (2001). (a) Adult human EEG. Figure adapted from Fig. 5 of Herrmann (2001) after cropping, changing aspect ratio to 1:1, and put-ting independent variable on the horizontal axis. (b) Model with ( νee,νei,νes,νse,νsr, νsn,νre,νrs)=(1. 7,-2. 8, 0. 19, 0. 84,-0. 62, 1. 0, 0. 28, 3. 0) m V s, ( α,β,αsr,βsr)=(80, 800, 10, 60)s-1,bϕn0>=18 s-1,σn=1s-1,Φn=2s-1; other parameters as in Breakspear et al. (2006). Spectra are smoothed with a 0. 4 Hz moving window to display sharp peaks more clearly. 1948 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
Corticothalamic model Neural ﬁeld theories have successfully reproduced many aspects of the dynamics of populations of neurons by averaging over small spatial and temporal scales to give a description at scales ≳0. 1 mm; see Deco et al. (2008) for a recent review. We use a recent corticothalamic model (Breakspear et al., 2006; Robinson et al., 1997, 2002, 2004 )t oa n a l y z e nonlinear SSVEPs ( Roberts, 2010; Robinson et al., 2002, 2008 ). We fol-low the exposition and notation of Robinson et al. (2002) here, except where otherwise noted. The neuronal populations included are (see Fig. 2 ) excitatory ( e) and inhibitory ( i) cortical neurons, and the speci ﬁc relay ( s) and reticular ( r) thalamic neurons. The relay in the visual sys-tem is the lateral geniculate nucleus, which has reciprocal connections with both the cortex and the reticular nucleus as shown in Fig. 2 (Sherman and Guillery, 1996; Steriade et al., 1997 ). External input from the brainstem and sensory systems ( n) drives the system via s. U n-structured stimuli presented to the entire visual ﬁeld drive a large area of cortex, so we concentrate on spatially uniform activity. A nonlinear sigmoidal function S(Va), given by ( Freeman, 1975; Wilson and Cowan, 1973 ) SVað Þ ¼Qmax 1þexp-Va-θ ð Þ =σ′/C2/C3 ; ð1Þ relates the mean ﬁring rate Qa=S(Va) of each neuronal population a to its mean cell body potential Va(relative to resting), where Qmaxis the maximum ﬁring rate, θis the mean threshold voltage and σ′π=ﬃﬃﬃ 3p is its population standard deviation. Here Va=∑b Vab, with Dab Vabr;tð Þ ¼ νabϕbr;t-τab ð Þ ; ð2Þ Dab¼1 αabβabd2 dt2þ1 αabþ1 βab/C18/C19d dtþ1; ð3Þ where Vabis the contribution due to input ϕbfrom neurons in popu-lation barriving after mean axonal delay τab,Dabgives the soma po-tential response allowing for synaptodendritic dynamics and soma capacitance, connection strength νab=Nabsab,Nabis the mean num-ber of synapses per neuron of type afrom neurons of type b,sabis the strength of response to a unit input from neurons of type b, and αabandβabare the inverse decay and rise times of the soma response to input b, respectively. For the system in Fig. 2, the only nonzero de-laysτabareτes=τis=τse=τre=t0/2 where t0is the total cor-ticothalamic loop delay. Average afferent ﬁring rates ϕapropagate as ﬁelds in the continu-um limit, approximately governed by a damped wave equation with source Qa(Jirsa and Haken, 1996; Nunez, 1995; Robinson et al., 1997 ), giving 1 γ2 a∂2 ∂t2þ2 γa∂ ∂tþ1-r2 a∇2"# ϕar;tð Þ ¼ SVar;tð Þ1/2/C138 ; ð4Þwhere γa=va/ra,vais the axonal conduction speed, and rais the char-acteristic axon length. We assume that the activity is spatially uni-form over the cortical area of interest, consistent with a large-scale mode driven by ﬂicker stimuli; thus the ∇2term is zero and for nota-tional simplicity we omit the rargument hereafter. We set γa=∞for a=i,r,sowing to the short ranges of these axons ( Robinson et al., 2002 ). We retain long-range excitatory intracortical connectivity via nonzero reandﬁniteγe. Note that, although we neglect position de-pendence of activity, we have not reduced our model to a neural mass model, since this would also require shrinking the cortical man-ifold to be zero-dimensional by setting re=0 and γe=∞. We assume random intracortical connectivity, which implies νib=νeb forb=e,i,sand hence Vi=Ve, as derived in detail previously ( Robinson et al., 1998 ). The inhibitory population is modeled explicitly ( νei,νii,νie, andνisare all nonzero); random connectivity implies that its mean-ﬁeld dynamics are closely related to (but not identical to) those of the excit-atory population with which it is spatially commingled (the excitatory neurons project much further, giving rise to the parameter γe). For sim-plicity we assume that all the synaptodendritic time constants are equal except for the inhibitory intrathalamic srconnection, which is mediated by GABA Bdynamics slower than the other channels ( Steriade et al., 1997 ). Hence we set αab=α,βab=β,Dab=Dαfor all abexcept for the srconnection; αandβare thus to be interpreted as effective values. We do not explicitly model the bursting mode of thalamic neurons, which tends to only be relevant to frequencies lower than those typical-ly observed in SSVEPs ( Steriade et al., 1997 ). We set parameters at or near previously published values (see ﬁgure captions) ( Breakspear et al., 2006 ), which enables uni ﬁcation of the ﬁndings here with those of previous studies. Here the parameters place the system near a stability boundary associated with the intrathalamic loop ( Robinson et al., 2002 ). All model parameters corre-spond to physiological quantities and have been rigorously constrained across numerous independent experimental measures ( Robinson et al., 2004 ). Thus for the connectivity in Fig. 2 and the above assumptions, the model equations are 1 γ2 ed2 dt2þ2 γed dtþ1"# ϕetð Þ¼SVetð Þ1/2/C138 ; ð5Þ DαVetð Þ¼ νeeϕetð Þ þνeiϕitð Þ þνesϕst-t0=2 ð Þ ; ð6Þ DαVrtð Þ¼ νreϕet-t0=2 ð Þ þ νrsϕs; ð7Þ Vstð Þ¼ Vsetð Þ þ Vsrtð Þ þ Vsntð Þ; ð8Þ DαVsetð Þ¼νseϕet-t0=2 ð Þ ; ð9Þ Dsr Vsrtð Þ¼ νsrϕrtð Þ; ð10Þ DαVsntð Þ¼ νsnϕntð Þ; ð11Þ with ϕb(t)=S[Vb(t)] for b=i,r,s. External stimuli ϕndrive the brain via the relay nuclei. Resting EEG spectra are successfully generated when driving the model with white noise ( Robinson et al., 1997, 2001a, 2001b, 2002, 2004 ), which assumes that the collective input from all sensory pathways is so complex that it does not favor any particular frequency in the range of interest ( Engel et al., 2001; Lopes da Silva et al., 1974; Stam et al., 1999 ). Impulsive stimuli yield evoked response potentials (Kerr et al., 2008; Robinson et al., 2001a, 2004 ). Sinusoidally modulat-edﬂicker stimuli generate SSVEPs ( Robinson et al., 2002, 2008 ) and in the linear regime the model yields cortical phase velocities for SSVEPs consistent with experiment ( Robinson et al., 2008 ). Here we concen-trate on the nonlinear regime, where the drive is strong.,rs sn se reesrelay nuclei reticular nucleus cortex ee,eisr Fig. 2. Connectivities of cortical (excitatory e, inhibitory i) and thalamic (relay s, retic-ular r) neural populations; arrowheads abdenote where ﬁeldsϕbproject, with connec-tion strength νabfor input to afrom b;ϕndenotes sensory input. 1949 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
To study interaction between ﬂicker and background activity we model ϕnas ϕntð Þ¼ ϕnoise n tð Þ þϕD ntð Þ; ð12Þ where ϕnnoiseis Gaussian white noise with mean ϕn(0)and standard de-viation σn, and ϕn Dis a periodic drive with zero mean. We use two forms for ϕn Din the Results section. For comparison with Fig. 1 (a) we use a square wave drive given by ϕD ntð Þ¼Φnsgn cos 2 πf Dt ð Þ 1/2/C138 ; ð13Þ where Φnis the drive amplitude and sgnindicates the sign function. This square wave consists of a sum of sinusoids, such that the drive contains power at all odd multiples of f D; system responses at even multiples are necessarily nonlinear. To study the simpler case of a sin-gle sinusoid we use the form ϕD ntð Þ¼ Φncos 2 πf Dt ð Þ ; ð14Þ for which all higher harmonics in the model output are necessarily generated nonlinearly. Our use of the square wave drive Eq. (13) matches the photic drive used in Herrmann (2001), but neglects possible nonlinearities in the retina that might have modi ﬁed the latter drive before it ar-rived at the corticothalamic system. We thus use a square wave so as to introduce the fewest additional assumptions, but note that it would be straightforward to implement other more complicated forms for the input. For example, the model could be explicitly coupled to models of retinal dynamics, which would enable it to compare and evaluate putative nonlinearities in the earliest parts of the visual stream. However, these topics are beyond the scope of this paper. To solve the model equations numerically we use a standard ex-plicit forward step integration scheme with a timestep of 0. 1 ms (Press et al., 1992 ), of the type used by Robinson et al. (2002), for ex-ample. We have checked that the results are numerically stable and that the timestep is suf ﬁciently short that they do not change signif-icantly when it is further reduced. Time series analysis To characterize the frequency content of the model's response to a range of stimulus parameters, we calculate power spectra from model time series. We estimate spectra from windowed Fourier transforms via Welch's method (as implemented in MATLAB R2008b), using 10 s windows overlapping by 50%. To display sharp peaks more clearly, we additionally smooth spectra with a 0. 4 Hz moving window. To elucidate the role of nonlinear frequency coupling in generating peaks in the model EEG spectra we calculate bicoherence from model time series ( Kim and Powers, 1979 ). Bicoherence is a nonlinear coher-ence measure that is sensitive to phase locking between Fourier compo-nents at different frequencies, and is derived from the bispectrum, a higher-order spectrum that generalizes the usual power spectrum (Kim and Powers, 1979 ). The bicoherence b2(ω1,ω2) between frequen-ciesω1andω2of a signal with Fourier transform X(ω)i sg i v e nb y( Kim and Powers, 1979 ) b2ω1;ω2 ð Þ ¼Xω1ð Þ Xω2ð Þ X/C3ω1þω2 ð Þ jj2 Xω1ð Þ Xω2ð Þ jj2Xω1þω2 ð Þjj2; ð15Þ where X*is the complex conjugate of X. W ee s t i m a t et h e X(ω) terms by averaging FFTs of windowed time series (16384-point Hanning win-dow, mean subtracted from each). To characterize nonlinear dynamics at large drive amplitudes we estimate the largest Lyapunov exponent (LLE) ( Parlitz, 1998 ) frommodel time series using the software package Open TSTOOL v1. 2 (Merkwirth et al., 2009 ). A positive LLE implies that nearby trajecto-ries diverge exponentially and is indicative of chaos; periodic orbits have an LLE of zero. We use the method of time-delay embedding on long (80 s) time series for ϕesampled at 1000 Hz, with the mini-mum embedding dimension chosen using the method of Cao (1997) with three nearest neighbors and 1000 reference points, and thedelay chosen using the ﬁrst minimum of the auto-mutual information function (see Open TSTOOL documentation for further details). A ben-eﬁt of using this method for LLE estimation is that it is also applicable to experimental time series (i. e., does not require access to the full underlying dynamical system). To analyze phase-locking regimes as a function of drive amplitude and frequency, we calculate for each point in parameter space the ratio of the drive's period to that of time series simulated in the ab-sence of noise. We use the sequence of turning points in ϕ eto calcu-late both the time series period (thus identifying 1: Nphase-locking regimes) and the number of oscillations per period (identifying N:1 regimes, where the drive and ϕehave the same period). Results We begin in the Results section by driving the system with a square wave photic input to compare the model output to the data of Herrmann (2001) in Fig. 1 (a). We analyze the driven dynamics in more detail using sinusoidal input in the Sinusoidal drive section. Square wave drive Fig. 1 (b) shows the model response spectrum vs. f Dfor a square wave drive given by Eq. (13), with each spectrum multiplied by ffor comparison with Fig. 1 (a). Key features are peaks at f D(1:1) and its har-monics ( N:1), in good agreement with Fig. 1 (a) (Herrmann, 2001; Rager and Singer, 1998 ). Most signi ﬁcantly, entrainment to f Dis evidenced by suppression of the background alpha activity fα≈11 Hz in favor of f Dfor f D≲15 Hz, consistent with Fig. 1 (a) where no strong alpha peaks are seen off the 1:1 diagonal. The alpha peak is suppressed without any changes to cortical or thalamic parameters; only the drive frequency changes. Further striking agreement is shown by 1:2 subharmonic en-trainment for f D≈15-24 Hz, where peaks follow f=f D/2. Suppression of background activity is necessarily nonlinear; in the linear regime the drive and background superpose without interacting. The suppres-sion observed experimentally cannot be attributed to averaging over trials, or else the alpha peak would not be observed for any ﬁxed drive frequency contradicting Fig. 1 (a). The extent to which the alpha activity is suppressed by the drive in the experimental data is not as clear as in the model; peaks in Fig. 1 (b) are narrower than in Fig. 1 (a) due to the FFT used here yielding better resolution than the autoregressive spec-trum in Fig. 1 (a). The experimental data also appears “lumpier ”than the model data due to being sampled on only a coarse grid of frequen-cies; similarly coarse sampling in the model would yield an even more similar appearance. Additional responses in the model at f=3f D/2 and f=5f D/2 are not seen in Fig. 1 (a), likely due to Fig. 1 (a) being an average over ten subjects, thus further blurring peaks via variability between subjects. Sinusoidal drive In this subsection we drive the system with a single sinusoid to avoid ambiguity between spectral peaks that would be present in a nonsinusoidal drive and peaks generated by nonlinear processes. Fig. 3 (a) shows that a sinusoidal drive given by Eq. (14) reproduces the main features of the EEG response to a square wave drive, demon-strating that entrainment and harmonic generation are not speci ﬁct o square wave input. Here and in Fig. 3 (b) additional features are re-vealed by plotting a wider dynamic range than in Fig. 1 (a) (here we1950 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
also show the actual spectrum without multiplication by f). The model parameters here are modi ﬁed slightly from those in Fig. 1 (b) to better maintain the resemblance to key features of the data, such as widths of entrainment regimes. This parameter adjustment is needed because removing harmonics from the drive in going from square to sine wave reduces power at higher frequencies. This mod-iﬁes the EEG spectra (such as by changing the mix of subharmonics generated), which can be approximately compensated for by small adjustment of model parameters. We stress that this step is carriedout solely to probe the underlying mechanisms with more clarity and does not affect the level of agreement between the square-wave case and experiment found in the Square wave drive section. Entrainment of the alpha frequency to 2 f Doccurs over a narrow range where 2 f D≈fα; just above this drive frequency there is an en-hancement where the background activity is shifted slightly down-wards. Sum and difference frequencies f±=\|f D±fα\| are generated by nonlinear interactions (other than entrainment) between the drive and background activity. This phenomenon is well-known as Fig. 3. Model results for ( νee,νei,νes,νse,νsr,νsn,νre,νrs)=(1. 3,-2. 9, 0. 13, 2. 9,-0. 57, 1. 0, 0. 67, 2. 9) m V s, Φn=2. 8s-1, other parameters as in Fig. 1 (b). (a) Spectral response for f D=0-50 Hz, labeled as in Fig. 1 (b) but with grayscale range extended to show lower-level peaks, and without multiplying by f. (b) Detail of (a); Features A-C are discussed in the text. (c) Bicoherence for SSVEP at f D=30 Hz and parameters of (a). (d) Turning points (after transients) of noise-free time series vs. Φnforf D=20 Hz and parameters of (a). Green and red points show bistable solutions obtained by continuation up and down in Φn, respectively; yellow shows where red would otherwise overplot green. (e) Phase-locking zones for noise-free driven system initially at a stable ﬁxed point for parameters of (a). (f) Largest Lyapunov exponent (LLE) vs. Φnfor time series corresponding to the forward continuation in (d). Dark points in (a)-(c) correspond to high values. 1951 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
mixing in nonlinear wave physics ( Robinson et al., 2001a; Whitham, 1974 ), and is seen in SSVEP studies using two stimulus frequencies (Regan, 1989 ). To brie ﬂy illustrate how mixing occurs in a general setting, consider a signal y(t)=sin( ω1t)+sin( ω2t) comprising two sinusoids with frequencies ω1andω2. If this signal is passed through a quadratic nonlinearity (i. e., is squared), the resulting signal (after using trigonometric identities) is y2(t)=1-cos(2 ω1t)/2-cos(2 ω2t)/ 2+cos[( ω1-ω2)t]-cos[(ω1+ω2)t], and thus contains second har-monics 2 ω1and 2 ω2, and sum and difference frequencies ω1+ω2and ω1-ω2, respectively. To have high power at frequencies f±requires a sharp alpha peak, so they are not observed in Fig. 1 (a) [except possibly near (0,10)Hz], where the alpha peak is broader and intersubject vari-ability tends to obscure this feature in the group average. Bicoherence (Kim and Powers, 1979 )i ss h o w ni n Fig. 3 (c) for f D=30Hz. Clear peaks at ( f1,f2)=( f D,fα) signify high phase coherence between f D,fα, and f D+fα. Peaks around ( f D,f D), (f D,0), and (0, f D) show phase coher-ence between f Dandf D±f D(2:1 harmonic generation; phase coherence with f1,2=0 represents the limiting case as f1,20). Enhancement is also seen on diagonals where f1+f2=f D,2f D,3f D, showing coupling be-tween frequencies that sum to f D,2f D,a n d3 f D. Entrainment involves more than just suppression of background peaks: enhancements A, B, and C in Fig. 3 (b) are precursors to phase-locked regions entered by increasing the amplitude Φn, as is explored further in Fig. 4. Enhancement A is an incipient 1:4 subharmonic; a small further increase in Φnis suf ﬁcient to enter this regime, yielding high power at f=f D/4, as seen by comparing Figs. 4 (c) and (d). Both B and C have two branches (those at B roughly have a forked shape on the end of the 1:2 line, but are blurred together here into a broad peak), and at each f Dthe sum of their (mean) frequencies equals that of the sharp peak immediately above in f(i. e., the frequencies sum to f Dat B and to f D/2 at C). Under changes in Φnthe branches at B always meet at the left endpoint of the subharmonic regime, shifting the back-ground activity away from f D. This is seen more clearly by comparing Figs. 4 (a)-(c), where increasing Φnleads to competition between the linear purely noise-driven fαresonance and a new resonance near onset of 1:2 entrainment. Such competition can result in incomplete en-trainment, where background activity is shifted in frequency to yield an enhancement at a new frequency different from f D; periodic pulling (Klinger et al., 1995 ) is a similar phenomenon seen in plasma physics. The noise-driven spectrum of the forced system results from interactionbetween background resonances (set by system parameters) and reso-nances due to the drive (set by both system and drive parameters). The system undergoes period doubling bifurcations as Φ nis in-creased for ﬁxed f Din intervals around f≈fα,2fα,....Fig. 3 (d) shows turning points (after transients) of sinusoidally driven time series at f D=20Hz vs. Φnin the absence of noise (i. e., σn=0). We stress that this analysis is of the interaction between the system and the periodic drive, not of the various frequency interactions possible with noise input that are better probed by spectra. For small Φnthe peak-to-peak amplitude (the maximum minus minimum shown) depends linearly onΦn; at larger Φnthe solution jumps to the period doubled branch(subharmonic entrainment). Successive period doubling with increas-ingΦnleads to aperiodic dynamics; further increases yield periodic so-lutions again. The LLE is positive for drive amplitudes in the range Φn≈5-9s-1(apart from narrow periodic windows), as shown in Fig. 3 (f), indicating that the dynamics are chaotic. Since the LLE can be estimated from time series without needing access to the full set of un-derlying dynamical variables, this model prediction should be testableusing experimental EEG time series. Another key feature is the existence of bistability between 1:1 and 1:2 entrainment, shown by coexistence of green and red points around Φ n≈1s-1. Thus both 1:1 and 1:2 entrainments (the latter always ex-hibits 1:1 activity concurrently) exist at the same point in parameter space. The implication is that experimental SSVEPs at given f DandΦn are not necessarily unique, yielding a new source of intertrial variability that warrants study in its own right. We also predict hysteresis in stud-ies where f Dis ramped slowly up and down through a zone where bis-table entrained states exist, which would give different spectra on the up and down paths. This should be relatively straightforward to test; one possible place to look is where the 1:1 and 1:2 entrainment zones meet in Fig. 1. Hysteresis effects would likely be obscured in studies where stimuli at the f Dof interest are presented in random order, as is commonly used to avoid habituation effects ( Regan, 1989 ). Spontane-ous transitions between bistable states during a long recording session might also contribute to intertrial variability. Extending the analysis of period doubling bifurcations in Fig. 3 (d) across a range of drive frequences reveals a rich pattern of entrainment zones. Fig. 3 (e) shows phase-locking regions vs. f DandΦnfor dynamics initially at a stable ﬁxed point. For comparison, Fig. 3 (d) lies at f D=20 Hz, and Fig. 3 (a) lies at Φn=2. 8s-1showing that 1:1 entrain-ment of fαoccurs near the 1:2 regime boundary. The 1:1 region is near-linear for small Φnbut increasingly nonlinear at larger Φn,f o re x-ample yielding regions of N:1 phase-locking at f D≈fα/N. The 1:1 regime bifurcates to 1: Nregimes over ranges of f Daround (and for minimal Φn at)f D≈Nfα, consistent with observed 1:3 and 1:8 entrainments (Herrmann, 2001 ) (in Figs. 5 and 4 therein, respectively). The 1: Nre-gions are always nonlinear, showing nested and overlapping period doublings toward dark blue regions where dynamics are variously of high period ( f D>80 f), quasiperiodic, or chaotic. Gaps in the 1:3 family of period doublings around f D=30Hz and Φn=14s-1reveal the locking regions underneath, hinting at a complicated structure reminis-cent of overlapping Arnold tongues ( Glass, 2001 ). The layout of entrain-ment zones in parameter space is also a new source of parameter constraints, because a given parameter set can be ruled out if it does not exhibit the correct entrainment properties. This is complementary to previous constraints in the linear regime, because satisfying these does not guarantee the same for the nonlinear regime. For example, the requirement that the system lie near a period doubling in the nonlinear dynamics is complementary to constraints previously eluci-dated for linear background EEG spectra ( Robinson et al., 2004 ). The structure in Fig. 3 (d) is for a driven system initially at a stable ﬁxed point, not a pre-existing limit cycle, and the entrainment in f D (Hz)f (Hz)a) 01 0 2 0 3 005101520 f D (Hz)f (Hz)b) 0 1 02 03 005101520 f D (Hz)f (Hz)c) 01 0 2 0 3 005101520 f D (Hz)f (Hz)d) 01 0 2 0 3 005101520 Fig. 4. Spectra vs. f Dfor increasing drive amplitude Φnwith other parameters as in Fig. 3 (a). (a) Φn=1s-1. (b)Φn=2s-1. (c)Φn=3s-1. (d)Φn=4s-1. Dark shading corre-sponds to high values. 1952 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
Fig. 1 (b) is for noisy perturbations of a ﬁxed point. Bifurcations of the ﬁxed point itself (under changes in parameters other than the drive) affect entrainment. Numerically we ﬁnd that entrainment requires the system to lie near a linear stability boundary, in accord with evi-dence that the brain operates near marginal stability ( Breakspear et al., 2006; Robinson et al., 1997; Stam, 2005; Stam et al., 1999 ), and also near a nonlinear stability boundary, as seen in Fig. 3 (e). Entrain-ment like that in Fig. 1 also requires that no other attractor coexist near the ﬁxed point; otherwise the spectrum would change marked-ly. For a bistable limit cycle associated with seizure dynamics, such as near the subcritical 10 Hz bifurcation in the model ( Breakspear et al., 2006 ), a sinusoidal drive near resonance perturbs the system away from the stable state onto a large-amplitude attractor, even for rela-tively weak stimuli ( Kim et al., 2009 ); a typical case in Fig. 5 shows a dramatic increase in power at all frequencies for drives near fα≈9 Hz and 2 fα, with a waveform similar to the 10 Hz seizure in Robinson et al. (2002) and Breakspear et al. (2006) ; such behavior is characteristic of photosensitive epilepsy ( Kim et al., 2009; Parra et al., 2005 ). The seizure frequency f≈10 Hz is little affected by the drive, which hardly perturbs the large-amplitude attractor. Numerical exploration reveals that entrainment of fαis favored for parameters yielding decreases (2-to 10-fold relative to typical alert resting values) in mean ﬁring rates ϕe,ϕi,a n dϕs,w i t h ϕrincreased (pos-sibly substantially) or unchanged. Such decreases are qualitatively con-sistent with decreased metabolic load in these structures during entrainment ( Parkes et al., 2004 ) (assuming that background activity is analogous to the aperiodic stimulation used there for comparison), suggesting that subnetworks activated by stimuli likely differ from those underlying resting activity, agreeing with recent experiments (Birca et al., 2006; Kerr et al., 2008 ). A relative increase in thalamic inhi-bition is plausibly a form of automatic gain control ( Schwartz and Simoncelli, 2001 ), attenuating strong sensory stimuli before they are re-layed to the cortex. Discussion We have studied driven nonlinear brain dynamics in a physiologically-based model of the cortex and thalamus, have compared its predictions with experiments from the literature, and have made a series of newpredictions that can be tested in future experimental work. The main re-sults are as follows: (i) Our theory reproduces the key features of entrainment of the alpha rhythm to periodic stimuli, including entrainment tosubharmonics of the drive, showing extensive agreement with experiment ( Herrmann, 2001 ). Further good agreement is shown by the presence of nonlinearly-generated harmonics of the drive. The comparison in Fig. 1 demonstrates that the model agrees with experiment over wide ranges of drive and response frequencies. (ii) The model predicts additional nonlinear dynamics to be found in future experiments. We predict bistability between different entrained states having the same drive amplitude and drive frequency, which can be tested experimentally by looking for hysteresis when slowly ramping the drive frequency up and down. Spontaneous transitions between bistable states during a long recording session might also contribute to intertrial var-iability, a possibility that could be tested experimentally. (iii) At large drive amplitudes we predict period doubling of phase-locked states leading to quasiperiodic and chaotic dynamics. The 1:1 state bifurcates to a 1: Nregime for minimal drive am-plitude when f D≈Nfα, consistent with experiment ( Herrmann, 2001 ). (iv) We predict nonlinear sum and difference frequency generation in cases where background activity is sharply peaked but not entrained by the drive. This has been observed in studies using two stimulus frequencies simultaneously ( Regan, 1989 ). (v) The model predicts that when a stable limit cycle coexists with the stable ﬁxed point corresponding to the resting state, peri-odic stimuli near the alpha frequency and its harmonics can drive the system into a seizure state. This plausibly explains seizures induced by high amplitude ﬂashing light ( Kim et al., 2009; Parra et al., 2005 ), and is open to further test. These ﬁndings provide new veri ﬁcations of the model, and of neu-ralﬁeld theory more generally, complementary to previous studies. Since the same model has previously successfully described many lin-ear and nonlinear dynamics ( Breakspear et al., 2006; Robinson et al., 1997, 2002, 2004 ), this paper uni ﬁes these phenomena with nonlinear SSVEPs within the same framework using compatible pa-rameters. Moreover, this work opens new interdisciplinary research avenues by predicting a rich variety of nonlinear dynamics that should be experimentally veri ﬁable in human subjects. Quantitative modeling is essential for distinguishing between potential physiolog-ical mechanisms in such studies, and a model that also covers multi-ple other phenomena is essential to enable uni ﬁcation and avoid ad hoc interpretations. It is worth pointing out that the data sets of Herrmann (2001) and Rager and Singer (1998) were acquired under quite different experi-mental conditions, yet our model reproduces the main features of both. Herrmann (2001) recorded scalp EEGs from awake human sub-jects, while Rager and Singer (1998) recorded intracortical multi-unit activity (MUA) and local ﬁeld potentials (LFPs) from anesthetized cats. While EEGs necessarily involve ﬁltering through the scalp, this is predominantly linear ( David et al., 2006; Nunez, 1995; Robinson et al., 2001a ) and so does not change the observed frequencies. In any case, EEGs and cortical LFPs are closely related because the same brain structures (mainly pyramidal cells) are ultimately respon-sible for both ( Nunez, 1995; Steriade et al., 1997 ), and indeed the re-sults of Herrmann (2001) and Rager and Singer (1998) are in good agreement. Two differences are that the cat LFP data do not show ev-idence of the subharmonics and gamma-band resonances seen in the human data; however, preliminary investigations indicate that our model can also reproduce this situation for plausible parameter values, though the exploration of appropriate parameters for the anesthetized cat is beyond the scope of this paper, and any such 50f (Hz) f D(Hz)45 40353025201510 50 10 20 30 40 50 Fig. 5. Spectrum vs. f Dfor parameters with a bistable seizure limit cycle attractor for the parameters used by Breakspear et al. (2006). Dark shading corresponds to high values. 1953 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
study should also account for the fact that the cat “alpha ”resonance is near 25 Hz ( Nunez, 1995; Rager and Singer, 1998 ). Parameter dependences of features such as entrainment regions, coupled with experiment, potentially enable calibration of parame-ters in the real brain, extending the utility of SSVEPs in probing phys-iology. For example, experiments could map out the phase locking zones as a function of drive frequency and amplitude for direct com-parison with Fig. 3 (e). This would be a particularly sharp test of the theory. Another measure that could be mapped experimentally is LLE; indeed Spiegler et al. (2011) recently performed such an explo-ration in their model. Formal model inversion schemes may also en-able parameter inference by ﬁtting directly to EEG data, particularly when coupled to model selection criteria ( David et al., 2006; Moran et al., 2009; Pinotsis et al., 2012 ), or by using state estimation tech-niques ( Valdes-Sosa et al., 2009 ) for real-time parameter estimation; such methods are highly nontrivial for nonlinear time series and an active area of research. We analyzed the driven dynamics in the vicinity of a stable ﬁxed point, whereas Spiegler et al. (2011) analyzed interactions between an external drive and a pre-existing limit cycle. Recent evidence points toward both types of dynamics occurring in the alpha band with erratic switching between low and high amplitude modes ( Freyer et al., 2009, 2011 ). Thus there is the potential for entrainment properties to ﬂuctu-ate on similar time scales, or perhaps for the mode-switching dynamics to themselves be altered by a periodic drive. Such possibilities could be explored experimentally by measuring the time dependence of entrain-ment and other nonlinear measures. Another important avenue for future theoretical work is to study phase relations between the drive and the ongoing and driven dynam-ics, thus complementing the frequency properties studied here. Phase-resetting of ongoing activity is thought to contribute to some ERP phe-nomena ( Klimesch, 1999 ), and phase coupling in the brain is an area of intense research interest ( Canolty et al., 2006; Varela et al., 2001 ). Retention of spatial variation in activity and intracortical connectiv-ity will enable study of spatiotemporal SSVEP dynamics ( Nunez, 1995; Robinson et al., 2008 ) and gamma activity ( Robinson, 2007 ), which is known to interact with ﬂicker ( Herrmann, 2001 ), although we have shown that neither is necessary to reproduce the entrainment proper-ties discussed here. Spatial variation in both the cortex and the stimuli will also enable exploration of the rich array of patterns associatedwith visual hallucinations ( Bressloff et al., 2002 ). In addition, spatially-varying stimuli such as stationary ( Muthukumaraswamy and Singh, 2008 ) and moving ( Swettenham et al., 2009 ) gratings evoke beta and gamma band ( ≳20 Hz) activity, and visual evoked responses to gratings differ in photosensitive subjects at both alpha and lower frequencies (Porciatti et al., 2000 ). These phenomena could all be explored in the model. Here we focused on visual stimulation, but the same model and analysis should be applicable to other periodic sensory stimuli that pass through the thalamus, with appropriate parameter changes to account for the different thalamic relay nuclei and cortical subnet-works activated. For example, cortical responses to auditory stimula-tion in schizophrenic subjects and healthy controls have been shown to differ in their frequency pro ﬁles ( Vierling-Claassen et al., 2008 ), and such differences could be explored and interpreted with the same techniques as used here. Another potential avenue lies in driv-ing cortical populations directly. This would enable modeling of non-sensory stimulation methods such as transcranial alternating current stimulation (t ACS), which has been shown to interact with ongoing cortical activity in a frequency-dependent manner ( Kanai et al., 2008 ). In summary, the model applied here exhibits the main features of observed EEG responses to periodic visual stimuli, reproducing key nonlinear features of entrainment and harmonic generation, and en-abling quantitative analysis in terms of underlying physiology. It does this by using an established approach that immediately relatesthese phenomena to ones in other ﬁelds to which the model has been applied. This uni ﬁed framework permits parameters deter-mined elsewhere to be used here; this is far more parsimonious than introducing an ad hoc approach with free parameters. Additional readily-veri ﬁable nonlinear dynamics are also predicted for strong drives, including period doubling, bistability, and chaos, thereby en-abling the theory to be further tested in future experiments. More-over, photosensitive seizures emerge within the same framework, illustrating the ability of neural ﬁeld models to unify a wide range of phenomena and opening new avenues for future work. Acknowledgments We thank J. W. Kim for helpful comments on the manuscript. The Australian Research Council and the Westmead Millennium Institute supported this work. References Amari, S., 1975. Homogeneous nets of neuron-like elements. Biol. Cybern. 17, 211-220. Baker, T. 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Dynamics of the brain at global and microscopic scales: Neural networks and the EEG. Behav. Brain Sci. 19, 285-320. 1955 J. A. Roberts, P. A. Robinson / Neuro Image 62 (2012) 1947-1955	Quantitative theory of driven nonlinear brain dynamics -- J_A_ Roberts P_A_ Robinson -- NeuroImage 3 62 pages 1947-1955 2012 sep -- Elsevier -- 10_1016_j_neuroimage_2012_05_054 -- dda3e87b2a5b9e0181a8
170mm x 244mm Collerton s03. tex V3-11/21/2014 4:01 P. M. Page 217 Section 3 Models and Theories	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 219 10 Geometric visual hallucinations and the structure of the visual cortex Jack D. Cowan Department of Mathematics, Department of Neurology, Committee on Computational Neuroscience, University of Chicago, Chicago Il 60637... the hallucination is... not a staticprocess but adynamic process, theinstabilityof which reflects an instabilityinitsconditions of origin Klüver(1966)inacommenton Mourgue(1932) 10. 1 Introduction Geometric visual hallucinations are more or less regular patterns seen in the visual field, that move with the eyes. They have been studied scientifically for almost 200 years, beginning with the work of Purkinje on flicker phosphenes in 1819, later reported in Purkinje (1918). However, the first major modern study was carried out by Klüver in the 1920's and summarized in Klüver (1966). Klüver observed first-hand the effects of peyote (otherwise known as mescal) and orga-nized the various images he saw into four classes, which he called form-constants (Figure10. 1). Theseclassescomprised(i)grating,lattice,fretwork,filigree,honey-comb,orchessboard;(ii)cobweb;(iii)tunnel,funnel,alley,cone,orvessel;and(d) spiral. Figures10. 2-10. 4showexamplesoftheseformconstants,whileexamplesof the other classes are shown in figures 10. 3-10. 5 Klüver also found that the colour, brightness, and movement of the imagery were particularly vivid and striking. The Neuroscienceof Visual Hallucinations, First Edition. Edited by Daniel Collerton,Urs Peter Mosimann and Elaine Perry. © 2015John Wiley& Sons, Ltd. Published 2015by John Wiley& Sons, Ltd.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 220 220 CH10 GEOMETRIC VISUAL HALLUCINATIONS Figure 10. 1 Heinrich Klüver. c. 1970. Source: Adapted from “Biographical Memoirs Volume 73,The National Academies Press,Washington DC 1998”. (a) (b) Figure 10. 2 (a) 'Phosphene' produced by deep binocular pressure on the eyeballs. Source: Redrawn from Tyler (1978). (b) Honeycomb hallucination generated by Marihuana. Source: Adapted from Clottes and Lewis-Williams (1998). Klüver was for many years a faculty member at the University of Chicago, retiring in 1963. But he remained active for several years, eventually developing a form of dementiaafewyearsbeforehisdeathin1979. Ibecamea Professoratthe University of Chicago in 1967, but only became interested in visual hallucinations in 1977. Unfortunately, by then it was no longer possible to discuss the topic with Klüver.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 221 10. 1 INTRODUCTION 221 Figure 10. 3 Funnel and spiral hallucinations generated by Lysergic acid diethylamide or LSD. Source: Reproduced withpermissionof Nature Publishing Group. Figure 10. 4 Funnel and Ssiral tunnel hallucinations generated by LSD. Source: Reproduced with permissionof Nature Publishing Group. Figure 10. 5 Cobweb petroglyph. Source: Reproduced with permission of Nature Publishing Group.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 222 222 CH10 GEOMETRIC VISUAL HALLUCINATIONS In this chapter I present the results of some 35 years work with my students and collaborators,onthedevelopmentofamathematicalmodeloftheoriginoftheform constants in theneural networks ofthe visual brain. 10. 1. 1 The nature of the neocortex If one examines the histological section of the neocortex shown in Figure 10. 6, at first sight it appears spatially homogeneous. That is, any one region looks more-or-less the same as anyother. Weformulatethismathematicallyasfollows:let H[x]denotetheregionlocatedat someposition x. Thentheconditionforspatialhomogeneityisexpressed(partially) as H[x]=H[x+L]. Wesaythesectionis translationallyinvariant ortranslationally symmetric on the length-scale L. In similar fashion if we move from one location to another, in any direction, the result is the same. The section is also rotationally invariant orrotationallysymmetric. Wewritethisresultas H[x]=H[R𝜃[x]],where 𝜃is the direction relative to horizontal in neocortical coordinates. Finally we also havereflectionsymmetry H [x]=H[-x]. Thus,onthelength-scale L,theneocortex is invariant under translations, rotations and reflections. Mathematically it is said to have the symmetries of the group 𝔾of rigid body motions in the plane-the Euclidean Group 𝔼(2). Mathematically we represent this by the equation H[x]= H[𝔾[x]]. Figure 10. 6 Histological section of neocortex. Source: Reproduced with permission of John Wiley &Sons.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 223 10. 1 INTRODUCTION 223 But what are the spatial dimensions of the human neocortex? If it were to be extracted from the skull and flattened out it would be about 1m square and about 3mm thick, and contain about 5 ×1010neurons. To a first approximation it is a planewith Euclidean (or Cartesian) coordinates xandy. Thus our picture of the neocortex is very simple, any one local region or patch looks just like any other patch,andtheunderlyingsymmetryofthispictureisthatofthegroupofrigidbody motions in the plane. Usingthisinformationwecancalculatewhatshouldbethenormalmodesofvibra-tion of such a neocortex. The answer is planewaves of the form cos(k⋅x)=cos(kxx+kyy) wherekxandkyarespatialwavenumbersinthe xandydirections. Thecentralthesis ofalltheresultsthatwillbedescribedinthischapteristhattheneocortexproduces spatiallyorganizedpatternsthatarecombinationsoftheseplanewaves,whenitloses the dynamical stability of its (postulated) spatially homogeneous resting state. 10. 1. 2 The retino-cortical map Sowhydoweseetheformconstants?Alargepartoftheansweriscontainedinthe factthatthereexistsa topographicmap fromtheeyestotheprimaryvisualcortexor V1, (and beyond with a coarser topology). Every retinal ganglion cell has an axon that connects topographically to the lateral geniculate nucleus LGN, and thence by optic radiation to layer IV of V1. [Note that topographic means that neighbouring locationsinthevisualfieldstimulateneighbouringlocationsin V1. ]However,there are important differences between the neuroanatomy of the retina and that of V1. First,theretinaisathinlayerofphotoreceptorsandneuronsontheinnersurfaceof theeyeball,andrepresentspointsinthevisualfield(approximately)in polarcoordi-nates(r,𝜃)wheretheradialcoordinate rrepresentsdegreesofvisualanglefromthe location of the fixation point, and the angular coordinate 𝜃represents the elevation oftheobjectfromthehorizon. But V1hasa Cartesiancoordinatesystemx,y. Sothe topographicmaphastobe {1∶1}oftheform (r,𝜃)(x,y). Thesituationismade more complex by the fact that the packing density of retinal ganglion cells is not uniform,butinfoveatedprimatesfallswithan(approximate)inversesquarelawof the form (Drasdo, 1977): 𝜌G(r)=4 𝜋k 𝜋(𝑤0+𝜖r)2(10. 1) where𝑤0=0. 087istheaveragediameteroffovealretinalganglioncells, 𝜖=0. 051 is the rate of increase with rof𝑤G(r), the receptive field width of retinal ganglion cells,andk=7isthenumberofothercell'sreceptivefieldsoverlappingonerecep-tive field. This is to be compared with the packing density of cells in V1, 𝜌Cwhich is uniform.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 224 224 CH10 GEOMETRIC VISUAL HALLUCINATIONS We can now calculate the retino-cortical map by assuming that the number of retinal ganglion cell axons in a single elementary retinal area d AR=𝜌Grdrd𝜃maps ina{1∶1}fashionontothesurfaceofasingleelementaryareaof V1 d Ccontaining 𝜌Cdxdycortical cells, that is, 𝜌Grdrd𝜃=𝜌Cdxdy (10. 2) We solve this equation using equation 10. 1, by letting √𝜌Cdx=√𝜌Gdr;√𝜌Cdy=√𝜌Grd𝜃 whence x=√ 4k 𝜋𝜌C1 𝜖ln[ 1+𝜖 𝑤0r] ;y=√ 4k 𝜋𝜌Cr𝜃 𝑤0+𝜖r(10. 3) We note that near the fovea x√ 4k 𝜋𝜌Cr 𝑤0;y√ 4k 𝜋𝜌Cr𝜃 𝑤0. whereas away from the fovea 𝑤√ 4k 𝜋𝜌C1 𝜖ln[ 𝜖r 𝑤0] ;y√ 4k 𝜋𝜌C𝜃 𝜖. Note also that 𝑤G(r)=𝑤0+𝜖rso that its rate of increase is 𝜖. This was mea-sured both in the retina and in V1 by Hubel and Wiesel (1974b) and is called the retino-corticalmagnificationfactor. Thus, away from the fovea there is an (approximately) logarithmic map from the retinal surface onto the surface of V1, that is, x∼lnr,y∼𝜃. Letz=x+iybe a complex number. This in turn leads to the expression z=lnr+i𝜃ez=elnr+i𝜃=elnr⋅ei𝜃=rei𝜃 Such a complex map is called the complex logarithm and was first used to model the retino-cortical map by Schwartz (1977). Figure 10. 7 shows details of the map. Evidently it expresses the effects of the retino-cortical magnification factor in that, away from the fovea, the width of cortical receptive fields Δ𝑤C=Δx∼1 𝜖Δr r=1 sinceΔr r=Δ𝑤G r=𝜖. Thuscorticalreceptivefieldshave constantwidthsincorticalcoordinates,whichis a requirement for the spatial homogeneity of V1. [See also Cowan (1977). ]	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 225 10. 1 INTRODUCTION 225 π/2 y π/2 ππ/2 3π/23π/2 π/2 π/2 π 3π/2 3π/23π/20 0 Visual field (a)Striate cortexx (b) Striate cortex transfor med (c)y x0π Figure 10. 7 The retino-cortical map. (a) Visual field, (b) the actual cortical map, comprising rightandlefthemispheretransforms,(c)atransformedversionofthecorticalmap. Thetwotrans-formsare realigned sothat both foveal regions correspondto x=0. It should now be clear that the logarithmic map from retina to V1 provides a constant area or number of cortical neurons for the analysis of retinal stimuli independent of their retinal location, that is, V1 is optimized for the analysis of retinal images. Similar properties arefound inall other sensory cortices. 10. 1. 3 Form constants in cortical coordinates We now apply the retino-cortical map to the form constants. Figure 10. 8 shows, for example, the funnel and spiral images of figure 10. 3 mapped into cortical coor-dinates. It will be seen that both images are mapped into noisy stripes in cortical coordinates. These correspond to the plane waves discussed earlier and reflect the symmetries of the Euclidean group 𝔼(2). This analysis and the resulting images in cortical coordinates were first worked out by Ermentrout and Cowan (1979). 10. 1. 4 The Turing mechanism Thestartingpointofthe Ermentrout-Cowanworkwastherecognitionthattheform constants were stripe patterns or combinations of stripe patterns that generated	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 226 226 CH10 GEOMETRIC VISUAL HALLUCINATIONS Figure 10. 8 Funnel and spiral hallucinations in retinal and cortical coordinates. Source: Redrawn from Bressloff etal. (2001). periodic blob-like patterns, and the realization that the Turing mechanism for the development of stripes or spot patterns in animal coat markings (Turing, 1952) had a neural analogue. The Turing mechanism is implemented in terms of two diffusion-coupledchemicalreactions. Thefirstreactionisautocatalyticandinvolves a large molecule with a small diffusion constant; the second is autoinhibitory and involves a small molecule with a large diffusion constant. The equations take the form𝜕 𝜕tu(x,t)=-bu+∇2u+f(u,𝑣) 𝜕 𝜕t𝑣(x,t)=-𝑣+D∇2𝑣+g(u,𝑣) (10. 4) whereuand𝑣are the concentrations of the large and small molecules, fandgare nonlinearfunctionsof uand𝑣,implyingthatthereactionsinvolvebinaryorternary combinationsofthemolecules, bisthedecayconstantofthe ureaction,and Disthe diffusionconstantofthe 𝑣reaction. Thediffusionconstantofthe ureactionissetto one,asisthedecayconstantofthe 𝑣reaction. Turingwasabletoshowbycomputer simulation that a planar system of such coupled reactions could generate periodic	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 227 10. 1 INTRODUCTION 227 patterns of stripes and spots. Ermentrout and I were able to show 27 years later thataplanararraycomprisingexcitatoryandinhibitoryneutralscouldalsogenerate Turing patterns. The starting point of our analysis was a neural field theory introduced by Wilson and Cowan (1973), the equations of which can be written in the form: 𝜕 𝜕tu(x,t)=-bu+∫𝑤EE(x-x′)f[u(x′)]dx′-∫𝑤EI(x-x′)g[𝑣(x′)]dx′ 𝜕 𝜕t𝑣(x,t)=-𝑣+∫𝑤IE(x-x′)f[u(x′)]dx′-∫𝑤II(x-x′)g[𝑣(x′)]dx′(10. 5) where the variables uand𝑣denote, respectively, excitatory and inhibitory neural densities, and the various functions 𝑤EEand so on denote the density and strength of synaptic connections from neighbouring neurons a distance x-x′away. The effects of each pair of excitatory and inhibitory weighting functions produce an effective “Mexican Hat” coupling function which is the neural analogue of Tur-ing's diffusion-coupled autocatalytic and auto inhibitory molecular interactions, as showninfigure10. 9. [Notethataneffectiveinverted Mexican Hatwithshortrange inhibition and longer range excitation will also work. ] Wecaneliminateoneoftheseequationsandrewriteequation10. 5intheabbrevi-ated form 𝜕 𝜕tu(x,t)=-𝛼u+𝜇∫𝑤lat(\|x-x′\|)f[u(x′)]dx′(10. 6) where𝑤latis the effective Mexican Hat weighting function shown in figure 10. 9, and𝜇is the coupling strength parameter. [Note that because 𝑤latdepends on the magnitude of the distance between xandx′it is invariant to rotations, and is thus isotropic. ] Ermentrout and I were able to show that these equations are an exact neural ana-logue of Turing's equations, so they can also generate stripe and spot patterns in +--+w(s) s Figure 10. 9 Theeffective “Mexican Hat” weighting function.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 228 228 CH10 GEOMETRIC VISUAL HALLUCINATIONS a planar array. However, the theory can account for only two of the four Klüver formconstants. Thusitcanaccountverywellforformconstants(c)tunnel,funnel, alley, cone or vessel, and (d) spiral; and partially for form constant (a) grating, and chessboard, but not for the other forms classed as (a) lattice, fretwork, filigree or honeycomb, and not for class (b) cobweb. In other words it works for thickbars or spots, butnotfor thinlineimagery. Itwasnotuntil1993that Ibegantounderstand how to obtain the thin line form constants classed as (a) or (b). 10. 2 A new mathematical formulation of V1 circuitry The Ermentrout-Cowan model essentially treats V1 as a cortical retina. But Hubel and Wiesel's Nobel Prize-winning discovery that most V1 neurons have a definite preferencefor orientated stimuliintheformofmovinggratingsorbars(Hubeland Wiesel, 1959, 1962) showed that V1 is functionally considerably more complex thantheretina. Theirsubsequentdiscoveryoftheregularityandspatialhomogene-ityof V1anditsmodulararchitecture(Hubeland Wiesel,1974a)indicatedahighly organized almost crystalline structure. However, it was not until the techniques of optical imaging were applied to V1 (Blasdel, 1992) that the real structure of V1 became apparent. Figure 10. 10a shows an optical image of a small patch of Macaque V1, and Figure 10. 10b shows another optical image of a smaller patch of Cat V1 combined with the image of a stained inhibitory neuron and its local connections. Similarly Figure 10. 11a shows an optical image of a patch of Owl Monkey V1, in which connections between iso-orientation patches have been stained. These connections are the longer-ranged horizontal patchy connections of V1 (Gilbert, (a) (b) Figure 10. 10 (a) Distribution of orientation preferences in Macaque V1 obtained via optical imaging. Differentorientationpreferencepatchesarepseudo-coloured. (b)Connectionsmadeby aninhibitoryinterneuronin Cat V1. Theinjectionsiteisdenotedbythesmallwhitestar. Thecell body of the inhibitory neuron is shown in white. Source: Reproduced with permission of Nature Publishing Group. ( Seeinsertforcolourrepresentationofthisfigure. )	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 229 10. 2 A NEW MATHEMATICAL FORMULATION OF V1 CIRCUITRY 229 (a) (b) Figure 10. 11 Lateralconnectionsmadeby V1cellsin Owl Monkey(a)and Tree Shrew(b)V1. A radioactive tracer is used to show the locations of all terminating axons from cells in a central injection site, superimposed on an orientation map obtained by optical imaging. Source: Repro-duced with permission of Nature Publishing Group. ( Seeinsertforcolourrepresentationofthis figure. ) 1992). Figure 10. 11b shows a similar arrangement in the Tree Shrew, where the anisotropy of the longer-ranged connections is more striking. We can conclude from this data that local connectivity in V1 is isotropic, and is predominantly inhibitory, whereas lateral connectivity is anisotropic, and from Gilbert's data that it is predominantly excitatory. Thus there is an anisotropy in the inverted Mexican Hat, representing the intrinsic connectivity of V1, so that iso-orientation preference patches connect mainly to other iso-orientation patches with the same preference. [Note that Mitchison and Crick (1982) conjectured that such was the case, based on (then) unpublished data collected by Rockland and Lund (1982, 1983). ] Figure 10. 12 shows a diagram of such a connectivity pattern between Hubel-Wiesel hypercolumns (Hubel and Wiesel, 1974b). Hypercolumn was the term introduced by Hubel and Wiesel to denote a patch of V1 containing two completesetsofiso-orientationpreferencepatchesrunningfrom0∘to180∘,oneset driven by images from the right eye, the other by images from the left eye. It turns out that the diameter of such a hyper column corresponds precisely to the 2-point resolutionlimitoftheeyes,thatis,the visualacuity ateachpointinthevisualfield. This limit increases with r, the radial component of the retinal coordinate system, and translates to the (constant) diameter of any hypercolum in V1. In the human cortex this diameter is approximately 1. 5mm. Thus the direction of the lateral connections between neighbouring hypercolumns is determined by the common orientation preference of the patches they connect. Interestingly,thissameanisotropywasintroducedby Zweckand Williams(2000, 2004)intwopaperson Computer Visionasamethodofpreserving Euclideansym-metry in transforming from a continuous planar sheet to a discrete lattice or array, which is precisely what we do inour analysis of hallucinations.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 230 230 CH10 GEOMETRIC VISUAL HALLUCINATIONS Lateral connections Hypercolu mn Local connections Figure 10. 12 Illustration of the local connections within a hypercolumn and the anisotropic lateral connections between hypercolumns. 10. 2. 1 Some more mathematical details The first point to note is that the lateral connectivity now depends on the orien-tation preference of a given patch. Let 𝜙be the variable representing orientation preference. Such a preference varies from 0∘to 18∘and is said to be 𝜋-periodic. [An example of such a 𝜋-periodic function is cos2 𝜙. ] We therefore need to repre-sentneuralactivityatanylocationin V1,notonlyintermsofthecoordinates xand ybut in terms of orientation preference 𝜙., that is, by the variables u(x,𝜙,t)and 𝑣(x,𝜙,t),respectively,forexcitationandinhibition. Thisraisesaninterestingmath-ematicalquestion:what spacedothesevariablesoccupy?The xcomponentlivesin the Euclideanplane,butwhataboutthevariable 𝜙?Theansweristhatitlivesonthe circle. Mathematically the Euclidean plane comprises all real number pairs xand y. Each by itself lives on the real line ℝ, but the(x,y)pair lives on the real plane ℝ×ℝ=ℝ2. The variable 𝜙lives on the circle 𝕊, so that the various points (x,𝜙) live onthe space ℝ2×𝕊.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 231 10. 2 A NEW MATHEMATICAL FORMULATION OF V1 CIRCUITRY 231 The second point to note is that the normal modes of vibration or eigenfunc-tionsof the plane are cos (x⋅k)as previously noted. What are the eigenfunctions orharmonics of the circle? The answer is that in our case they are the 𝜋-periodic functions cos2 𝜙. It follows that we expect the normal modes of vibration of our extended model of V1 to be of the form cos2𝜙⋅cos(x⋅k) that is, plane waves whose amplitude is modulated by a sinusoidal function. The third point is a little more complicated. The question we need to ask is how is the postulated Euclidean symmetry of V1 affected by the introduction of ori-entation preference? The answer to this question is as follows: we first write the Ermentrout-Cowanequationmodifiedandextendedtodealwithorientationprefer-ence in the form: 𝜕 𝜕tu(x,𝜙,t)=-𝛼u(x,𝜙,t)+𝜇∫𝜋 0∫R2𝑤(x,𝜙\|x′,𝜙′)f[u(x′,𝜙′,t)]dx′d𝜙′ 𝜋(10. 7) where𝑤(x,𝜙\|x′,𝜙′)is the weight of connections between neurons at xtuned to the orientation 𝜙, and neurons at x′tuned to𝜙′. We take the weighting function to decompose as: 𝑤(x,𝜙\|x′,𝜙′)=𝑤loc(𝜙-𝜙′)𝛿(x-x′)+𝛽𝑤lat(x-x′,𝜙′)𝛿(𝜙-𝜙′)(10. 8) where𝑤loc(-𝜙)=𝑤loc(𝜙),and𝛽isthestrengthofthelateralcoupling. Theeffects ofthe Diracdeltafunctions 𝛿(x-x′)and𝛿(𝜙-𝜙′)actinginsidetheintegralsareto localizetheactionof 𝑤loctoxand𝑤latto𝜙. Theweightingfunctionthencontainsa localpartthatis isotropic in𝜙(andinhibitory),andalateralpartthatis anisotropic (andexcitatory). Inaddition,becauselateralconnectionslieonlyalongthedirection of the orientation preference, wecan see that 𝑤lat(x,𝜙)=𝑤(R-𝜙x) (10. 9) where R𝜃is the matrix operation that rotates the vector xby the angle 𝜃. The effect of this rotation isthat we can rewrite 𝑤lat(x-x′,𝜙)as𝑤lat(R-𝜙(x-x′)). The net effect of all this is that we can preserve Euclidean symmetry by redefin-ing the rotation operation to express the above property of the weighting func-tion, that is by defining the rotation operator to be not only R𝜃x, but also a label change of the orientation variable from 𝜙to𝜙+𝜃. This combination of a direc-tionalchangecombinedwithanorientationpreferencechangepreserves Euclidean symmetryinthelatticeversionoftheextendedspace ℝ2×𝕊,andwasreferredtoas a Shift-Twist symmetry by Zweck and Williams (2000). Thus, although I first rec-ognized the underlying shift-twist symmetry of the lateral connections a few years earlier(Cowan,1997),Ididnotthenseethatitwasamechanismforpreservingthe Euclidean symmetry ofthe neocortical plane.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 232 232 CH10 GEOMETRIC VISUAL HALLUCINATIONS 10. 2. 2 A little group theory We can now go a little more deeply into the consequences of recognizing such a symmetryin V1. Wenotethatallthesymmetry-preservingoperationswehaveintro-duced have the effect that equation 10. 7 is still left invariant under the modified action of the Euclidean group 𝔼(2). How can we use this property? The answer is that we can use the fact that every symmetry group has subgroups, each of which has some of the symmetries of the group, but not all of them. For example, the group of rotations and reflections of a circle, referred to as 𝕆(2)is a continuous group,inthatanyrotation,howeversmall,isagroupelement. Butonecaninscribe squares,rectangles,rhombuses,trianglesandhexagonsinsideacircle. Eachofthese objects has a finite set of symmetries, some rotations and some reflections which preserveitsappearance. Eachsetdefinesafinite subgroup of𝕆(2)knownasa dihe-dralgroupwithnelements,referredtoas 𝔻n. Thesearethesymmetrygroupsofthe square, rectangle or rhombus, triangle and hexagon. Interestingly, these geometric shapes are the only ones which can be used to tile a plane in repeating or periodic patterns. How can we relate these facts to neocortical neurodynamics? It turns out that there is a piece of mathematics proved by one of our collaborators on this work, Martin Golubitsky,andhiscolleagues,knownasthe Equivariant Branching Lemma (Golubitsky Stewart,and Schaeffer,1988)whichshowsthatwhenadynamicalsys-tem whose resting state has a certain symmetry becomes unstable, new stable or unstablestatesemergewithsymmetriesthataredescribedbyfinitesubgroupsofthe symmetry group of the resting state. In particular, the new states have symmetries oftheaxialsubgroupsofsuchasymmetrygroup. [Anaxialsubgroupisasubgroup that leaves only one unique planform invariant. ] We are able to use this lemma and thedemonstrationthatourmathematicalmodelof V1dynamicshas Euclideansym-metry to predict the planforms of the new stable patterns which should form when the homogeneous resting state of V1 is destabilized by the action, for example, of hallucinogens. 10. 3 Conditions for the loss of stability of the homogeneous state Before calculating such planforms we first analyze the conditions under which the stable homogeneous solution of equation 10. 7 loses its stability. We do this by lin-earizingtheequationaboutthehomogeneousstationarysolution u(x,𝜙,t)=0[The complete details can be found in Bressloff et al. (2001). ] Essentially we assume thesolutiontobeintheform u(x,𝜙,t)=e𝜆tu(x,𝜙),andsubstitutethissolutioninto equation 10. 7. The result is the eigenvalue equation 𝜆u(x,𝜙)=-𝛼u(x,𝜙)+f′𝜇[ ∫𝜋 0𝑤lat(𝜙-𝜙′)u(x,𝜙′)d𝜙′ 𝜋	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 233 10. 3 CONDITIONS FOR THE LOSS OF STABILITY OF THE HOMOGENEOUS STATE 233 +𝛽∫R2𝑤lat(x-x′,𝜙)u(x′,𝜙)dx′] (10. 10) Because of translation symmetry the solutions of equation 10. 10 can be written in the form u(x,𝜙)=𝑣(𝜙-𝜑)eik⋅x+𝑣(𝜙-𝜑)e-ik⋅x(10. 11) where k=q(cos𝜑,sin𝜑)and 𝜆𝑣(𝜙)=-𝛼𝑣(𝜙)+f′𝜇[ ∫𝜋 0𝑤loc(𝜙-𝜙′)𝑣(𝜙′)d𝜙′ 𝜋+𝛽̃𝑤lat(k,𝜙+𝜑)𝑣(𝜙)] (10. 12) and̃𝑤(k,𝜙)isthe Fouriertransformof 𝑤lat(x,𝜙). Themainresultthatcanbederived from these formulae is that solutions take the form 𝑣(𝜙)⋅cos(k⋅x) in which (a) 𝑣(𝜙)=cos2𝜙,( b )𝑣(𝜙)=sin2𝜙,( c )𝑣(𝜙)=1. The first solution cos2𝜙⋅cos(k⋅x)generatesevencontouredplanforms,becausecos2 𝜙=cos(-2𝜙) is even, the second sin2 𝜙⋅cos(k⋅x)generatesoddcontoured planforms, because sin2𝜙=-sin(-2𝜙)is odd, and the third generates non-contoured planforms cos(k⋅x). Figure 10. 13 shows the dispersion curves (the eigenvalue 𝜆plotted as a function of the wave number q), derived from equation 10. 12. It will be seen that theinstabilityfirstoccursatthelowestvalueof 𝜆,forevenplanforms. [Thisoccurs in case there is some scatterin the lateral connectivity 𝑤lat(x,𝜙). ] 0. 98 odd even qcλ 0. 96 0. 94 0. 92 0. 90 12345q Figure 10. 13 Dispersion curves 𝜆(q)obtained from equation 10. 12 giving the conditions for marginal stability. The critical wavenumber at which spontaneous pattern formation occurs is q=qc.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 234 234 CH10 GEOMETRIC VISUAL HALLUCINATIONS 10. 3. 1 Doubly periodic planforms Howeverthereisstillaproblemthatneedstobeaddressed. Thesymmetrywehave introducediscontinuous. Thisimpliesthatthereexistinfinitelymanysolutions. But the neocortex is not really a continuous plane. In fact, the way we have interpreted the data provided by optical imaging and cell staining suggests that the neocortex, and particularly the visual cortex, is approximately a doubly periodic array whose symmetriesarethoseofthesymmetrygroups 𝔻n,withn=2,3,4and6,androtation angles that are multiples of 𝜋∕n. It follows that the relevant planforms are doubly periodic. Figures 10. 14 and 10. 15 show some doubly periodic even planforms. 10. 3. 2 What do such planforms look like in the visual ﬁeld? We now turn to the question of what such platforms look like in visual field coordinates (r,𝜃). In order to compute their appearance we need to apply an inverse cortico-retinal transformation. It should be clear from the derivation of the retino-corticalmapthatthiswillinvolvetheorientationpreferencevariable 𝜙. Thus the retino-cortical map needs to be extended to deal with such a preference, that is, it needs to be extended to cover the mapping of local contours in the visual field. Let𝜙Rbe the orientation of such a contour in the visual field, and 𝜙its image in V1. Wenowask:Whatisthemap 𝜙R𝜙thatmustbeaddedtotheretino-cortical mapz Rzdescribedearlier?Notefirstthatalinein V1ofconstantslopegivenby tan𝜙is alevelcurve of the equation f(x,y)=ycos𝜙-xsin𝜙 (a) (b) Figure 10. 14 (a) Non-contoured even axial planform on the square lattice. (b) Non-contoured even axial planform on thehexagonal lattice.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 235 10. 3 CONDITIONS FOR THE LOSS OF STABILITY OF THE HOMOGENEOUS STATE 235 (a) (b) Figure 10. 15 (a)Contouredevenaxialplanformonthesquarelattice. (b)Contouredevenaxial planform onthehexagonal lattice. [Whenf(x,y)=0,y∕x=tan𝜙. ] Such a line has a constant tangent vector, whose magnitude is 𝑣=cos𝜙𝜕 𝜕x+sin𝜙𝜕 𝜕y The image of such a line in the visual field is obtained by changing to retinal coor-dinates via the inverse of the complex logarithmic map-the complex exponential so that f(x,y)𝜃Rcos𝜙-logr Rsin𝜙 thelevelcurvesofwhichare logarithmicspirals oftheformr R(𝜃R)∝exp(cot(𝜙)𝜃R). Itcanbeshownthatthetangentvectorofsuchacurvehasamagnitudeoftheform ̃𝑣=r Rcos(𝜙+𝜃R)𝜕 𝜕x R+r Rsin(𝜙+𝜃R)𝜕 𝜕y R Sotheretinalvectorfieldinducedbytheconstantvectorfieldin V1twistswiththe retinal angle 𝜃Rand shifts (stretches) with the retinal radius r R. Thus it performs a twist-shift of the cortical vector field. A direct consequence of this is that if 𝜙Ris the orientation of a line in the visual field, then 𝜙=𝜙R-𝜃R (10. 13) that is, local orientation in V1 is relative to the angular coordinate of position in thevisualfield. Consequentlythelocalorientationsofcircles,rays,andlogarithmic	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 236 236 CH10 GEOMETRIC VISUAL HALLUCINATIONS π/2π/2 3π/2 π/2 3π/2π/2-π/2π/2 x x-π/2 Single map Double map Striate cortex Visual fieldyy 3π/2 Figure 10. 16 Actionoftheretino-corticalpoint(single)andvector-field(double)Mapsonlog-arithmic spiralcontours inthevisual field. (a) (b) Figure 10. 17 (a)Non-contouredrollpatternonthesquarelattice. (b)Non-contouredhexagonal pattern onthehexagonal lattice. spirals in the visual field, measured in such relative terms, all lie along the cortical images of such forms, as shown in figure 10. 16 Given such a double map we can now compute the appearance in the visual field of the doubly periodic even V1 planforms shown in figures 10. 14 and 10. 15. The results are shown in figures 10. 17and 10. 18.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 237 10. 3 CONDITIONS FOR THE LOSS OF STABILITY OF THE HOMOGENEOUS STATE 237 (a) (b) Figure 10. 18 (a) Contoured square pattern on the square lattice. (b) Contoured hexagonal pat-ternon thehexagonal lattice. [Therearesomeotherdetailsthathavetobeworkedouttoobtaintheseplanforms withthecorrectscalings. See Bressloff etal. (2001)forsuchdetails. ]Thedetailslead to about 30-40 repetitions of the roll pattern around a circumference of the visual fieldcorrespondingtolength-scalesofabout2. 4-3. 2mm,sothecriticalwavelength 2𝜋∕qcat which patterns form is in this range. [This length-scale is consistent with a lattice spacing Lof some 0. 4-0. 5mm. ] The striking feature of these images is that each is an example of one of the four classes of form constants used by Klüver to classify hallucinatory images. Thus the above model has provided an answer as to why there are four, and only four, classes of form constants, that is, they correspond to only those V1 planforms with thesymmetries oftheaxialsubgroups 𝔻nofthe Euclidean group 𝔼(2)intheplane. 10. 3. 3 Some comments There are several aspects of this work that require further comment. 1. The analysis indicates that under certain conditions the planforms are either contoured or else non-contoured, depending on the strength of inhibition between neighbouring isoorientation patches. Ifsuchinhibitionisweak,individualhypercolumnsdonotexhibitanytendency to amplify any particular orientation. In normal circumstances such a prefer-ence would have to be supplied by inputs from the LGN. In this case, V1 can besaidtooperateinthe Hubel-Wieselmode. Ifthehorizontalinteractionsare stilleffective,thenplanewavesofcorticalactivitycanemerge,withnolabelfor	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 238 238 CH10 GEOMETRIC VISUAL HALLUCINATIONS orientation preference. The resulting planforms are called noncontoured, and correspond to a subset of the Klüver form constants: tunnels and funnels, and spirals. Conversely, if there is strong inhibition between neighbouring isoori-entation patches, even weakly biased inputs to a hypercolumn can trigger a sharplytunedresponsesuchthat,underthecombinedactionofmanyinteract-inghypercolumns,planewaveslabelledfororientationpreferencecanemerge. The resulting planforms correspond to contoured patterns and to the remain-ingformconstantsdescribedby Klüver-honeycombsandcheckerboards,and cobwebs. Interestingly, all but the square planforms are stable. But there do exist hallucinatory images that correspond to square planforms. It is possible that these are just transitional forms. 2. Another conclusion to be drawn from this analysis is that the circuits in V1 whicharenormallyinvolvedinthedetectionoforientededgesandtheforma-tion and processing of contours, are also responsible for the generation of the hallucinatory form constants. Thus, we introduced earlier a V1 model circuit in which the lateral connectivity is anisotropic and excitatory. However 20% of these connections end on inhibitory interneurons. So the overall interaction of the lateral connections between hyper columns can become inhibitory, par-ticularlyathighlevelsofactivity. Themathematicalconsequenceofthisisthe selection of odd planforms. But these do not form continuous contours. [This isconsistentwiththepossibilitythatsuchconnectionsareinvolvedintheseg-mentationofvisualimages(Li,1999). ]Inordertoselectevenplanforms,which arecontour-formingandcorrespondtoseenformconstants,itprovedsufficient toallowfordeviationsawayfromthevisuotopicaxisbyatleast45∘inthepat-tern of lateral connections between isoorientation patches. These results are consistentwithobservationsthatsuggestthattherearetwocircuitsin V1,one dealingwithcontrastedges,inwhichtherelevantlateralconnectionshavethe anisotropy found by Sincich and Blasdel (2001) and Bosking et al. (1997), and another that might be involved with the processing of textures, surfaces and colour contrast, and which has a much more isotropic lateral connectivity (Livingstone and Hubel, 1984). One can interpret the less anisotropic pattern needed to generate even planforms as a composite ofthe two circuits. 10. 4 Extensions of the model These additional comments suggest that it might be interesting to try to formulate a model that can deal with hallucinations involving additional features reported in hallucinatory images, for example, those mentioned above involving textures, sur-faces, and colour contrast, which suggest the need to involve spatial frequency or barwidthtuning,binoculardisparitytuningleadingtodepthperception,andfinally tuning for colour contrast, andfor directional motion.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 239 10. 4 EXTENSIONS OF THE MODEL 239 Figure 10. 19 Relationshipbetweenspatialfrequencyandorientationmaps. Grayregionsdenote low spatial frequency. Note that iso-orientation contours tend to cross the border of spatial fre-quency domains at right angles, and that the pinwheel centers or singularities of orientation preference are often located at the centers of either low or high spatial frequency domains (not highlighted). Source: Reproduced with permission of Nature Publishing Group. ( See insert for colourrepresentationofthisfigure. ) 10. 4. 1 Spatial frequency preferences A first step in this direction is contained in Bressloff and Cowan (2003b). This is basedonapaperby Hübener etal. (1997)containingtheresultsofanopticalimag-ing study of the Cat V1, in which spatial frequency preference patches are plotted relative to the contours of orientation preference patches. Figure 10. 19 reproduces one of their illustrations. The observation that there may be an orthogonal relation-ship between orientation and spatial frequency is interesting but not unexpected, giventhatorientationandspatialfrequencyare,infact,the(orthogonal)components of two-dimensional spatial frequency. This suggested to us the need to include the spatialfrequencyorwidthofastripeorbar,inadditiontoitsorientationinthevisual field, asanother featuretobeincorporated intothe functional geometryof V1. The formulation we introduced is shown in figure 10. 20. Itwillbeseenthatinsteadofeigenfunctionscos2 𝜙whichliveonacircle,wenow need eigenfunctions which also live on 𝜃, the polar angle specifying the latitude of a point(𝜃,𝜙)of a sphere, where 𝜙is the azimuthal angle specifying the longitude. We therefore choose 𝜃=log(p∕pmin) log(pmax∕pmin)(10. 14)	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 240 240 CH10 GEOMETRIC VISUAL HALLUCINATIONS Orientation ϕ pmaxpmin Spatial frequency p Figure 10. 20 Sphericalnetworktopology. Orientationandspatialfrequencylabelsaredenoted by(𝜙,p)with0 ≤𝜙<𝜋andpmin≤p≤pmax. so that𝜃varies linearly with log p, [This is consistent with observations which show that log pvaries linearly with cortical separation (Issa et al. 2000)], and the eigenfunctions we need now live on the sphere 𝕊2, instead of on the circle 𝕊. Thus the space our extended model of V1 lives in is now ℝ2×𝕊2, and the symmetry group𝕆(2)now generalizes to 𝕊𝕆(3)the group of rotations of the sphere, and the eigenfunctionsorharmonicsarenownolongersimplesinusoids,butinsteadarethe sphericalharmonics Ym n(𝜃,𝜙)∝Pm n(cos𝜃)e2im𝜙 where Pm n(cos𝜃)are associated Legendre polynomials. Thefirstfewoftheseare P0 0(cos𝜃)=1,P1 0(cos𝜃)=cos𝜃,P1 1(cos𝜃)=-sin𝜃,so thatsphericalharmonicsaremadeupofsumsofproductsofbothsinesandcosines of both the angles 𝜃and𝜙. It follows that whereas two points on a circle have an angular separation 𝛼such that cos𝛼=cos(𝜙-𝜙′), two points on asphere havean angular separation such that cos𝛼=cos𝜃cos𝜃′+sin𝜃sin𝜃cos2(𝜙-𝜙′). These angular separations are invariant under the actions, respectively, of the rota-tion groups 𝕆(2)and𝕊𝕆(3).	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 241 10. 4 EXTENSIONS OF THE MODEL 241 An important consequence of these properties is that if we choose weighting functions that are invariant under such group actions, then, as we have described earlier, the connectivity pattern shows the shift-twist property for orientation pref-erence𝜙, but now in the case of pairs of angles (𝜃,𝜙), the spherical representation hassingularities at the two poles of the sphere at which the orientation preferences disappear. We can expect contoured patterns to be produced only in regions well away from the poles, where the horizontal connectivity pattern is isotropic. This is consistentwiththeobservationsof Livingstoneand Hubel(1984). Wecanconclude fromthisanalysisthatourextensionoftheearliertheorywillagaingenerateallthe form constants, but the mechanism is not the same as in the earlier study. In this modelitistheanatomyandnotthedynamicswhichgeneratesthefullrepertoireof all the imagery. 10. 4. 2 Pinning There is however a problem that requires further work. The V1 planforms we have shownarethosegeneratedbytheequationsofthemodelswehaveintroduced. How-ever when we attempt to simulate the generation of these patterns by computer simulation,wefindthattheplanformsarequiteirregular. Onewaytosolvethisprob-lemistoincorporatemorefullyintothe V1modelitscrystallinestructure(Hubeland Wiesel, 1974a). A first version of this was published by Bressloff (2002), and later morefullydevelopedin Bressloff(2003),and Bressloffand Cowan(2003a). Essen-tially,thebasicideaofthesepaperswastousethephysicist'sapproachtoanalyzethe properties of crystalline structures using Blochwaves. This methodology was later usedby Bakerand Cowan(2009)tosolvetheaboveirregularityproblem. Consider, forexample,theirregularstripesshowninfigure10. 21,producedbysimulatingthe Ermentrout-Cowanequations. Howcanwestraightenoutsuchanirregularpattern? One answer is that the irregular pattern is pinnedto a planar lattice generated by thehorizontalpatchyconnectionsthatcoupleeachvertexofthelattice(i. e. ahyper-column) to its neighbors [See figure 10. 12. ] The mechanism of pinning is the spa-tial analogue of the mechanism of synchronization of coupled nonlinear oscillators (Pikovsky,2003). Itcanbeexplainedinsimpletermsusingalittle Fourieranalysis. Consider figure 10. 22 showing (a) the connectivity of the lattice and of the Mexi-can Hatweightingfunctionofthe Turingmechanismusedinthe Ermentrout-Cowan model, and their associated Fourier spectra. Itwillbeseenthatthe Mexicanhatfitsintothelatticeina commensurate fashion, inthatthepeaksandvalleysofthetwodistributionsarealignedinspatial-frequency space. This is the prerequisite for pinning. The effects of such a pinning are shown in Figure 10. 21b. Of course V1 is not, in fact, a perfect crystal, but as Figure 10. 10 shows, it is quite irregular so that the stripes that form as a consequence of pinning should still be somewhat irregular, as is indicated in Figure 10. 8.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 242 242 CH10 GEOMETRIC VISUAL HALLUCINATIONS (a) (b) Figure 10. 21 (a) Irregular stripes generated by simulating the Ermentrout-Cowan equations with only a local Mexican Hat connectivity, and no long-range connections. (b) Effects of cou-plingthisconnectivitypatterntoaplanarlatticeofhorizontalconnectionsbetweenhypercolumns. Lattice sites,indicated bythedots,represent iso-orientationpatches of oneunique preference. 2 1 0-1-2-3-2-1 0 1 2 3 Wavenu mber k (in units of 2 π/d0) (b)-3 -2-1 0 1 2 3 Cortical position x (in units of d0). 15. 1. 05 0-. 05-. 1 (a) Figure 10. 22 (a)Aone-dimensionallatticewithconnectionsthatweakenwithdistancebetween latticepoints,togetherwiththeprofileofaone-dimensional Mexican Hatconnectivityusedinthe Ermentrout-Cowan model. (b) The Fourier spectra of the profiles shown in (a). The dark grey curves represent thesuperpositionof the twoweighting functions. 10. 4. 3 Intrinsic ﬂuctuations We now consider the question of what triggers the hallucinations. The calculations we have carried out on the stability of the spatially homogeneous state of V1 indi-cate that when such a state loses stability to small fluctuations, it is usually the result of changes in the coupling parameters of the underlying circuitry, or by the	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 243 10. 4 EXTENSIONS OF THE MODEL 243 effects of external visual stimuli, or by chemical stimulation. Here we consider anotherpossibility:thatstabilitychangesarecausedbytheeffectsofintrinsicnoise in V1 circuits. But to consider such a possibility required extending the underlying mathematical model of large-scale neocortical activity, that is, the Wilson-Cowan equations (Wilson and Cowan, 1972, 1973) to deal with such effects. This was accomplished in a series of papers which were summarized in Buice and Cowan (2009). The resulting Stochastic Wilson-Cowan equations were then used to analyze the effects of intrinsic fluctuations on the triggering of hallucinations (Butler et al. 2012). Theresultsareveryinformative. Itturnsoutthatsuchfluctuationscanindeed trigger hallucinations. Indeed, in a randomly organized network with as many long-range inhibitory connections as long-range excitatory ones, hallucinations wouldbeubiquitous. Howeverthecircuitryof V1is,aswehaveindicated,farfrom random. In fact it is highly organized with only sparse direct long-range inhibitory connections, compared with the excitatory long-range connections (Stepanyants etal. 2009). Sucharestrictionofconnectivityincreasesthestabilityofthehomoge-neous state so that spontaneous pattern formation in the form of the hallucinations occurs, more or less, only under the conditions given above, which we refer to as mean-field pattern formation. Such conditions correspond exactlyto those reported by many observers viahallucinogens, or sensory deprivation and so on. Thus the anatomy we introduced to trigger the form constants is precisely that required to stabilize V1 against the effects of intrinsic fluctuations. Interestingly, this same anatomy was used by Kaschube etal. (2010) in a study of the development of the iso-orientation patches in V1 shown in figure 10. 10. It turns out that the horizontal connections between such patches, shown in figures 10. 11, play a key role, not only in generating the form constants, but also to stabilizethe development of the functional architecture of V1. 10. 4. 4 Flicker phosphenes Sofarwehaveconsideredthetriggeringofformconstantstobetheresultofchanges in thecontrolparameters caused by, for example, ingesting psychedelic substances or by the after-effects of anesthetics such as ketamine. However, the early work by Purkinje (1918) on flicker phosphenes indicated that flickering external stim-uli can trigger some of the form constants. Purkinje's 1819 observations were the subject of a recent paper by Rule etal. (2011), in which they added the flickering external stimulus h(t)∝Θ[ sin2𝜋t T-𝜏] (10. 15) whereΘis the Heaviside step function, and 𝜏is a constant, to a form of equations 10. 5, equivalent to a stimulus-driven version of equations 10. 6, the Ermentrout-Cowan equations.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 244 244 CH10 GEOMETRIC VISUAL HALLUCINATIONS (a) (b) Figure 10. 23 (a) Non-contoured roll pattern. Flicker frequency 18Hz. (b) Non-contoured hexagonal pattern. Flicker frequency 8Hz. Source: Redrawnfrom Rule etal. (2011). The reader will recall that these equations can only generate non-contoured form constants. However, the results fit Purkinje's observations very well. Figure 10. 23 shows the main results (in V1 coordinates). It should be noted that low frequency stimuli (10Hz or less) generate Hexagonal planforms,whilehigherfrequencystimuligenerateperiodicstripeplanforms. There are at least two mechanisms involved in such results. The first relates to a property observedinsimulationsby Wilsonand Cowan(1973),thatof frequencydemultipli-cation, or equivalently, period doubling. This phenomenon occurs in periodically driven nonlinear equations like those introduced by Wilson and Cowan. It follows that solutions of such equations follow a low frequency flicker in a {1∶1}fash-ion,butahigherfrequencyin {2∶1}fashion,andsoon. Thesecondmechanismis more subtle. It turns out that the pattern forming mechanism that generates stripes or hexagons is controlled by the way in which the homogeneous V1 resting state losesitsstability. Thedetailsofallthisareoutsidethescopeofthisreview,butwere worked out for the Wilson-Cowan equations in Ermentrout and Cowan (1980), and elaborated in Ermentrout (1991). Ruleet al. also note that the flicker phosphenes are observed in vivid colours, so that the figures in their paper are displayed as blue stripes or blobs on yellow, orangeorgreenbackgrounds. Infactmanyoftheformconstantswehaveshownin this chapter are also seen by subjects as vividly coloured. It remains a problem to formulate and developa theory of coloured form constants. 10. 4. 5 Fortiﬁcation patterns Thefinaltopicwewillrevueistheoriginoftheimageryseenbymanypeopleduring the course of a migraine episode. The subject is well documented (Sacks, 1999).	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 245 10. 4 EXTENSIONS OF THE MODEL 245 Figure 10. 24 The development of a scintillating scotoma experienced while reading a book. Thesequencestartswiththeupper-leftpanelandproceedsclockwise. Theentireepisodelastsfor about 20-30min. Source: Reproduced withpermissionof J& A Churchill. Figure 10. 24 shows the development of a scintillating scotoma while the subject was reading a book. From this and similar sequences of images, it was possible to estimatethepropagationvelocityin V1asabout2-3mm/min. Myowncalculations onsimilardatakindlysenttomeby G. Baumgartnerand B. Hassenstein,someyears ago,provideanestimateof1. 5mm/min. Toobtainsuchestimatesisstraightforward, one simply maps the imagery, which is in retinal or visual field coordinates (r,𝜃) into V1 coordinates (x,y)using equation 10. 3, the retino-cortical map. It turns out that the forms obtained from such a transformation are propagating circular wave patterns (Grüsser, 1995). Therehavebeenanumberofattemptstomodelthespreadofsuchpatterns,essen-tiallybypropagatingatravellingwaveofspreadingdepressionoveramodelof V1, basedlargelyonanearlypaperby Tuckwelland Miura(1978). Severalpapersalong theselineshavesubsequentlybeenpublished,forexample,Reggiaand Montgomery (1996) and Dahlem and Müller (2004). The paper by Reggia and Montgomery (1996) tries to take into account V1 circuitry but lacks any representation of orien-tation preference and therefore fails to generate the appropriate local patterns seen by observers. It remains to develop a model including orientation preference in V1 circuitry that iscapable of generating the patterns shown in Figures 10. 24. 10. 4. 6 Complex hallucinatory images There are many more complex visual images that do not fit easily into the theoretical framework we have presented here. Figure 10. 25 shows two examples. Figure10. 25ashowsanimagegeneratedbybinocularpressureontheeyeballs,and	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 246 246 CH10 GEOMETRIC VISUAL HALLUCINATIONS (a) (b) Figure 10. 25 (a) Binocular pressure phosphene. Source: Redrawn from Tyler (1978). (b) Lattice tunnel hallucination. Source: Reproduced from Siegel (1977), with permission from Alan D. Iselin. is a more complex form of the imagery. Figure 10. 25b shows an artistic rendering of many reports describing tunnelimagery. It shows contours varying in thickness, colour, and even a suggestion of depth(although this may be an optical illusion). It is evident that such complex imagery requires a deeper and more comprehensive theory than that presented here. In particular, it requires a theory that has a random element, and covers the remaining visual features, that is, preferences for colour, depth and motion. [A paper on such a theory is in preparation. ] There is another source of data concerning more complex visual hallucinations, and that source is cave art. Figure 10. 26 shows a painting from the Peche-Merle cave, and Figure 10. 27 shows one from the older Chauvet cave. It will be seen that both paintings contain elements of some of the imagery seen in the Klüver form constants. A study of such paintings led Clottes and Lewis-Williams (1998) to propose that they were the creation of Shamans who painted what they saw on the cave walls deep underground, (perhaps triggered by flickering light from torches). Thus they could be the earliest known examples of the generation of flicker phosphenes in Homo sapiens. But the cave paintings, and other imagery painted or carved on rocksin South Africa,led Lewis-Williamstoproposeaclassificationoftheimagery as hallucinogenic in origin and comprising three stages of increasing complexity (Lewis-Williamsand Dowson,1988). Figure10. 28showsdetailsofthethreestages in the evolution of complexity inthe observed imagery. 10. 4. 7 In which neocortical areas is the imagery located? It seems clear that the Turing mechanism that generates the Klüver form constants is first triggered in V1, but then it moves progressively forward from V1 to higher	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 247 10. 4 EXTENSIONS OF THE MODEL 247 Figure 10. 26 Pech-Merle cave art (c25,000BP). Figure 10. 27 Chauvet Cave Art (c 30,000BP). Source: Reproduced with permission of Kersti Nebelsiek/Wikimedia Commons/Public Domain.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 248 248 CH10 GEOMETRIC VISUAL HALLUCINATIONS Transition Stage two Stage one Stage three Figure 10. 28 Stages in the development of complex hallucinations. Source: Reproduced with permissionof HTO/Wikimedia Commons/Public Domain. visual areas and beyond. Such areas are known to carry out computations on data provided by V1, and so on, generating ever more complex representations of the imagestherein. Thewholeprocessultimatelytravelstoprefrontalcorticalareassuch as area 46, and beyond; and of course there are feedbacks from such areas to lower levels. It is striking that there is an almost direct path through V4 directly to area 46 (Young, 1992) and back. There is some data that supports the proposed trigger location in V1, provided by Ganis etal. (2004) in a functional magnetic resonance imaging study. However it should be noted that the BOLD signal found in V1 and closely related regions is small compared with signals recorded in other regions during the hallucinatory experience. See for example ffytche etal. (1998); ffytche (2008), and Allen etal. (2008). But all the data is consistent with the idea that the initialtriggeringeventislocatedin V1andproceedsfromtheretohigherlocations. 10. 5 Summary and concluding remarks 10. 5. 1 Summary It should now be clear that all the results reviewed in this chapter have a common origin based on the idea that in confronting the immense complexity of neural cir-cuitry,somesimplificationisnecessary. Wethereforestartedwithtwofundamental ideas. First, despite the complexity, the neocortex is approximately homogeneous, thatis,anyonelocalregionlooks,moreorless,thesameasanyother,sothatifthe neocortex were to be removed from the skull and flattened out, it would look like a thinsheetorslabofhomogeneousneuraltissue. Theseobservationshavemathemat-icalconsequences. Theyindicatethatthe spacethecortex'livesin'canbespecified bythe Cartesiancoordinates (x,y,z)ofsuchaslab. Buttheslabdepth(3mm)isvery small compared with the lateral extent (1m) of each side, so that the zcoordinate can be neglected. We can therefore consider the space the cortex lives in to be the 2-dimensionalrealnumberplane ℝ×ℝ=ℝ2. Second,becauseofthehomogeneity, wecanmoveanywhereintheplane,andthecircuitrywillstilllookthesame. Math-ematicallywesaythattheneocorticalplaneisinvarianttorigidbodymotionsofthe	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 249 10. 5 SUMMARY AND CONCLUDING REMARKS 249 plane, and therefore it has the symmetry of the Euclidean Group in the plane, 𝔼(2). It followed from this mathematical formulation that we could immediately predict thatthenormalmodesofvibrationoftheneocortexwillbe planewaves oftheform cos(k⋅x) where kis the wave number or spatial frequency of the waves, and x=(x,y). W e then used the fact that the primary visual cortex, V1, is connected via the LGN directly to the retina, in such a fashion that the retino-cortical map is topographic. Fromsuchamap Ermentroutand Iwereabletocomputetheappearanceoftheplane waves in the visual field, using the inverse cortico-retinal map, and found that they generated the non-contoured form constants described by Klüver. It took another decade or so before Blasdel's pioneering work on imaging the details of the distribution of iso-orientation preference patches in V1 appeared in print. From this and related work combining such images with the staining of indi-vidualneuronalconnectivity,Iwasabletoworkouttheshift-twistsymmetryofthe connections between such patches. This led to the next model in which the orien-tation preference of a patch, 𝜙, was taken to be another coordinate of the space in which V1 lives. Thus the coordinate system became (x,y,𝜙)and the space ℝ2×𝕊, where𝕊isthecircle. Thepredictionfromthiswasthatthenormalmodesofvibration of V1 should now be of the form cos2𝜙⋅cos(k⋅x) that is, plane waves whose amplitude is modulated by the 𝜋-periodic function cos2𝜙. After suitable modification of the retino-cortical map to incorporate its actiononthe 𝜙variable,wewereabletocomputetheappearanceofsuchmodified plane waves in the visual field. We were also able to make use of another piece of mathematics, that is, the equivariant branching lemma worked out by Golubitsky et al., which enabled us to predict that our model of V1's functional architecture couldgenerateexactlyfourcombinationsoftheabovenormalmodes,andthatunder the action of the extended cortico-retinal map, these combinations corresponded exactlytothefour Klüverformconstants. Thusgeometry,symmetry,andnonlinear dynamics led us to an exact theory of the generation of the Klüver form constants, both contoured, andnoncontoured. The remaining papers described in this chapter contain various elaborations and extensions of these results. Thus Bressloff and I extended the theory to incorpo-rateanothervisualfeaturepreferenceencodedin V1,thatof spatialfrequency,first reported by Hübener et al. The main assumption we made is that orientation and spatialfrequencypreferencesareencodedin V1insuchawaythattheycanberep-resentedmathematicallyasthetwoangularcomponentsofthesurfaceofa sphere,𝜙, and𝜃,sothatthespace V1livesinisnow ℝ2×𝕊2. Itfollowsthatthenormalmodes of vibration of V1 should now be plane waves modulated by sphericalharmonics,	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 250 250 CH10 GEOMETRIC VISUAL HALLUCINATIONS the lowest order of which generates the normal mode form cos2𝜙cos𝜃⋅cos(k⋅x) Such a form can be either contoured or noncontoured. Shortly after this work, Bressloff and I introduced a way to model V1 as a crystal-like structure, and Bressloff later applied the techniques of condensed matter physics to such a structure, whence Baker and I showed how such a structure could regularize the rather irregular patterns that the Ermentrout-Cowan mechanism actually generates. More recently, we were able to make use of a new development in the theory of large-scale neocortical activity that incorporates the effects of intrinsic fluctuations in Stochastic Wilson-Cowanequations,toshowthat V1circuitrybasedonthatused in the Ermentrout-Cowan paper, but with short-range inhibition, and long-range excitationbetweenhypercolumns, can suppresstheeffectsofintrinsicfluctuations, so that they do not by themselves trigger hallucinations. Finally, we have also describedinthischapter,workinprogressonamodelforthetriggeringofmigraine auras. 10. 5. 2 Concluding remarks Despite the progress reported above, it is clear that there are many unsolved prob-lems,andmanyunansweredquestionsconcerningvisualhallucinationsthathave,as yet,norealanswers. Thereaderisreferredto Billockand Tsou(2012)forarecent, very comprehensive review of much of the literature on the Klüver form constants. Howeveritdoesseemthatthetheorydescribedinthischapterhasonlyjustscratched the surface of the problems posed by our seeing visual hallucinations. 10. 6 References Allen, P., Larøi, F., Mc Guire, P. K. and Aleman, A. (2008) The hallucinating brain: a review of structural and functional neuroimaging studies of hallucinations. Neuro-scienceand Biobehavioral Reviews,32, 175-191. Baker, T. I. and Cowan, J. D. (2009) Spontaneous pattern formation and pinning in the primaryvisual cortex. Journalof Physiology(Paris),103, 52-68. Billock,V. A. and Tsou,B. H. (2012)Elementaryvisualhallucinationsandtheirrelation-ships toneural pattern-forming mechanisms. Psychological Bulletin,138, 744-774. Blasdel,G. G. (1992)Orientationselectivity,preference,andcontinuityinmonkeystriate cortex. Journalof Neuroscience,12, 3139-3161. Bosking,W. H.,Zhang,Y.,Schofield,B. and Fitzpatrick,D. (1997)Orientationselectiv-ityandthearrangementofhorizontalconnectionsintreeshrewstriatecortex. Journal of Neuroscience,17, 2112-2127.	The Neuroscience of Visual Hallucinations Collerton_The -- Collerton Daniel Mosimann Urs Peter Perry Elaine -- 10_1002_9781118892794 pages -- 10_1002_9781118892794_ch10 -- 6bc5b4c16a4c580883529036a975
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Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 Behavioural Brain Research xxx (2014) xxx-xxx Contents lists available at Science Direct Behavioural Brain Research jou rn al hom epage: www. elsevier. com/locate/bbr Research report Recent advances in the neuropsychopharmacology of serotonergic hallucinogens Adam L. Halberstadt∗Q1 Department of Psychiatry, University of California San Diego, La Jolla, CA, United States h i g h l i g h t s Serotonergic hallucinogens are classiﬁed as phenylalkylamines and indoleamines. The two classes of hallucinogens produce similar subjective effects in humans and show cross-tolerance. Hallucinogen effects are primarily mediated by the serotonin 5-HT2A receptor. Many effects of hallucinogens are mediated in the prefrontal cortex. a r t i c l e i n f o Article history: Received 18 April 2014 Received in revised form 7 July 2014 Accepted 8 July 2014 Available online xxx Keywords:Psychedelic5-HT2A receptor Head twitch Prefrontal cortex Visual effectsa b s t r a c t Serotonergic hallucinogens, such as (+)-lysergic acid diethylamide, psilocybin, and mescaline, are some-what enigmatic substances. Although these drugs are derived from multiple chemical families, they all produce remarkably similar effects in animals and humans, and they show cross-tolerance. This article reviews the evidence demonstrating the serotonin 5-HT 2Areceptor is the primary site of hallucinogen action. The 5-HT 2Areceptor is responsible for mediating the effects of hallucinogens in human subjects, as well as in animal behavioral paradigms such as drug discrimination, head twitch response, prepulse inhibition of startle, exploratory behavior, and interval timing. Many recent clinical trials have yielded important new ﬁndings regarding the psychopharmacology of these substances. Furthermore, the use of modern imaging and electrophysiological techniques is beginning to help unravel how hallucinogens work in the brain. Evidence is also emerging that hallucinogens may possess therapeutic efﬁcacy. © 2014 Published by Elsevier B. V. 1. Introduction Hallucinogenic drugs have been used by humans for thousands of years, but western scientists only became interested in these substances beginning in the late 1800s. These agents produce pro-found changes in consciousness. Because other drug classes can sometimes produce effects that overlap with those of the hallu-cinogens, it has been important to develop a formal deﬁnition for these compounds. This has turned out to be a difﬁcult and con-tentious task. Hallucinogens have been deﬁned as agents that alter thought, perception, and mood without producing memory impair-ment, delirium, or addiction [1,2]. However, this deﬁnition is overly broad because it fails to exclude a wide-range of agents that are generally not classiﬁed as hallucinogens, such as cannabinoids and NMDA antagonists. It is now recognized that hallucinogens produce ∗Correspondence to: Department of Psychiatry, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0804, United States. Tel. : +1 619 471 0525. E-mail addresses: ahalberstadt@ucsd. edu, ahalbers@ucsd. edusimilar discriminative stimulus effects [3] and act as agonists of the serotonin-2A (5-HT 2A) receptor [4]. Therefore, it has been proposed [5] that in addition to having the characteristics listed above, hal-lucinogens should also bind to the 5-HT 2Areceptor and produce full substitution in animals trained to discriminate the prototyp-ical hallucinogen 2,5-dimethoxy-4-methylamphetamine (DOM). For this reason, hallucinogens are often categorized as classical hal-lucinogens or serotonergic hallucinogens. This article will review the pharmacology of hallucinogens, including their mechanism-of-action, their effects in animals and humans, and recent ﬁndings regarding how they interact with speciﬁc brain regions. 2. Pharmacology of hallucinogens 2. 1. Receptor interactions Classical hallucinogens can be divided into two main struc-tural classes: indoleamines and phenylalkylamines [6]. Indoleamines include the tetracyclic ergoline (+)-lysergic acid diethylamide (LSD) http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016 0166-4328/© 2014 Published by Elsevier B. V. 1 2 3 4 5 6 7 8 9101112 13 14 15 1617181920 21 22232425262728 29 30 31323334353637383940414243 44454647484950515253 54 55 56 5758	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 2 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx Fig. 1. Chemical structures of indolealkylamine, phenylalkylamine, and ergoline hallucinogens. and the chemically simpler indolealkylamines, which includes N,N-dimethyltryptamine (DMT), N,N-dipropyltryptamine (DPT), 5-methoxy-DMT (5-Me O-DMT), and psilocybin (4-phosphoryloxy-DMT) and its active O-dephosphorylated metabolite psilocin (4-hydroxy-DMT). DMT is found in several hallucinogenic snuffs used in the Caribbean and in South America. It is also a com-ponent of ayahuasca, an infusion or decoction prepared from DMT-containing plants in combination with species of Banisteriop-sis containing /H9252-carboline alkaloids that act as monoamine oxidase inhibitors [7]. Psilocybin and its metabolite psilocin are the active components of hallucinogenic teonanácatl mushrooms belonging to the genus Psilocybe. The phenylalkylamines can be subdivided into phenethy-lamines, such as mescaline from the peyote cactus (Lophophora williamsii ), 2,5-dimethoxy-4-bromophenethylamine (2C-B), and 2,5-dimethoxy-4-iodophenethylamine (2C-I); and phenylisopropylamines (“amphetamines”), including DOM, 2,5-dimethoxy-4-iodoamphetamine (DOI), and 2,5-dimethoxy-4-bromoamphetamine (DOB). Although N-alkyl substituted phenylalkylamines are usually inactive as hallucinogens, the addition of a N-benzyl group to phenethylamines can dramat-ically increase their activity, and N-benzylphenethylamines are a new class of potent hallucinogenic compounds [8]. Examples of N-benzylphenethylamine hallucinogens include N-(2-methoxybenzyl)-2,5-dimethoxy-4-iodophenethylamine(25I-NBOMe) and N-(2-methoxybenzyl)-2,5-dimethoxy-4-bromophenethylamine (25B-NBOMe). The chemical structures of many of these hallucinogens are illustrated in Fig. 1. Nichols and colleagues have also developed conformationally restricted derivatives of phenylalkylamine hallucinogens: bromo-Dragon FLY (1-(8-bromobenzo[1,2-b;4,5-b]difuran-4-yl)-2-amino-propane; [9]); TCB-2 (4-bromo-3,6-dimethoxybenzocyclobuten-1-yl)methylamine; [10]; and 2S,6S-DMBMPP ((2 S,6S)-2-(2,5-dimethoxy-4-bromobenzyl)-6-(2-methoxyphenyl)piperidine; [11]). Likewise, lysergic acid 2,4-dimethylazetidide was developed as a rigid analog of LSD that shows similar in vivo potency [12]. Fig. 2 shows examples of rigid hallucinogen analogs. Phenylalkylamine hallucinogens are selective for 5-HT 2recep-tors, including 5-HT 2A, 5-HT 2B, and 5-HT 2Csites [13-15]. The indolealkylamines, by contrast, bind non-selectively to 5-HT recep-tors. Certain indolealkylamines, most notably DMT and some of its derivatives, bind to /H92681receptors [16] and the trace amine receptor [17], and are substrates for the 5-HT trans-porter (SERT) [18,19]. However, compared with /H92681and SERT, tryptamines are more potent at 5-HT 1Aand 5-HT 2Areceptors by several orders of magnitude, so the former sites probably do not contribute to the hallucinogenic response. LSD and other ergoline hallucinogens display high afﬁnity for 5-HT receptors, as well as dopaminergic and adrenergic receptors (reviewed by: [6,20] ). 59 60616263646566676869707172737475767778798081828384 858687888990919293949596979899100101102103104105106107108	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 3 Fig. 2. Chemical structures of conformationally restricted hallucinogens. 2. 2. Pharmacology of the 5-HT 2Areceptor The neurotransmitter serotonin (5-hydroxytryptamine, 5-HT, see Fig. 3) has potent contractile effects upon smooth muscle, espe-cially rat uterus and guinea pig ileum. The ﬁrst indication that there are multiple 5-HT receptor subtypes came from studies conducted by Gaddum and Picarelli [21]. They reported that treatment with either dibenzyline or morphine alone could only partially block the effect of 5-HT on guinea pig ileum. However, in tissue exposed to dibenzyline for 30 min, morphine markedly antagonized 5-HT-induced contraction, and dibenzyline acted as a full 5-HT antagonist in tissue previously exposed to morphine. These ﬁndings demon-strated that 5-HT was acting through two different receptor classes (type D and type M) to induce contraction of guinea pig ileum. Soon after the development of radioreceptor techniques to demonstrate receptor binding, this methodology was applied to the investigation of 5-HT receptors. The ﬁrst radioligands utilized were [3H]LSD and [3H]5-HT [22,23]. Both of those radioligands bind to rat brain membranes with high-afﬁnity in a reversible, saturable, and stereoselective manner, suggesting they are inter-acting with speciﬁc recognition sites. After introduction of the dopamine antagonist radioligand [3H]spiperone, it was recog-nized that [3H]spiperone binds to 5-HT receptors distinct from the sites labeled by [3H]5-HT [24]. The sites labeled by [3H]5-HT and [3H]spiperone were designated as 5-HT 1and 5-HT 2receptors, respectively, and it was recognized that [3H]LSD labeled both sites. The D receptor was eventually shown to be equivalent to the 5-HT2receptor, whereas the M receptor is pharmacologically distinct from 5-HT 1sites and was later classiﬁed by Bradley and coworkers [25] as the 5-HT 3receptor. The 5-HT 2receptor class was later reor-ganized to include three subtypes: 5-HT 2A(equivalent to the site known historically as the 5-HT 2receptor or the D receptor), 5-HT 2B Fig. 3. Structure of serotonin. (formerly known as the 5-HT 2Freceptor), and 5-HT 2C(formerly known as the 5-HT 1Creceptor) [26]. The 5-HT 2Areceptor couples to Gq and activates phospholi-pase C/H9252 (PLC/H9252) signaling, resulting in the hydrolysis of membrane phospholipids to inositol triphosphate (IP3) and diacylglycerol, and mobilization of intracellular Ca2+(see Fig. 4). There is evi-dence that 5-HT 2Ais coupled to several non-canonical signaling pathways, including /H9252-arrestin-2, Src (potentially involving Gi/o-associated G/H9252/H9253 subunits), extracellular-regulated kinase (ERK), p38 mitogen-activated protein (MAP) kinase, phospholipase A2 (downstream from ERK 1,2 and p38 MAP kinase), Akt, and phos-pholipase D (dependent on the small G protein ADP-ribosylation factor-1 (ARF1)) [27-30]. However, the signaling pathways respon-sible for mediating the characteristic effects of hallucinogens have not been conclusively identiﬁed. Activation of the canonical Gq-PLC/H9252 signaling pathway is apparently not sufﬁcient to produce hallucinogen-like behavioral effects in animal models [28,31,32]. Multiple signaling pathways may be involved because the behav-ioral response to DOI is partially blunted in Gq knockout mice [33]. Schmid and colleagues have reported that /H9252-arrestin-2 is not required for the behavioral effects of DOI and 5-Me O-DMT [29,34]. There also does not appear to be a direct relationship between phospholipase A2activation and generation of hallucinogen effects [32]. 3. Evidence that serotonergic hallucinogens belong to a unitary class 3. 1. Subjective effects Despite having different chemical structures, phenylalkylamine, tryptamine, and ergoline hallucinogens produce remarkably simi-lar subjective effects [35-42]. It is very difﬁcult for hallucinogen-experienced subjects to distinguish between psilocybin and LSD if those substances are administered in a blinded fashion, with the only apparent difference being the duration of action [41]. Similar ﬁndings have been reported when mescaline, LSD, and psilocy-bin are compared in the same subjects [37-39]. By contrast, the effects of hallucinogens can be distinguished from those of other drug classes. The effects of classical hallucinogens and anticholin-ergic agents are qualitatively distinct [43,44]. Studies using the Addiction Research Center Inventory (ARCI) instrument [45] have conﬁrmed that the effects of LSD are dissimilar from those of (+)-amphetamine [46] and /Delta19-tetrahydrocannabinol [47]. The ARCI can also distinguish between the subjective responses to 20 mg (+)-amphetamine and an ayahuasca preparation containing the equivalent of a 1 mg/kg dose of DMT [48]. Although it does not appear that any studies have directly compared the experiences produced by classical hallucinogens and the /H9260-opioid receptor ago-nist salvinorin A from Salvia divinorum, there is evidence that the phenomenology of salvinorin A is unique [49], and the ARCI is rel-atively insensitive to the effects of salvinorin A [50]. Several recent studies have compared the effects of hallu-cinogens and other drug classes using psychometrically validated instruments. One instrument that has been widely used to assess the subjective response to hallucinogens is the Altered States of Consciousness Questionnaire (APZ), as well as well as APZ vari-ants such as the APZ-OAV and the 5D-ASC. These rating scales are designed to assess altered states of consciousness indepen-dent of their etiology [51,52]. The APZ and APZ-OAV include three core dimensions: Oceanic Boundlessness (OB), Anxious Ego Dissolu-tion (AED) and Visionary Restructuralization (VR). The OB dimension reﬂects a pleasant state of depersonalization and derealization, the AED dimension measures dysphoric effects such as ego disintegra-tion, delusions, loss of self-control, thought disorder, and anxiety,109 110 111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140 141142143144145146147148149150151152153154155156157158159160161162163 164 165 166 167 168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 4 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx Fig. 4. Signaling pathways coupled to the 5-HT 2Areceptor. Abbreviations : AA, arachidonic acid; 2-AG, 2-arachidonoylglycerol; ARF, ADP-ribosylation factor-1; DAG, diacyl-glycerol; DGL, diacylglycerol lipase; ERK1/2, extracellular-regulated kinases 1 and 2; GRB, growth factor receptor-bound protein 2; IP3, inositol triphosphate; p38 MAPK, p38 mitogen-activated protein kinase; MEK1/2, mitogen/extracellular signal-regulated kinases 1 and 2; MKK3/6, MAPK kinases 3and 6; MKK4, MAPK kinase 4; MEKK, MAPK kinase kinase; PA, phosphatidic acid; PC, phosphatidyl choline; PIP 2, phosphatidylinositol 4,5-biphosphate; PKC, protein kinase C; PKN, protein kinase N; PL, phospholipids; PLC/H9252, phospholipase C/H9252; PLD, phospholipase D; SHC, Src homology 2 domain containing transforming factor; SOS, son of sevenless homolog. and the VR dimension involves elementary and complex visual hallucinations and perceptual illusions (see Table 1). Mescaline, psilocybin, and DMT produce profound increases in OB, AED and VR scores [52-56]. Another instrument is the Hallucinogen Rat-ing Scale (HRS), which was speciﬁcally designed to measure the effects of parenteral DMT [57] Double-blind studies have conﬁrmed the APZ and the HRS can distinguish the effects of psilocybin and mescaline from those of (+)-methamphetamine, methylphenidate, and 3,4-methylenedioxyethylamphetamine [53,55,58]. Ayahuasca also elicited signiﬁcantly greater effects than (+)-amphetamine on 4 of 6 subscales of the HRS [48]. A double-blind crossover study comparing DMT and the NMDA antagonist (S)-ketamine found DMT produces effects that more closely resemble the positive symptoms of schizophrenia, whereas the effects of (S)-ketamine are more similar to the negative and catatonic symptoms of schizophrenia [59]. Subjects experienced vivid visual hallucinations after treatment with DMT but not with (S)-ketamine; this difference was reﬂected by scores in the VR dimension of the APZ-OAV, which was more strongly affected by DMT than by (S)-ketamine. Another notable difference between Table 1 Core dimensions of the APZ [52]. Dimension Symptoms assessed Oceanic Boundlessness (OB) Positive derealization Positive depersonalization Altered sense of time Positive mood Mania-like experience Anxious Ego Dissolution (AED) Anxious derealization Thought disorder Delusion Fear of loss of control Visionary Restructuralization (VR) Elementary hallucinations Visual pseudohallucinations Synesthesia Changed meaning of percepts Facilitated recollection Facilitated imaginationketamine and serotonergic hallucinogens is that ketamine does not produce mystical experiences [60], whereas hallucinogens induce these states with some reliability [58,61-64]. Vollenweider and colleagues have conducted a psycho-metric assessment of APZ-OAV data pooled from 43 studies with psilocybin, (S)-ketamine, and the entactogen 3,4-methylenedioxymethamphetamine (MDMA, “Ecstasy”) [65]. Examination of the factorial structure of the APZ-OAV revealed the OB, AED and VR scales are multidimensional, and Vollenweider et al. were able to extract 11 new homogenous APZ-OAV scales that are very effective at differentiating the subjective effects of psilocybin, (S)-ketamine, and MDMA. There are clear differences in the relative magnitude of drug effects on several of the new scales; for example MDMA has strong effects on blissful state, (S)-ketamine produces the largest increase in disembodiment, and complex imagery and elementary imagery are most strongly inﬂuenced by psilocybin Fig. 5 compares the effects of psilocybin and placebo on the new homogeneous APZ-OAV subscales. In summary, even though there are some similarities between the subjective effects of serotonergic hallucinogens, NMDA antago-nists, psychostimulants, and entactogens, the effects produced by the latter three drug classes are clearly distinct from those elicited by classical hallucinogenic drugs. 3. 2. Tolerance and cross-tolerance Tachyphylaxis (tolerance) develops rapidly to the effects of classical hallucinogens. If LSD and DOM are administered repeat-edly at daily intervals tolerance is observed after 1-3 days and there is eventually nearly a complete loss of response [66-69]. Tolerance occurs with a variety of phenylalkylamine, indolealky-lamine, and ergoline hallucinogens, and compounds from these classes exhibit symmetrical cross-tolerance [37,41,42,68,70-72]. Importantly, cross-tolerance does not occur between LSD and (1) (+)-amphetamine [46], (2) the anticholinergic N-methyl-3-piperidyl benzilate [73], or (3) /Delta19-tetrahydrocannabinol [47]. Similar ﬁndings have been reported by parallel studies in laboratory animals [74-79]. The fact that serotonergic hallucinogens produce202 203204205206207208209210211212213214215216217218219220221222 223224225226227228229230231232233234235236237238239240241242243244 245 246 247248249250251252253254255256257	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 5 Fig. 5. Subjective effects of psilocybin as measured by the 5-Dimension Altered States of Consciousness instrument (5D-ASC). The values reported by Grob et al. [56] were re-analyzed using the 11 new homogenous APZ subscales developed by Studerus et al. [65]. Values are the mean (SEM) percentages of the total possible score. The placebo was niacin. similar experiences and induce cross-tolerance indicates that these compounds share a common mechanism of action. 4. Involvement of the 5-HT 2Areceptor in hallucinogen effects 4. 1. Evidence from human studies Multiple, converging lines of evidence point to 5-HT 2Arecep-tor activation as the unitary mechanism responsible for mediating hallucinogenesis. Indoleamine and phenylalkylamine hallucino-gens bind to 5-HT 2sites with moderate to high afﬁnity [80-83]. Although indoleamine hallucinogens show relatively promiscu-ous binding proﬁles, phenylisopropylamine hallucinogens such as DOM and DOB are highly selective for 5-HT 2receptors [13,15] and therefore it is likely that their effects are mediated by a member of the 5-HT 2family. Additionally, there is a very strong correla-tion (r = 0. 90-0. 97) between 5-HT 2Areceptor afﬁnity and human hallucinogenic potency [13,82,84]. Another compelling ﬁnding is that 5-HT 2Areceptor blockade ameliorates most of the effects of psilocybin in human subjects. A series of studies conducted by Franz Vollenweider and colleagues at the University Hospital of Psychiatry in Zürich have shown that the effects of psilocybin (215-260 /H9262g/kg, p. o. ) on the OB, AED, and VR dimensions of the APZ-OAV and 5D-ASC are completely blocked by pretreatment with either the 5-HT 2A/2C antagonist ketanserin or the mixed 5-HT 2A/D2 antagonist risperidone [85-90]. By contrast, pretreatment with the dopamine D2antagonist haloperidol had no effect on psilocybin-induced VR scores and actually intensiﬁed the effect of psilocybin on scores in the AED dimension [85]. Ketanserin also blocks the effects of psilocybin on a variety of neurophysiological measures in humans, including tests of spatial working memory [85], prepulse inhibition of acoustic startle [90], N170 visual-evoked potentials [89], semantic interference in the Stroop test [90], and recogni-tion of emotional facial cues in a go/nogo task [88]. Furthermore, a positron emission tomography (PET) study with the 5-HT 2Aradio-tracer [18F]altanserin has shown that the intensity of the responseto psilocybin is directly correlated with the level of 5-HT 2Aoccupa-tion [91]. 4. 2. Evidence from animal behavioral models Because of regulatory constraints on human studies, animal behavioral models are the primary methodology used to study hallucinogens in vivo. Although it has been difﬁcult to develop appropriate models of hallucinogenic activity because of the vari-ability and complexity of their effects, several animal models have made important contributions to our understanding of hal-lucinogen pharmacology. Importantly, although there are some exceptions, almost all the behavioral effects of hallucinogens stud-ies in laboratory animals are mediated by the 5-HT 2Areceptor. 4. 2. 1. Drug discrimination Laboratory animals can be trained to discriminate hallucino-gens from saline using operant conditioning techniques. Rats are the species most commonly employed, although mice and mon-keys have also been used. Many classical hallucinogens have been used as training drugs, including LSD, mescaline, DOM, DOB, DOI, psilocybin, 5-Me O-DMT, DMT, and DPT [3,92-102]. All of these hallucinogens produce cross-generalization, suggesting that they evoke similar interoceptive stimulus cues. By contrast, drugs from other pharmacological classes do not produce hallucinogen-like stimulus effects [3,101,103]. There is a great deal of evidence that the discriminative stimulus effects of hallucinogens are mediated by the 5-HT 2Areceptor. For example, Glennon and colleagues con-ducted substitution tests with 22 hallucinogens in rats trained to discriminate 1 mg/kg DOM from saline and found that the ED50 values for stimulus generalization are highly correlated (r = 0. 938) with 5-HT 2Abinding afﬁnity [84]. Another study with 18 hallu-cinogens found a strong correlation (r = 0. 90) between ED50values for stimulus generalization to 1 mg/kg DOM and afﬁnity at 5-HT2Areceptors labeled with [3H]DOB [13]. The stimulus effects of hallucinogens can be blocked by the selective 5-HT 2antag-onists ketanserin and pirenperone [4,96,104-106]. Blockade by ketanserin and pirenperone, however, does not eliminate the pos-sibility of 5-HT 2Creceptor involvement because those antagonists are relatively nonselective for 5-HT 2Aversus 5-HT 2Csites. Impor-tantly, M100907, a 5-HT 2Aantagonist with high selectivity versus the 5-HT 2Creceptor, blocks stimulus control in animals trained with DOI [97,107-109], DOM [101,110], R-(-)-DOM [111], LSD [98,112-114], and psilocybin [99]. Conversely, neither the selective 5-HT 2Cantagonist SB 242,084 nor the mixed 5-HT 2C/2B antagonists SB 200,646A and SB 206,553 block stimulus control induced by DOI, LSD, or psilocybin [99,107-109,114]. Furthermore, Fiorella et al. [115] tested eleven 5-HT 2antagonists and found the rank order of potencies for blocking R-(-)-DOM substitution in LSD-trained rats parallels their afﬁnities for 5-HT 2A(r = 0. 95) but not for 5-HT 2C (r = -0. 29). Although most phenalkylamines are relatively nonselective for 5-HT 2Aversus 5-HT 2C, 2S,6S-DMBMPP displays 124-fold selectiv-ity for 5-HT 2Areceptors [11]. Although racemic trans-DMBMPP is less selective, it still shows 98-fold higher afﬁnity for 5-HT2Aover 5-HT 2Creceptors. Importantly, trans-DMBMPP fully substitutes in rats trained to discriminate 0. 08 mg/kg LSD. By contrast, several studies have demonstrated that 5-HT 2Cago-nists fail to mimic the hallucinogen discriminative stimulus. Neither 1-(3-triﬂuoromethylphenyl)piperazine (TFMPP) nor m-chlorophenylpiperazine (m CPP) substitute for DOM, DOI, or LSD [103,116,117]. These ﬁndings demonstrate that 5-HT 2Aactivation is sufﬁcient to produce hallucinogen-like stimulus effects. Further-more, 5-HT 2Cactivation does not play a role in mediating the hallucinogen discriminative stimulus cue. The available data pro-vide strong support for the conclusion that hallucinogens evoke a258 259 260 261 262 263 264265266267268269270271272273274275276277278279280281282283284285286287288289290291292 293 294 295 296297298299300301302303 304 305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344345346347348349350351352353354	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 6 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx Fig. 6. Chemical structure of lisuride. uniform discriminative stimulus cue that is mediated by the 5-HT 2A receptor. Although it is clear that the 5-HT 2Areceptor is primarily responsible for generating hallucinogen-induced stimulus control, interactions with other receptors may contribute to or mod-ify the stimulus effects of hallucinogens. This appears to be especially true for indoleamines, which are much less selec-tive than phenylalkylamines for 5-HT 2Asites. For example, there appears to be a time-dependent dopaminergic component to the LSD discriminative stimulus in rats [118,119]. There is evidence that the 5-HT 1Areceptor also contributes to the discriminative stimulus effects of LSD. 5-HT 1Aagonists such as 8-hydroxy-2-(di-n-propylamino)tetralin (8-OH-DPAT) and ipaspirone produce partial substitution in rats and mice trained with LSD [98,120-122]. The 5-HT 1Aantagonist WAY-100635 does not alter LSD discrimina-tion in rats [114,122,123], but the 5-HT 1Areceptor may make an more prominent contribution to the LSD cue in mice because dis-crimination can be partially blocked by administration of either WAY-100635 or M100907 [98]. However, the ability of R-(-)-DOB to substitute for LSD in mice is completely blocked by M100907 but not by WAY-100635, demonstrating the stimulus element gener-ated by 5-HT 1Ais a non-essential component of the LSD cue and not a shared aspect of hallucinogen pharmacology. Although cer-tain indolealkylamines produce compound stimulus cues involving both 5-HT 1A- and 5-HT 2A-mediated components [100,124,125], 5-HT1Areceptors do not play a role in the interoceptive effects of psilocybin [99] or 5-methoxy-N,N-diisopropyltryptamine [126]. A potential confound associated with drug discrimination stud-ies is the possibility of “false positive” results. False-positives occur where an animal trained to discriminate a hallucinogen general-izes to a drug that is known to be non-hallucinogenic in humans. Lisuride is one example of drug that can produce false-positive results. Lisuride is an isolysergic acid derivative that is struc-turally similar to LSD (see Fig. 6), and acts as an agonist at a variety of serotonergic, dopaminergic, and adrenergic receptors [12,14,127-130]. Despite the fact that lisuride has high afﬁnity for the 5-HT 2Areceptor and acts as an agonist [32,128,131], it is not hallucinogenic in humans [132-135] and has been used clini-cally to treat migraine and Parkinson's disease. Some studies have found that lisuride produces full substitution in rats trained with either LSD, DOI, or DOM [136-139], but in other studies it pro-duced only partial substitution [129,140]. Although clearly some degree of similarity exists between the stimulus cues evoked by lisuride and classical hallucinogens, there are also subtle differ-ences because rats can be trained to discriminate between lisuride and LSD using three-choice (drug-drug-vehicle) discrimination procedures [141]. Discrimination studies where animals are trained to discriminate between LSD and another drug such as pentobar-bital or cocaine also appear to be less sensitive to lisuride-induced false-positive responses [139]. González-Maeso et al. [28] have proposed that the behavioral differences between LSD and lisuride are due to 5-HT 2Afunctional selectivity. They found LSD and lisuride both activate Gq/11signal-ing via the 5-HT 2Areceptor, but only LSD increases the cortical expression of the immediate early genes egr-1 and egr-2 by acti-vating Gi/oand Src [28]. Therefore, they hypothesized that LSD is hallucinogenic because it is capable of activating speciﬁc signaling mechanisms that are not recruited by lisuride. Alternatively, the reason why lisuride fails to recruit Gi/omay have nothing to do with functional selectivity, and could be a consequence of its low intrin-sic efﬁcacy at 5-HT 2A[31,32,131]. Although animals trained with DOM will generalize to lisuride [137,138], the response to DOM is attenuated when it is co-administered with lisuride [142]. The fact that lisuride induces a response when administered alone but act as an antagonist in the presence of a full agonist (DOM) is consistent with the behavior of a partial agonist. 4. 2. 2. Head twitch response Many mammalian species display a paroxysmal rotational shaking of the head in response to mechanical or chemical irri-tation of the pinna. Mice show a similar behavior, known as the head twitch response (HTR), after administration of hallucinogens ([143] ;[144,145] ). Hallucinogens also induce head twitches in rats, but in that species the behavior often involves both the head and the trunk [146,147]. The responses made by rats are sometimes called wet-dog shakes because they resemble the behavior of a dog drying itself after emerging from the water. It is important to rec-ognize that the HTR can occur in response to administration of 5-HT precursors (e. g., l-tryptophan and l-5-hydroxytryptophan) and drugs that increase 5-HT release (e. g., fenﬂuramine and p-chloroamphetamine), and therefore the behavior is not speciﬁc to hallucinogens [148-151]. Nonetheless, the HTR has gained promi-nence as a behavioral proxy in rodents for human hallucinogen effects because the HTR is one of only a few behaviors that can reliably distinguish hallucinogenic and non-hallucinogenic 5-HT 2A agonists [28]. Indeed, even high doses of lisuride fail to induce the HTR in mice [28,152]. It is well-established that phenylisopropylamine and indoleamine hallucinogens induce the HTR (reviewed by: [20]), but the literature is less clear with regard to phenethylamine hallucinogens. Many studies have demonstrated that mescaline produces head twitch behavior in rats and mice [144,146,153]. It has also been reported that the hallucinogen 2,5-dimethoxy-4-n-propylthiophenethylamine (2C-T-7) induces the HTR in mice [154]. Studies in rats, however, have shown 2C-I, 2C-B, and 2,5-dimethoxy-4-methylphenethylamine (2C-D) do not induce the HTR [155]. In contrast to those ﬁndings, we recently reported 2C-I and the N-benzyl derivatives 25I-NBOMe and N-(2,3-methylenedioxybenzyl)-2,5-dimethoxy-4-iodophenethylamine(25I-NBMD) produce dose-dependent increases in HTR behavior in C57BL/6J mice [156]. 25B-NBOMe also induces the HTR in mice [157]. The discrepant ﬁndings with regard to 2C-I and other phenethylamines may reﬂect the fact that mice are more sensitive than rats to the HTR induced by 5-HT 2Apartial agonists. 2C-I has relatively low intrinsic activity at the 5-HT 2Areceptor [155,158], and it may not have sufﬁcient efﬁcacy to provoke head twitches in rats. Nevertheless, we are not aware of any serotonergic hallucinogens that do not produce the HTR in mice. The kinematics of the HTR induced by DOI have been charac-terized in C57BL/6J mice and Sprague-Dawley rats [152]. When mice make a head twitch, the head rapidly twists from side-to-side. Each HTR consists of 5-11 head movements, with the head movements occurring at 78-98 Hz (i. e., each head movement lasts approximately 11 msec). The behavior is similar in rats but in that species the frequency of head movement is lower. One drawback to traditional HTR studies is that they require direct behavioral355 356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405 406407408409410411412413414415416417418419420 421 422423424425426427428429430431432433434435436437438439440441442443444445446447448449450451452453454455456457458459460461462463464465466467468469	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 7 Fig. 7. Effect of pretreatment with the selective 5-HT 2Aantagonist M100907 on the head twitch response induced by 0. 3 mg/kg 25I-NBOMe in C57BL/6J mice. Data are presented as group means ± SEM for 20-min test sessions. **p < 0. 01, signiﬁcant difference from. 25I-NBOMe alone. Data from Ref. [156]. observation that can be extremely time-consuming. However, as we have recently demonstrated, it is possible to detect the behavior with a head-mounted magnet and a magnetometer coil, provid-ing a highly sensitive, semi-automated assessment of the behavior [152,156]. The HTR induced by hallucinogens and other 5-HT agonists is closely linked to 5-HT 2Aactivation. It was proposed in 1982 that the mescaline-induced HTR is mediated by the 5-HT 2Areceptor, based on the fact that the relative potency of 5-HT antagonists to block the behavior is correlated (r = 0. 875) with their 5-HT 2Aafﬁn-ity [159]. Similar ﬁndings were later reported for the HTR induced by DOI [160,161]. Numerous studies have shown M100907 blocks the HTR induced by hallucinogens (Table 2). For example, we found M100907 blocks the HTR induced by the hallucinogen 25I-NBOMe with an ID50= 6. 2 /H9262g/kg (Fig. 7; [156] ). Based on ex vivo binding data it is unlikely M100907 produces any appreciable occupation of 5-HT 2Creceptors at that dose level [162]. Studies have also demon-strated that the highly selective 5-HT 2Aantagonist MDL 11,939 blocks the HTR induced by DOI and TCB-2 in mice [163,164]. Mice lacking the 5-HT 2Areceptor gene do not produce head twitches in response to mescaline, DOI, DOM, LSD, DMT, 5-Me O-DMT, psilocin, or 1-methylpsilocin [28,165,166], although the response can be res-cued by selectively restoring the 5-HT 2Areceptor gene to cortical regions [28]. By contrast, 1 mg/kg DOI produces a signiﬁcant (albeit somewhat blunted) HTR in 5-HT 2Cknockout mice [167]. The fact that DOI can provoke head twitches in 5-HT 2Cknockout mice but not in 5-HT 2Aknockout mice strongly indicates the 5-HT 2Arecep-tor is the member of the 5-HT 2family responsible for mediating the HTR. Similarly, there is a consensus in the literature that the ability of DOI to induce the HTR is not blocked by selective 5-HT 2C antagonists or mixed 5-HT 2C/2B antagonists [160,168-171]. Although it has been conclusively established that the 5-HT2Creceptor is not required for generation of the HTR, there is some evidence that 5-HT 2Csites may play a modulatory role. 5-HT 2agonists that are selective for 5-HT 2Csites, such as (S)-6-chloro-5-ﬂuoro-/H9251-methyl-1 H-indole-1-ethanamine (Ro 60-0175), 6-chloro-2-(1-piperazinyl)pyrazine (MK-212), and m CPP, do not induce the HTR in rats unless administered in combination with the 5-HT 2Cantagonist SB 242,084 [170]. There is also evidence thatthe ability of DOI to induce the HTR is signiﬁcantly attenuated by pretreatment with selective 5-HT 2Cagonists, including Ro 60-0175, CP-809,101, and m CPP [160,171-173]. These ﬁndings indicate 5-HT2Cactivation suppresses expression of the HTR. Likewise, DOI produces a biphasic dose-response curve in NIH Swiss and Swiss-Webster mice, and SB 242084 reportedly shifts the descending arm of the DOI response to the right [171]. Here again there is evi-dence that the 5-HT 2Creceptor can inhibit the HTR. On the other hand, as was noted above, Canal and colleagues have reported that 5-HT 2Cknockout mice show a blunted HTR to 1 mg/kg DOI [167]. Furthermore, in contrast to many other reports, the same investigators found pretreatment with SB 242,084 or SB 206,553 diminished the magnitude of the HTR induced by 1 mg/kg DOI in C57BL/6J and DBA/2J mice [167,173]. It is not clear why the 5-HT 2C receptor attenuates the HTR in certain studies and augments the response in others, but Fantegrossi et al. [171] have argued these differences may be strain dependent. For example, there are strain differences in the editing of 5-HT 2Cm RNA [174,175]. Since 5-HT 2C editing can inﬂuence the downstream coupling of the receptor [176], the nature of the interactions between 5-HT 2Aand 5-HT 2C could potentially vary by mouse strain. 4. 2. 3. Prepulse inhibition of startle Prepulse inhibition (PPI) refers to the phenomenon where a weak prestimulus presented prior to a startling stimulus will atten-uate the startle response; PPI is often used as an operational measure of sensorimotor gating, and reﬂects central mechanisms that ﬁlter out irrelevant or distracting sensory stimuli [177]. Rats treated with DOI [178,179], DOB [180], LSD [181,182], mescaline [183], and 2C-B [184] show reductions in PPI. These effects can be blocked by M100907 and MDL 11,939 [179,181,182,185]. By con-trast, neither SB 242,084 nor the 5-HT 2C/2B antagonist SER-082 are effective. Although one study found haloperidol can block the PPI disruption produced by hallucinogens [178], this was not replicated by subsequent investigations [181,186]. Lisuride also disrupts PPI in rats, but this effect is blocked by the D2/3antagonist raclopride and not by MDL 11,939 [182]. 4. 2. 4. Interval timing Temporal perception can be markedly altered by hallucinogens. Subjects under the inﬂuence of mescaline and LSD often report that their sense of time appears to speed up or slow down, or they may experience a sensation of timelessness [187-191]. Psilocybin also alters performance on laboratory measures of timing [192]. Temporal perception can be assessed in rodents using interval timing paradigms. For example, in the free-operant psychophys-ical task, animals are trained to respond on two levers, and they must respond on one lever during the ﬁrst half of the trial and on the other lever during the second half [193]. In the discrete-trials task, animals are trained to press one lever in response to short duration stimuli and another lever in response to long duration stimuli, and are then challenged with a variety of stimulus dura-tions [194]. DOI disrupts the performance of rats in both of these tasks [195-197]. Although DOI affects performance in the discrete trials task, it does not affect performance in a similar task where rats have to discriminate different light intensities, indicating that DOI is speciﬁcally inﬂuencing temporal perception and not disrupting stimulus control or attentional processes [198]. The effect of DOI in the discrete-trials task and that free-operant task are blocked by ketanserin and M100907 [196,197], demonstrating the involve-ment of 5-HT 2A. 4. 2. 5. Exploratory and investigatory behavior Measures of locomotor activity are often used to character-ize the effects of psychoactive drugs on exploratory behavior. Locomotion alone, however, is not necessarily a reliable measure470 471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509 510511512513514515516517518519520521522523524525526527528529 530 531532533534535536537538539540541542543544 545 546547548549550551552553554555556557558559560561562563564565566567 568 569570571	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 8 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx Table 2 The selective 5-HT 2Aantagonist M100907 blocks the head twitch response induced by hallucinogens in rats and mice. Hallucinogen M100907 Species Reference Drug Dose Routea Potencyb Effective dosec Routea 5-Me O-DMT 30 mg/kg IP ID50= 0. 03 IP Mouse [448] 5-Me O-DMT 10 mg/kg IP 0. 05 mg/kg IP Mouse [29] DPT 3 mg/kg IP 0. 01 mg/kg IP Mouse [100] DOI 2. 5 mg/kg IP ID50= 0. 005 0. 04 mg/kg SC Rat [160] DOI 3 mg/kg IP 1 mg/kg IP Rat [169] R-(-)-DOI 3 mg/kg IP ID50= 0. 01 0. 1 mg/kg SC Mouse [449] DOI 2. 5 mg/kg IP 0. 25 mg/kg IP Mouse [33] DOI 2 mg/kg IP 0. 3 mg/kg IP Mouse [450] DOI 1 mg/kg IP 0. 05 mg/kg IP Mouse [34] DOI 1 mg/kg IP 0. 25 mg/kg SC Mouse [167] DOI 1 mg/kg IP 0. 025 mg/kg SC Mouse [173] 2C-I 3 mg/kg SC ID50= 0. 0045 0. 1 mg/kg SC Mouse [156] 25I-NBOMe 0. 3 mg/kg SC ID50= 0. 0062 0. 1 mg/kg SC Mouse [156] 25I-NBMD 3 mg/kg SC ID50= 0. 0015 0. 1 mg/kg SC Mouse [156] a IP, intraperitoneal; SC, subcutaneous b ID50= inhibitory dose 50in mg/kg. c Dose of M100907 that produced 90-100% blockade of the head twitch response. of exploration because it includes does not distinguish speciﬁc exploratory responses to environmental stimuli from other types of motor activity [199]. Given the complexity of hallucinogen effects, it is not surprising that hallucinogens cannot be distinguished from other drug classes using traditional open ﬁeld locomotor mea-sures [144]. However, multivariate assessment methods have been more successful. One example is the Behavioral Pattern Monitor (BPM), which combines features from activity chambers and hole-boards and provides quantitative as well as qualitative measures of the spatial and temporal structure of activity [200,201]. BPM studies have shown hallucinogens produce a very characteristic proﬁle of behavioral effects. When rats are tested in unfamiliar BPM chambers after administration of hallucinogens (including mesca-line, DOM, DOI, LSD, DMT, 5-Me O-DMT, and psilocin), the animals display reduced amounts of locomotor activity, rearings, and hole-pokes at the beginning of the test session, and avoidance of the center of the BPM chamber is increased [202-205]. Most of these effects are markedly diminished in animals habituated to the BPM chambers, indicating that hallucinogens act by enhancing neopho-bia. The ability of hallucinogens to increase the avoidance of novel (and potentially threatening) test chambers by rats may be anal-ogous to the enhanced sensitivity and reactivity to environmental stimuli that occurs in humans [206]. Extensive testing has conﬁrmed this pattern of effects in the BPM is highly speciﬁc to hallucinogens [200,207-210]. For example, although 8-OH-DPAT and other selective 5-HT 1Aagonists reduce locomotor activity, rearings, and holepokes in rats, these effects are not inﬂuenced by environmental familiarity and hence are likely to reﬂect sedation [208]. When Adams and Geyer [211] compared lisuride and LSD in the BPM, they found the two com-pounds produce markedly different patterns of effects. Lisuride produces effects that are similar to those of apomorphine and other dopamine agonists, with sedative effects occurring at low doses and perseverative patterns of hyperactivity occurring at higher doses. The 5-HT 2Areceptor is responsible for mediating most of the effects of hallucinogens in the rat BPM. It was ﬁrst shown that ritanserin and ketanserin block the effects of mescaline, DOM, and DOI in the BPM, indicating 5-HT 2involvement [204]. Later studies demonstrated that the effects of DOI are blocked by M100907 but not by SER-082 [212], conﬁrming mediation by 5-HT 2A. The action of indoleamine hallucinogens in the BPM is more complex mech-anistically, with 5-HT 1Aand 5-HT 2Areceptors contributing to the effects of LSD and 5-Me O-DMT [205,213-215]. Hallucinogens have also been tested in a version of the BPM designed for mice [216]. In contrast to rats, phenylalkylamineand indolealkylamine hallucinogens produce disparate effects on exploratory and investigatory behavior in C57BL/6J mice. Phenylalkylamines, including DOI, mescaline, and TCB-2, inhibit investigatory behavior and alter locomotor activity in a dose-dependent manner, increasing activity at low to moderate doses and reducing activity at high doses [217,218]. Other groups have reported similar ﬁndings with DOM and DOI in mice [146,219-221]. The increase in locomotor activity induced by 1 mg/kg DOI, 25 mg/kg mescaline, or 3 mg/kg TCB-2 is blocked by low doses of M100907 and is absent in 5-HT 2Aknockout mice. By contrast, the reduction of locomotor activity induced by 10 mg/kg DOI is attenu-ated by SER-082. Taken together, it appears that 5-HT 2Aand 5-HT 2C receptors have countervailing effects on locomotor activity, with 5-HT2Aactivation increasing activity and 5-HT 2Cactivation reducing activity. Administration of psilocin and 5-Me O-DMT to C57BL/6J mice reduces locomotor activity and investigatory behavior [166]. These effects are blocked by WAY-100635 but are unaffected by SB 242,084 or by 5-HT 2Agene deletion. Similarly, 5-Me O-DMT has no effect on activity in 5-HT 1Aknockout mice [222]. Hence, whereas the phenylalkylamines act through 5-HT 2sites to alter behavior in the mouse BPM, indoleamine hallucinogens appear to act via the 5-HT 1Areceptor. 4. 3. Tolerance studies As noted in Section 3. 2, serotonergic hallucinogens produce a profound degree of tolerance and cross-tolerance in animals and humans. Although very little is known about the mechanisms lead-ing to the development of tolerance to hallucinogens in humans, there is evidence in animals that tolerance is linked to 5-HT 2A downregulation. Rats treated repeatedly with DOM, LSD, or psilocin show a signiﬁcantly lowered density of 5-HT 2Areceptors in sev-eral brain regions [223-225]. Binding to 5-HT 1A, 5-HT 1B, /H92512, /H92521, or D2receptors is unaffected. Another study demonstrated that treat-ment with 1 mg/kg DOI for 8 days produced a signiﬁcant reduction in the density of 5-HT 2Areceptors in the cortex, but there was no change in 5-HT 2Creceptor expression [109]. An identical treatment regimen caused tolerance to develop in rats trained to discriminate DOI. Likewise, there is a signiﬁcant reduction of 5-HT 2A-stimulated [35S]GTP /H9253S binding in the medial prefrontal cortex (m PFC) and anterior cingulate cortex in rats treated with LSD (0. 13 mg/kg/day) for 5 days [226] ; this indicates tolerance to LSD is associated with a reduction of 5-HT 2Asignaling. Although most hallucinogens produce tolerance in humans, DMT seems to be the exception. It has been reported that DMT572 573574575576577578579580581582583584585586587588589590591592593594595596597598599600601602603604605606607608609610611612613614615616617 618619620621622623624625626627628629630631632633634635636637638 639 640 641642643644645646647648649650651652653654655656657658659	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 9 does not evoke tolerance in man, even after an intramuscular (IM) dosage regimen of 0. 7 mg/kg twice daily for ﬁve days [227]. More recently, Strassman et al. [228] found there was no toler-ance to the subjective effects of DMT in volunteers who received four intravenous (i. v. ) injections of 0. 3 mg/kg at 30 min inter-vals. In vitro experiments have shown that exposure to LSD or DOI desensitizes 5-HT 2Aand 5-HT 2Creceptors in transfected cell lines [108,229]. However, after exposure to DMT, 5-HT 2Creceptors showed desensitization but there was no change in the response to 5-HT 2Aactivation [108]. These observations suggest that DMT fails to induce tolerance because it does not desensitize the 5-HT 2A receptor. 5. Hallucinogen effects on neuronal activity 5. 1. Locus coeruleus The locus coeruleus (LC), located in the dorsal pons, is the source of almost all noradrenergic projections in the CNS. LC neurons are responsive to sensory stimuli, especially of a novel or arousing nature, and the ﬁring of LC neurons is markedly increased by nox-ious stimulation (reviewed by: [230] ). Intravenous administration of mescaline (2 mg/kg), LSD (5-10 /H9262g/kg), DOM (20-80 /H9262g/kg), DOB (50-100 /H9262g/kg), or DOI (16-50 /H9262g/kg) profoundly enhances the responses of LC neurons to sensory stimuli while simultaneously depressing their spontaneous ﬁring [231-234]. After administra-tion of hallucinogens, the enhancement of responsiveness is so pronounced that even innocuous sensory stimuli normally inef-fective at driving LC cell ﬁring will evoke a response [231]. The ability to produce opposite effects upon spontaneous and sensory-evoked LC ﬁring is a speciﬁc property of LSD-like drugs, as other pharmacological agents that alter the basal activity of LC cells (e. g., (+)-amphetamine, clonidine, desipramine, or idazoxan) do not alter evoked LC ﬁring [231,232,234]. The observation that hallucinogens decrease the spontaneous activity of LC cells is supported by the work of Done and Sharp [235] who found that DOI and DOB lower the concentration of NE in hippocampal dialysates, which indicates those compounds decrease tonic NE release from LC projections. The effects of hallucinogens upon LC unit activity appear to be mediated by 5 HT2Areceptors. The 5-HT 2antagonists ketanserin and ritanserin have been shown to block the actions of hallucino-gens in the LC [232,233]. Furthermore, Szabo and Blier [236] found that the ability of DOI to alter the activity of LC neurons is abol-ished by M100907. Nonetheless, 5-HT 2Areceptors are sparsely distributed within the LC (e. g., [237] ), and application of the 5-HT2A/5-HT 3agonist quipazine or hallucinogens such as DOI directly into the LC does not mimic the effects of their systemic administra-tion [232-234,238]. Intravenous administration of mescaline and LSD also had no effect on the ability of locally applied acetylcholine, glutamate (Glu), or substance P to excite LC neuronal activity [231]. Presumably then, hallucinogens act upon LC afferents, altering the ﬁring of LC cells indirectly by modulating the activity of one or more input pathways. Chiang and Aston-Jones [234] reported that the decrease in LC spontaneous ﬁring induced by DOI could be blocked by the GABA Areceptor antagonists bicuculline and picrotoxin, whereas the ability of DOI to enhance sensory-evoked LC responses was blocked by the NMDA receptor antagonist 2-amino-5-phosphonopentanoic acid but not by the AMPA receptor antagonist 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX). Thus, hallucino-gens appear to tonically activate GABAergic input to LC and concomitantly facilitate glutamatergic sensory input. It is likely that the nucleus prepositus hypoglossi (Pr H), an area known to provide direct GABAergic inhibitory input into the LC [239,240], mediates the hallucinogen-induced inhibition of spontaneous LC activity. Although one group reported that microinjection of quipazine directly into the Pr H did not alter LC unit activity in the rat [238], subsequent work conﬁrmed that DOI depolarizes Pr H neurons [241]. Moreover, electrolytic lesions of Pr H signiﬁcantly attenuate the ability of systemic quipazine injections to reduce the frequency of LC unit discharge [238]. This strongly implicates the Pr H or one of its afferents as the site through which 5-HT 2Aagonists modulate spontaneous LC ﬁring. The identity of the speciﬁc LC afferent(s) responsible for the hallucinogen-induced facilitation of LC gluta-matergic sensory input is currently unknown. Although the nucleus paragigantocellularis in the ventrolateral rostral medulla is a major source of excitatory input into the LC [234,242], the ability of somatosensory stimuli to excite the LC is unaffected by lesions of nucleus paragigantocellularis [243]. The LC also receives excitatory input from the prefrontal cortex (PFC), both directly and indirectly [244-246], and the excitatory effects of hallucinogens on the LC may be mediated by those pathways. As will be discussed below in Section 5. 2, hallucinogens increase the ﬁring of PFC projection neurons. The LC projects heavily to cortex, where there is overlap between the distribution of /H92511-adrenoceptors and 5-HT 2Arecep-tors [247]. Interestingly, in the PFC, /H92511-adrenoceptors and 5-HT 2A receptors have similar effects on the activity of layer V pyrami-dal neurons [248]. Hallucinogens increase the intensity of sensory experiences and affective responses, and it is tempting to speculate that the LC may contribute to these effects. Indeed, the ability of LSD to potentiate neophobia in rats in the Behavioral Pattern Monitor is diminished by depletion of norepinephrine from LC projections [249]. 5. 2. Prefrontal cortex (PFC) 5. 2. 1. Effects on PFC network activity in vitro It is now recognized that the PFC is an important site of action for hallucinogens. The 5-HT 2Areceptor is expressed heavily in the PFC and adjacent cortical regions, particularly in lamina V [237,250-252]. In situ hybridization histochemistry has conﬁrmed that most of the cells in monkey and human PFC express 5-HT 2A m RNA [253]. Likewise, in rats, a large percentage of the cells in the superﬁcial, middle, and deep layers of the secondary motor, anterior cingulate (ACA), prelimbic (Pr L), and infralimbic (IL) areas express 5-HT 2Am RNA [254,255]. Almost all prefrontal pyramidal neurons express the 5-HT 2Areceptor, with the receptor localized primarily to the proximal apical dendrites [237,252,256,257]. In addition to pyramidal neurons, 5-HT 2Areceptors are also expressed by subsets of parvalbumin- and calbindin-positive interneurons [253,255,256,258-260]. Approximately 20-25% of the glutamic acid decarboxylase-positive cells in PFC express 5-HT 2Am RNA [253]. From their morphology these interneurons appear to be basket cells and chandelier cells [258]. GABAergic interneurons expressing parvalbumin and calbindin are sources of perisomatic inhibition that synchronize the oscillatory ﬁring of large ensembles of pyramidal neurons [261-263]. Therefore, 5-HT 2Areceptors are likely to have direct and indirect effects on the activity of pyramidal cells (see Fig. 8). Electrophysiological studies have shown that 5-HT 2Aactivation (with DOB or DOI) produces several effects on the membrane properties of layer V pyramidal neurons: there is a moderate depolarization, spike-frequency accommodation is reduced, and the afterhyperpolarization (AHP) that normally follows a burst of spikes is replaced by a slow depolarizing afterpotential (s ADP) [264-266]. The effect on AHP is mediated by activation of PLC/H9252 signaling, which inhibits one of the currents (Is AHP) underlying the AHP [267,268] ; the induction of s ADP is probably a conse-quence of activating a Ca2+-dependent nonselective cation channel (ICAN). Both of these effects increase the excitability of pyramidal660 661662663664665666667668669670671 672 673 674 675676677678679680681682683684685686687688689690691692693694695696697698699700701702703704705706707708709710711712713714715716717718719720721722 723724725726727728729730731732733734735736737738739740741742743744745746747748749750 751 752 753754755756757758759760761762763764765766767768769770771772773774775776777778779780781782783784785	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 10 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx Fig. 8. Distribution of 5-HT 2Areceptors in neurons in layer V of the prefrontal cortex. 5-HT 2Areceptors are expressed by glutamatergic pyramidal neurons and GABAergic basket cells and chandelier cells. Hallucinogens increase the frequency of spontaneous EPSCs and IPSCs in layer V pyramidal neurons by enhancing recurrent glutamatergic and GABAergic network activity. neurons [269]. DOI also produces a 5-HT 2A-dependent inhibition of voltage-dependent Na+-currents and L-type Ca2+-currents in PFC pyramidal cells via the PLC/H9252-IP3-protein kinase C and PLC/H9252-IP3-calcineurin signaling cascades, respectively, effects that would likely inﬂuence dendritic integration [270,271]. Hallucinogens have profound effects on excitatory and inhibitory transmission in medial PFC (m PFC) in vitro. Recordings from brain slices have shown that DOI and other 5-HT 2Aagonists produce a marked enhancement of the frequency and ampli-tude of spontaneous excitatory postsynaptic potentials/currents (EPSPs/EPSCs) in most layer V pyramidal neurons in m PFC [272,273] (Zhou and Hablitz, 1999). These effects are mediated Q2 by an increase in Glu release and subsequent activation of post-synaptic AMPA receptors [272,274]. Because these studies failed to locate any glutamatergic m PFC neurons that were driven to ﬁre action potentials by 5-HT 2Aactivation, it was initially thought that the increase in Glu release was caused by local activation of the terminals of glutamatergic thalamocortical afferents [275,276]. However, although the ability of 5-HT to induce EPSCs is lost after deletion of the 5-HT 2Agene (htr2A-/-mice), the effect can be res-cued by selective restoration of 5-HT 2Areceptors to pyramidal neurons in the forebrain [277]. The htr2A-/-mice used by Weis-staub et al. were generated by inserting a ﬂoxed Neo-stop cassette between the promoter and the coding region, so the gene could be rescued by crossing the mice with Emx1-Cre+/-mice (which selectively expresses Cre recombinase in the forebrain). The fact that the EPSCs were rescued in htr2A-/-× Emx1-Cre+/-mice shows that projections from thalamus and other subcortical structures are not being directly excited by 5-HT 2Areceptors. More recent work has identiﬁed a subpopulation of pyramidal neurons in m PFC deep layer V that are depolarized and excited by DOI [278], indicating hallucinogens induce spontaneous EPSCs by increasing recurrentglutamatergic network activity. 5-HT 2Areceptor activation also increases the frequency of spontaneous IPSCs in pyramidal neurons (Zhou and Hablitz, 1999), an effect that is mediated by activa-tion of neighboring GABAergic interneurons [260,279]. Therefore, it appears hallucinogens recruit glutamatergic and GABAergic neu-rons, which produces a marked enhancement of excitatory and inhibitory recurrent network activity in m PFC [280,281]. This conclusion is supported by microdialysis data showing that hallu-cinogens increase extracellular levels of Glu [282-284] and GABA [285] in m PFC. There is evidence that enhancement of glutamatergic activ-ity in m PFC plays an important role in mediating the effects of hallucinogens. Manipulations that suppress the facilitation of recurrent glutamatergic network activity, including the use of m Glu 2/3agonists, /H9262-opioid agonists, adenosine A1agonists, and AMPA antagonists [273,286-290], block many of the neurochemi-cal and behavioral effects of hallucinogens. These interactions have been demonstrated most extensively for the HTR (see Table 3), a 5-HT 2A-mediated behavior that can be provoked by infusion of DOI directly into the m PFC [291,292]. Likewise, the discrimina-tive stimulus effects of LSD are attenuated by the m Glu 2/3agonist LY379268 and augmented by the m Glu 2/3antagonist LY341495 [112], and there is evidence that the LSD stimulus cue is mediated by activation of 5-HT 2Areceptors in the ACA [114]. Another exam-ple is the ability of DOI to increase impulsive responding in rats, which is attenuated by administration of LY379268 systemically or directly into m PFC [293,294]. In addition to 5-HT 2Aantagonists, m Glu 2/3agonists and AMPA antagonists also block the ability of DOI to increase cortical expression of BDNF and the immediate-early genes c-fos, erg-2, and Arc [289,294-298]. Evidence has emerged that m Glu 2and 5-HT 2Areceptors can form heteromeric com-plexes in cortex [298,299], and these complexes may mediate the crosstalk that occurs between these receptors. It is important to note, however, that it has not been conclusively demonstrated that the heterodimers are responsible for the interactions between 5-HT2Aand m Glu 2[300,301], and it is possible the crosstalk is purely functional and occurs at the circuit level. m Glu 2receptors func-tion predominantly as presynaptic autoreceptors [302], so m Glu 2 activation could potentially suppress 5-HT 2A-induced spontaneous EPSCs by reducing Glu release from axon terminals. 5. 2. 2. Effects on PFC network activity in vivo Recent studies have examined the effects of hallucinogens on PFC activity in vivo. Extracellular recordings from anesthetized rats have shown that DOI (0. 05-0. 8 mg/kg, i. v. ) and 5-Me O-DMT (0. 1 mg/kg, i. v., in combination with the monoamine oxidase inhibitor clorgyline) produce a net excitatory effect on pyrami-dal neurons in the Pr L, IL, and ACA regions of m PFC [303-305]. Individual pyramidal neurons are either excited (38-53%), inhib-ited (27-35%), or show no response. It appears that these effects are mediated by recruitment of glutamatergic and GABAergic neurons because the excitatory response to DOI is blocked by LY379268 and the inhibitory response is blocked by the GABA A antagonist picrotoxinin [303,304]. These effects are also blocked by 5-HT 2Aantagonists. In contrast to those ﬁndings, another group has reported that higher doses of DOI (3-5 mg/kg, i. p. ) tend to inhibit the ﬁring of pyramidal cells in ACA and the ventral, dorsolateral, and lateral orbitofrontal cortices of behaving rats [306]. Despite the discrepant ﬁndings outlined above, hallucinogens produce strikingly similar effects on cortical network activity in anesthetized and freely moving rats. Under anesthesia or dur-ing slow-wave sleep, cortical networks display slow (0. 5-1 Hz) and delta (1-4 Hz) oscillations [307-309] that reﬂect alternations between periods of silence (DOWN states) and periods of depo-larization with repetitive spiking (UP states). This contrasts with the active waking state, which is characterized by fast rhythms in786 787788789790791792793794795796797798799800801802803804805806807808809810811812813814815816817818 819820821822823824825826827828829830831832833834835836837838839840841842843844845846847848849850851852853854855856857 858 859860861862863864865866867868869870871872873874875876877878879880881882	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 11 Table 3 Receptor agonists and antagonists that modulate the electrophysiological effects of 5-HT 2Aactivation in the m PFC also alter the head twitch response in rats and mice. Receptor Ligandpharmacology5-HT 2A-induced s EPSCs in layer V pyramidal neuronsa DOI-induced head twitch responsea 5-HT 2A Antagonist ↓ M100907 [272] Beique et al., 2007 [290]↓ M100907 [160] 5-HT 2C Antagonist Ø SB242084 [248] Beique et al., 2007Ø SB242084 [170] [171] AMPA Antagonist ↓ LY293558 [272][274] ↓ LY300164 [274] ↓ CNQX [273] Beique et al., 2007 [290]↓ LY293558 [274] ↓ GYKI 52466 [274] ↓ DNQX [451] ↓ NBQX [452] /H9262-opioid Agonist ↓ DAMGO [286] ↓ endomorphin-1 [286]↓ morphine [453] ↓ buprenorphine [454] ↓ fentanyl [454] m Glu 2/3 Agonist ↓ LY354740 [287] ↓ LY379268 [287] [273]↓ LY354740 [455][273] ↓ LY379268 [273] Antagonist ↑ LY341495 [287]↑ LY341495 [455] Adenosine A1 Agonist ↓ N6-cyclopentyladenosine [288]↓ N6-cyclohexyladenosine Marek et al., 2009 CNQX, 6-cyano-7-nitroquinoxaline-2,3-dione; DAMGO, [d-Ala2, N-Me Phe4, Gly-ol5]-enkephalin; DNQX, 6,7-dinitroquinoxaline-2,3-dione; NBQX, 2,3-dioxo-6-nitro-1,2,3,4-tetrahydrobenzo[ f]quinoxaline-7-sulfonamide; s EPSCs, spontaneous excitatory postsynaptic currents. a The speciﬁed ligand reduces the response (↓), has no effect (Ø), or enhances the response (↑). the gamma range (30-80 Hz) that play a putative role in a multi-tude of perceptual and cognitive functions [310-314]. Recordings of local ﬁeld potentials (LFPs) from the PFC have shown DOI reduces low-frequency oscillations in anesthetized rats [315], and dampens gamma oscillations in freely moving rats [306]. DOI also desynchro-nizes the ﬁring of pyramidal neurons so that their activity is no longer coupled to LFPs [306,315]. 5-Me O-DMT has similar effects on low-frequency PFC network activity in anesthetized rats [305]. Taken together, these ﬁndings demonstrate that hallucinogens dis-rupt the oscillatory activity of cortical networks and reduce the likelihood that individual pyramidal neurons will ﬁre in synchrony. Similar to the LFP data in rats, magnetoencephalographic (MEG) recordings in humans have shown that psilocybin (2 mg, i. v. ) pro-duces broadband reductions in cortical oscillatory power [316]. Dynamic causal modeling of the MEG data indicates that psilo-cybin reduces cortical synchrony by increasing the excitability of deep-layer pyramidal neurons. Likewise, electroencephalographic studies have reported that ayahuasca (containing the equivalent of 0. 85 mg/kg DMT) reduces cortical oscillatory power across multiple frequency bands [317,318]. Since cortical oscillations play a funda-mental role in a diverse set of mental processes and are required for the coordination of neural processing [319-324], it is tempting to speculate that the reduction of neuronal synchrony by hallu-cinogens could be responsible for mediating many of their effects on perception and cognition. Along these lines, there is evidence that schizophrenia patients show deﬁcits of gamma oscillations and synchrony [325-328] and reductions in slow-wave sleep [329], and it has been hypothesized that these abnormalities play an impor-tant role in the pathophysiology of psychosis. As was noted earlier, neuroimaging studies have demonstrated that hallucinogens alter activity in the frontal cortex. Studies using PET and single-photon emission computed tomography(SPECT) have consistently found that hallucinogens produce frontal hyperactivity. Administration of mescaline sulfate (500 mg, p. o. ) produces a hyperfrontal metabolic pattern, especially in the right hemisphere [53]. PET studies with [18F]ﬂuorodeoxyglucose ([18F]FDG) have shown that psilocybin (0. 20-0. 36 mg/kg, p. o. ) also produces a hyperfrontal pattern, with robust metabolic increases in frontolateral and frontomedial cortical regions and ACA [54,330]. Similar patterns of brain activation were found in subjects administered ayahuasca as part of a SPECT study [331]. By contrast, it has been argued, based on functional MRI (f MRI) data, that psilocybin reduces resting-state brain activity [332]. In that study, volunteers received 2 mg i. v. psilocybin and regional blood ﬂow and venous oxygenation were assessed using arterial spin labeling and blood-oxygen level-dependent (BOLD) f MRI scans. Psilocybin reduced blood ﬂow and BOLD signal in ACA and m PFC, and there was evidence of reduced coupling between m PFC and the posterior cingulate cortex. Based on those results, Carhart-Harris, Nutt, and colleagues concluded that psilocybin reduces activity and connectivity in core nodes of the default-mode network, brain regions that are active during the resting state and potentially involved in introspective processes (for more information, see: [333-335] ). It remains to be determined why psilocybin produces such discrepant effects in PET and f MRI studies. One potential explanation is that the hemodynamic responses measured by f MRI are actually better correlated with cortical oscillatory activity than with neuronal ﬁring [336-340]. Indeed, recent work by Artigas and co-workers conﬁrms the decoupling of BOLD measures and spiking in rats [305]. According to their report, 5-Me O-DMT (0. 1 mg/kg, i. v. ) increased the ﬁring rate of m PFC pyramidal cells by 215%, but signiﬁcantly reduced the BOLD signal (measured by f MRI) and the power of low-frequency oscillations (measured by LFP recordings). Therefore, PET and f MRI studies may be assessing different types883 884885886887888889890891892893894895896897898899900901902903904905906907908909910911912913914915 916917918919920921922923924925926927928929930931932933934935936937938939940941942943944945946	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 12 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx of neurophysiological responses to psilocybin, with PET measuring effects on neuronal ﬁring (reﬂected by changes in metabolic activity and [18F]FDG uptake) and f MRI measuring effects on cortical oscillatory activity. Alternatively, it is possible that the hemodynamic changes induced by psilocybin are unrelated to its hallucinogenic effects. Psilocybin and its O-dephosphorylated metabolite psilocin activate the 5-HT 1Areceptor in vivo [20,166], and 5-HT 1Aagonists are known to alter hemodynamic measures in cingulate cortex and other brain regions [341]. 5. 2. 3. Interactions of the PFC with other structurescortical and subcortical sites Since most of the projections from PFC to cortical and sub-cortical regions originate from pyramidal neurons in deep layers V-VI, hallucinogens could potentially profoundly alter how the PFC regulates activity in downstream regions. Indeed, there is some evi-dence that hallucinogens excite efferent projections from the PFC. For example, DOI activates serotonergic neurons in the dorsal raphe nucleus indirectly by exciting the projection from m PFC [303,342]. Similar ﬁndings have been reported for the projection to the ventral tegmental area [303]. Additionally, a recent study by Mocci et al. [284], Mocci et al. [284] assessed whether 5-HT 2Areceptors modu-late the activity of the projection from m PFC to nucleus accumbens (NAc). Retrodialysis of DOI into the m PFC increased the extracel-lular level of Glu in the NAc by 174%, indicating that DOI activates NAc-projecting m PFC neurons. According to another report, 5-HT 2A receptors excite cortico-cortical projections originating from m PFC [343]. In that study, microiontophretic application of 5-HT excited pyramidal neurons with commissural/callosal projections. Because 5-HT had no effect in the presence of the selective 5-HT 2Aantag-onist MDL 11,939, the most reasonable interpretation is that the excitation is mediated by 5-HT 2Areceptors, but this needs to be conﬁrmed using a selective agonist. The PFC exerts top-down executive control over processing in temporal and parietal cortices [344-347]. As shown by FDG-PET imaging, psilocybin increases absolute cerebral metabolic rates in the parietal and temporal cortices [54,348]. It is conceivable that hallucinogens could enhance the activity of neuronal ensembles in those regions by driving the ﬁring of glutamatergic projections from the PFC. Moreover, 5-HT 2Areceptors are expressed at high to mod-erate densities in temporal and parietal cortical areas [349-353], so the inﬂuence exerted by the PFC would act in concert with any local response induced by hallucinogens. Hallucinogenic drugs produce body image changes, derealization, and depersonalization [354,355], effects that are speciﬁcally linked to altered activity in fronto-parietal cortex and occipital cortex [356]. This is not sur-prising because the posterior parietal cortex is part of the dorsal visual stream and generates multiple egocentric representations of space [357-359]. Likewise, hallucinogens enhance memory recall [360], sometimes producing extremely vivid recollections. Since the medial temporal lobe plays a crucial role in the storage and recall of autobiographical memories [361], it has been proposed that hallucinogen effects on memory recall may be linked to acti-vation of this region. The amygdala, which is involved in generating fear responses and processing the emotional context of sensory input [362], is another subcortical structure potentially affected by changes in the activity of m PFC projections. An f MRI BOLD study in healthy volunteers revealed that psilocybin (0. 16 mg/kg p. o. ) reduces activation of the amygdala by negative and neutral pictures, and the BOLD signal change was inversely correlated with reports of increased positive mood [363]. Likewise, an electrical neuroimag-ing study conducted by the same group found psilocybin impairs processing of facial expression valence in the amygdala and other limbic regions [364]. In healthy subjects, there is an inverse correlation between the density of m PFC 5-HT 2Abinding and theresponsiveness of the amygdala to threatening stimuli [365], sug-gesting processing in the amygdala is regulated by 5-HT 2Areceptors in m PFC. Hence, the ability of psilocybin to reduce emotional pro-cessing in the amygdala could potentially be a consequence of increased inhibitory top-down control from the PFC [364]. The IL subregion of m PFC impairs fear conditioning by inhibit-ing central amygdaloid nucleus output neurons, which project to brainstem and hypothalamic sites responsible for expressing fear responses [366]. Although it was not initially clear how m PFC inhibits the amygdala because the projection is glutamatergic [367,368], the mechanism is now believed to involve excitation of GABAergic neurons in the intercalated nuclei of the amygdala [369-371]. Psilocybin and TCB-2 have been shown to facilitate the extinction of fear conditioning in C57BL/6J mice [372,373], which could be a consequence of activating the projection from IL to the intercalated nuclei. However, it has not been ruled out that psilo-cybin and TCB-2 are acting directly in the amygdala; excitatory and inhibitory neurons in the amygdala express 5-HT 2Areceptors [374,375], and DOI and other 5-HT 2Aagonists act locally to produce direct excitatory and indirect inhibitory effects in the amygdala [376-378]. 5. 2. 4. Interactions of the PFC with other structures: effects on cortico-striato-thalamo-cortical (CSTC) loops It has been theorized that hallucinogen-induced altered states may arise in part through effects on cortico-striato-thalamo-cortical (CSTC) feedback loops [348,356,379]. CSTC loops are parallel, anatomically segregated circuits relaying information between the basal ganglia, thalamus, and cortex [380,381]. In each circuit, projections from multiple cortical regions converge in spe-ciﬁc subregions of the striatum. The striatum, in turn, projects to the pallidum, which sends feedback to the cortex via the thala-mus. In this regime, the thalamus serves as a ﬁlter that restricts or gates the ﬂow of sensory and cognitive information to the cortex. There has been some debate about the exact number of CSTC loops [382,383], but at least ﬁve have been putatively identiﬁed, each serving a different function. The limbic loop, for example, receives input from the temporal lobe, ACA, and medial orbitofrontal cortex, and links the ventral striatum (including NAc, lateral caudate, and ventromedial putamen), ventral pallidum (VP), and mediodorsal thalamus. Vollenweider and Geyer [356] have proposed that psilo-cybin reduces thalamic ﬁltering by activating 5-HT 2Areceptors in the limbic CSTC loop, resulting in excessive stimulation of frontal regions, hyperfrontality, and symptoms such as sensory overload and hallucinations. Although involvement of CSTC loops in the effects of hallucino-gens is admittedly speculative, it does receive some support from the fact that hallucinogens disrupt PPI in humans and in animal models [90,178,179,182,183,384]. Importantly, PPI is regulated by components of the limbic CSTC loop, including m PFC, NAc, and VP [385]. The VP appears to be responsible for the disruption of PPI by hallucinogens [386]. DOI disrupts PPI when infused directly into the VP, but not when infused into the NAc. Likewise, infusion of M100907 into the VP prevents systemically administered DOI from disrupting PPI. It is important to note, however, that the PPI-disruptive effects of DOI are partially blocked when M100907 is infused into the dorsal striatum, so it is not entirely certain that the VP is the only site of action for DOI. 5. 3. Visual cortex Hallucinogens produce profound effects on visual perception. This includes visual distortions such as micropsia or macropsia, kinetopsia, pareidolias, hyperchromatopsia, dysmorphopsia, and polyopia-like trailing phenomena; elementary imagery composed of multicolored geometric patterns; and complex imagery with947 948949950951952953954955 956 9579589599609619629639649659669679689699709719729739749759769779789799809819829839849859869879889899909919929939949959969979989991000100110021003100410051006100710081009101010111012 10131014101510161017101810191020102110221023102410251026102710281029103010311032 1033 10341035103610371038103910401041104210431044104510461047104810491050105110521053105410551056105710581059106010611062106310641065106610671068 1069 1070 1071107210731074	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx 13 scenes, objects, and people (see Fig. 5). The visual imagery induced by hallucinogens is extremely vivid and can be observed with the eyes open or closed. When scientists began to experiment with mescaline at the end of the nineteenth century almost all of their work focused on the visual phenomenology [387-392]. Despite its highly subjective nature, the drug-induced imagery has been characterized in great detail [393,394]. Heinrich Klüver [393] was the ﬁrst to recognize that all of the elementary geometric hallu-cinations induced by mescaline are elaborated variations of four basic forms, which he called form constants : (a) tunnels and fun-nels, (b) spirals, (c) lattices and checkerboards, and (d) cobwebs. The form constants are not unique to hallucinogens and can occur during a variety of hallucinatory states, including migraine aura [395], epilepsy (Horowitz et al., 1967), sensory isolation [396], viewing ﬂickering light [397,398], and electrical cortical stimula-tion [399,400]. Several theoretical explanations for geometric visual halluci-nations have been proposed based on retinocortical mapping and the architecture of V1 [401-405]. According to these mathemat-ical models, excitation of V1 neurons produces self-organizing patterns of activity that correspond to Klüver's form constants. The excitation of V1 is presumably driven by 5-HT 2Areceptors because ketanserin blocks the visual hallucinations induced by psilocybin [85,89]. There are moderate to high densities of 5-HT2Areceptors in V1 [349,350,353,406], with the highest level occurring in geniculorecipient sublayer IVc/H9252 [350]. Similar to other cortical regions, almost all glutamatergic pyramidal neu-rons and very few GABAergic interneurons in V1 express 5-HT 2A m RNA [407,408]. A recent electrophysiology study conducted in anesthetized macaque monkeys revealed that DOI produces a com-bination of excitatory and inhibitory effects in V1, exciting neurons with low ﬁring rates and inhibiting neurons with high ﬁring rates [407]. Since neuronal ﬁring in V1 is driven by visual stimuli, one possible interpretation is that DOI reduces the response to visual input while enhancing spontaneous internally driven activity. It is fairly well-established that hallucinogens reduce retinocortical transmission [409-411]. Indeed, psilocybin inhibits N170 visu-ally evoked potentials in human subjects via 5-HT 2A[89,412]. Visual input stabilizes network activity in V1 by driving inhibitory interneurons [413]. Therefore, a reduction of visual input, cou-pled with an increase in the excitability of pyramidal neurons, could destabilize network activity in area V1, generating patterns of neuronal ﬁring that are perceived as geometric form con-stants. In contrast to the elementary visual hallucinations, which are linked to area V1, complex visual hallucinations probably arise from 5-HT 2Aactivation in higher level visual areas. There is evi-dence that excitation of Brodmann area (BA) 19 and BA 37 can produce complex visual hallucinations [414-416]. Among patients with Parkinson's disease, approximately 22% experience complex visual hallucinations [417]. Their visual hallucinations are linked to elevated levels of 5-HT 2Areceptor binding in ventral visual pathway [418,419], and can be ameliorated by blocking 5-HT 2Areceptors. For example, a PET imaging study with [18F]setoperone found that visual hallucinations in Parkinson's patients are associated with unusually high levels of 5-HT 2Abinding in the inferooc-cipital gyrus (BA 19), fusiform gyrus (BA 20 and BA 37), and inferotemporal gyrus (BA 20) [418]. According to another study conducted post-mortem, Parkinson's patients with visual halluci-nations show elevated levels of 5-HT 2Abinding in the inferolateral temporal cortex (BA 21) [419]. Two clinical trials have shown that the selective 5-HT 2Ainverse agonist pimavanserin reduces the severity of hallucinations in Parkinson's disease [420,421]. The atypical antipsychotics clozapine and risperidone, which block the 5-HT 2Areceptor, are also effective against the visual hallucinations [422-424]. 6. Summary Despite the complexity of hallucinogen effects, we are beginning to understand how these substances work in the brain. The 5-HT 2A receptor was ﬁrst identiﬁed about thirty years ago as a possible site of action of hallucinogens. It is now clear that most of the effects of hallucinogens are mediated by 5-HT 2Aactivation. Although the vast majority of this evidence was derived from studies in animals, the resumption of human studies with hallucinogens has provided additional support. Recent clinical trials have provided a highly detailed charac-terization of hallucinogen effects. However, most of this work has focused on one hallucinogen (psilocybin). By comparison, very lit-tle is known about the effects of other agents. This is especially true for ergoline and phenylalkylamine hallucinogens. One of the most characteristic properties of hallucinogens is how unpredictable their effects can be. The exact nature of the experience is highly variable and depends on the mood and expectations of the subject (the “set”) as well as the environment in which the drug is ingested (the “setting”) (Bercel et al., 1955) [425-427]. Depending on the cir-cumstances, the effects of hallucinogens may be perceived as being highly pleasurable or highly aversive (e. g., Aldous Huxley's descrip-tion of mescaline as “heaven and hell”). Although hallucinogens act in a relatively unspeciﬁc manner [428], and hence a broad range of experiences are possible, previous clinical studies have conﬁrmed that there is also a great deal of similarity between the effects of different hallucinogens. In other words, although it is impossible to predict exactly what type of experience will be produced by, for example, LSD or psilocybin, it appears that for the most part any experience produced by LSD can also occur with psilocybin. Thus, volunteers could not identify any clear differences between the subjective effects of those two compounds when administered by blind dosing [37-39,41]. However, those studies need to be repeated using modern psychometric assessment methods. Addi-tionally, it is not clear to what extent those ﬁndings extend to other hallucinogens, or even to higher doses of LSD and psilocybin. One potentially unique aspect of the LSD experience is that it reportedly occurs in two distinct temporal phases [206,427,429,430], but this needs to be conﬁrmed by future investigations. It appears that 5-HT 2Aactivation is a common characteristic of serotonergic hallucinogens and is responsible for mediating their shared effects, but this does not eliminate the possibility that other receptors may play an ancillary role. There are pharmaco-logical differences between the phenalkylamine, tryptamine, and ergoline classes, as well as between speciﬁc compounds within each class, and these differences could potentially inﬂuence the subjective effects [20]. The receptors activated by hallucinogens may be analogous to individual musical notes that can be played in combination to generate chords associated with unique subjective impressions [431], with 5-HT 2Areceptor activation being akin to the root note. Extramural investigations have attempted to categorize the existence of subtle subjective differences between the effects of different hallucinogens (e. g., [432,433] ). However, it is not clear to what extent the apparent differences between individual compounds are inﬂuenced by expectation and by other factors. There are also dose- and route-dependent variations in the effects of hallucinogens, which can alter both the intensity and the qualitative nature of the response. Furthermore, even individual subjects may experience markedly different responses to the same drug on different occasions [434]. The possibility exists that for hallucinogen effects, there may be just as much intra-drug variabil-ity as there is inter-drug variability. Only detailed, well-controlled clinical trials comparing multiple compounds over a wide range of doses will answer these questions. Nevertheless, it seems to be fairly well established that there are marked qualitative differences between the effects produced by serotonergic hallucinogens and by1075 10761077107810791080108110821083108410851086108710881089109010911092109310941095109610971098109911001101110211031104110511061107110811091110111111121113111411151116111711181119112011211122112311241125112611271128112911301131113211331134113511361137113811391140 1141 114211431144114511461147114811491150115111521153115411551156115711581159116011611162116311641165116611671168116911701171117211731174117511761177117811791180118111821183118411851186118711881189119011911192119311941195119611971198119912001201120212031204	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
Please cite this article in press as: Halberstadt AL. Recent advances in the neuropsychopharmacology of serotonergic hallucinogens. Behav Brain Res (2014), http://dx. doi. org/10. 1016/j. bbr. 2014. 07. 016ARTICLE IN PRESSG Model BBR 9035 1-22 14 A. L. Halberstadt / Behavioural Brain Research xxx (2014) xxx-xxx members of other drug classes. Although it was recently reported that subjects administered high doses of the NMDA antagonist dextromethorphan under double-blind conditions identiﬁed it as a classical hallucinogen when they were asked to classify it pharma-cologically [435], there are major confounds associated with this study. First, Reissig et al. [435] acknowledged that most if not all of the study participants were expecting to receive psilocybin, and this may have inﬂuenced their response to dextromethorphan. Second, the subjects did not receive a hallucinogen as an active con-trol, so the study did not actually quantify the similarity between the effects of dextromethorphan and hallucinogens. It is also surprising that none of the subjects classiﬁed dextromethorphan as a dissociative anesthetic, since dextromethorphan is abused for its dissociative-like effects [436] and produces phencyclidine- and ketamine-like discriminative stimulus effects in rats [437,438]. Over the last decade, there has been renewed interest into the potential therapeutic uses for hallucinogens. Psilocybin can induce highly meaningful spiritual experiences [58], and some subjects have reported experiencing positive changes in mood and behav-ior that persist for many months [62]. It may be possible to exploit these effects therapeutically. Recent clinical trials have investigated whether psilocybin has efﬁcacy against anxiety in terminal can-cer patients [56], and LSD has been tested as a potential adjunct for psychotherapy [439]. Several follow-up studies are currently in progress. It is anticipated that these and other studies will yield important insights into the psychopharmacology of hallucinogens, as well as showing whether there are potential medical uses for these drugs. Uncited references Q3 [440-447]. Acknowledgements This work was supported by grants from NIMH (K01 Q4 MH100644), NIDA (R01 DA002925), and the Brain and Behavior Research Foundation. References Q5 [1] Hollister LE. Chemical psychoses: LAD and related drugs. Springﬁled, IL: Charles C. Thomas; 1968. [2] Grinspoon L, Bakalar J. Psychedelic drugs reconsidered. New York: Basic Books; 1979. [3] Glennon RA, Rosecrans JA, Young R. The use of the drug discrimination paradigm for studying hallucinogenic agents. A review. 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Comprehens Psychiatry 1962;3:235-41. 1205 120612071208120912101211121212131214121512161217121812191220122112221223122412251226122712281229123012311232 1233 1234 1235 1236 12371238 1239 1240 1241124212431244124512461247124812491250125112521253125412551256125712581259126012611262126312641265126612671268126912701271 1272127312741275127612771278127912801281128212831284128512861287128812891290129112921293129412951296129712981299130013011302130313041305130613071308130913101311131213131314131513161317131813191320132113221323132413251326132713281329133013311332133313341335133613371338133913401341134213431344134513461347134813491350135113521353135413551356	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046
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Life Sci 1986;39:825-30. 1357 1358135913601361136213631364136513661367136813691370137113721373137413751376137713781379138013811382138313841385138613871388138913901391139213931394139513961397139813991400140114021403140414051406140714081409141014111412141314141415141614171418141914201421142214231424142514261427142814291430143114321433143414351436143714381439144014411442 144314441445144614471448144914501451145214531454145514561457145814591460146114621463146414651466146714681469147014711472147314741475147614771478147914801481148214831484148514861487148814891490149114921493149414951496149714981499150015011502150315041505150615071508150915101511151215131514151515161517151815191520152115221523152415251526	Recent advances in the neuropsychopharmacology of -- Halberstadt Adam L_ -- Behavioural Brain Research 277 pages 99-120 2015 jan -- Elsevier -- 10_1016_j_bbr_2014_07_016 -- 95e2af2df9463e8c4ed63dee046