Geometricvisualhallucinations,Euclidean ...bresslof/publications/01-1.pdfGeometricvisualhallucinations,Euclidean symmetryandthefunctionalarchitecture ofstriatecortex PaulC.Bresslo¡1,JackD.Cowan2*,MartinGolubitsky3,

Geometric visual hallucinations Euclideansymmetry and the functional architecture

of striate cortex

Paul C Bressloiexcl1 Jack D Cowan2 Martin Golubitsky3Peter J Thomas4 and Matthew C Wiener5

1Department of Mathematics University of Utah Salt Lake City UT 84112 USA2Department of Mathematics University of Chicago Chicago IL 60637 USA

3Department of Mathematics University of Houston HoustonTX 77204-3476 USA4Computational Neurobiology Laboratory Salk Institute for Biological Studies PO Box 85800 San Diego CA 92186-5800 USA

5Laboratory of Neuropsychology National Institutes of Health Bethesda MD 20892 USA

This paper is concerned with a striking visual experience that of seeing geometric visual hallucinationsHallucinatory images were classicented by Klulaquo ver into four groups called form constants comprising(i) gratings lattices fretworks centligrees honeycombs and chequer-boards (ii) cobwebs (iii) tunnelsfunnels alleys cones and vessels and (iv) spirals This paper describes a mathematical investigation oftheir origin based on the assumption that the patterns of connection between retina and striate cortex(henceforth referred to as V1)oumlthe retinocortical map oumland of neuronal circuits in V1 both local andlateral determine their geometry

In the centrst part of the paper we show that form constants when viewed in V1 coordinates essentiallycorrespond to combinations of plane waves the wavelengths of which are integral multiples of the widthof a human Hubel^Wiesel hypercolumn ca 133^2 mm We next introduce a mathematical description ofthe largescale dynamics of V1 in terms of the continuum limit of a lattice of interconnected hyper-columns each of which itself comprises a number of interconnected iso-orientation columns We thenshow that the patterns of interconnection in V1 exhibit a very interesting symmetry ie they are invariantunder the action of the planar Euclidean group E(2)oumlthe group of rigid motions in the planeoumlrotations repoundections and translations What is novel is that the lateral connectivity of V1 is such that anew group action is needed to represent its properties by virtue of its anisotropy it is invariant withrespect to certain shifts and twists of the plane It is this shift^twist invariance that generates newrepresentations of E(2) Assuming that the strength of lateral connections is weak compared with that oflocal connections we next calculate the eigenvalues and eigenfunctions of the cortical dynamics usingRayleigh^Schrolaquo dinger perturbation theory The result is that in the absence of lateral connections theeigenfunctions are degenerate comprising both even and odd combinations of sinusoids in iquest the corticallabel for orientation preference and plane waves in r the cortical position coordinate `Switching-onrsquo thelateral interactions breaks the degeneracy and either even or else odd eigenfunctions are selected Theseresults can be shown to follow directly from the Euclidean symmetry we have imposed

In the second part of the paper we study the nature of various even and odd combinations of eigen-functions or planforms the symmetries of which are such that they remain invariant under the particularaction of E(2) we have imposed These symmetries correspond to certain subgroups of E(2) the so-calledaxial subgroups Axial subgroups are important in that the equivariant branching lemma indicates thatwhen a symmetrical dynamical system becomes unstable new solutions emerge which have symmetriescorresponding to the axial subgroups of the underlying symmetry group This is precisely the case studiedin this paper Thus we study the various planforms that emerge when our model V1 dynamics becomeunstable under the presumed action of hallucinogens or poundickering lights We show that the planformscorrespond to the axial subgroups of E(2) under the shift^twist action We then compute what suchplanforms would look like in the visual centeld given an extension of the retinocortical map to include itsaction on local edges and contours What is most interesting is that given our interpretation of thecorrespondence between V1 planforms and perceived patterns the set of planforms generates represent-atives of all the form constants It is also noteworthy that the planforms derived from our continuummodel naturally divide V1 into what are called linear regions in which the pattern has a near constantorientation reminiscent of the iso-orientation patches constructed via optical imaging The boundaries ofsuch regions form fractures whose points of intersection correspond to the well-known pinwheelsrsquo

Phil Trans R Soc Lond B (2001) 356 299^330 299 copy 2001 The Royal SocietyReceived 18 April 2000 Accepted 11 August 2000

doi 101098rstb20000769

Author for correspondence (cowanmathuchicagoedu)

To complete the study we then investigate the stability of the planforms using methods of nonlinearstability analysis including Liapunov^Schmidt reduction and Poincare^Lindstedt perturbation theoryWe centnd a close correspondence between stable planforms and form constants The results are sensitive tothe detailed specicentcation of the lateral connectivity and suggest an interesting possibility that the corticalmechanisms by which geometric visual hallucinations are generated if sited mainly in V1 are closelyrelated to those involved in the processing of edges and contours

Keywords hallucinations visual imagery poundicker phosphenes neural modellinghorizontal connections contours

` the hallucination is not a static process but adynamic process the instability of which repoundects aninstability in its conditions of originrsquo (Klulaquo ver (1966)p 95 in a comment on Mourgue (1932))

1 INTRODUCTION

(a) Form constants and visual imageryGeometric visual hallucinations are seen in many situa-tions for example after being exposed to poundickeringlights (Purkinje 1918 Helmholtz 1924 Smythies 1960)after the administration of certain anaesthetics (Winters1975) on waking up or falling asleep (Dybowski 1939)following deep binocular pressure on onersquos eyeballs(Tyler 1978) and shortly after the ingesting of drugs suchas LSD and marijuana (Oster 1970 Siegel 1977) Patternsthat may be hallucinatory are found preserved in petro-glyphs (Patterson 1992) and in cave paintings (Clottes ampLewis-Williams 1998) There are many reports of suchexperiences (Knauer amp Maloney 1913 pp 429^430)

Ìmmediately before my open eyes are a vast number ofrings apparently made of extremely centne steel wire allconstantly rotating in the direction of the hands of aclock these circles are concentrically arranged the inner-most being incentnitely small almost pointlike theoutermost being about a meter and a half in diameterThe spaces between the wires seem brighter than thewires themselves Now the wires shine like dim silver inparts Now a beautiful light violet tint has developed inthem As I watch the center seems to recede into thedepth of the room leaving the periphery stationary tillthe whole assumes the form of a deep tunnel of wire ringsThe light which was irregularly distributed among thecircles has receded with the center into the apex of thefunnel The center is gradually returning and passing theposition when all the rings are in the same vertical planecontinues to advance till a cone forms with its apextoward me The wires are now poundattening into bandsor ribbons with a suggestion of transverse striation andcolored a gorgeous ultramarine blue which passes inplaces into an intense sea green These bands move rhyth-mically in a wavy upward direction suggesting a slowendless procession of small mosaics ascending the wall insingle centles The whole picture has suddenly receded thecenter much more than the sides and now in a momenthigh above me is a dome of the most beautiful mosiacs The dome has absolutely no discernible pattern Butcircles are now developing upon it the circles arebecoming sharp and elongated now they are rhombicsnow oblongs and now all sorts of curious angles areforming and mathematical centgures are chasing each otherwildly across the roof rsquo

Klulaquo ver (1966) organized the many reported images intofour classes which he called form constants (I) gratingslattices fretworks centligrees honeycombs and chequer-

boards (II) cobwebs (III) tunnels and funnels alleyscones vessels and (IV) spirals Some examples of class Iform constants are shown in centgure 1 while examples ofthe other classes are shown in centgures 2^4

Such images are seen both by blind subjects and insealed dark rooms (Krill et al 1963) Various reports(Klulaquo ver 1966) indicate that although they are dicurrencult tolocalize in space and actually move with the eyes theirpositions relative to each other remain stable with respectto such movements This suggests that they are generatednot in the eyes but somewhere in the brain One clue ontheir location in the brain is provided by recent studies ofvisual imagery (Miyashita 1995) Although controversialthe evidence seems to suggest that areas V1 and V2 thestriate and extra- striate visual cortices are involved invisual imagery particularly if the image requires detailedinspection (Kosslyn 1994) More precisely it has beensuggested that (Ishai amp Sagi 1995 p 1773)

`[the] topological representation [provided by V1] mightsubserve visual imagery when the subject is scrutinizingattentively local features of objects that are stored inmemoryrsquo

Thus visual imagery is seen as the result of an interactionbetween mechanisms subserving the retrieval of visualmemories and those involving focal attention In thisrespect it is interesting that there seems to be competitionbetween the seeing of visual imagery and hallucinations(Knauer amp Maloney 1913 p 433)

` after a picture had been placed on a background andthen removed ` I tried to see the picture with open eyesIn no case was I successful only [hallucinatory] visionaryphenomena covered the groundrsquorsquorsquo

Competition between hallucinatory images and after-images was also reported Klulaquo ver 1966 p 35)

Ìn some instances the [hallucinatory] visions preventedthe appearance of after-images entirely [however] inmost cases a sharply outlined normal after-imageappeared for a while while the visionary phenomenawere stationary the after-images moved with the eyesrsquo

As pointed out to us by one of the referees the fusedimage of a pair of random dot stereograms also seems tobe stationary with respect to eye movements It has alsobeen argued that because hallucinatory images are seenas continuous across the midline they must be located athigher levels in the visual pathway than V1 or V2(R Shapley personal communication) In this respectthere is evidence that callosal connections along the V1V2 border can act to maintain continuity of the imagesacross the vertical meridian (Hubel amp Wiesel 1967)

All these observations suggest that both areas V1 andV2 are involved in the generation of hallucinatory

300 P C Bressloiexcl and others Geometric visual hallucinations

PhilTrans R Soc Lond B (2001)

images In our view such images are generated in V1 andstabilized with respect to eye movements by mechanismspresent in V2 and elsewhere It is likely that the action ofsuch mechanisms is rapidly fed back to V1 (Lee et al1998) It now follows because all observers report seeingKlulaquo verrsquos form constants or variations that thoseproperties common to all such hallucinations should yieldinformation about the architecture of V1 We thereforeinvestigate that architecture ie the patterns of connec-tion between neurons in the retina and those in V1together with intracortical V1 connections on thehypothesis that such patterns determine in large partthe geometry of hallucinatory form constants and wedefer until a later study the investigation of mechanismsthat contribute to their continuity across the midline andto their stability in the visual centeld

(b) The human retinocortical mapThe centrst step is to calculate what visual hallucinations

look like not in the standard polar coordinates of thevisual centeld but in the coordinates of V1 It is well estab-lished that there is a topographic map of the visual centeldin V1 the retinotopic representation and that the centralregion of the visual centeld has a much bigger representationin V1 than it does in the visual centeld (Sereno et al 1995)The reason for this is partly that there is a non-uniform

Geometric visual hallucinations P C Bressloiexcl and others 301

Phil Trans R Soc Lond B (2001)

(a)

(b)

Figure 1 (a) `Phosphenersquo produced by deep binocularpressure on the eyeballs Redrawn from Tyler (1978)(b) Honeycomb hallucination generated by marijuanaRedrawn from Clottes amp Lewis-Williams (1998)

(a)

(b)

Figure 2 (a) Funnel and (b) spiral hallucinations generatedby LSD Redrawn from Oster (1970)

(a)

(b)

Figure 3 (a) Funnel and (b) spiral tunnel hallucinationsgenerated by LSD Redrawn from Siegel (1977)

distribution of retinal ganglion cells each of whichconnects to V1 via the lateral geniculate nucleus (LGN)This allows calculation of the details of the map (Cowan1977) Let raquoR be the packing density of retinal ganglioncells per unit area of the visual centeld raquo the correspondingdensity per unit surface area of cells in V1 and permilrR sup3RŠretinal or equivalently visual centeld coordinates ThenraquoR rR drR dsup3R is the number of ganglion cell axons in aretinal element of area rR drR dsup3R By hypothesis theseaxons connect topographically to cells in an element ofV1 surface area dx dy ie to raquo dx dy cortical cells (V1 isassumed to be locally poundat with Cartesian coordinates)Empirical evidence indicates that raquo is approximatelyconstant (Hubel amp Wiesel 1974ab) whereas raquoR declinesfrom the origin of the visual centeld ie the fovea with aninverse square law (Drasdo 1977)

raquoR ˆ1

(w0 Dagger erR)2

where w0 and e are constants Estimates of w0 ˆ 0087and e ˆ 0051 in appropriate units can be obtained frompublished data (Drasdo 1977) From the inverse squarelaw one can calculate the Jacobian of the map and henceV1 coordinates fx yg as functions of visual centeld or retinalcoordinates frR sup3Rg The resulting coordinate transforma-tion takes the form

x ˆnot

eln 1Dagger

e

w0

rR

y ˆshy rRsup3R

w0 Dagger erR

where not and shy are constants in appropriate unitsFigure 5 shows the map

The transformation has two important limiting cases(i) near the fovea erR5w0 it reduces to

x ˆnotrR

w0

y ˆshy rRsup3R

w0

and (ii) sucurrenciently far away from the fovea erR frac34 w0 itbecomes

x ˆnot

eln

erR

w0

y ˆshy sup3R

e

Case (i) is just a scaled version of the identity map and case(ii) is a scaled version of the complex logarithm as was centrstrecognized by Schwartz (1977) To see this letzR ˆ xR Dagger iyR ˆ rR exppermilisup3RŠ be the complex representation



Figure 4 Cobweb petroglyph Redrawn from Patterson(1992)

0

2

x

y2

0

0

2

(b)(a)

(c)

p

p

p

23p

23p

2p

23p

2p

p

23p

23p

p

p

y

x

Figure 5 The retinocortical map (a) visual centeld (b) the actual cortical map comprising right and left hemisphere transforms(c) a transformed version of the cortical map in which the two transforms are realigned so that both foveal regions correspond tox ˆ 0

of a retinal point (xR yR) ˆ (rR sup3R) then z ˆ x Dagger iyˆ ln( rR exppermilisup3RŠ) ˆ ln rR Dagger isup3R Thus x ˆ ln rR y ˆ sup3R

(c) Form constants as spontaneous cortical patternsGiven that the retinocortical map is generated by the

complex logarithm (except near the fovea) it is easy tocalculate the action of the transformation on circles raysand logarithmic spirals in the visual centeld Circles ofconstant rR in the visual centeld become vertical lines in V1whereas rays of constant sup3R become horizontal linesInterestingly logarithmic spirals become oblique lines inV1 the equation of such a spiral is just sup3R ˆ a ln rR

whence y ˆ ax under the action of zR z Thus formconstants comprising circles rays and logarithmic spiralsin the visual centeld correspond to stripes of neural activityat various angles in V1 Figures 6 and 7 show the mapaction on the funnel and spiral form constants shown incentgure 2

A possible mechanism for the spontaneous formation ofstripes of neural activity under the action of hallucinogenswas originally proposed by Ermentrout amp Cowan (1979)They studied interacting populations of excitatory andinhibitory neurons distributed within a two-dimensional(2D) cortical sheet Modelling the evolution of the systemin terms of a set of Wilson^Cowan equations (Wilson ampCowan 1972 1973) they showed how spatially periodicactivity patterns such as stripes can bifurcate from ahomogeneous low-activity state via a Turing-likeinstability (Turing 1952) The model also supports theformation of other periodic patterns such as hexagonsand squaresoumlunder the retinocortical map these

generate more complex hallucinations in the visual centeldsuch as chequer-boards Similar results are found in areduced single-population model provided that the inter-actions are characterized by a mixture of short-rangeexcitation and long-range inhibition (the so-called`Mexican hat distributionrsquo)

(d) Orientation tuning in V1The Ermentrout^Cowan theory of visual hallucinations

is over-simplicented in the sense that V1 is represented as if itwere just a cortical retina However V1 cells do muchmore than merely signalling position in the visual centeldmost cortical cells signal the local orientation of a contrastedge or baroumlthey are tuned to a particular local orienta-tion (Hubel amp Wiesel 1974a) The absence of orientationrepresentation in the Ermentrout^Cowan model meansthat a number of the form constants cannot be generatedby the model including lattice tunnels (centgure 42) honey-combs and certain chequer-boards (centgure 1) and cobwebs(centgure 4) These hallucinations except the chequer-boards are more accurately characterized as lattices oflocally orientated contours or edges rather than in terms ofcontrasting regions of light and dark

In recent years much information has accumulatedabout the distribution of orientation selective cells in V1and about their pattern of interconnection (Gilbert 1992)Figure 8 shows a typical arrangement of such cellsobtained via microelectrodes implanted in cat V1 The centrstpanel shows how orientation preferences rotate smoothlyover V1 so that approximately every 300 mm the same



(a)

(b)

Figure 6 Action of the retinocortical map on the funnel formconstant (a) Image in the visual centeld (b) V1 map of the image

(a)

(b)

Figure 7 Action of the retinocortical map on the spiral formconstant (a) Image in the visual centeld (b) V1 map of the image

preference reappears ie the distribution is ordm-periodic inthe orientation preference angle The second panel showsthe receptive centelds of the cells and how they change withV1 location The third panel shows more clearly therotation of such centelds with translation across V1

How are orientation tuned cells distributed and inter-connected Recent work on optical imaging has made itpossible to see how the cells are actually distributed inV1(Blasdel 1992) and a variety of stains and labels hasmade it possible to see how they are interconnected(G G Blasdel and L Sincich personal communication)(Eysel 1999 Bosking et al 1997) Figures 9 and 10 showsuch data Thus centgure 9a shows that the distribution oforientation preferences is indeed roughly ordm-periodic inthat approximately every 05 mm (in the macaque) thereis an iso-orientation patch of a given preference andcentgure 10 shows that there seem to be at least two length-scales

(i) localoumlcells less than 05 mm apart tend to makeconnections with most of their neighbours in aroughly isotropic fashion as seen in centgure 9b and

(ii) lateraloumlcells make contacts only every 05 mm orso along their axons with cells in similar iso-orientation patches

In addition centgure 10 shows that the long axons whichsupport such connections known as intrinsic lateral orhorizontal connections and found mainly in layers II and

III of V1 and to some extent in layer V (Rockland ampLund 1983) tend to be orientated along the direction oftheir cellrsquos preference (Gilbert 1992 Bosking et al 1997)ie they run parallel to the visuotopic axis of their cellrsquosorientation preference These horizontal connections arisealmost exclusively from excitatory neurons (Levitt ampLund 1997 Gilbert amp Wiesel 1983) although 20terminate on inhibitory cells and can thus have signicentcantinhibitory eiexclects (McGuire et al 1991)

There is some anatomical and psychophysical evidence(Horton 1996 Tyler 1982) that human V1 has severaltimes the surface area of macaque V1 with a hypercolumnspacing of ca 133^2 mm In the rest of this paper wework with this length-scale to extend the Ermentrout^Cowan theory of visual hallucinations to include orienta-tion selective cells A preliminary account of this wasdescribed in Wiener (1994) and Cowan (1997)

2 A MODEL OF V1 WITH ANISOTROPIC LATERAL

CONNECTIONS

(a) The modelThe state of a population of cells comprising an iso-

orientation patch at cortical position r 2 R2 at time t is



x

1

2

3

3

21(a)

(b)

(c) y

x

y

Figure 8 (a) Orientation tuned cells in V1 Note theconstancy of orientation preference at each cortical location(electrode tracks 1 and 3) and the rotation of orientationpreference as cortical location changes (electrode track 2)(b) Receptive centelds for tracks 1 and 3 (c) Expansion of thereceptive centelds of track 2 to show the rotation of orientationpreference Redrawn from Gilbert (1992)

(a)

(b)

Figure 9 (a) Distribution of orientation preferences inmacaque V1 obtained via optical imaging Redrawn fromBlasdel (1992) (b) Connections made by an inhibitoryinterneuron in cat V1 Redrawn from Eysel (1999)

characterized by the real-valued activity variablea(r iquest t) where iquest 2 permil0 ordm) is the orientation preference ofthe patch V1 is treated as an (unbounded) continuous 2Dsheet of nervous tissue For the sake of analytical tract-ability we make the additional simplifying assumptionthat iquest and r are independent variablesoumlall possibleorientations are represented at every position A moreaccurate model would need to incorporate detailsconcerning the distribution of orientation patches in thecortical plane (as illustrated in centgure 9a) It is known forexample that a region of human V1 ca 267 mm2 on itssurface and extending throughout its depth contains atleast two sets of all iso-orientation patches in the range04iquest5ordm one for each eye Such a slab was called ahypercolumn by Hubel amp Wiesel (1974b) If human V1 asa whole (in one hemisphere) has a surface area of ca3500 mm2 (Horton 1996) this gives approximately 1300such hypercolumns So one interpretation of our model

would be that it is a continuum version of a lattice ofhypercolumns However a potential dicurrenculty with thisinterpretation is that the eiexclective wavelength of many ofthe patterns underlying visual hallucinations is of theorder of twice the hypercolumn spacing (see for examplecentgure 2) suggesting that lattice eiexclects might be impor-tant A counter-argument for the validity of thecontinuum model (besides mathematical convenience) isto note that the separation of two points in the visualcenteldoumlvisual acuityouml(at a given retinal eccentricity ofr0

R) corresponds to hypercolumn spacing (Hubel ampWiesel 1974b) and so to each location in the visual centeldthere corresponds to a representation in V1 of that loca-tion with centnite resolution and all possible orientations

The activity variable a(r iquest t) evolves according to ageneralization of the Wilson^Cowan equations (Wilson ampCowan 1972 1973) that takes into account the additionalinternal degree of freedom arising from orientationpreference

a(r iquest t)t

ˆ iexcl nota(r iquest t) Dagger middotordm

0 R2w(r iquestjr0 iquest0)

pound frac14permila(r0 iquest0 t)Šdr0diquest0

ordmDagger h(r iquest t) (1)

where not and middot are decay and coupling coecurrencientsh(r iquest t) is an external input w(r iquestjr0 iquest0) is the weightof connections between neurons at r tuned to iquest andneurons at r0 tuned to iquest0 and frac14permilzŠ is the smooth nonlinearfunction

frac14permilzŠ ˆ1

1 Dagger eiexclreg(ziexclplusmn) (2)

for constants reg and plusmn Without loss of generality we maysubtract from frac14permilzŠ a constant equal to permil1 Dagger eregplusmn Šiexcl1 to obtainthe (mathematically) important property that frac14permil0Š ˆ 0which implies that for zero external inputs the homoge-neous state a(r iquest t) ˆ 0 for all r iquest t is a solution toequation (1) From the discussion in frac12 1(d) we take thepattern of connections w(r iquestjr0 iquest0) to satisfy thefollowing properties (see centgure 1)

(i) There exists a mixture of local connections within ahypercolumn and (anisotropic) lateral connectionsbetween hypercolumns the latter only connectelements with the same orientation preference Thusin the continuum model w is decomposed as

w(r iquestjr0 iquest0) ˆ wloc(iquest iexcl iquest0)macr(r iexcl r0)

Dagger wlat(r iexcl r0 iquest)macr(iquest iexcl iquest0) (3)

with wloc( iexcl iquest) ˆ wloc(iquest)(ii) Lateral connections between hypercolumns only

join neurons that lie along the direction of their(common) orientation preference iquest Thus in thecontinuum model

wlat(r iquest) ˆ w(Riexcliquestr) (4)

with

w(r) ˆ1

0

g(s)permilmacr(r iexcl sr0) Dagger macr(r Dagger sr0)Šds (5)

where r0 ˆ (1 0) and Rsup3 is the rotation matrix



(a)

(b)

Figure 10 Lateral connections made by cells in (a) owlmonkey and (b) tree shrew V1 A radioactive tracer is used toshow the locations of all terminating axons from cells in acentral injection site superimposed on an orientation mapobtained by optical imaging Redrawn from G G Blasdel andL Sincich (personal communication) and Bosking et al (1997)

Rsup3

xy

ˆcos sup3 iexcl sin sup3

sin sup3 cos sup3

xy

The weighting function g(s) determines how thestrength of lateral connections varies with the distanceof separation We take g(s) to be of theparticular form

g(s) ˆ permil2ordmsup12latŠiexcl1=2exp iexcl s22sup12

lat iexclAlatpermil2ordmsup12latŠiexcl1=2

pound exp iexcl s22sup12lat (6)

with sup1lat5sup1lat and Alat4 1 which represent a combi-nation of short-range excitation and long-rangeinhibition This is an example of the Mexican hatdistribution (Note that one can view the short-range excitatory connections as arising from patchylocal connections within a hypercolumn)

It is possible to consider more general choices of weightdistribution w that (i) allow for some spread in thedistribution of lateral connections (see centgure 12) and(ii) incorporate spatially extended isotropic local inter-actions An example of such a distribution is given by thefollowing generalization of equations (3) and (4)

w(r iquestjr0 iquest0) ˆ wloc(iquest iexcl iquest0)centloc(jr iexcl r0j)Dagger w(Riexcliquestpermilr0 iexcl rŠ)centlat(iquest iexcl iquest0) (7)

with centlat(iexcl iquest) ˆ centlat(iquest) centlat(iquest) ˆ 0 for jiquestj4iquest0 andcentloc(jrj) ˆ 0 for r4sup10 Moreover equation (5) is modi-cented according to

w(r) ˆsup30

iexclsup30

p(sup3)1

0

g(s)permilmacr(r iexcl srsup3) Dagger macr(r Dagger srsup3)Šdsdsup3

(8)

with rsup3 ˆ (cos (sup3) sin (sup3)) and p(iexcl sup3) ˆ p(sup3) The para-meters iquest0 and sup30 determine the angular spread of lateralconnections with respect to orientation preference and

space respectively whereas sup10 determines the (spatial)range of the isotropic local connections

(b) Euclidean symmetrySuppose that the weight distribution w satiscentes

equations (7) and (8) We show that w is invariant underthe action of the Euclidean group E(2) of rigid motions inthe plane and discuss some of the important conse-quences of such a symmetry

(i) Euclidean group actionThe Euclidean group is composed of the (semi-direct)

product of O(2) the group of planar rotations and repoundec-tions with R2 the group of planar translations Theaction of the Euclidean group on R2 pound S1 is generated by

s cent (r iquest) ˆ (r Dagger s iquest) s 2 R2

sup3 cent (r iquest) ˆ (Rsup3 r iquest Dagger sup3) sup3 2 S1

micro cent (r iquest) ˆ (micror iexcl iquest)(9)

where micro is the repoundection (x1 x2) 7 (x1 iexcl x2) and Rsup3 is arotation by sup3

The corresponding group action on a functionaR2 pound S1 R where P ˆ (r iquest) is given by

reg cent a(P) ˆ a(regiexcl1 cent P) (10)

for all reg 2 O(2) _Dagger R2 and the action on w(PjP 0) is

reg cent w(P jP 0) ˆ w(regiexcl1 cent P jregiexcl1 cent P 0)

The particular form of the action of rotations inequations (9) repoundects a crucial feature of the lateralconnections namely that they tend to be orientated alongthe direction of their cellrsquos preference (see centgure 11) Thusif we just rotate V1 then the cells that are now connectedat long range will not be connected in the direction oftheir preference This dicurrenculty can be overcome bypermuting the local cells in each hypercolumn so thatcells that are connected at long range are againconnected in the direction of their preference Thus inthe continuum model the action of rotation of V1 by sup3corresponds to rotation of r by sup3 while simultaneouslysending iquest to iquest Dagger sup3 This is illustrated in centgure 13 Theaction of repoundections is justicented in a similar fashion

(ii) Invariant weight distribution wWe now prove that w as given by equations (7) and (8) is

invariant under the action of the Euclidean group decentnedby equations (9) (It then follows that the distribution



hypercolumn

lateral connections

local connections

Figure 11 Illustration of the local connections within ahypercolumn and the anisotropic lateral connections betweenhypercolumns

q 0

0f

0f

0f

Figure 12 Example of an angular spread in the anisotropiclateral connections between hypercolumns with respect toboth space (sup30) and orientation preference (iquest0)

satisfying equations (3)^(5) is also Euclidean invariant)Translation invariance of w is obvious ie

w(r iexcl s iquestjr0 iexcl s iquest0) ˆ w(r iquestjr0iquest0)

Invariance with respect to a rotation by sup3 follows from

w(Riexclsup3r iquest iexcl sup3jRiexclsup3r0 iquest0 iexcl sup3)

ˆ wloc(iquest iexcl iquest0)centloc(jRiexclsup3permilr iexcl r0Šj)Dagger w(RiexcliquestDaggersup3Riexclsup3(r iexcl r0))centlat(iquest iexcl iquest0)

ˆ wloc(iquest iexcl iquest0)centloc(jr iexcl r0j) Dagger w(Riexcliquestr)centlat(iquest iexcl iquest0)

ˆ w(r iquestjr0 iquest0)

Finally invariance under a repoundection micro about the x-axisholds because

w(micror iexcl iquestjmicror0 iexcl iquest0) ˆ wloc( iexcl iquest Dagger iquest0)centloc(jmicropermilr iexcl r0Šj)Dagger w(Riquestmicro(r iexcl r0))centlat( iexcl iquest Dagger iquest0)

ˆ wloc(iquest iexcl iquest0)centloc(jr iexcl r0j)Dagger w(microRiquest(r iexcl r0))centlat(iquest iexcl iquest0)

ˆ wloc(iquest iexcl iquest0)centloc(jr iexcl r0j)Dagger w(Riexcliquest(r iexcl r0))centlat(iquest iexcl iquest0)


We have used the identity microRiexcliquest ˆ Riquestmicro and the conditionswloc( iexcl iquest) ˆ wloc(iquest) centlat( iexcl iquest) ˆ centlat(iquest) w(micror) ˆ w(r)

(iii) Implications of Euclidean symmetryConsider the action of reg on equation (1) for h(r t) ˆ 0

a(regiexcl1P t)t

ˆ iexcl nota(regiexcl1P t)Dagger middotR2poundS1

w(regiexcl1PjP0)frac14permila(P0 t)ŠdP0

ˆ iexcl nota(regiexcl1P t) Dagger middotR2poundS1

w(PjregP0)frac14permila(P 0 t)ŠdP 0


w(PjP00)frac14permila(regiexcl1P00 t)ŠdP00

since dpermilregiexcl1PŠ ˆ sectdP and w is Euclidean invariant If werewrite equation (1) as an operator equation namely

FpermilaŠ sup2 dadt

iexcl GpermilaŠ ˆ 0

then it follows that regFpermilaŠ ˆ FpermilregaŠ Thus F commutes withreg 2 E(2) and F is said to be equivariant with respect to the

symmetry group E(2) (Golubitsky et al 1988) The equiv-ariance of the operator F with respect to the action ofE(2) has major implications for the nature of solutionsbifurcating from the homogeneous resting state Let middot bea bifurcation parameter We show in frac12 4 that near a pointfor which the steady state a(r iquest middot) ˆ 0 becomes unstablethere must exist smooth solutions to the equilibrium equa-tion Gpermila(r iquest middot)Š ˆ 0 that are identicented by theirsymmetry (Golubitsky et al 1988) We centnd solutions thatare doubly periodic with respect to a rhombic square orhexagonal lattice by using the remnants of Euclideansymmetry on these lattices These remnants are the (semi-direct) products iexcl of the torus T 2 of translations modulothe lattice with the dihedral groups D2 D4 and D6 theholohedries of the lattice Thus when a(r iquest middot) ˆ 0becomes unstable new solutions emerge from theinstability with symmetries that are broken comparedwith iexcl Sucurrenciently close to the bifurcation point thesepatterns are characterized by (centnite) linear combinationsof eigenfunctions of the linear operator L ˆ D0Gobtained by linearizing equation (1) about the homoge-neous state a ˆ 0 These eigenfunctions are derived in frac12 3

(c) Two limiting casesFor the sake of mathematical convenience we restrict

our analysis in this paper to the simpler weight distribu-tion given by equations (3) and (4) with w satisfyingeither equation (5) or (8) The most important property ofw is its invariance under the extended Euclidean groupaction (9) which is itself a natural consequence of theanisotropic pattern of lateral connections Substitution ofequation (3) into equation (1) gives (for zero externalinputs)

a(r iquest t)t

ˆ iexcl nota(r iquest t)Dagger middotordm

0

wloc(iquest iexcliquest0)frac14permila(r iquest0 t)Šdiquest0

ordm

Dagger shyR2

wlat(r iexcl r0 iquest)frac14permila(r0 iquest t)Šdr0 (11)

where we have introduced an additional coupling para-meter shy that characterizes the relative strength of lateralinteractions Equation (11) is of convolution type in thatthe weighting functions are homogeneous in their respec-tive domains However the weighting function wlat(r iquest)is anisotropic as it depends on iquest Before proceeding to



- 1 05 0 05 1- 1

05-

- -

-

0

05

1two points P Q Icirc R2 acute [0 p )

- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f

Figure 13 Action of a rotation bysup3 (r iquest) (r0 iquest0) ˆ (Rsup3r iquest Dagger sup3)

analyse the full model described by equation (11) it isuseful to consider two limiting cases namely the ringmodel of orientation tuning and the Ermentrout^Cowanmodel (Ermentrout amp Cowan 1979)

(i) The ring model of orientation tuningThe centrst limiting case is to neglect lateral connections

completely by setting shy ˆ 0 in equation (11) Each point rin the cortex is then independently described by the so-called ring model of orientation tuning (Hansel ampSompolinsky 1997 Mundel et al 1997 Ermentrout 1998Bressloiexcl et al 2000a)

a(r iquest t)t


0wloc(iquest iexcl iquest0)

pound frac14permila(r iquest0 t)Š diquest0

ordm (12)

Linearizing this equation about the homogeneous statea(r iquest t) sup2 0 and considering perturbations of the forma(r iquest t) ˆ elta(r iquest) yields the eigenvalue equation

la(r iquest) ˆ iexclnota(r iquest) Dagger middotordm

0

wloc(iquest iexcl iquest0)a(r iquest0)diquest0

ordm

Introducing the Fourier series expansion a(r iquest)ˆ m zm(r) e2imiquest Dagger cc generates the following discretedispersion relation for the eigenvalue l

l ˆ iexclnot Dagger frac141middotWm sup2 lm (13)

where frac141 ˆ dfrac14permilzŠdz evaluated at z ˆ 0 and

wloc(iquest) ˆn2Z

Wne2niiquest (14)

Note that because wloc(iquest) is a real and even function of iquestWiexclm ˆ Wm ˆ Wm

Let Wp ˆ maxfWn n 2 ZDagger g and suppose that p isunique with Wp40 and p51 It then follows fromequation (13) that the homogeneous state a(r iquest) ˆ 0 isstable for sucurrenciently small middot but becomes unstable whenmiddot increases beyond the critical value middotc ˆ notfrac141Wp due toexcitation of linear eigenmodes of the forma(r iquest) ˆ z(r)e2ipiquest Dagger z(r)eiexcl2ipiquest where z(r) is an arbitrarycomplex function of r It can be shown that the saturatingnonlinearities of the system stabilize the growing patternof activity (Ermentrout 1998 Bressloiexcl et al 2000a) Interms of polar coordinates z(r) ˆ Z(r)e2iiquest(r) we havea(r iquest) ˆ Z(r) cos (2ppermiliquest iexcl iquest(r)Š) Thus at each point r inthe plane the maximum (linear) response occurs at theorientations iquest(r) Dagger kordmp k ˆ 0 1 p iexcl 1 when p 6ˆ 0

Of particular relevance from a biological perspectiveare the cases p ˆ 0 and p ˆ 1 In the centrst case there is abulk instability in which the new steady state shows noorientation preference Any tuning is generated in thegenicocortical map We call this the `Hubel^Wieselrsquomode (Hubel amp Wiesel 1974a) In the second case theresponse is unimodal with respect to iquest The occurrence ofa sharply tuned response peaked at some angle iquest(r) in alocal region of V1 corresponds to the presence of a localcontour there the orientation of which is determined bythe inverse of the double retinocortical map described infrac12 5(a) An example of typical tuning curves is shown in

centgure 14 which is obtained by taking wloc(iquest) to be adiiexclerence of Gaussians over the domain permiliexclordm2 ordm2Š

wloc(iquest) ˆ permil2ordmsup12locŠiexcl1=2exp(iexcl iquest22sup12

loc) iexcl Alocpermil2ordmsup12locŠiexcl1=2

pound exp(iexcl iquest22sup12loc) (15)

with sup1loc5sup1loc and Aloc4 1The location of the centre iquest(r) of each tuning curve is

arbitrary which repoundects the rotational equivariance ofequation (12) under the modicented group actionsup3 (r iquest) (r iquest Dagger sup3) Moreover in the absence oflateral interactions the tuned response is uncorrelatedacross diiexclerent points in V1 In this paper we show howthe presence of anisotropic lateral connections leads toperiodic patterns of activity across V1 in which the peaksof the tuning curve at diiexclerent locations are correlated

(ii) The Ermentrout^Cowan modelThe other limiting case is to neglect the orientation

label completely Equation (11) then reduces to a one-population version of the model studied by Ermentrout ampCowan (1979)

ta(r t) ˆ iexclnota(r t) Dagger cedil

O

wlat(riexclr0)frac14 a(r0 t) dr0 (16)

In this model there is no reason to distinguish any direc-tion in V1 so we assume that wlat(r iexcl r0) wlat(jr iexcl r0j)ie wlat depends only on the magnitude of r iexcl r0 It canbe shown that the resulting system is equivariant withrespect to the standard action of the Euclidean group inthe plane

Linearizing equation (16) about the homogeneous stateand taking a(r t) ˆ elta(r) gives rise to the eigenvalueproblem

la(r) ˆ iexclnota(r) Dagger cedilfrac141O

wlat(jr iexcl r0j)a(r0)dr0



orientation f

(a)

s

1

2

3

4

p0 2p

Figure 14 Sharp orientation tuning curves for a Mexican hatweight kernel with sup1 loc ˆ 208 sup1 loc ˆ 608 and Aloc ˆ 1 Thetuning curve is marginally stable so that the peak activity a ateach point in the cortical plane is arbitrary The activity istruncated at frac14 ˆ 0 in line with the choice of frac14permil0Š ˆ 0

which upon Fourier transforming generates a dispersionrelation for the eigenvalue l as a function of q ˆ jkj ie

l ˆ iexclnot Dagger cedilfrac141W(q) sup2 l(q)

where W(q) ˆ wlat(k) is the Fourier transform ofwlat(jrj) Note that l is real If we choose wlat(jrj) to be inthe form of a Mexican hat function then it is simple toestablish that l passes through zero at a critical para-meter value cedilc signalling the growth of spatially periodicpatterns with wavenumber qc where W(qc)ˆ maxqfW(q)g Close to the bifurcation point thesepatterns can be represented as linear combinations ofplane waves

a(r) î

ci exp(iki cent r)

with jki j ˆ qc As shown by Ermentrout amp Cowan (1979)and Cowan (1982) the underlying Euclidean symmetryof the weighting function together with the restriction todoubly periodic functions then determines the allowablecombinations of plane waves comprising steady- state solu-tions In particular stripe chequer-board and hexagonalpatterns of activity can form in the V1 map of the visualcenteld In this paper we generalize the treatment byErmentrout amp Cowan (1979) to incorporate the eiexclects oforientation preferenceoumland show how plane waves ofcortical activity modulate the distribution of tuningcurves across the network and lead to contoured patterns

3 LINEAR STABILITY ANALYSIS

The centrst step in the analysis of pattern-forminginstabilities in the full cortical model is to linearize equa-tion (11) about the homogeneous solution a(r iquest) ˆ 0 andto solve the resulting eigenvalue problem In particularwe wish to centnd conditions under which the homogeneoussolution becomes marginally stable due to the vanishingof one of the (degenerate) eigenvalues and to identify themarginally stable modes This will require performing aperturbation expansion with respect to the small para-meter shy characterizing the relative strength of the aniso-tropic lateral connections

(a) LinearizationWe linearize equation (11) about the homogeneous state

and introduce solutions of the form a(r iquest t) ˆ elta(r iquest)This generates the eigenvalue equation

la(r iquest) ˆ iexcl nota(r iquest) Dagger frac141middotordm

0


ordm

Dagger shyR2

wlat(r iexcl r0 iquest)a(r0 iquest)dr0 (17)

Because of translation symmetry the eigenvalue equation(17) can be written in the form

a(r iquest) ˆ u(iquest iexcl rsquo)eikcentr Dagger cc (18)

with k ˆ q(cos rsquo sin rsquo) and

lu(iquest) ˆ iexcl notu(iquest) Dagger frac141middotordm

0

wloc(iquest iexcl iquest0)u(iquest0)diquest0

ordm

Dagger shy wlat(k iquest Dagger rsquodaggeru(iquest) (19)

Here wlat(k iquest) is the Fourier transform of wlat(r iquest)Assume that wlat satiscentes equations (4) and (5) so that

the total weight distribution w is Euclidean invariant Theresulting symmetry of the system then restricts the struc-ture of the solutions of the eigenvalue equation (19)

(i) l and u(iquest) only depend on the magnitude q ˆ jkj ofthe wave vector k That is there is an incentnite degen-eracy due to rotational invariance

(ii) For each k the associated subspace of eigenfunctions

Vk ˆfu(iquest iexcl rsquo)eikcentr Dagger cc u(iquest Dagger ordm)ˆ u(iquest) and u 2 Cg(20)

decomposes into two invariant subspaces

Vk ˆ VDaggerk copy Viexcl

k (21)

corresponding to even and odd functions respec-tively

VDaggerk ˆ fv 2 Vk u(iexcl iquest) ˆ u(iquest)g

and

Viexclk ˆ fv 2 Vk u( iexcl iquest) ˆ iexclu(iquest)g (22)

As noted in greater generality by Bosch Vivancoset al (1995) this is a consequence of repoundection invar-iance as we now indicate That is let microk denoterepoundections about the wavevector k so that microkk ˆ kThen microka(r iquest)ˆ a(microkr 2rsquoiexcl iquest) ˆ u(rsquo iexcl iquest)eikcentrDagger ccSince microk is a repoundection any space that it acts ondecomposes into two subspaces one on which it actsas the identity I and one on which it acts as iexclI

Results (i) and (ii) can also be derived directly fromequation (19) For expanding the ordm-periodic function u(iquest)as a Fourier series with respect to iquest

u(iquest) ˆn2Z

Ane2niiquest (23)

and setting wlat(r iquest) ˆ w(Riexcliquestr) leads to the matrixeigenvalue equation

lAm ˆ iexclnotAm Dagger frac141middot WmAm Dagger shyn2Z

Wmiexcln(q)An (24)

with Wn given by equation (14) and

Wn(q) ôrdm

0eiexcl2iniquest

R2eiexcliqpermilx cos (iquest)Dagger y sin (iquest)Šw(r)dr

diquest

ordm

(25)

It is clear from equation (24) that item (i) holds Thedecomposition of the eigenfunctions into odd and eveninvariant subspaces (see equation (21) of item (ii)) is aconsequence of the fact that w(r) is an even function of xand y (see equation (5)) and hence Wn(q) ˆ Wiexcln(q)



(b) Eigenfunctions and eigenvaluesThe calculation of the eigenvalues and eigenfunctions

of the linearized equation (17) and hence the derivationof conditions for the marginal stability of the homoge-neous state has been reduced to the problem of solvingthe matrix equation (24) which we rewrite in the moreconvenient form

l Dagger not

frac141middotiexcl Wm Am ˆ shy

n2Z

Wmiexcln(q)An (26)

We exploit the experimental observation that the intrinsiclateral connections appear to be weak relative to the localconnections ie shy W frac12 W Equation (26) can then besolved by expanding as a power series in shy and usingRayleigh^Schrolaquo dinger perturbation theory

(i) Case shy ˆ 0In the limiting case of zero lateral interactions equation

(26) reduces to equation (13) Following the discussion ofthe ring model in frac12 2(c) let Wp ˆ maxfWn n 2 ZDagger g40and suppose that p ˆ 1 (unimodal orientation tuningcurves) The homogeneous state a(r iquest) ˆ 0 is then stablefor sucurrenciently small middot but becomes marginally stable atthe critical point middotc ˆ notfrac141W1 due to the vanishing ofthe eigenvalue l1 In this case there are both even andodd marginally stable modes cos(2iquest) and sin(2iquest)

(ii) Case shy 40If we now switch on the lateral connections then there

is a q-dependent splitting of the degenerate eigenvalue l1

that also separates out odd and even solutions Denotingthe characteristic size of such a splitting by macrl ˆ OO(shy )we impose the condition that macrl frac12 middotfrac141centW where

centW ˆ minfW1 iexcl Wm m 6ˆ 1g

This ensures that the perturbation does not excite statesassociated with other eigenvalues of the unperturbedproblem (see centgure 15) We can then restrict ourselves tocalculating perturbative corrections to the degenerateeigenvalue l1 and its associated eigenfunctions Thereforewe introduce the power series expansions

l Dagger not

frac141middotˆ W1 Dagger shy l(1) Dagger shy 2l(2) Dagger (27)

and

An ˆ zsect 1macrn sect 1 Dagger shy A(1)n Dagger shy 2A(2)

n Dagger (28)

where macrn m is the Kronecker delta function We substitutethese expansions into the matrix eigenvalue equation (26)and systematically solve the resulting hierarchy ofequations to successive orders in shy using (degenerate)perturbation theory This analysis which is carried out inAppendix A(a) leads to the following results (i) l ˆ lsect

for even (Dagger ) and odd (iexcl) solutions where to OO(shy 2)

l sect Dagger not

frac141middotˆW1 Dagger shy W0(q) sect W2(q)

Dagger shy 2

m5 0 m 6ˆ1

permilWmiexcl1(q) sect WmDagger 1(q)Š2

W1 iexcl Wm

sup2 Gsect (q) (29)

and (ii) u(iquest) ˆ usect (iquest) where to OO(shy )

uDagger (iquest) ˆ cos(2iquest) Dagger shym5 0 m 6ˆ1

uDaggerm (q) cos(2miquest) (30)

uiexcl(iquest) ˆ sin(2iquest) Dagger shym41

uiexclm (q) sin(2miquest) (31)

with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ

Wmiexcl1(q) sect WmDagger 1(q)W1 iexcl Wm

m41

(32)

(c) Marginal stabilitySuppose that Gsect (q) has a unique maximum at

q ˆ qsect 6ˆ 0 and let qc ˆ qDagger if GDagger (qDagger )4Giexcl(qiexcl) andqc ˆ qiexcl if Giexcl(qiexcl)4GDagger (qDagger ) Under such circumstancesthe homogeneous state a(r iquest) ˆ 0 will become margin-ally stable at the critical point middotc ˆ notfrac141Gsect (qc) and themarginally stable modes will be of the form

a(r iquest) ˆN

iˆ1

cieiki centru(iquest iexcl rsquoi) Dagger cc (33)

where ki ˆ qc(cos rsquoi sin rsquoi) and u(iquest) ˆ usect (iquest) for qc ˆ qsect The incentnite degeneracy arising from rotation invariancemeans that all modes lying on the circle jkj ˆ qc becomemarginally stable at the critical point However this canbe reduced to a centnite set of modes by restricting solutionsto be doubly periodic functions The types of doubly peri-odic solutions that can bifurcate from the homogeneousstate will be determined in frac12 4

As a specicentc example illustrating marginal stability letw(r) be given by equation (5) Substitution into equation(25) gives

Wn(q) ôrdm

0eiexcl2iniquest

1

0

g(s) cos(sq cos iquest)dsdiquest

ordm

Using the JacobiÂnger expansion

cos(sq cos iquest) ˆ J0(sq) Dagger 21

mˆ1

( iexcl 1)mJ2m(sq) cos(2miquest)



D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo

Figure 15 Splitting of degenerate eigenvalues due toanisotropic lateral connections between hypercolumns

with Jn(x) the Bessel function of integer order n wederive the result

Wn(q) ˆ (iexcl 1)n1

0

g(s)J2n(sq)ds (34)

Next we substitute equation (6) into (34) and use standardproperties of Bessel functions to obtain

Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4

iexcl Alatexpiexclsup1 2

latq2

4In

sup1 2latq

2

4 (35)

where In is a modicented Bessel function of integer order nThe resulting marginal stability curves middot ˆ middotsect (q)ˆ

notfrac141Gsect (q) are plotted to centrst order in shy in centgure 16a Theexistence of a non-zero critical wavenumber qc ˆ qiexcl atmiddotc ˆ middotiexcl(qc) is evident indicating that the marginallystable eigenmodes are odd functions of iquest The inclusion ofhigher-order terms in shy does not alter this basic result atleast for small shy If we take the fundamental unit oflength to be ca 400 mm then the wavelength of a patternis 2ordm(0400)qc mm ca 266 mm at the critical wave-number qc ˆ 1 (see centgure 16b)

An interesting question concerns under what circum-stances can even patterns be excited by a primaryinstability rather than odd in the regime of weak lateralinteractions One example occurs when there is a sucurren-cient spread in the distribution of lateral connectionsalong the lines shown in centgure 12 In particular supposethat w(r) is given by equation (8) with p(sup3) ˆ 1 for sup34 sup30

and zero otherwise Equation (34) then becomes

Wn(q) ˆ (iexcl 1)n sin(2nsup30)2nsup30

1

0

g(s)J2n(sq)ds (36)

To centrst order in shy the size of the gap between the oddand even eigenmodes at the critical point qc is deter-mined by 2W2(qc) (see equation 29) It follows that ifsup304ordm4 then W2(q) reverses sign suggesting that evenrather than odd eigenmodes become marginally stablecentrst This is concentrmed by the marginal stability curvesshown in centgure 17

(i) Choosing the bifurcation parameterIt is worth commenting at this stage on the choice of

bifurcation parameter middot One way to induce a primaryinstability of the homogeneous state is to increase theglobal coupling parameter middot in equation (29) until thecritical point middotc is reached However it is clear fromequation (29) that an equivalent way to induce such aninstability is to keep middot centxed and increase the slope frac141 ofthe neural output function frac14 The latter could be achievedby keeping a non-zero uniform input h(r iquest t) ˆ h0 inequation (1) so that the homogeneous state is non-zeroa(r iquest t) ˆ a0 6ˆ 0 with frac141 ˆ frac14 0(a0) Then variation of theinput h0 and consequently frac141 corresponds to changingthe eiexclective neural threshold and hence the level ofnetwork excitability Indeed this is thought to be one ofthe possible eiexclects of hallucinogens In summary themathematically convenient choice of middot as the bifurcationparameter can be reinterpreted in terms of biologicallymeaningful parameter variations It is also possible thathallucinogens act indirectly on the relative levels of inhi-bition and this could also play a role in determining thetype of patterns that emergeoumla particular example isdiscussed below



1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)

Figure 16 (a) Plot of marginal stability curves middotsect (q) for g(s)given by the diiexclerence of Gaussians (equation (6)) withsup1 lat ˆ 1 sup1 lat ˆ 3 Alat ˆ 1 and shy ˆ 04W1 Also setnotfrac141W1 ˆ 1 The critical wavenumber for spontaneouspattern formation is qc The marginally stable eigenmodes areodd functions of iquest (b) Plot of critical wavenumber qsect formarginal stability of even ( + ) and odd (7) patterns as afunction of the strength of inhibitory coup ling Alat If theinhibition is too weak then there is a bulk instability withrespect to the spatial domain

m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090

Figure 17 Same as centgure 16 except that W(q) satiscentesequation (36) with sup30 ˆ ordm3 rather than equation (34) It canbe seen that the marginally stable eigenmodes are now evenfunctions of iquest

(d) The Ermentrout^Cowan model revisitedThe marginally stable eigenmodes (equation (33))

identicented in the analysis consist of spatially periodicpatterns of activity that modulate the distribution oforientation tuning curves across V1 Examples of thesecontoured cortical planforms will be presented in frac12 4 andthe corresponding hallucination patterns in the visualcenteld (obtained by applying an inverse retinocortical map)will be constructed in frac12 5 It turns out that the resultingpatterns account for some of the more complicated formconstants where contours are prominent includingcobwebs honeycombs and lattices (centgure 4) Howeverother form constants such as chequer-boards funnels andspirals (centgures 6 and 7) comprise contrasting regions oflight and dark One possibility is that these hallucinationsare a result of higher-level processes centlling in thecontoured patterns generated in V1 An alternative expla-nation is that such regions are actually formed in V1 itselfby a mechanism similar to that suggested in the originalErmentrout^Cowan model This raises the interestingissue as to whether or not there is some parameter regimein which the new model can behave in a similar fashionto the cortical retinarsquo of Ermentrout amp Cowan (1979)that is can cortical orientation tuning somehow beswitched oiexcl One possible mechanism is the followingsuppose that the relative level of local inhibition which isspecicented by the parameter Aloc in equation (15) isreduced (eg by the possible (indirect) action of halluci-nogens) Then W0 ˆ maxfWn n 2 ZDagger g rather than W1and the marginally stable eigenmodes will consist ofspatially periodic patterns that modulate bulk instabiliti-ties with respect to orientation

To make these ideas more explicit we introduce theperturbation expansions

l Dagger not


and

An ˆ zmacrn0 Dagger shy A(1)n Dagger shy 2A(2)

n Dagger (38)

Substituting these expansions into the matrix eigenvalueequation (26) and solving the resulting equations tosuccessive orders in shy leads to the following results

l Dagger not

frac141middotˆW0 Dagger shy W0(q) Dagger shy 2

m40

Wm(q)2

W0 iexcl WmDagger O(shy 3)

sup2 G0(q)

(39)

and

u(iquest) ˆ 1 Dagger shym40

u0m(q) cos(2miquestdagger Dagger O(shy 2) (40)

with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)

Substituting equation (40) into equation (33) shows thatthe marginally stable states are now only weakly depen-dent on the orientation iquest and to lowest order in shy simplycorrespond to the spatially periodic patterns of theErmentrout^Cowan model The length-scale of thesepatterns is determined by the marginal stability curvemiddot0(q) ˆ notfrac141G0(q) an example of which is shown incentgure 18

The occurrence of a bulk instability in orientationmeans that for sucurrenciently small shy the resulting corticalpatterns will be more like contrasting regions of light anddark rather than a lattice of oriented contours (see frac12 4)However if the strength of lateral connections shy wereincreased then the eigenfunctions (40) would develop asignicentcant dependence on the orientation iquest This couldthen provide an alternative mechanism for the generationof even contoured patternsoumlrecall from frac12 3(c) that onlyodd contoured patterns emerge in the case of a tunedinstability with respect to orientation unless there issignicentcant angular spread in the lateral connections

4 DOUBLY PERIODIC PLANFORMS

As we found in frac12 3 (c) and frac12 3 (d) rotation symmetryimplies that the space of marginally stable eigenfunctionsof the linearized Wilson^Cowan equation is incentnite-dimensional that is if u(iquest)eikcentr is a solution then so isu(iquest iexcl rsquo)eiRrsquokcentr However translation symmetry suggeststhat we can restrict the space of solutions of the nonlinearWilson^Cowan equation (11) to that of doubly periodicfunctions This restriction is standard in many treatmentsof spontaneous pattern formation but as yet it has noformal justicentcation There is however a wealth ofevidence from experiments on convecting pounduids andchemical reaction^diiexclusion systems (Walgraef 1997) andsimulations of neural nets (Von der Malsburg amp Cowan1982) which indicates that such systems tend to generatedoubly periodic patterns in the plane when the homoge-neous state is destabilized Given such a restriction theassociated space of marginally stable eigenfunctions isthen centnite-dimensional A centnite set of specicentc eigen-functions can then be identicented as candidate planformsin the sense that they approximate time-independentsolutions of equation (11) sucurrenciently close to the criticalpoint where the homogeneous state loses stability In thissection we construct such planforms



1 2 3 4 5

09

092

094

096

098

10

m

q

qc

Figure 18 Plot of marginal stability curve middot0(q) for a bulkinstability with respect to orientation and g(s) given by thediiexclerence of Gaussians (equation (6)) with sup1lat ˆ 1 sup1lat ˆ 3Alat ˆ 10 shy ˆ 04W0 and notfrac141W0 ˆ 1 The critical wave-number for spontaneous pattern formation is qc

(a) Restriction to doubly periodic solutionsLet be a planar lattice that is choose two linearly

independent vectors 1 and 2 and let

ˆ f2ordmm1 1 Dagger 2ordmm2 2 m1 m2 2 Zg

Note that is a subgroup of the group of planar trans-lations A function f R2 pound S1 R is doubly periodicwith respect to if

f (x Dagger iquest) ˆ f (x iquest)

for every 2 Let sup3 be the angle between the two basisvectors 1 and 2 We can then distinguish three types oflattice according to the value of sup3 square lattice (sup3 ˆ ordm2)rhombic lattice (05sup35ordm2 sup3 6ˆ ordm3) and hexagonal(sup3 ˆ ordm=3) After rotation the generators of the planarlattices are given in table 1 (for unit lattice spacing)

Restriction to double periodicity means that theoriginal Euclidean symmetry group is now restricted tothe symmetry group of the lattice iexcl ˆ H _Dagger T 2 whereH is the holohedry of the lattice the subgroup of Ohellip2)that preserves the lattice and T 2 is the two torus ofplanar translations modulo the lattice Thus the holo-hedry of the rhombic lattice is D2 the holohedry of thesquare lattice is D4 and the holohedry of the hexagonallattice is D6 Observe that the corresponding space ofmarginally stable modes is now centnite-dimensionaloumlwecan only rotate eigenfunctions through a centnite set ofangles (for example multiples of ordm2 for the square latticeand multiples of ordm3 for the hexagonal lattice)

It remains to determine the space K of marginallystable eigenfunctions and the action of iexcl on this spaceIn frac12 3 we showed that eigenfunctions either reside in VDagger

k(the even case) or Viexcl

k (the odd case) where the length ofk is equal to the critical wavenumber qc In particularthe eigenfunctions have the form u(iquest iexcl rsquo)eikcentr where u iseither an odd or even eigenfunction We now choose thesize of the lattice so that eikcentr is doubly periodic withrespect to that lattice ie k is a dual wave vector for thelattice In fact there are incentnitely many choices for thelattice size that satisfy this constraintoumlwe select the onefor which qc is the shortest length of a dual wave vectorThe generators for the dual lattices are also given intable 1 with qc ˆ 1 The eigenfunctions corresponding todual wave vectors of unit length are given in table 2 Itfollows that KL can be identicented with the m-dimensionalcomplex vector space spanned by the vectors(c1 cm) 2 Cm with m ˆ 2 for square or rhombiclattices and m ˆ 3 for hexagonal lattices It can be shownthat these form iexcl -irreducible representations Theactions of the group iexcl on K can then be explicitlywritten down for both the square or rhombic and hexa-gonal lattices in both the odd and even cases Theseactions are given in Appendix A(b)

(b) PlanformsWe now use an important result from bifurcation

theory in the presence of symmetries namely the equi-variant branching lemma (Golubitsky et al 1988) For ourparticular problem the equivariant branching lemmaimplies that generically there exists a (unique) doublyperiodic solution bifurcating from the homogeneous statefor each of the axial subgroups of iexcl under the action(9)oumla subgroup S raquo iexcl is axial if the dimension of thespace of vectors that are centxed by S is equal to unity Theaxial subgroups are calculated from the actions presentedin Appendix A(b) (see Bressloiexcl et al (2000b) for details)and lead to the even planforms listed in table 3 and theodd planforms listed in table 4 The generic planformscan then be generated by combining basic properties ofthe Euclidean group action (equation (9)) on doubly peri-odic functions with solutions of the underlying lineareigenvalue problem The latter determines both thecritical wavenumber qc and the ordm-periodic function u(iquest)In particular the perturbation analysis of frac12 3(c) andfrac12 3(d) shows that (in the case of weak lateral interactions)u(iquest) can take one of three possible forms

(i) even contoured planforms (equation (30)) withu(iquest) ordm cos(2iquest)

(ii) odd contoured planforms (equation (31)) withu(iquest) ordm sin(2iquest)

(iii) even non-contoured planforms (equation (40)) withu(iquest) ordm 1

Each planform is an approximate steady-state solutiona(r iquest) of the continuum model (equation (11)) decentned onthe unbounded domain R2 pound S1 To determine how thesesolutions generate hallucinations in the visual centeld wecentrst need to interpret the planforms in terms of activitypatterns in a bounded domain of V1 which we denote byM raquo R Once this has been achieved the resultingpatterns in the visual centeld can be obtained by applyingthe inverse retinocortical map as described in frac12 5(a)

The interpretation of non-contoured planforms isrelatively straightforward since to lowest order in shy thesolutions are iquest-independent and can thus be directlytreated as activity patterns a(r) in V1 with r 2 M At thesimplest level such patterns can be represented ascontrasting regions of light and dark depending onwhether a(r)40 or a(r)50 These regions form squaretriangular or rhombic cells that tile M as illustrated incentgures 19 and 20

The case of contoured planforms is more subtle At agiven location r in V1 we have a sum of two or threesinusoids with diiexclerent phases and amplitudes (see tables 3and 4) which can be written as a(r iquest) ˆ A(r) cospermil2iquestiexcl 2iquest0(r)Š The phase iquest0(r) determines the peak of the



Table 1 Generators for the planar lattices and their duallattices

lattice 1 2 k1 k2

square (1 0) (0 1) (1 0) (0 1)hexagonal (1 1 3

p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p

)rhombic (1 iexclcot sup2) (0 cosec sup2) (1 0) (cos sup2 sin sup2)

Table 2 Eigenfunctions corresponding to shortest dual wavevectors

lattice a(r iquest)

square c1u(iquest)eik1 cent r Dagger c2u(iquest iexcl ordm2)eik2 cent r Dagger cchexagonal c1u(iquest)eik1 cent r Dagger c2u(iquest iexcl 2ordm3)eik3 cent r

Dagger c3u(iquest Dagger 2ordm3)eik3 cent rDagger ccrhombic c1u(iquest)eik1 cent r Dagger c2u(iquestiexcl sup2)eik2 cent r Dagger cc

orientation tuning curve at r (see centgure 14) Hence thecontoured solutions generally consist of iso-orientationregions or patches over which iquest0(r) is constant but theamplitude A(r) varies As in the non-contoured case thesepatches are either square triangular or rhombic in shapeHowever we now show each patch to be represented by alocally orientated contour centred at the point of maximalamplitude A(rmax) within the patch The resulting odd andeven contoured patterns are shown in centgures 21 and 22 forthe square latttice in centgures 23 and 24 for the rhombiclatttice and in centgures 25 and 26 for the hexagonal latticeNote that our particular interpretation of contoured plan-forms breaks down in the case of an odd triangle on ahexagonal lattice the latter comprises hexagonal patchesin which all orientations are present with equal magni-tudes In this case we draw a starrsquo shape indicating thepresence of multiple orientations at a given point seecentgure 26b Note that this planform contains the well-known pinwheelsrsquodescribed by Blasdel (1992)

5 FROM CORTICAL PATTERNS TO VISUAL

HALLUCINATIONS

In frac12 4 we derived the generic planforms that bifurcatefrom the homogeneous state and interpreted them interms of cortical activity patterns In order to computewhat the various planforms look like in visual centeldcoordinates we need to apply an inverse retinocorticalmap In the case of non-contoured patterns this can becarried out directly using the (single) retinocortical mapintroduced in frac12 1(b) However for contoured planforms itis necessary to specify how to map local contours in thevisual centeld as well as positionoumlthis is achieved byconsidering a so-called double retinocortical mapAnother important feature of the mapping between V1

and the visual centeld is that the periodicity of the angularretinal coordinate sup3R implies that the y-coordinate in V1satiscentes cylindrical periodic boundary conditions (seecentgure 5) This boundary condition should be commensu-rate with the square rhombic or hexagonal latticeassociated with the doubly periodic planforms

(a) The double retinocortical mapAn important consequence of the introduction of orien-

tation as a cortical label is that the retinocortical mapdescribed earlier needs to be extended to cover themapping of local contours in the visual centeldoumlin eiexclect totreat them as a vector centeld Let iquestR be the orientation ofsuch a local contour and iquest its image in V1 What is theappropriate map from iquestR to iquest that must be added to themap zR z described earlier We note that a line in V1of constant slope tan iquest is a level curve of the equation

f (x y) ˆ y cos iquest iexcl x sin iquest

where (x y) are Cartesian coordinates in V1 Such a linehas a constant tangent vector

v ˆ cos iquest

xDagger sin iquest

y

The pre-image of such a line in the visual centeld assumingthe retinocortical map generated by the complexlogarithm is obtained by changing from cortical to retinalcoordinates via the complex exponential is

f (x y) ~f (rR sup3R) ˆ sup3R cos iquest iexcl log rR sin iquest

the level curves of which are the logarithmic spirals

rR(sup3R) ˆ A exp(cot (iquest)sup3R)



Table 3 Even planforms with u(iexcl iquest) ˆ u(iquest)

(The hexagon solutions (0) and (ordm) have the same isotropy subgroup but they are not conjugate solutions)

lattice name planform eigenfunction

square even square u(iquest) cos x Dagger u(iquest iexcl ordm2) cos yeven roll u(iquest) cos x

rhombic even rhombic u(iquest) cos(k1 cent r) Dagger u(iquest iexcl sup2) cos(k2 cent r)even roll u(iquest) cos(k1 cent r)

hexagonal even hexagon (0) u(iquest) cos(k1 cent r) Dagger u(iquest Dagger ordm3) cos(k2 cent r) Dagger u(iquest iexcl ordm3) cos(k3 cent r)even hexagon (ordm) u(iquest) cos(k1 cent r) Dagger u(iquest Dagger ordm3) cos(k2 cent r) iexcl u(iquest iexcl ordm3) cos(k3 cent r)even roll u(iquest) cos(k1 cent r)

Table 4 Odd planforms with u(iexcl iquest) ˆ iexclu(iquest)


square odd square u(iquest) cos x iexcl u(iquest iexcl ordm2) cos yodd roll u(iquest) cos x

rhombic odd rhombic u(iquest) cos(k1 cent r) Dagger u(iquest iexcl sup2) cos(k2 cent r)odd roll u(iquest) cos(k1 cent r)

hexagonal odd hexagon u(iquest) cos(k1 cent r) Dagger u(iquest Dagger ordm3) cos(k2 cent r) Dagger u(iquest iexcl ordm3) cos(k3 cent r)triangle u(iquest) sin(k1 cent r) Dagger u(iquest Dagger ordm3) sin(k2 cent r) Dagger u(iquest iexcl ordm3) sin(k3 cent r)patchwork quilt u(iquest Dagger ordm3) cos(k2 cent r) iexcl u(iquest iexcl ordm3) cos(k3 cent r)odd roll u(iquest) cos(k1 cent r)



(b)

(a)

Figure 21 Contours of even axial eigenfunctions on thesquare lattice (a) square (b) roll

(b)

(a)

Figure 19 Non-contoured axial eigenfunctions on the squarelattice (a) square (b) roll

(b)

(a)

Figure 20 Non-contoured axial eigenfunctions on rhombicand hexagonal lattices (a) rhombic (b) hexagonal

(b)

(a)

Figure 22 Contours of odd axial eigenfunctions on the squarelattice (a) square (b) roll



(a)

(b)

Figure 25 Contours of even axial eigenfunctions on thehexagonal lattice (a) ordm-hexagonal (b) 0-hexagonal

(a)

(b)

Figure 26 Contours of odd axial eigenfunctions on thehexagonal lattice (a) triangular (b) 0-hexagonal

(b)

(a)

Figure 23 Contours of even axial eigenfunctions on therhombic lattice (a) rhombic (b) roll

(a)

(b)

Figure 24 Contours of odd axial eigenfunctions on therhombic lattice (a) rhombic (b) roll

It is easy to show that the tangent vector correspondingto such a curve takes the form

~v ˆ rR cos(iquest Dagger sup3R)

xRDagger rR sin(iquest Dagger sup3R)

yR

Thus the retinal vector centeld induced by a constant vectorcenteld in V1 twists with the retinal angle sup3R and stretcheswith the retinal radius rR It follows that if iquestR is theorientation of a line in the visual centeld then

iquest ˆ iquestR iexcl sup3R (42)

ie local orientation in V1 is relative to the angular coor-dinate of visual centeld position The geometry of the abovesetup is shown in centgure 27

The resulting double map fzR iquestRg fz iquestg has veryinteresting properties As previously noted the mapzR z takes circles rays and logarithmic spirals intovertical horizontal and oblique lines respectively Whatabout the extended map Because the tangent to a circle ata given point is perpendicular to the radius at that pointfor circles iquestR ˆ sup3R Dagger ordm2 so that iquest ˆ ordm2 Similarly forrays iquestR ˆ sup3R so iquest ˆ 0 For logarithmic spirals we canwrite either sup3R ˆ a ln rR or rR ˆ exppermilbsup3RŠ In retinal coor-dinates we centnd the somewhat cumbersome formula

tan iquestR ˆbrR sin sup3R Dagger ebsup3R cos sup3R

brR cos sup3R iexcl ebsup3R sin sup3R

However this can be rewritten as tan (iquestR iexcl sup3R) ˆ a sothat in V1 coordinates tan iquest ˆ a Thus we see that thelocal orientations of circles rays and logarithmic spiralsmeasured in relative terms all lie along the corticalimages of such forms Figure 28 shows the details

(b) Planforms in the visual centeldIn order to generate a visual centeld pattern we split our

model V1 domain M into two pieces each running72 mm along the x direction and 48 mm along the ydirection representing the right and left hemicentelds in thevisual centeld (see centgure 5) Because the y coordinate corre-sponds to a change from iexclordm2 to ordm2 in 72 mm whichmeets up again smoothly with the representation in theopposite hemicenteld we must only choose scalings androtations of our planforms that satisfy cylindrical periodicboundary conditions in the y direction In the x directioncorresponding to the logarithm of radial eccentricity weneglect the region immediately around the fovea and also

the far edge of the periphery so we have no constraint onthe patterns in this direction

Recall that each V1 planform is doubly periodic withrespect to a spatial lattice generated by two lattice vectors1 2 The cylindrical periodicity is thus equivalent to

requiring that there be an integral combination of latticevectors that spans Y ˆ 96 mm in the y direction with nochange in the x direction

096

ˆ 2ordmm1 1 Dagger 2ordmm2 2 (43)

If the acute angle of the lattice sup20 is specicented then thewavevectors ki are determined by the requirement

ki cent j ˆ1 i ˆ j0 i 6ˆ j (44)

The integral combination requirement limits which wave-lengths are permitted for planforms in the cortex Thelength-scale for a planform is given by the length of thelattice vectors j 1j ˆ j 2 j ˆ j j

j j ˆ96

m21 Dagger 2m1m2 cos(sup20) Dagger m2

2

(45)

The commonly reported hallucination patterns usuallyhave 30^40 repetitions of the pattern around a circumfer-ence of the visual centeld corresponding to length-scalesranging from 24^32 mm Therefore we would expectthe critical wavelength 2ordmqc for bifurcations to be in thisrange (see frac12 3(c)) Note that when we rotate the planformto match the cylindrical boundary conditions we rotatek1 and hence the maximal amplitude orientations iquest0(r)by

cosiexcl1 m2j jY

sin(sup20) Dagger sup20 iexcl ordm

2

The resulting non-contoured planforms in the visual centeldobtained by applying the inverse single retinocorticalmap to the corresponding V1 planforms are shown incentgures 29 and 30

Similarly the odd and even contoured planformsobtained by applying the double retinocortical map areshown in centgures 31 and 32 for the square lattice incentgures 33 and 34 for the rhombic lattice and incentgures 35 and 36 for the hexagonal lattice

One of the striking features of the resulting (contoured)visual planforms is that only the even planforms appearto be contour completing and it is these that recover theremaining form constants missing from the originalErmentrout^Cowan model The reader should comparefor example the pressure phosphenes shown in centgure 1with the planform shown in centgure 35a and the cobwebof centgure 4 with that of centgure 31a

6 THE SELECTION AND STABILITY OF PATTERNS

It remains to determine which of the various planformswe have presented above in our model are actually stablefor biologically relevant parameter sets So far we haveused a mixture of perturbation theory and symmetry toconstruct the linear eigenmodes (equation (33)) that arecandidate planforms for pattern forming instabilities To



yR

xRRq

f

Rf

Rr

Figure 27 The geometry of orientation tuning

determine which of these modes are stabilized by thenonlinearities of the system we use techniques such asLiapunov^Schmidt reduction and Poincare^Lindstedtperturbation theory to reduce the dynamics to a set ofnonlinear equations for the amplitudes ci appearing inequation (33) (Walgraef 1997) These amplitude equa-tions which eiexclectively describe the dynamics on a centnite-dimensional centre manifold then determine the selec-tion and stability of patterns (at least sucurrenciently close to

the bifurcation point) The symmetries of the systemseverely restrict the allowed forms (Golubitsky et al1988) however the coecurrencients in this form are inher-ently model dependent and have to be calculated expli-citly

In this section we determine the amplitude equation forour cortical model up to cubic order and use this to



(a)

(b)

Figure 29 Action of the single inverse retinocortical map onnon-contoured square planforms (a) square (b) roll

(a)

(b)

Figure 30 Action of the single inverse retinocortical mapon non-contoured rhombic and hexagonal planforms(a) rhombic (b) hexagonal

y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)

Figure 28 Action of the singleand double maps on logarithmicspirals Dashed lines show thelocal tangents to a logarithmicspiral contour in the visual centeldand the resulting image in V1Since circle and ray contours inthe visual centeld are just specialcases of logarithmic spirals thesame result holds also for suchcontours (a) Visual centeld (b)striate cortex (i) single map(ii) double map

investigate the selection and stability of both odd patternssatisfying u(iexcl iquest) ˆ iexclu(iquest) and even patterns satisfyingu(iexcl iquest) ˆ u(iquest) A more complete discussion of stabilityand selection based on symmetrical bifurcation theorywhich takes into account the possible eiexclects of higher-order contributions to the amplitude equation will bepresented elsewhere (Bressloiexcl et al 2000b)

(a) The cubic amplitude equationAssume that sucurrenciently close to the bifurcation point at

which the homogeneous state a(r iquest) ˆ 0 becomesmarginally stable the excited modes grow slowly at a rateO(e2) where e2 ˆ middot iexcl middotc One can then use the method ofmultiple-scales to perform a Poincare^Lindstedt pertur-bation expansion in e First we Taylor expand thenonlinear function frac14permilaŠ appearing in equation (11)

frac14permilaŠ ˆ frac141a Dagger frac142a2 Dagger frac143a

3 Dagger

where frac141 ˆ frac140permil0Š frac142 ˆ frac1400permil0Š2 frac143 ˆ frac14000permil0Š3 etc Thenwe perform a perturbation expansion of equation (11)with respect to e by writing

a ˆ ea1 Dagger e2a2 Dagger

and introducing a slow time-scale t ˆ e2t Collectingterms with equal powers of e then generates a hierarchyof equations as shown in Appendix A(c) The OO(e)

equation is equivalent to the eigenvalue equation (10)with l ˆ 0 middot ˆ middotc and jkj ˆ qc so that

a1(r iquest t) ˆN

jˆ1

cj(t)eikj centru(iquest iexcl rsquoj) Dagger cc (46)

with kj ˆ qc( cos rsquoj sin rsquoj) Requiring that the OO(e2)and OO(e3) equations in the hierarchy be self-consistentthen leads to a solvability condition which in turngenerates evolution equations for the amplitudes cj(t)(see Appendix A(c))

(i) Square or rhombic latticeFirst consider planforms (equation (46)) corresponding

to a bimodal structure of the square or rhombic type(N ˆ 2) That is take k1 ˆ qc(1 0) and k2 ˆ qc( cos(sup3)sin(sup3)) with sup3 ˆ ordm2 for the square lattice and 05sup35ordm2sup3 6ˆ ordm3 for a rhombic lattice The amplitudes evolveaccording to apair of equations of the form

dc1

dtˆ c1permilL iexcl reg0 jc1j2 iexcl 2regsup3 jc2 j2Š

dc2


(47)

where L ˆ middot iexcl middotc measures the deviation from thecritical point and

regrsquo ˆ3notjfrac143 j

frac141iexcl(3)(rsquo) (48)



(a)

(b)

Figure 31 Action of the double inverse retinocortical map oneven square planforms (a) square (b) roll

(a)

(b)

Figure 32 Action of the double inverse retinocortical map onodd square planforms (a) square (b) roll



(a)

(b)

Figure 33 Action of the double inverse retinocortical map oneven rhombic planforms (a) rhombic (b) roll

(a)

(b)

Figure 34 Action of the double inverse retinocortical map onodd rhombic planforms (a) rhombic (b) roll

(a)

(b)

Figure 35 Action of the double inverse retinocortical map oneven hexagonal planforms (a) ordm-hexagonal (b) 0-hexagonal

(b)

(a)

Figure 36 Action of the double inverse retinocortical map onodd hexagonal planforms (a) triangular (b) 0-hexagonal

for all 04 rsquo5ordm with

iexcl(3)(rsquo) ôrdm

0

u(iquest iexcl rsquo)2u(iquest)2 diquest

ordm (49)

(ii) Hexagonal latticeNext consider planforms on a hexagonal lattice with

N ˆ 3 rsquo1 ˆ 0 rsquo2 ˆ 2ordm3 rsquo3 ˆ iexcl2ordm3 The cubic ampli-tude equations take the form

dcj

dtˆ cjpermilL iexcl reg0 jcj j2 iexcl 2reg2ordm=3(jcjDagger 1 j2 Dagger jcjiexcl1 j2)Š Dagger sup2cjiexcl1cjDagger 1

(50)

where j ˆ 1 2 3 mod 3 reg2ordm=3 is given by equation (49)for rsquo ˆ 2ordm3 and

sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0

u(iquest)u(iquest iexcl 2ordm3)u(iquest Dagger 2ordm3)diquest

ordm (52)

In deriving equation (50) we have assumed that theneurons are operating close to threshold such thatfrac142 ˆ OO(e)

The basic structure of equations (47) and (50) isuniversal in the sense that it only depends on the under-lying symmetries of the system and on the type of bifurca-tion that it is undergoing In contrast the actual values ofthe coecurrencients regrsquo and sup2 are model-dependent and haveto be calculated explicitly Moreover these coecurrencients arediiexclerent for odd and even patterns because they havedistinct eigenfunctions u(iquest) Note also that because ofsymmetry the quadratic term in equation (50) mustvanish identically in the case of odd patterns

(iii) Even contoured planformsSubstituting the perturbation expansion of the eigen-

function (equation (30)) for even contoured planformsinto equations (52) and (49) gives

iexcl(2) ˆ 34 shy permiluDagger

2 (qc) iexcl uDagger0 (qc)Š Dagger OO(shy 2) (53)

iexcl(3)(sup3) ˆ 18permil2 Dagger cos(4sup3) Dagger 4shy uDagger

3 (qc) cos(4sup3) Dagger OO(shy 2)Š(54)

with the coecurrencients uDaggern decentned by equation (32)

(iv) Odd contoured planformsSubstituting the perturbation expansion of the eigen-

function (equation (31)) for odd contoured planforms intoequations (52) and (49) gives respectively

iexcl(2) ˆ 0 (55)

and

iexcl(3)(sup3) ˆ 18 permil2 Dagger cos(4sup3) iexcl 4shy uiexcl


with the coecurrencients uiexcln decentned by equation (32) Note

that the quadratic term in equation (50) vanishes identi-cally in the case of odd patterns

(v) Even non-contoured planformsSubstituting the perturbation expansion of the eigen-

function (equation (40)) for even non-contoured plan-forms into equations (52) and (49) gives respectively

iexcl(2) ˆ 1 Dagger 32 shy 2

m40

permilu0m(qc)Š2 cos(2mordm3) Dagger OO(shy 3) (57)

and

iexcl(3)(sup3) ˆ 1 Dagger shy 2

m40

permilu0m(qc)Š2permil1 Dagger 2 cos(2msup3)Š Dagger OO(shy 3)

(58)

with the coecurrencients u0n decentned by equation (41)

(b) Even and odd patterns on square or rhombiclattices

We now use equation (47) to investigate the selectionand stability of odd or even patterns on square orrhombic lattices Assuming that regsup340 and L40 thefollowing three types of steady state are possible for arbi-trary phases Aacute1 Aacute2

(i) the homogeneous state c1 ˆ c2 ˆ 0(ii) rolls c1 ˆ Lreg0

peiAacute1 c2 ˆ 0 or c1 ˆ 0 c2 ˆ L=reg0e

iAacute2(iii) squares or rhombics c1 ˆ Lpermilreg0 Dagger 2regsup3ŠeiAacute1

c2 ˆ Lpermilreg0 Dagger 2regsup3ŠeiAacute2

The non-trivial solutions correspond to the axial plan-forms listed in tables 3 and 4 A standard linear stabilityanalysis shows that if 2regsup34reg0 then rolls are stablewhereas the square or rhombic patterns are unstable Theopposite holds if 2regsup35reg0 These stability propertiespersist when higher-order terms in the amplitudeequation are included (Bressloiexcl et al 2000b)

Using equations (48) (54) (56) and (58) with 3notjfrac143jfrac141

ˆ 1 we deduce that for non-contoured patterns

2regsup3 ˆ reg0 Dagger 1 Dagger OO(shy )

and for (odd or even) contoured patterns

2regsup3 ˆ reg0 Dagger1 Dagger 2 cos(4sup3)

8Dagger OO(shy )

Hence in the case of a square or rhombic lattice we havethe following results concerning patterns bifurcating fromthe homogeneous state close to the point of marginalstability (in the limit of weak lateral interactions)

(i) For non-contoured patterns on a square or rhombiclattice there exist stable rolls and unstable squares

(ii) For (even or odd) contoured patterns on a squarelattice there exist stable rolls and unstable squares Inthe case of a rhombic lattice of angle sup3 6ˆ ordm2 rollsare stable if cos(4sup3)4iexcl12 whereas sup3-rhombics arestable if cos(4sup3)5iexcl12 that is if ordm65sup35ordm3

It should be noted that this result diiexclers from that obtainedby Ermentrout amp Cowan (1979) in which stable squareswere shown to occur for certain parameter ranges (see alsoErmentrout (1991)) We attribute this diiexclerence to theanisotropy of the lateral connections incorporated into thecurrent model and the consequent shift^twist symmetry ofthe Euclidean group action The eiexclects of this anisotropypersist even in the limit of weak lateral connections andpreclude the existence of stable square patterns



(c) Even patterns on a hexagonal latticeNext we use equations (50) and (51) to analyse the

stability of even planforms on a hexagonal lattice Ondecomposing ci ˆ CieiAacutei it is a simple matter to show thattwo of the phases Aacutei are arbitrary while the sumAacute ˆ 3

iˆ1 Aacutei and the real amplitudes Ci evolve accordingto the equations

dCi

dtˆ LCi Dagger sup2CiDagger 1Ciiexcl1 cos Aacuteiexcl reg0C

3i iexcl2reg2ordm=3(C

2iDagger 1 Dagger C2

iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)

with i j ˆ 1 2 3 mod 3 It immediately follows fromequation (60) that the stable steady-state solution willhave a phase Aacute ˆ 0 if sup240 and a phase Aacute ˆ ordm if sup250

From equations (48) (54) and (58) with 3notjfrac143 jfrac141 ˆ 1we see that

2reg2ordm=3 ˆ reg0 Dagger 1 Dagger OO(shy 2)

for even non-contoured patterns and

2reg2ordm=3 ˆ reg0 Dagger shy uiexcl3 (qc) Dagger OO(shy 2)

for even contoured patterns In the parameter regimewhere the marginally stable modes are even contouredplanforms (such as in centgure 17) we centnd that uDagger

3 (qc)40This is illustrated in centgure 37

Therefore 2reg2ordm=34reg0 for both the contoured and non-contoured cases Standard analysis then shows that (tocubic order) there exists a stable hexagonal patternCi ˆ C for i ˆ 1 2 3 with amplitude (Busse 1962)

C ˆ1

2permilreg0 Dagger 4reg2ordm=3Šjsup2j Dagger sup22 Dagger 4permilreg0 Dagger 4reg2ordm=3ŠL (61)

over the parameter range

iexclsup22

4permilreg0 Dagger 4reg2ordm=3Š5L5

2sup22permilreg0 Dagger reg2ordm=3Špermilreg0 iexcl 2reg2ordm=3Š2

The maxima of the resulting hexagonal pattern arelocated on an equilateral triangular lattice for sup240

(0-hexagons) whereas the maxima are located on anequilateral hexagonal lattice for sup250 (ordm-hexagons)Both classes of hexagonal planform have the same D6

axial subgroup (up to conjugacy) see table A2 inAppendix A(b) One can also establish that rolls areunstable versus hexagonal structures in the range

05L5sup22

permilreg0 iexcl 2reg2ordm=3Š2 (62)

Hence in the case of a hexagonal lattice we have thefollowing result concerning the even patterns bifurcatingfrom the homogeneous state close to the point of marginalstability (in the limit of weak lateral interactions) Foreven (contoured or non-contoured) patterns on ahexagonal lattice stable hexagonal patterns are the centrstto appear (subcritically) beyond the bifurcation pointSubsequently the stable hexagonal branch exchangesstability with an unstable branch of roll patterns asshown in centgure 38

Techniques from symmetrical bifurcation theory can beused to investigate the eiexclects of higher order terms in theamplitude equation (Buzano amp Golubitsky 1983) in thecase of even planforms the results are identical to thoseobtained in the analysis of Benard convection in theabsence of midplane symmetry For example one centndsthat the exchange of stability between the hexagons androlls is due to a secondary bifurcation that generatesrectangular patterns

(d) Odd patterns on a hexagonal latticeRecall that in the case of odd patterns the quadratic

term in equation (50) vanishes identically The homoge-neous state now destabilizes via a (supercritical) pitchforkbifurcation to the four axial planforms listed in tables 3and 4 In this particular case it is necessary to includehigher-order (quartic and quintic terms) in the amplitudeequation to completely specify the stability of these



1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+

Figure 37 Plot of the even eigenfunction coecurrencient uDagger3 (q) of

equation (32) as a function of wavenumber q Also plotted isthe OO(shy ) contribution to the even eigenvalue expansionequation (29) wDagger (q) ˆ W0(q) Dagger W2(q) The peak of wDagger (q)determines the critical wavenumber qc (to centrst order in shy )Same parameter values as centgure 17

c

-hexagonsp

mm

0-hexagons

rolls

C

RA

Figure 38 Bifurcation diagram showing the variation of theamplitude C with the parameter middot for even hexagonal and rollpatterns with sup240 Solid and dashed curves indicate stableand unstable solutions respectively Also shown is a secondarybranch of rectangular patterns RA Higher-order terms inthe amplitude equation are needed to determine its stability

various solutions and to identify possible secondarybifurcations Unfortunately one cannot carry overprevious results obtained from the study of the Benardconvection problem with midplane symmetry eventhough the corresponding amplitude equation is identicalin structure at cubic order (Golubitsky et al 1984) higher-order contributions to the amplitude equation will diiexclerin the two problems due to the radically diiexclerent actionsof the Euclidean group and the resulting diiexclerences inthe associated axial subgroups1 The eiexclects of such contri-butions on the bifurcation structure of odd (and even)cortical patterns will be studied in detail elsewhere(Bressloiexcl et al 2000b) Here we simply describe the morelimited stability results that can be deduced at cubicorder A basic question concerns which of the four oddplanforms on a hexagonal lattice (hexagons trianglespatchwork quilts and rolls) are stable It turns out that if2reg2ordm=34reg0 then rolls are stable whereas if 2reg2ordm=35reg0

then either hexagons or triangles are stable (dependingupon higher-order terms) Equations (49) and (56) with3notjfrac143 jfrac141 ˆ 1 imply that

2reg2ordm=3 ˆ reg0 Dagger shy uiexcl3 (qc) Dagger OO(shy 2) (63)

In the parameter regime where the marginally stablemodes are odd contoured planforms (such as in centgure 16)we centnd that uiexcl

3 (qc)50 and thus 2reg2ordm=35reg0 This isillustrated in centgure 39 Hence in the case of a hexagonallattice we have the following result concerning the oddpatterns bifurcating from the homogeneous state close tothe point of marginal stability (in the limit of weaklateral interactions) For odd (contoured) patterns on ahexagonal lattice there exist four primary bifurcationbranches corresponding to hexagons triangles patchworkquilts and rolls Either the hexagons or the triangles arestable (depending on higher-order terms through asecondary bifurcation) and all other branches areunstable This is illustrated in centgure 40

7 DISCUSSION

This paper describes a new model of the spontaneousgeneration of patterns in V1 (seen as geometric hallucina-tions) Whereas the earlier work of Ermentrout and

Cowan started with a general neural network and soughtthe minimal restrictions necessary to produce hallucina-tion patterns the current model incorporates data gath-ered over the past two decades to show that commonhallucinatory images can be generated by a biologicallyplausible architecture in which the connections betweeniso-orientation patches in V1 are locally isotropic butnon-locally anisotropic As we and Ermentrout andCowan before us show the Euclidean symmetry of suchan architecture ie the symmetry with respect to rigidmotions in the plane plays a key role in determiningwhich patterns of activation of the iso-orientation patchesappear when the homogeneous state becomes unstablepresumed to occur for example shortly after the actionof hallucinogens on those brain stem nuclei that controlcortical excitability

There are however two important diiexclerences betweenthe current work and that of Ermentrout and Cowan inthe way in which the Euclidean group is implemented

(i) The group action is diiexclerent and novel and so theway in which the various subgroups of the Euclideangroup are generated is signicentcantly diiexclerent Inparticular the various planforms corresponding tothe subgroups are labelled by orientation preferenceas well as by their location in the cortical plane Itfollows that the eigenfunctions that generate suchplanforms are also labelled in such a fashion Thisadds an additional complication to the problem ofcalculating such eigenfunctions and the eigenvaluesto which they belong from the linearized corticaldynamics Assuming that the non-local lateral orhorizontal connections are modulatory and weakrelative to the local connections we show how themethods of Rayleigh^Schrolaquo dinger degenerate pertur-bation theory can be used to compute to someappropriate level of approximation the requisite



1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-

Figure 39 Plot of the odd eigenfunction coecurrencient uiexcl3 (q) of

equation (32) as a function of wavenumber q Also plotted isthe OO(shy ) contribution to the odd eigenvalue expansionequation (29) wiexcl(q) ˆ W0(q) iexcl W2(q) The peak of wiexcl(q)determines the critical wavenumber qc (to centrst order in shy )Same parameter values as centgure 16

c mm

R

C

PQ

HT

Figure 40 Bifurcation diagram showing the variation of theamplitude C with the parameter middot for odd patterns on ahexagonal lattice Solid and dashed curves indicate stable andunstable solutions respectively Either hexagons (H) ortriangles (T) are stable (depending on higher-order terms inthe amplitude equation) whereas patchwork quilts (PQ ) androlls (R) are unstable Secondary bifurcations (not shown)may arise from higher-order terms (Bressloiexcl et al 2000b)

eigenvalues and eigenfunctions and therefore theplanforms Given such eigenfunctions we then makeuse of Poincare^Lindstedt perturbation theory tocompute the stability of the various planforms thatappear when the homogeneous state becomesunstable

(ii) Because we include orientation preference in theformulation we have to consider the action of theretinocortical map on orientated contours or edgesIn eiexclect we do this by treating the local tangents tosuch contours as a vector centeld As we discussed thisis carried out by the tangent map associated withthe complex logarithm one consequence of which isthat iquest the V1 label for orientation preference is notexactly equal to orientation preference in the visualcenteld iquestR but diiexclers from it by the angle sup3R thepolar angle of receptive centeld position We called themap from visual centeld coordinates frR sup3R iquestRg to V1coordinates fx y iquestg a double map Its possiblepresence in V1 is subject to experimental vericentcationIf the double map is present then elements tuned tothe same relative angle iquest should be connected withgreater strength than others if only the single mapfrR sup3Rg fx yg obtains then elements tuned to thesame absolute angle iquestR should be so connected Ifin fact the double map is present then elementstuned to the same angle iquest should be connectedalong lines at that angle in V1 This would supportMitchison and Crickrsquos hypothesis on connectivity inV1 (Mitchison amp Crick 1982) and would be consis-tent with the observations of G G Blasdel (personalcommunication) and Bosking et al (1997) In thisconnection it is of interest that from equation (42) itfollows that near the vertical meridian (where mostof the observations have been made) changes in iquestclosely approximate changes in iquestR However aprediction of the double map is that such changesshould be relatively large and detectable with opticalimaging near the horizontal meridian

The main advance over the Ermentrout^Cowan workis that all the Klulaquo ver form constants can now be obtainedas planforms associated with axial subgroups of theEuclidean group in the plane generated by the newrepresentations we have discovered There are severalaspects of this work that require comment

(i) The analysis indicates that under certain conditionsthe planforms are either contoured or else non-contoured depending on the strength of inhibitionbetween neighbouring iso-orientation patches Ifsuch inhibition is weak individual hypercolumns donot exhibit any tendency to amplify any particularorientation In normal circumstances such a prefer-ence would have to be supplied by inputs from theLGN In this case V1 can be said to operate in theHubel^Wiesel mode (see frac12 2(c)) If the horizontalinteractions are still eiexclective then plane waves ofcortical activity can emerge with no label for orien-tation preference The resulting planforms are callednon-contoured and correspond to a subset of theKlulaquo ver form constants tunnels and funnels andspirals Conversely if there is strong inhibitionbetween neighbouring iso-orientation patches even

weakly biased inputs to a hypercolumn can trigger asharply tuned response such that under thecombined action of many interacting hypercolumnsplane waves labelled for orientation preference canemerge The resulting planforms correspond tocontoured patterns and to the remaining formconstants described by Klulaquo veroumlhoneycombs andchequer-boards and cobwebs Interestingly all butthe square planforms are stable but there do existhallucinatory images that correspond to squareplanforms and it is possible that these are just transi-tional forms

(ii) Another conclusion to be drawn from this analysis isthat the circuits in V1 which are normally involvedin the detection of oriented edges and the formationand processing of contours are also responsible forthe generation of the hallucinatory form constantsThus we introduced in frac12 2(a) a V1 model circuit inwhich the lateral connectivity is anisotropic andinhibitory (We noted in frac12 1(d) that 20 of the(excitatory) lateral connections in layers II and IIIof V1 end on inhibitory inter-neurons so the overallaction of the lateral connections could becomeinhibitory especially at high levels of activity) As wedemonstrated in frac12 3(c) the mathematical conse-quences of this is the selection of odd planforms butthese do not form continuous contours (see frac12 5(b))This is consistent with the possibility that suchconnections are involved in the segmentation ofvisual images (Li 1999) In order to select even plan-forms which are contour forming and correspond toseen form constants it proved sucurrencient to allow fordeviation away from the visuotopic axis by at least458 in the pattern of lateral connections betweeniso-orientation patches These results are consistentwith observations that suggest that there are twocircuits in V1 one dealing with contrast edges inwhich the relevant lateral connections have theanisotropy found by G G Blasdel and L Sincich(personal communication) and Bosking et al (1997)and another that might be involved with the proces-sing of textures surfaces and colour contrast andwhich has a much more isotropic lateral connec-tivity Livingstone amp Hubel 1984) One can interpretthe less anisotropic pattern needed to generate evenplanforms as a composite of the two circuits

There are also two other intriguing possiblescenarios that are consistent with our analysis Thecentrst was referred to in frac12 3(d) In the case where V1 isoperating in the Hubel^Wiesel mode with nointrinsic tuning for orientation and if the lateralinteractions are not as weak as we have assumed inour analysis then even contoured planforms canform The second possibility stems from the observa-tion that at low levels of V1 activity lateralinteractions are all excitatory (Hirsch amp Gilbert1991) so that a bulk instability occurs if the homo-geneous state becomes unstable followed bysecondary bifurcations to patterned planforms at thecritical wavelength of 24^32 mm when the level ofactivity rises and the inhibition is activated Inmany cases secondary bifurcations tend to be asso-ciated with complex eigenvalues and are therefore



Hopf bifurcations (Ermentrout amp Cowan 1980) thatgive rise to oscillations or propagating waves Insuch cases it is possible for even planforms to beselected by the anistropic connectivity and oddplanforms by the isotropic connectivity In additionsuch a scenario is actually observed many subjectswho have taken LSD and similar hallucinogensreport seeing bright white light at the centre of thevisual centeld which then explodes into a hallucinatoryimage (Siegel amp Jarvik 1975) in ca 3 s corre-sponding to a propagation velocity in V1 of ca24 cm siexcl1 suggestive of slowly moving epileptiformactivity (Milton et al 1995 Senseman 1999)

(iii) One of the major aspects described in this paper isthe presumed Euclidean symmetry of V1 Manysystems exhibit Euclidean symmetry but what isnovel here is the way in which such a symmetry isgenerated Thus equation (9) shows that thesymmetry group is generated in large part by atranslation or shift fr iquestg fr Dagger s iquestg followed by arotation or twist fr iquestg fRsup3r iquest Dagger sup3g It is the centnaltwist iquest iquest Dagger sup3 that is novel and which is requiredto match the observations of G G Blasdel andL Sincich (personal communication) and Boskinget al (1997) In this respect it is of considerableinterest that Zweck amp Williams (2001) haveintroduced a set of basis functions with the sameshift^twist symmetry as part of an algorithm toimplement contour completion Their reason fordoing so is to bind sparsely distributed receptivecentelds together functionally to perform Euclideaninvariant computations It remains to explicate theprecise relationship between the Euclidean invariantcircuits we have introduced here and the Euclideaninvariant receptive centeld models introduced byZweck and Williams

(iv) Finally it should also be emphasized that manyvariants of the Klulaquo ver form constants have beendescribed some of which cannot be understood interms of the simple model we have introduced Forexample the tunnel image shown in centgure 41exhibits a reversed retinocortical magnicentcation andcorresponds to images described in Knauer ampMaloney (1913) It is possible that some of thecircuits beyond V1 for example those in the dorsal

segment of medial superior temporal cortex (MSTd)that process radial motion are involved in thegeneration of such images via a feedback to V1(Morrone et al 1995)

Similarly the lattice^tunnel shown in centgure 42ais more complicated than any of the simple formconstants shown earlier One intriguing possibility isthat such images are generated as a result of amismatch between the planform corresponding toone of the Klulaquo ver form constants and the underlyingstructure of V1 We have (implicitly) assumed that V1has patchy connections that endow it with latticeproperties It should be clear from centgures 9 and 10that such a cortical lattice is somewhat disorderedThus one might expect some distortions to occur



Figure 41 Tunnel hallucination generated by LSD Redrawnfrom Oster (1970)

(a)

(b)

Figure 42 (a) Lattice^tunnel hallucination generated bymarijuana Reproduced from Siegel (1977) with permissionfrom Alan D Eiselin (b) A simulation of the lattice tunnel

Figure 43 Complex hallucination generated by LSDRedrawn from Oster (1970)

when planforms are spontaneously generated in sucha lattice Figure 42b shows a computation of theappearance in the visual centeld of a hexagonal roll on asquare lattice when there is a slight incommensur-ability between the two

As a last example we show in centgure 43 anotherhallucinatory image triggered by LSD Such animage does not centt very well as a form constantHowever there is some secondary structure along themain (horizontal) axis of the its major components(This is also true of the funnel and spiral imagesshown in centgure 2 also triggered by LSD) Thissuggests the possibility that at least two diiexcleringlength-scales are involved in their generation butthis is beyond the scope of the model described in thecurrent paper It is of interest that similar imageshave been reported following stimulation with poundick-ering light (Smythies 1960)

The authors wish to thank Dr Alex Dimitrov Dr Trevor Mundeland Dr Gary Blasdel for many helpful discussions The authorsalso wish to thank the referees for a number of helpful commentsand Alan D Eiselin for permission to reproduce his artwork incentgure 41 This work was supported in part by grant 96-24 fromthe James S McDonnell Foundation to JDC The research ofMG was supported in part by NSF grant DMS-9704980 MGwishes to thank the Center for Biodynamics Boston Universityfor its hospitality and support The research of PCB was sup-ported by a grant from the Leverhulme Trust PCB wishes tothank the Mathematics Department University of Chicago forits hospitality and support PCB and JDC also wish to thankProfessor Geoiexclrey Hinton FRS and the Gatsby ComputationalNeurosciences Unit University College London for hospitalityand support PJT was supported in part by NIH grant T-32-MH20029

APPENDIX A

(a) Perturbation expansion of the eigenfunctionsWe summarize here the derivation of equations (29)^

(31) This involves solving the matrix equation

l Dagger not


n2Z

Wmiexcln(q)An (A1)

using a standard application of degenerate perturbationtheory That is we introduce the perturbation expansions

l Dagger not

frac141middotˆ W1 Dagger shy l(1) Dagger shy 2l(2) Dagger (A2)


n Dagger (A3)

and substitute these into the eigenvalue equation (26) Wethen systematically solve the resulting hierarchy ofequations to successive orders in shy

(i) (shy ) termsSetting m ˆ 1 in equation (A1) yields the hellipshy ) equation

W0(q)z1 Dagger W2(q)ziexcl1 ˆ l(1)z1

Combining this with the conjugate equation m ˆ iexcl1 weobtain the matrix equation

W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)

Equation (A4) has solutions of the form

l(1) ˆ W0(q) sect W2(q) (A5)

ziexcl1 ˆ sectz1 (A6)

where plus and minus denote the even and odd solutionsWe have used the result Wiexcl2 ˆ W2 The (shy ) terms inequation (A1) for which m 6ˆ sect1 generate the corre-sponding centrst-order amplitudes

A(1)m ˆ

Wmiexcl1(q)z1 Dagger WmDagger 1(q)ziexcl1

W1 iexcl Wm (A7)

(ii) (shy 2) termsThe (shy 2) contribution to equation (A1) for m ˆ 1 is

n 6ˆ sect 1

W1iexcln(q)A(1)n Dagger permilW0(q)iexcl l(1)ŠA(1)

1 Dagger W2(q)A(1)iexcl1 ˆ l(2)z1

Combining with the analogous equation for m ˆ iexcl1 yieldsthe matrix equation

W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where

B1(q) ˆ l(2)z1 iexcln 6ˆ sect 1

W1iexcln(q)A(1)n (A9)

Multiplying both sides of equation (A8) on the left byhellipziexcl1 z1) and using equation (A4) implies that B1(q) ˆ 0This together with equation (A7) determines the second-order contribution to the eigenvalue

l(2) ˆm6ˆ1m5 0

permilW1iexclm(q) sect W1Dagger m(q)Š2

W1 iexcl Wm (A10)

Having obtained l(2) we can then use equations (A8) and(A5) to obtain the result

A(1)iexcl1 ˆ sectA(1)

1 (A11)

The unknown amplitudes z1 and A(1)1 are determined by

the overall normalization of the solutionFinally combining equations (A2) (A5) (A6) and

(A10) generates equation (29) Similarly combining equa-tions (A3) (A6) (A7) (A11) and (23) yields the pair ofequations (30) and (31)

(b) Construction of axial subgroupsWe sketch how to construct the axial subgroups from

the irreducible representations of the holohedry H corre-sponding to the shortest dual wave vectors as given intable 2 By rescalings we can assume that the criticalwavenumber qc ˆ 1 and that the doubly periodic func-tions are on a lattice whose dual lattice curren is generatedby wave vectors of length unity There are two types ofirreducible representations for each lattice correspondingto the cases u(iquest) odd and u(iquest) even We derive the explicit



action of iexcl on these subspaces and determine the axialsubgroups

The action of the torus T 2 on the subspace KLL isderived as follows write sup3 2 T 2 as

sup3 ˆ 2ordmsup31 1 Dagger 2ordmsup32 2

Using the fact that ki cent j ˆ macrij the result of the translationaction is given in table A1

The holohedries H are D4 D6 and D2 for the squarehexagonal and rhombic lattices respectively In eachcase the generators for these groups are a repoundection anda rotation For the square and hexagonal lattices therepoundection is micro the repoundection across the x-axis wherer ˆ hellipx y) For the rhombic lattice the repoundection is microsup2The counterclockwise rotation sup1 through angles ordm2 ordm3and ordm is the rotation generator for the three lattices Theaction of HLL for the various lattices is given in table A2

Finally for each of the six types of irreducible represen-tations we compute the axial subgroupsoumlthose isotropysubgroups S that have one-dimensional centxed-pointsubspaces centx(S) in each irreducible representation Thecomputations for the square and rhombic lattices arestraightforward because we can use the T 2 action intable A1 to assume after conjugacy that c1 and c2 are realand non-negative The computations on the hexagonallattice are more complicated (Bressloiexcl et al 2000b) Theresults up to conjugacy are listed in tables A3 and A4

(c) Derivation of the amplitude equationAssume that sucurrenciently close to the bifurcation point at

which the homogeneous state a(r iquest) ˆ 0 becomes margin-

ally stable the excited modes grow slowly at a rate (e2)where e2 ˆ middot iexcl middotc We use the method of multiple scales toderive the cubic amplitude equations (47) and (50)

(i) Multiple-scale analysisWe begin by rewriting equation (11) in the more

compact form

dadt

ˆ iexclnota Dagger middotw curren frac14permilaŠ (A12)

with

w curren frac14permilaŠ ôrdm

0

wloc(iquest iexcl iquest0)frac14permila(r iquest0 t)Šdiquest0

Dagger shyR2

wlat(r iexcl r0 iquest)frac14permila(r0 iquest t)Šdr0 (A13)

middot ˆ middotc Dagger e2 Taylor expanding the nonlinear function frac14permilaŠappearing in equation (A12) gives


3 Dagger

where frac141 ˆ frac140permil0Š frac142 ˆ frac1400permil0Š2 frac143 ˆ frac14000permil0Š3 etc Thenwe perform a perturbation expansion of equation (A12)with respect to e by writing




Table A1 Torus action on iexclLL-irreducible representation

lattice torus action

square (e2ordmisup31 c1 e2ordmisup32 c2)hexagonal (e2ordmisup31 c1 e2ordmisup31 c2 eiexcl2ordmi(sup31Dagger sup32)c3)rhombic (e2ordmisup31 c1 e2ordmisup32 c2)

Table A2 Left D2 Dagger T2 action on rhombic lattice (centreD4 _Dagger T2 action on square lattice right D6 Dagger T 2 action onhexagonal lattice)

(For u(iquest) even e ˆ Dagger 1 for u(iquest) odd e ˆ iexcl1)

D2 action D4 action D6 action

1 (c1 c2) 1 (c1 c2) 1 (c1 c2 c3)sup1 (c1 c2) sup1 (c2 c1) sup1 (c2 c3 c1)microsup2 e(c2 c1) sup12 (c1 c2) sup12 (c3 c1 c2)microsup2sup1 e(c2 c1) sup13 (c2 c1) sup13 (c1 c2 c3)

micro e(c1 c2) sup14 (c2 c3 c1)microsup1 e(c2 c1) sup15 (c3 c1 c2)microsup12 e(c1 c2) micro e(c1 c3 c2)microsup13 e(c2 c1) microsup1 e(c2 c1 c3)

microsup12 e(c3 c2 c1)microsup13 e(c1 c3 c2)microsup14 e(c2 c1 c3)microsup15 e(c3 c2 c1)

[sup31 sup32] (eiexcl2ordmisup3i1 c1 eiexcl2ordmisup32 c2) (eiexcl2ordmisup31 c1 eiexcl2ordmisup32 c2eiexcl2ordmi(sup31Dagger sup32 )c3)

Table A3 Axial subgroups when u(iexcl iquest) ˆ u(iquest)

(O(2) is generated by [0 sup32]2 T2 and rotation by ordm (sup1 on therhombic lattice sup12 on the square lattice and sup13 on thehexagonal lattice) The points (1 1 1) and (iexcl1 iexcl 1 iexcl 1)have the same isotropy subgroup (D6) but are not conjugatedby a group element therefore the associated eigenfunctionsgenerate diiexclerent planforms)

lattice subgroup S centx(S) name

square D4(micro sup1) (1 1) even squareO(2) copy Z2(micro) (1 0) even roll

rhombic D2(microsup2 sup1) (1 1) even rollO(2) (1 0) even rhombic

hexagonal D6(micro sup1) (1 1 1) even hexagon (0)D6(micro sup1) (iexcl1 iexcl 1 iexcl 1) even hexagon (ordm)O(2) copy Z2(micro) (1 0 0) even roll


(O(2) is generated by [0 sup32]2 T2 and rotation by ordm (sup1 on therhombic lattice sup12 on the square lattice and sup13 on thehexagonal lattice))


square D4(micropermil12 1

2Š sup1) (1 iexcl1) odd squareO(2) copy Z2(sup12micropermil1

2 0Š) (1 0) odd rollrhombic D2(microsup2 permil1

2 12Š sup1) (1 1) odd rhombic

O(2) copy Z2(sup12micropermil12 0Š) (1 0) odd roll

hexagonal Z6(sup1) (1 1 1) odd hexagonD3(microsup1 sup12) (i i i) triangleD2(micro sup13) (0 1 iexcl1) patchwork quiltO(2) copy Z4(sup13micropermil1

2 0Š) (1 0 0) odd roll

and introducing a slow time-scale t ˆ e2t Collectingterms with equal powers of e then generates a hierarchyof equations of the form

La1 ˆ 0

Lan ˆ bn n41(A14)

where

La ˆ nota iexcl middotcfrac141w curren a

and

b2 ˆ middotcfrac142w curren a21 (A15)

b3 ˆ middotcfrac143w curren a31 Dagger 2middotcfrac142w curren a1a2 iexcl da1

dtiexcl frac141w curren a1

(A16)

(ii) Solvability conditionsThe centrst equation in the hierarchy is equivalent to the

eigenvalue equation (26) with l ˆ 0 middot ˆ middotc and jkj ˆ qcTherefore the relevant classes of solution are of the form(equation (46))

a1(r iquest t) ˆN

jˆ1

cj(t)eikj centru(iquest iexcl rsquoj) Dagger cc (A17)

Following frac12 4 we restrict solutions to the space of doublyperiodic functions on a square or rhombic lattice (N ˆ 2)or a hexagonal lattice (N ˆ 3) Next we decentne the innerproduct of two arbitrary functions a(r iquest) b(r iquest)according to

hajbi Ô

ordm

0

a(r iquest)b(r iquest)diquest

ordmdr

where O is a fundamental domain of the periodicallytiled plane (whose area is normalized to unity) Thelinear operator L is self-adjoint with respect to this innerproduct ie hajLbi ˆ hLajbi Therefore decentning

vl(r iquest) ˆ eikl centru(iquest iexcl rsquol)

we have hvl jLani ˆ hLvljani ˆ 0 for n ˆ 2 3 SinceLan ˆ bn we obtain a hierarchy of solvability conditions

hvljbni ˆ 0

From equation (A15) the lowest order solvability conditionis middotcfrac142hvl jw curren a2

1i ˆ 0 It turns out that in the presence oflateral interactions the inner product hvl jw curren a2

1i can benon-vanishing (in the case of even patterns) which leadsto a contradiction when frac142 6ˆ 0 This can be remedied byassuming that frac142 ˆ efrac140

2 Dagger OO(e2) and considering themodicented solvability condition hvl jeiexcl1b2 Dagger b3i ˆ 0 Thisgenerates the equation

vlda1

dtiexcl frac141w curren a1 ˆ middotcfrac143hvl jw curren a3

1i Dagger middotcfrac1402hvl jw curren a2

1i

(A18)

An alternative approach to handling the non-vanishing ofthe inner product hvl jw curren a2

1i would be to expand thebifurcation parameter as middot ˆ middotc Dagger emiddot1 Dagger e2middot2 Dagger This

would then give a quadratic (rather than a cubic)amplitude equation describing the growth of unstablehexagonal patterns In the case of odd patternshvl jw curren a2

1i sup2 0 and no restriction on frac142 is requiredHowever for ease of exposition we treat the odd and evencases in the same way

(iii) Amplitude equationsIn order to explicitly derive the amplitude equations

(47) and (50) from the solvability condition (A18) weneed to evaluate inner products of the form hvl jw curren an

1iSince vl is a solution to the linear equations (A14) itfollows that

hvl jw curren an1i ˆ hw curren vl jan

1i ˆnot

middotcfrac141

hvl jan1i (A19)

Thus substituting equation (A17) into the left-hand sideof equation (A18) and using (A19) shows that

vlda1

dtiexcl frac141w curren a1 ˆ permil1 Dagger iexcl(1)Š dcl

dtiexcl permilfrac141W01 Dagger iexcl(1)Šcl

(A20)

with iexcl(1) iexcl(1) ˆ OO(shy )The shy -dependent factors appearingon the right-hand side of equation (A20) are eliminatedfrom the centnal amplitude equations by an appropriaterescaling of the time t and a global rescaling of the ampli-tudes cj Similarly

hvl ja21i ˆ iexcl(2)

3

i jˆ1

cicjmacr(ki Dagger kj Dagger kl) (A21)

and

hvl ja31iˆ 3cl iexcl(3)(0)jcl j2 Dagger 2

j 6ˆ l

iexcl(3)(rsquoj iexcl rsquol)jcj j2 (A22)

with iexcl(2) and iexcl(3) given by equations (57) and (58) Notefrom equation (A21) that the inner product hvl ja2

1i is onlynon-vanishing when N ˆ 3 (corresponding to hexagonalplanforms) since we require N

jˆ1 kj ˆ 0 One possible setof wave vectors is kj ˆ qc(cos(rsquoj) sin(rsquoj)) withrsquo1 ˆ 0 rsquo2 ˆ 2ordm3 rsquo3 ˆ iexcl2ordm3 Also note that if u(iquest) isan odd eigenfunction then it immediately follows thatiexcl(2) ˆ 0

Finally we substitute equations (A19) (A20) (A21) and(A22) into (A18) and perform the rescalingecl

p(frac141W1 Dagger iexcl(1))cl After an additional rescaling of

time we obtain the amplitude equations (47) for N ˆ 2and (50) for N ˆ 3

ENDNOTE

1Interestingly there does exist an example from pounduiddynamics where the modicented Euclidean group action(equation (9)) arises (BoschVivancos et al 1995)

REFERENCES

Blasdel G G 1992 Orientation selectivity preference andcontinuity in monkey striate cortex J Neurosci 123139^3161

Bosch Vivancos I Chossat P amp Melbourne I 1995 Newplanforms in systems of partial diiexclerential equations withEuclidean symmetry Arch Ration Mech Analysis 131 199^224



Bosking W H Zhang Y Schocenteld B amp Fitzpatrick D1997 Orientation selectivity and the arrangement of hori-zontal connections in tree shrew striate cortex J Neurosci17 2112^2127

Bressloiexcl P C Bressloiexcl N W amp Cowan J D 2000aDynamical mechanism for sharp orientation tuning in anintegrate-and-centre model of a cortical hypercolumn NeuralComput 12 2473^2512

Bressloiexcl P C Cowan J D Golubitsky M amp Thomas P J2000b Scalar and pseudoscalar bifurcations pattern forma-tion in the visual cortex (Submitted)

Busse F H 1962 Das stabilitalaquo tsverhalten der zellular konvektion beiendlicher amplitude PhD dissertation University of MunichGermany Rand Report LT-66-19 [English translation byS H Davis]

Buzano E amp Golubitsky M 1983 Bifurcation involving thehexagonal lattice and the planar Benard problem Phil TransR Soc Lond A 308 617^667

Clottes J amp Lewis-Williams D 1998 The shamans of prehistorytrance and magic in the painted caves NewYork Abrams

Cowan J D 1977 Some remarks on channel bandwidths forvisual contrast detection Neurosci Res Program Bull 15 492^517

Cowan J D 1982 Spontaneous symmetry breaking in largescale nervous activity Int J Quantum Chem 22 1059^1082

Cowan J D 1997 Neurodynamics and brain mechanisms InCognition computation and consciousness (ed M Ito Y Miyashitaamp E T Rolls) pp205^233 Oxford University Press

Drasdo N 1977 The neural representation of visual spaceNature 266 554^556

Dybowski M 1939 Conditions for the appearance ofhypnagogic visions Kwart Psychol 11 68^94

Ermentrout G B 1991 Spots or stripes Proc R Soc Lond A 434413^428

Ermentrout G B 1998 Neural networks as spatial patternforming systems Rep Prog Phys 61 353^430

Ermentrout G B amp Cowan J D 1979 A mathematical theoryof visual hallucination patterns Biol Cyber 34 137^150

Ermentrout G B amp Cowan J D 1980 Secondary bifurcationin neuronal nets SIAM J Appl Math 39 323^340

Eysel U 1999 Turning a corner in vision research Nature 399641^644

Gilbert C D 1992 Horizontal integration and corticaldynamics Neuron 9 1^13

Gilbert C D amp Wiesel T N 1983 Clustered intrinsic connec-tions in cat visual cortex J Neurosci 3 1116^1133

Golubitsky M Swift J W amp Knobloch E 1984 Symmetriesand pattern selection in Rayleigh^Benard convection PhysicaD10 249^276

Golubitsky M Steward I amp Schaeiexcler D G 1988 Singularitiesand groups in bifurcation theory II Berlin Springer

Hansel D amp Sompolinsky H 1997 Modeling feature selectivityin local cortical circuits In Methods of neuronal modeling 2ndedn (ed C Koch amp I Segev) pp499^567 Cambridge MAMIT Press

Helmholtz H 1924 Physiological optics vol 2 Rochester NYOptical Society of America

Hirsch J D amp Gilbert C D 1991 Synaptic physiology ofhorizontal connections in the catrsquos visual cortex J Neurosci11 1800^1809

Horton J C 1996 Pattern of ocular dominance columns inhuman striate cortex in strabismic amblyopia Vis Neurosci 13787^795

Hubel D H amp Wiesel T N 1967 Cortical and callosal connec-tions concerned with the vertical meridian in the cat JNeurophysiol 30 1561^1573

Hubel D H amp Wiesel T N 1974a Sequence regularity andgeometry of orientation columns in the monkey striate cortexJ Comp Neurol 158 267^294

Hubel D H amp Wiesel T N 1974b Uniformity of monkey striatecortex a parallel relationship between centeld size scatter andmagnicentcation factor J Comp Neurol 158 295^306

Ishai A amp Sagi D 1995 Common mechanisms of visualimagery and perception Science 268 1772^1774

Klulaquo ver H 1966 Mescal and mechanisms and hallucinationsUniversity of Chicago Press

Knauer A amp MaloneyW J1913 Apreliminary note on thepsychicaction of mescalin with special reference to the mechanism ofvisual hallucinations J Nerv Ment Dis 40 425^436

Kosslyn S M 1994 Image and brain Cambridge MA MITPress

Krill A E Alpert H J amp Ostcenteld A M 1963 Eiexclects of ahallucinogenic agent in totally blind subjects Arch Ophthalmol69 180^185

Lee T S Mumford D Romero R amp Lamme V A F 1998The role of primary visual cortex in higher level vision VisionRes 38 2429^2454

Levitt J B amp Lund J 1997 Contrast dependence of contextualeiexclects in primate visual cortex Nature 387 73^76

Li Z 1999 Pre-attentive segmentation in the primary visualcortex SpatialVision 13 25^50

Livingstone M S amp Hubel D H 1984 Specicentcity of intrinsicconnections in primate primary visual cortex J Neurosci 42830^2835

McGuire B A Gilbert C D Rivlin P K amp Wiesel T N1991 Targets of horizontal connections in macaque primaryvisual cortex J Comp Neurol 305 370^392

Milton J Mundel T an der Heiden U Spire J-P ampCowan J D 1995 Traveling activity waves InThe handbook ofbrain theory and neural networks (ed M A Arbib) pp 994^997Cambridge MA MIT Press

Mitchison G amp Crick F 1982 Long axons within the striatecortex their distribution orientation and patterns of connec-tion Proc Natl Acad Sci USA 79 3661^3665

Miyashita Y 1995 How the brain creates imagery projection toprimary visual cortex Science 268 1719^1720

Morrone M C Burr D C amp Vaina L M 1995 Two stages ofvisual processing for radial and circular motion Nature 376507^509

Mourgue R 1932 Neurobiologie de lrsquohallucination BrusselsLamertin

Mundel T Dimitrov A amp Cowan J D 1997 Visual cortexcircuitry and orientation tuning In Advances in neural informa-tion processing systems vol 9 (ed M C Mozer M I Jordan ampT Petsche) pp 887^893 Cambridge MA MIT Press

Oster G 1970 Phosphenes Sci Am 222 83^87Patterson A 1992 Rock art symbols of the greater southwest Boulder

CO Johnson BooksPurkinje J E 1918 Opera omnia vol 1 pp 1^62 Prague Society

for Czech PhysiciansRockland K S amp Lund J 1983 Intrinsic laminar lattice

connections in primate visual cortex J Comp Neurol 216303^318

Schwartz E 1977 Spatial mapping in the primate sensoryprojection analytic structure and relevance to projectionBiol Cyber 25 181^194

Senseman D M 1999 Spatiotemporal structure of depolariza-tion spread in cortical pyramidal cell populations evoked bydiiexcluse retinal light poundashesVis Neurosci 16 65^79

Sereno M I Dale A M Reppas J B Kwong K KBelliveau J W Brady T J Rosen B R amp TootellR B H 1995 Borders of multiple visual areas in humansrevealed by functional magnetic resonance imaging Science268 889^893

Siegel R K 1977 Hallucinations Sci Am 237 132^140Siegel R K amp Jarvik M E 1975 Drug-induced hallucinations

in animals and man In Hallucinations behavior experience and



theory (ed R K Siegel amp L J West) pp 81^161 New YorkWiley

Smythies J R 1960 The stroboscopic patterns III Furtherexperiments and discussion Br J Psychol 51 247^255

Turing A M 1952 The chemical basis of morphogenesis PhilTrans R Soc Lond B 237 32^72

Tyler C W 1978 Some new entoptic phenomena Vision Res 1811633^1639

Tyler C W 1982 Do grating stimuli interact with the hyper-column spacing in the cortex Suppl Invest Ophthalmol 222254

Von der Malsburg C amp Cowan J D 1982 Outline of a theoryfor the ontogenesis of iso-orientation domains in visualcortex Biol Cyber 45 49^56

Walgraef D 1997 Spatio-temporal pattern formation New YorkSpringer-Verlag

Wiener M C 1994 Hallucinations symmetry and the structureof primary visual cortex a bifurcation theory approach PhDthesis University of Chicago IL USA

Wilson H R amp Cowan J D 1972 Excitatory and inhibitoryinteractions in localized populations of model neuronsBiophys J 12 1^24

Wilson H R amp Cowan J D 1973 A mathematical theory ofthe functional dynamics of cortical and thalamic nervoustissue Kybernetik 13 55^80

Winters W D 1975 The continuum of CNS excitatory statesand hallucinosis In Hallucinations behavior experience andtheory (ed R K Siegel amp L J West) pp53^70 New YorkWiley

Zweck J W amp Williams L R 2001 Euclidean group invariantcomputation of stochastic completion centelds using shiftable^twistable functions Neural Comput (In the press)



To complete the study we then investigate the stability of the planforms using methods of nonlinearstability analysis including Liapunov^Schmidt reduction and Poincare^Lindstedt perturbation theoryWe centnd a close correspondence between stable planforms and form constants The results are sensitive tothe detailed specicentcation of the lateral connectivity and suggest an interesting possibility that the corticalmechanisms by which geometric visual hallucinations are generated if sited mainly in V1 are closelyrelated to those involved in the processing of edges and contours

Keywords hallucinations visual imagery poundicker phosphenes neural modellinghorizontal connections contours

` the hallucination is not a static process but adynamic process the instability of which repoundects aninstability in its conditions of originrsquo (Klulaquo ver (1966)p 95 in a comment on Mourgue (1932))

1 INTRODUCTION

(a) Form constants and visual imageryGeometric visual hallucinations are seen in many situa-tions for example after being exposed to poundickeringlights (Purkinje 1918 Helmholtz 1924 Smythies 1960)after the administration of certain anaesthetics (Winters1975) on waking up or falling asleep (Dybowski 1939)following deep binocular pressure on onersquos eyeballs(Tyler 1978) and shortly after the ingesting of drugs suchas LSD and marijuana (Oster 1970 Siegel 1977) Patternsthat may be hallucinatory are found preserved in petro-glyphs (Patterson 1992) and in cave paintings (Clottes ampLewis-Williams 1998) There are many reports of suchexperiences (Knauer amp Maloney 1913 pp 429^430)

Ìmmediately before my open eyes are a vast number ofrings apparently made of extremely centne steel wire allconstantly rotating in the direction of the hands of aclock these circles are concentrically arranged the inner-most being incentnitely small almost pointlike theoutermost being about a meter and a half in diameterThe spaces between the wires seem brighter than thewires themselves Now the wires shine like dim silver inparts Now a beautiful light violet tint has developed inthem As I watch the center seems to recede into thedepth of the room leaving the periphery stationary tillthe whole assumes the form of a deep tunnel of wire ringsThe light which was irregularly distributed among thecircles has receded with the center into the apex of thefunnel The center is gradually returning and passing theposition when all the rings are in the same vertical planecontinues to advance till a cone forms with its apextoward me The wires are now poundattening into bandsor ribbons with a suggestion of transverse striation andcolored a gorgeous ultramarine blue which passes inplaces into an intense sea green These bands move rhyth-mically in a wavy upward direction suggesting a slowendless procession of small mosaics ascending the wall insingle centles The whole picture has suddenly receded thecenter much more than the sides and now in a momenthigh above me is a dome of the most beautiful mosiacs The dome has absolutely no discernible pattern Butcircles are now developing upon it the circles arebecoming sharp and elongated now they are rhombicsnow oblongs and now all sorts of curious angles areforming and mathematical centgures are chasing each otherwildly across the roof rsquo

Klulaquo ver (1966) organized the many reported images intofour classes which he called form constants (I) gratingslattices fretworks centligrees honeycombs and chequer-

boards (II) cobwebs (III) tunnels and funnels alleyscones vessels and (IV) spirals Some examples of class Iform constants are shown in centgure 1 while examples ofthe other classes are shown in centgures 2^4

Such images are seen both by blind subjects and insealed dark rooms (Krill et al 1963) Various reports(Klulaquo ver 1966) indicate that although they are dicurrencult tolocalize in space and actually move with the eyes theirpositions relative to each other remain stable with respectto such movements This suggests that they are generatednot in the eyes but somewhere in the brain One clue ontheir location in the brain is provided by recent studies ofvisual imagery (Miyashita 1995) Although controversialthe evidence seems to suggest that areas V1 and V2 thestriate and extra- striate visual cortices are involved invisual imagery particularly if the image requires detailedinspection (Kosslyn 1994) More precisely it has beensuggested that (Ishai amp Sagi 1995 p 1773)

`[the] topological representation [provided by V1] mightsubserve visual imagery when the subject is scrutinizingattentively local features of objects that are stored inmemoryrsquo

Thus visual imagery is seen as the result of an interactionbetween mechanisms subserving the retrieval of visualmemories and those involving focal attention In thisrespect it is interesting that there seems to be competitionbetween the seeing of visual imagery and hallucinations(Knauer amp Maloney 1913 p 433)

` after a picture had been placed on a background andthen removed ` I tried to see the picture with open eyesIn no case was I successful only [hallucinatory] visionaryphenomena covered the groundrsquorsquorsquo

Competition between hallucinatory images and after-images was also reported Klulaquo ver 1966 p 35)

Ìn some instances the [hallucinatory] visions preventedthe appearance of after-images entirely [however] inmost cases a sharply outlined normal after-imageappeared for a while while the visionary phenomenawere stationary the after-images moved with the eyesrsquo

As pointed out to us by one of the referees the fusedimage of a pair of random dot stereograms also seems tobe stationary with respect to eye movements It has alsobeen argued that because hallucinatory images are seenas continuous across the midline they must be located athigher levels in the visual pathway than V1 or V2(R Shapley personal communication) In this respectthere is evidence that callosal connections along the V1V2 border can act to maintain continuity of the imagesacross the vertical meridian (Hubel amp Wiesel 1967)

All these observations suggest that both areas V1 andV2 are involved in the generation of hallucinatory








(a)

(b)


(a)

(b)


(a)

(b)



raquoR ˆ1

(w0 Dagger erR)2


x ˆnot

eln 1Dagger

e

w0

rR

y ˆshy rRsup3R

w0 Dagger erR



x ˆnotrR

w0

y ˆshy rRsup3R

w0


x ˆnot

eln

erR

w0

y ˆshy sup3R

e





0

2

x

y2

0

0

2

(b)(a)

(c)

p

p

p

23p

23p

2p

23p

2p

p

23p

23p

p

p

y

x













(a)

(b)


(a)

(b)










CONNECTIONS





x

1

2

3

3

21(a)

(b)

(c) y

x

y


(a)

(b)






a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES














































































(a)

(b)


(a)

(b)


(a)

(b)



raquoR ˆ1

(w0 Dagger erR)2


x ˆnot

eln 1Dagger

e

w0

rR

y ˆshy rRsup3R

w0 Dagger erR



x ˆnotrR

w0

y ˆshy rRsup3R

w0


x ˆnot

eln

erR

w0

y ˆshy sup3R

e





0

2

x

y2

0

0

2

(b)(a)

(c)

p

p

p

23p

23p

2p

23p

2p

p

23p

23p

p

p

y

x













(a)

(b)


(a)

(b)










CONNECTIONS





x

1

2

3

3

21(a)

(b)

(c) y

x

y


(a)

(b)






a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES










































































raquoR ˆ1

(w0 Dagger erR)2


x ˆnot

eln 1Dagger

e

w0

rR

y ˆshy rRsup3R

w0 Dagger erR



x ˆnotrR

w0

y ˆshy rRsup3R

w0


x ˆnot

eln

erR

w0

y ˆshy sup3R

e





0

2

x

y2

0

0

2

(b)(a)

(c)

p

p

p

23p

23p

2p

23p

2p

p

23p

23p

p

p

y

x













(a)

(b)


(a)

(b)










CONNECTIONS





x

1

2

3

3

21(a)

(b)

(c) y

x

y


(a)

(b)






a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES




















































































(a)

(b)


(a)

(b)










CONNECTIONS





x

1

2

3

3

21(a)

(b)

(c) y

x

y


(a)

(b)






a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES

















































































CONNECTIONS





x

1

2

3

3

21(a)

(b)

(c) y

x

y


(a)

(b)






a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES













































































a(r iquest t)t






frac14permilzŠ ˆ1









with

w(r) ˆ1

0





(a)

(b)


Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES









































































Rsup3

xy


sin sup3 cos sup3

xy









w(r) ˆsup30

iexclsup30

p(sup3)1

0


(8)




















hypercolumn

lateral connections

local connections


q 0

0f

0f

0f
















a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES























































































a(regiexcl1P t)t














a(r iquest t)t


0


ordm

Dagger shyR2





- 1 05 0 05 1- 1

05-

- -

-

0

05


- 1 05 0 05 1- 1

05

0

05

1rotation by = 6pq

q

(rP P)f

(rQ Q)f(r Q)Q

(r p) p

f

f





a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































a(r iquest t)t




ordm (12)



0


ordm




wloc(iquest) ˆn2Z

Wne2niiquest (14)













O








orientation f

(a)

s

1

2

3

4

p0 2p





a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































a(r) î

ci exp(iki cent r)







0


ordm

Dagger shyR2






0


ordm









k (21)



and




u(iquest) ˆn2Z

Ane2niiquest (23)



Wmiexcln(q)An (24)


Wn(q) ôrdm

0eiexcl2iniquest


diquest

ordm

(25)






l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































l Dagger not


n2Z

Wmiexcln(q)An (26)









l Dagger not


and


n Dagger (28)



l sect Dagger not


Dagger shy 2

m5 0 m 6ˆ1


W1 iexcl Wm

sup2 Gsect (q) (29)






with

uDagger0 (q) ˆ

W1(q)W1 iexcl W0

usectm (q) ˆ


m41

(32)



a(r iquest) ˆN

iˆ1




Wn(q) ôrdm

0eiexcl2iniquest

1

0


ordm



mˆ1




D W

d l

= 0 ltlt 1b b

m

m s 1

0

1

m

l

l

l rsquo




0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































0

g(s)J2n(sq)ds (34)


Wn(q) ˆ( iexcl 1)n

2exp

iexclsup1 2latq

2

4In

sup1 2latq

2

4


latq2

4In

sup1 2latq

2

4 (35)






1

0

g(s)J2n(sq)ds (36)






1 2 3 4 5

09

092

094

096

098

10

m

odd

even

q c

02 04 06 08 1

02

04

06

08

1

q

q

-q

q +

Alat

(a)

(b)


m

q

qc

1 2 3 4 5

094

096

098

092

even

odd

090





l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































l Dagger not


and


n Dagger (38)


l Dagger not


m40

Wm(q)2


sup2 G0(q)

(39)

and



with

u0m(q) ˆ

Wm(q)W0 iexcl Wm

(41)







1 2 3 4 5

09

092

094

096

098

10

m

q

qc























lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES






























































































lattice 1 2 k1 k2


p) (0 2 3

p) (1 0) 1

2 (iexcl1 3p



lattice a(r iquest)





HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































HALLUCINATIONS







v ˆ cos iquest

xDagger sin iquest

y




















(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































(b)

(a)


(b)

(a)


(b)

(a)


(b)

(a)




(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































(a)

(b)


(a)

(b)


(b)

(a)


(a)

(b)





yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































yR













096





j j ˆ96


2

(45)


cosiexcl1 m2j jY


2








yR

xRRq

f

Rf

Rr







(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES














































































(a)

(b)


(a)

(b)


y

2p

p

p

2p

p

p

p3 2

3 2

p3 2

y

x

x

2

p 2

- 2

p- 2

2

(a) (b)(i)

(ii)






3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES













































































3 Dagger





a1(r iquest t) ˆN

jˆ1





dc1


dc2


(47)






(a)

(b)


(a)

(b)




(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































(a)

(b)


(a)

(b)


(a)

(b)


(b)

(a)




0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES











































































0


ordm (49)



dcj


(50)


sup2 ˆnotfrac142

frac141 frac141W1

p iexcl(2) (51)

with

iexcl(2) ôrdm

0


ordm (52)












iexcl(2) ˆ 0 (55)

and








m40


and


m40


(58)














8Dagger OO(shy )










dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































dCi




iiexcl1)Ci

(59)and

dAacute

dtˆ iexclsup2

3

iˆ1

CiDagger1Ciiexcl1

Cisin Aacute (60)









C ˆ1



iexclsup22






05L5sup22








1 2 3 4 5- 005

01

02

q

qc

(q)

f

w+(q)

u3(q)+



c

-hexagonsp

mm

0-hexagons

rolls

C

RA







7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES














































































7 DISCUSSION







1 2 3 4 5

- 04

- 02

02

04

q

qc

(q)

f

w - (q)

u3(q)-



c mm

R

C

PQ

HT



















(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES


























































































(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES

















































































(a)

(b)






APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES












































































APPENDIX A



l Dagger not


n2Z

Wmiexcln(q)An (A1)


l Dagger not



n Dagger (A3)





W0(q)

Wiexcl2(q)

W2(q)

W0(q)

z1

ziexcl1

ˆ l(1) z1

ziexcl1 (A4)


l(1) ˆ W0(q) sect W2(q) (A5)



A(1)m ˆ


W1 iexcl Wm (A7)


n 6ˆ sect 1




W0(q) iexcl l(1)

Wiexcl2(q)

W2(q)

W0(q) iexcl l(1)

A(1)1

A(1)iexcl1

ˆB1(q)Biexcl1(q)

(A8)

where




l(2) ˆm6ˆ1m5 0


W1 iexcl Wm (A10)



1 (A11)


















compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES



















































































compact form

dadt


with


0


Dagger shyR2




3 Dagger






























2 0Š) (1 0 0) odd roll


La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES










































































La1 ˆ 0

Lan ˆ bn n41(A14)

where


and




(A16)



a1(r iquest t) ˆN

jˆ1



hajbi Ô

ordm

0


ordmdr




hvljbni ˆ 0





vlda1



1i

(A18)









1i ˆnot

middotcfrac141

hvl jan1i (A19)


vlda1



(A20)



3

i jˆ1


and


j 6ˆ l








ENDNOTE


REFERENCES



























































































































































Documents

Geometricvisualhallucinations,Euclidean ...bresslof/publications/01-1.pdfGeometricvisualhallucinations,Euclidean symmetryandthefunctionalarchitecture ofstriatecortex PaulC.Bresslo¡1,JackD.Cowan2*,MartinGolubitsky3,