paul c. bressloff et al- geometric visual hallucinations, euclidean symmetry and the functional...

8/3/2019 Paul C. Bressloff et al- Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striat

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Geometric visual hallucinations, Euclideansymmetry and the functional architecture

of striate cortex

Paul C. Bresslo1, Jack D. Cowan2*, Martin Golubitsky3,

Peter 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, Houston,TX 772 04- 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 hallucinations.Hallucinatory images were classied by Klu ver into four groups called form constants comprising(i) gratings, lattices, fretworks, ligrees, honeycombs and chequer-boards, (ii) cobwebs, (iii) tunnels,funnels, 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)the retinocortical mapand of neuronal circuits in V1, both local andlateral, determine their geometry.

In the rst p art of the p aper 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. 1.33^2 mm. We next introduce a mathematical description ofthe large- scale 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, i .e. they are invariantunder the action of the planar Euclidean group E(2)the group of rigid motions in the planerotations, reections 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 withresp ect 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^Schro 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 , the corticallabel for orientation preference, and plane waves in r, the cortical position coordinate. `Switching-on 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 ickering l ights. 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 eld, 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 `pinwheels.

Phil.Trans. R. Soc. Lond. B (2001) 356, 299^330 299 2001 The Royal SocietyReceived18 April 2000 Accepted 11 August 2000

doi 10.1098/rstb.2000.0769

* Author for correspondence ([email protected]).


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To complete the study we then investigate the stability of the planforms, using methods of nonlinearstability analysis, including Liapunov^Schmidt reduction and Poincare^Lindstedt p erturbation theory.We nd a close correspondence between stable planforms and form constants. The results are sensitive tothe detailed specication 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; icker phosphenes; neural modelling;

horizontal connections; contours

`. . . the hallucination is . . . not a static p rocess but a

dynamic process, the instability of which reects an

instability in its conditions of origin (Kluver (1966),p. 95, in a comment on Mourgue (1932)).

1. INTRODUCTION

(a) Form constants and visual imagery

Geometric visual hallucinations are seen in many situa-tions, for example, after being exposed to ickering

lights (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 ones 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 &Lewis-Williams 1998). There are many reports of suchexperiences (Knauer & Maloney 1913, pp. 429^430):

Ìmmediately before my open eyes are a vast number ofrings, apparently made of extremely ne steel wire, all

constantly rotating in the direction of the hands of aclock; these circles are concentrically arranged, the inner-most being innitely small, almost pointlike, the

outermost being about a meter and a half in diameter.The spaces between the wires seem brighter than the

wires themselves. Now the wires shine like dim silver inparts. Now a beautiful light violet tint has developed in

them. As I watch, the center seems to recede into the

depth of the room, leaving the periphery stationary, tillthe whole assumes the form of a deep tunnel of wire rings.

The light, which was irregularly distributed among thecircles, has receded with the center into the apex of the

funnel. The center is gradually returning, and passing the

position when all the rings are in the same vertical plane,continues to advance, till a cone forms with its apex

toward me . . . . The wires are now attening into bandsor ribbons, with a suggestion of transverse striation, and

colored a gorgeous ultramarine blue, which passes in

places into an intense sea green. These bands move rhyth-mically, in a wavy upward direction, suggesting a slow

endless procession of small mosaics, ascending the wall insingle les. The whole picture has suddenly receded, the

center much more than the sides, and now in a moment,high above me, is a dome of the most beautiful m osiacs,

. . . . The dome has absolutely no discernible pattern. But

circles are now developing upon it; the circles arebecoming sharp and elongated . . . now they are rhombics

now oblongs; and now all sorts of curious angles areforming; and mathematical gures are chasing each other

wildly across the roof. . . .

Klu ver (1966) organized the many reported images intofour classes, which he called form constants: (I) gratings,lattices, fretworks, ligrees, honeycombs and chequer-

boards; (II) cobwebs; (III) tunnels and funnels, alleys,cones, vessels; and (IV) spirals. Some examples of class Iform constants are shown in gure 1, while examples ofthe other classes are shown in gures 2^4.

Such i mages are seen both by blind subjects and insealed dark rooms (Krill et al. 1963). Various reports(Klu ver 1966) indicate that although they are dicult tolocalize in space, and actually move with the eyes, theirpositions relative to each other remain stable with respect

to 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 controversial,the 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 & Sagi 1995, p. 1773)

`[the] topological representation [provided by V1] might

subserve visual imagery when the subject is scrutinizingattentively local features of objects that are stored in

memory.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 & Maloney 1913, p. 433):

`. . . after a picture had been placed on a background andthen removed `Ì tried to see the picture with open eyes.

In no case was I successful; only [hallucinatory] visionary

phenomena covered the ground.

Competition between hallucinatory images and after-

images was also reported Klu ver 1966, p. 35):

Ìn some instances, the [hallucinatory] visions preventedthe appearance of after-images entirely; [however] in

most cases a sharply outlined normal after-imageappeared for a while . . . while the visionary phenomena

were stationary, the after-images moved with the eyes.

As p ointed 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 V1/

V2 border can act to maintain continuity of the imagesacross the vertical meridian (Hubel & Wiesel 1967).

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

300 P. C. Bresslo and others Geometric visual hallucinations

Phil.Trans. R. Soc. Lond. B (2001)


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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 t hat the action ofsuch mechanisms is rapidly fed back to V1 (Lee et al.1998). It now follows, because all observers report seeingKlu vers form constants or variations, that thoseproperties common to all such hallucinations should yield

information about the architecture of V1. We thereforeinvestigate that architecture, i.e. the patterns of connec-tion between neurons in the retina and those in V1,together with intracortical V1 connections, on thehypothesis that such patterns determine, in large p art,the 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 eld.

(b) The human retinocortical map

The rst step is to calculate what visual hallucinationslook like, not in the standard p olar coordinates of thevisual eld, but in the coordinates of V1. It is well estab-lished that there is a topographic map of the visual eldin V1, the retinotopic representation, and that the centralregion of the visual eld has a much bigger representationin V1 than it does in the visual eld (Sereno et al. 1995).The reason for this is partly that there is a non-uniform

Geometric visual hallucinations P. C. Bresslo and others 301


(a)

(b)

Figure 1. (a) `Phosp hene p roduced by deep binocularp ressure on the eyeballs. Redrawn from Tyler ( 1978).

(b) Honeycomb hallucination generated by marijuana.

Redrawn from Clottes & Lewis-Williams (1998).

(a)

(b)

Figure 2. (a) Funnel and (b) spiral hallucinations generated

by LSD. Re drawn from Oster (1970).

(a)

(b)

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


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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 R be the packing density of retinal ganglioncells per unit area of the visual eld, the correspondingdensity per unit surface area of cells in V1, and rR; Rretinal or equivalently, visual eld coordinates. ThenR rR drR dR is the number of ganglion cell axons in aretinal element of area rR drR dR. By hyp othesis theseaxons connect topographically to cells in an element ofV1 surface area dx dy, i.e. to dx dy cortical cells. (V1 isassumed to be locally at with Cartesian coordinates.)

Empirical evidence indicates that is approximatelyconstant (Hubel & Wiesel 1974a,b), whereas R declinesfrom the origin of the visual eld, i.e. the fovea, with aninverse square law (Drasdo 1977)

R 1

(w0 erR)2,

where w0 and e are constants. Estimates of w0 0:087and e 0:051 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,y

gas functions of visual eld or retinal

coordinates frR, Rg. The resulting coordinate transforma-tion takes the form

x e

ln 1 ew0

rR ,

y -rRRw0 erR

,

where and - are constants in appropriate units.Figure 5 shows the map.

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

x rRw0

,

y -rRRw0

,

and (ii), suciently far away from the fovea, erR w0, itbecomes

x e

lnerRw0

,

y -Re

.

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



Figure 4. Cobweb p etroglyph . Redrawn from Patterson(1992).

0

/2

x

y/2

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 eld; (b) th e 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 corresp ond to

x 0.


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of a retinal point (xR, yR) (rR, R ), then z x iy ln( rR expiR) ln rR iR. Thus x ln rR, y R.

(c) Form constants as spontaneous cortical patterns

Given that the retinocortical map is generated by thecomplex logarithm (except near the fovea), it is easy tocalculate the action of the transformation on circles, rays,and logarithmic spirals in the visual eld. Circles ofconstant rR in the visual eld become vertical lines in V1,whereas rays of constant R become horizontal lines.Interestingly, logarithmic spirals become oblique lines inV1: the equation of such a spiral is just R a ln rRwhence y ax under the action of zR ! z. Thus formconstants comprising circles, rays and logarithmic spiralsin the visual eld 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 ingure 2.

A possible mechanism for the spontaneous formation ofstripes of neural activity under the action of hallucinogenswas originally proposed by Ermentrout & Cowan (1979).They studied interacting p opulations 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 &

Cowan 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 squaresunder the retinocortical map these

generate more complex hallucinations in the visual eldsuch 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 distribution).

(d) Orientation tuning in V1

The Ermentrout^Cowan theory of visual hallucinationsis over-simplied in the sense that V1 is represented as if itwere just a cortical retina. However, V1 cells do much

more than merely signalling position in the visual eld:most cortical cells signal the local orientation of a contrastedge or barthey are tuned to a particular local orienta-tion (Hubel & 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 (gure 42), honey-combs and certain chequer-boards (gure 1), and cobwebs(gure 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 V1,and about their pattern of interconnection (Gilbert 1992).Figure 8 shows a typical arrangement of such cells,obtained via microelectrodes implanted in cat V1. The rstpanel 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 form

constant. (a) Image in the visual eld; (b) V1 map of the image.

(a)

(b)

Figure 7. Action of the retinocortical map on the spiral form

constant. (a) Image in the visual eld; (b) V1 map of the image.


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preference reappears, i.e. the distribution is -periodic inthe orientation preference angle. The second panel showsthe receptive elds of the cells, and how they change withV1 location. The third panel shows more clearly therotation of such elds 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 has

made it p ossible 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, gure 9a shows that the distribution oforientation p references is indeed roughly -p eriodic, inthat approximately every 0.5 mm (in the macaque) thereis an iso-orientation patch of a given preference, andgure 10 shows that there seem to be at least two length-scales:

(i) localcells less than 0.5 mm apart tend to makeconnections with most of their neighbours in aroughly isotropic fashion, as seen in gure 9b, and

(ii) lateralcells make contacts only every 0.5 mm orso along their axons with cells in similar iso-orientation patches.

In addition, gure 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 &Lund 1983), tend to be orientated along the direction oftheir cells preference (Gilbert 1992; Bosking et al. 1997),i.e. they run parallel to the visuotopic axis of their cellsorientation preference. These horizontal connections arise

almost exclusively from excitatory neurons (Levitt &Lund 1997; Gilbert & Wiesel 1983), although 20%terminate on inhibitory cells and can thus have signicantinhibitory eects (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. 1.33^2 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 p reliminary account of this wasdescribed in Wiener (1994) and Cowan (1997).

2. A MODEL OF V1 WITH ANISOTROPIC LATERAL

CONNECTIONS

(a) The model

The 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 orientationp reference as cortical location changes ( electrode track 2).

(b) Receptive elds for tracks 1 and 3. (c) Expansion of the

receptive elds of track 2 to show the rotation of orientationp reference. Redrawn from G ilbert (1992).

(a)

(b)

Figure 9 (a) Distribution of orientation preferences in

macaque V1 obtained via optical imaging. Redrawn fromBlasdel (1992). (b) Connections made by an inhibitory

interneuron in cat V1. Redrawn f rom Eysel (1999).


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characterized by the real-valued activity variablea(r, , t), where 2 0, ) 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 and r are independent variablesall possibleorientations are represented at every p osition. A moreaccurate model would need to incorporate detailsconcerning the distribution of orientation patches in thecortical plane (as illustrated in gure 9a). It is known, forexample, that a region of human V1 ca. 2:67 mm2 on itssurface and extending throughout its depth contains atleast two sets of all iso-orientation patches in the range045, one for each eye. Such a slab was called ahypercolumn by Hubel & Wiesel (1974b). If human V1 asa whole (in one hemisphere) has a surface area of ca.3500 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 p otential diculty with thisinterpretation is that the eective wavelength of many ofthe p atterns underlying visual hallucinations is of theorder of twice the hypercolumn spacing (see, for example,gure 2), suggesting that lattice eects might be impor-tant. A counter-argument for the validity of the

continuum model (besides mathematical convenience) isto note that the separation of two points in the visualeldvisual acuity(at a given retinal eccentricity ofr0R), corresponds to hypercolumn spacing (Hubel &Wiesel 1974b), and so to each location in the visual eldthere corresponds to a representation in V1 of that loca-tion with nite resolution and all possible orientations.

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

@a(r, , t)@t

a(r, , t)

0 R2w(r, jr0, 0)

a(r0, 0, t) dr0d0

h(r, , t), (1)

where and are decay and coupling coecients,h(r, , t) is an external input, w(r, jr0, 0) is the weightof connections between neurons at r tuned to andneurons at r0 tuned to 0, and z is the smooth nonlinearfunction

z 11

e(z)

, (2)

for constants and . Without loss of generality we maysubtract from z a constant equal to 1 e1 to obtainthe (mathematically) important property that 0 0,which implies that for zero external inputs the homoge-neous state a(r, , t) 0 for all r, , t is a solution toequation (1). From the discussion in 1(d), we take thepattern of connections w(r, jr0, 0) to satisfy t hefollowing p roperties (see gure 1).

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

w(r, jr0, 0) wloc ( 0)(r r0) wlat (r r0, )( 0), (3)

with wloc ( ) wloc().(ii) Lateral connections between hypercolumns only

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

wlat (r, )

w(R

r), (4)

with

w(r) 1

0

g(s)(r sr0) (r sr0)ds, (5)

where r0 (1, 0) and R is the rotation matrix



(a)

(b)

Figure 10. Lateral connections made by cells in (a) owl

monkey and (b) tree shrew V1. A radioactive tracer is used toshow the locations of all terminating axons from cells in a

central injection site, superimposed on an orientation map

obtained by optical imaging. Redrawn from G. G. Blasdel andL. Sincich (p ersonal communication) and Bosking et al. (1997).


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Rx

y cos sin

sin cos

x

y.

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

g(s) 22lat 1=2exp s2/22lat Alat 22lat 1=2

exp s2/22lat , (6)with lat5lat and Alat41, 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 gure 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, jr0, 0) wloc ( 0)loc(jr r0 j) w(Rr0 r)lat ( 0), (7)

with lat ( ) lat (), lat () 0 for jj40, andloc(jrj) 0 for r40. Moreover, equation (5) is modi-ed according to

w(r) 0

0p()

1

0g(s)(r sr) (r sr)dsd,

(8)

with r (cos (), sin()) and p( ) p(). The para-meters 0 and 0 determine the angular spread of lateralconnections with respect to orientation p reference and

space, respectively, whereas 0 determines the (spatial)range of the isotropic local connections.

(b) Euclidean symmetry

Suppose that the weight distribution w satisesequations (7) and (8). We show that w is invariant under

the 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 action

The Euclidean group is composed of the (semi-direct)product of O(2), the group of planar rotations and reec-tions, with R2, the group of p lanar translations. Theaction of the Euclidean group on R2 S1 is generated by

s (r, ) (r s, ) s 2 R2 (r, ) (R r, ) 2 S1

(r, )

(r,

),

(9)

where is the reection (x1, x2)7!(x1, x2) and R is arotation by .

The corresponding group action on a functiona:R2 S1 ! R where P (r, ) is given by

a(P) a(1 P), (10)

for all 2 O(2) _ R2 and the action on w(PjP0) i s w(PjP0) w(1 Pj1 P0).

The particular form of the action of rotations inequations (9) reects a crucial feature of the lateralconnections, namely that they tend to be orientated alongthe direction of their cells preference (see gure 11). Thus,if we just rotate V1, then the cells that are now connectedat long range will not be connected in the direction oftheir preference. This diculty 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 corresponds to rotation of r by while simultaneouslysending to

. This is illustrated in gure 13. The

action of reections is justied in a similar fashion.

(ii) Invariant weight distribution w

We now prove that w as given by equations (7) and (8) isinvariant under the action of the Euclidean group denedby equations (9). (It then follows that the distribution



hypercolumn

lateral connections

local connections

Figure 11. Illustration of the local connections within a

hypercolumn and the anisotropic lateral connections betweenhypercolumns.

q0

0f

0f

0f

Figure 12. Examp le of an angular spread in the anisotropic

lateral connections between hyp ercolumns with respect toboth space (0 ) and orientation p reference (0).


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satisfying equations (3)^(5) is also Euclidean invariant.)Translation invariance ofw is obvious, i.e.

w(r s, jr0 s, 0) w(r, jr0,0).

Invariance with respect to a rotation by follows from

w(Rr, jRr0, 0 ) wloc ( 0)loc (jRr r0j)

w(RR(r r0))lat ( 0) wloc ( 0)loc (jr r0j) w(Rr)lat ( 0) w(r, jr0, 0).

Finally, invariance under a reection about the x-axisholds because

w(r,

jr0,

0)

wloc(

0)loc (

j

r

r0

j)

w(R(r r0))lat ( 0) wloc( 0)loc (jr r0j)

w(R(r r0))lat ( 0) wloc( 0)loc (jr r0j)

w(R(r r0))lat( 0) w(r, jr0, 0).

We have used the identity R R and the conditionswloc ( ) wloc(), lat ( ) lat (), w(r) w(r).

(iii) Implications of Euclidean symmetry

Consider the action of on equation (1) for h(r, t) 0:

@a(1 P, t)@t

a(1P, t) R2S1

w(1PjP0)a(P0, t)dP0

a(1P, t) R2S1

w(PjP0)a(P0, t)dP0

a(1P, t) R2S1

w(PjP00)a(1 P00, t)dP00,

since d

1P

dP and w is Euclidean invariant. If we

rewrite equation (1) as an operator equation, namely,

Fa dadt

Ga 0,

then it follows that Fa Fa. Thus F commutes with 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 be

a bifurcation parameter. We show in 4 that near a pointfor which the steady state a(r, , ) 0 becomes unstable,there must exist smooth solutions to the equilibrium equa-tion Ga(r, , ) 0 that are identied by theirsymmetry (Golubitsky et al. 1988). We nd 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 of the torus T2 of translations modulothe lattice with the dihedral groups D2, D4 and D6, theholohedries of the lattice. Thus, when a(r, , ) 0becomes unstable, new solutions emerge from the

instability with symmetries that are broken comparedwith . Suciently close to the bifurcation point thesepatterns are characterized by (nite) linear combinationsof eigenfunctions of the linear operator L D0Gobtained by linearizing equation (1) about the homoge-neous state a 0. These eigenfunctions are derived in 3.

(c) Two limiting cases

For the sake of mathematical convenience, we restrictour analysis in this p aper 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, , t)

@t

a(r, , t)

0

wloc ( 0)a(r, 0, t)d0

-R2

wlat(r r0, )a(r0, , t)dr0 , (11)

where we have introduced an additional coupling para-meter - 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, )is anisotropic, as it depends on . Before proceeding to



-1 0.5 0 0.5 1-1

0.5-

- -

-

0

0.5

1two points P, QR

2 [0,p)

-1 0.5 0 0.5 1-1

0.5

0

0.5

1rotation by = /6pq

q

(rP, P)

f

(rQ, Q)f

(r , Q)Q

(r , p)p

f

f

Figure 13. Action of a rotation by

: (r, ) ! (r0, 0) (Rr, ).


10/32

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 & Cowan 1979).

(i) The ring model of orientation tuning

The rst limiting case is to neglect lateral connections

completely by setting - 0 in equation (11). Each point rin the cortex is then independently described by the so-called ring model of orientation tuning (Hansel &Sompolinsky 1997; Mundel et al. 1997; Ermentrout 1998;Bressloet al. 2000a):

@a(r, , t)

@t a(r, , t)

0

wloc ( 0)

a(r, 0, t) d0

. (12)

Linearizing this equation about the homogeneous statea(r, , t)

0 and considering p erturbations of the form

a(r, , t) elta(r, ) yields the eigenvalue equation

la(r, ) a(r, )

0

wloc( 0)a(r, 0)d0

.

Introducing the Fourier series expansion a(r, )

mzm(r) e

2im c:c: generates the following discretedispersion relation for the eigenvalue l :

l 1Wm lm, (13)

where 1 dz/dz evaluated at z 0 and

wloc () n2Z

Wne2ni. (14)

Note that because wloc() is a real and even function of ,Wm Wm Wm.

Let Wp maxfWn, n 2 Z g and suppose that p isunique with Wp40 and p51. It then follows fromequation (13) that the homogeneous state a(r, ) 0 isstable for suciently small , but becomes unstable when increases beyond the critical value c /1Wp due toexcitation of l inear eigenmodes of the forma(r, ) z(r)e2ip z(r)e2ip , where z(r) is an arbitrary

complex function of r. It can be shown that the saturatingnonlinearities of the system stabilize the growing patternof activity (Ermentrout 1998; Bresslo et al. 2000a). Interms of polar coordinates z(r) Z(r)e2i(r) we havea(r, ) Z(r)cos(2p (r)). Thus at each point r inthe plane the maximum (linear) response occurs at theorientations (r) k/p, k 0, 1, : : :, p 1 when p 6 0.

Of p articular relevance from a biological p erspectiveare the cases p 0 and p 1. In the rst 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^Wieselmode ( Hubel & Wiesel 1974a). In the second case theresponse is unimodal with resp ect to . The occurrence ofa sharply tuned response peaked at some angle (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 in 5(a). An example of typical tuning curves is shown in

gure 14, which is obtained by taking wloc () t o be adierence of Gaussians over the domain /2, /2:wloc() 22loc 1=2exp( 2/22loc ) Aloc 22loc 1=2

exp( 2/22loc ), (15)

with loc5^

loc and Aloc41.The location of the centre (r) of each tuning curve is

arbitrary, which reects the rotational equivariance ofequation (12) under the modied group action

: (r, ) ! (r, ). Moreover, in the absence oflateral interactions the tuned response is uncorrelatedacross dierent 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 dierent locations are correlated.

(ii) The Ermentrout^Cowan model

The other limiting case is to neglect the orientationlabel completely. Equation (11) then reduces to a one-population version of the model studied by Ermentrout &Cowan (1979):

@

@ta(r, t) a(r, t)

O

wlat (r r0) 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 r0) ! wlat (jr r0 j),i.e. wlat depends only on the magnitude of r r0. It canbe shown that the resulting system is equivariant withrespect to the standard action of the Euclidean group in

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

la(r) a(r) 1O

wlat (jr 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 loc 208, loc 608 and Aloc 1. Thetuning curve is marginally stable so that the p eak activity a at

each point in th e cortical p lane is arbitrary. The activity istruncated at 0 in line with the choice of 0 0.


11/32

which upon Fourier transforming generates a dispersionrelation for the eigenvalue l as a function of q jkj, i.e.

l 1 W(q) l(q),

where W(q) wlat(k) is the Fourier transform ofwlat (

jr

j). Note that l is real. If we choose wlat (

jr

j) to be in

the form of a Mexican hat function, then it is simple toestablish that l passes through zero at a critical para-meter value c signalling the growth of spatially periodicpatterns with wavenumber qc , where W(qc) maxqfW(q)g. Close to the bifurcation point thesepatterns can be represented as linear combinations ofplane waves

a(r) i

ci exp(iki r),

with

jki

j qc. As shown by Ermentrout & 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 visualeld. In this paper we generalize the treatment byErmentrout & Cowan (1979) to incorporate the eects oforientation preferenceand show how plane waves ofcortical activity modulate the distribution of tuningcurves across the network and lead to contoured patterns.

3. LINEAR STABILITY ANALYSIS

The rst step in the analysis of pattern-forminginstabilities in the full cortical model is to linearize equa-tion (11) about the homogeneous solution a(r, ) 0 andto solve the resulting eigenvalue problem. In particular,we wish to nd 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 - characterizing the relative strength of the aniso-tropic lateral connections.

(a) Linearization

We linearize equation (11) about the homogeneous stateand introduce solutions of the form a(r, , t) elta(r, ).This generates the eigenvalue equation

la(r, ) a(r, ) 1

0

wloc ( 0)a(r, 0)d0

-R2

wlat (r r0, )a(r0, )dr0 . (17)

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

a(r, ) u( )eikr c:c: (18)

with k q(cos , sin ) and

lu() u() 1

0

wloc ( 0)u(0)d0

-wlat (k, u() . (19)

Here wlat (k, ) is the Fourier transform of wlat (r, ).Assume that wlat satises 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() only depend on the magnitude q jkj ofthe wave vector k. That is, there is an innite degen-eracy due to rotational invariance.

(ii) For each k the associated subspace of eigenfunctions

Vk fu( )eikr c:c: : u( ) u() and u 2 Cg(20)

decomposes into two invariant subspaces

Vk Vk Vk , (21)corresponding to even and odd functions, respec-tively

Vk fv 2 Vk : u( ) u()gand

Vk fv 2 Vk : u( ) u()g: (22)As noted in greater generality by Bosch Vivancoset al. (1995), this is a consequence of reection invar-iance, as we now indicate. That is, let k denotereections about the wavevector k so that kk k.Then ka(r, ) a(kr, 2 ) u( )eikr c:c.Since k is a reection, any space that it acts ondecomposes into two subspaces, one on which it actsas the identity I and one on which it acts as I.

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

u() n2Z

Ane2ni , (23)

and setting wlat (r, ) w(Rr) leads to the matrixeigenvalue equation

lAm Am 1 WmAm -n2Z

Wmn(q)An , (24)

with Wn given by equation (14) and

Wn(q)

0

e2inR2

eiqx cos()y sin()w(r)drd

.

(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) Wn(q).




12/32

(b) Eigenfunctions and eigenvalues

The calculation of the eigenvalues and eigenfunctionsof the linearized equation (17), and hence the derivation

of conditions for the marginal stability of the homoge-neous state, has been reduced to t he problem of solvingthe matrix equation (24), which we rewrite in the moreconvenient form

l 1

Wm Am -n2Z

Wmn(q)An. (26)

We exploit the experimental observation that the intrinsiclateral connections appear to be weak relative to the localconnections, i.e. -W W. Equation (26) can then besolved by expanding as a p ower series in - and usingRayleigh^Schro dinger p erturbation theory.

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

(26) reduces to equation (13). Following the discussion ofthe ring model in 2(c), let Wp maxfWn, n 2 Z g40and suppose that p 1 (unimodal orientation tuningcurves). The homogeneous state a(r, ) 0 is then stablefor suciently small , but becomes marginally stable atthe critical point c /1W1 due to the vanishing ofthe eigenvalue l1. In this case there are both even andodd marginally stable modes cos(2) and sin(2).

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

is a q-dependent splitting of the degenerate eigenvalue l1that also separates out odd and even solutions. Denotingthe characteristic size of such a splitting by l OO(-),we impose the condition that l 1W, whereW minfW1 Wm, m 6 1g:This ensures that the perturbation does not excite statesassociated with other eigenvalues of the unperturbedproblem (see gure 15). We can then restrict ourselves tocalculating perturbative corrections to the degenerate

eigenvalue l1 and its associated eigenfunctions. Therefore,we introduce the power series expansions

l 1

W1 -l (1) -2l(2) . . . , (27)

and

An z 1n, 1 -A(1)n -2 A(2)n . . . , (28)

where n, 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 - using (degenerate)perturbation theory. This analysis, which is carried out in

Appendix A(a), leads to the following results: (i)l

l

for even ( ) and odd () solutions where to OO(-2)l

1W1 - W0(q) W2(q)

-2m50, m6 1

Wm1(q) Wm 1(q)2W1 Wm

G (q), (29)

and (ii) u() u () where to OO(-)u ()

cos(2)

-

m50, m6 1um (q) cos(2m), (30)

u() sin(2) -m41

um (q) sin(2m), (31)

with

u0 (q) W1(q)

W1 W0,

um (q) Wm1(q) Wm 1(q)

W1 Wm, m41.

(32)

(c) Marginal stability

Suppose that G (q) has a unique maximum atq q 6 0 and let qc q if G (q )4G(q) andqc q if G(q)4G (q ). Under such circumstances,the homogeneous state a(r, ) 0 will become margin-ally stable at the critical point c /1G (qc) and themarginally stable modes will be of the form

a(r, ) N

i 1cie

iki ru( i) c:c:, (33)

where ki qc(cos i, sin i) and u() u () for qc q .The innite degeneracy arising from rotation invariancemeans that all modes lying on the circle jkj qc becomemarginally stable at the critical point. However, this canbe reduced to a nite 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 4.

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

^Wn(q)

0 e2in

1

0 g(s) cos(sq cos )ds

d

.

Using the Jacobi^Anger expansion

cos(sq cos ) J0(sq) 21

m1( 1)mJ2m(sq) cos(2m),



DW

dl

= 0


13/32

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

Wn(q) ( 1)n1

0

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

Next we substitute equation (6) into (34) and use standard

properties of Bessel functions to obtain

Wn(q) ( 1)n

2exp

2lat q24

In2lat q

2

4

Alat exp2lat q2

4In

2lat q2

4, (35)

where In is a modied Bessel function of integer order n.The resulting marginal stability curves (q)

/1G (q) are plotted to rst order in - in gure 16a.Theexistence of a non-zero critical wavenumber qc q at

c

(q

c

) is evident, indicating that the marginallystable eigenmodes are odd functions of . The inclusion ofhigher-order terms in - does not alter this basic result, atleast for small -. If we take the fundamental unit oflength to be ca. 400 mm; then the wavelength of a patternis 2(0:400)/qc mm, ca. 2:66 mm at the critical wave-number qc 1 (see gure 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 su-cient spread in the distribution of lateral connectionsalong the lines shown in gure 12. In particular, supposethat w(r) is given by equation (8) with p() 1 for 4 0and zero otherwise. Equation (34) then becomes

Wn(q) ( 1)nsin(2n0)

2n0

1

0

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

To rst order in - the size of the gap between the oddand even eigenmodes at the critical p oint qc is deter-mined by 2W2(qc) (see equation 29). It follows that if

04/4 then W2(q) reverses sign, suggesting that evenrather than odd eigenmodes become marginally stablerst. This is conrmed by the marginal stability curvesshown in gure 17.

(i) Choosing the bifurcation parameter

It is worth commenting at this stage on the choice ofbifurcation parameter . One way to induce a primaryinstability of the homogeneous state is to increase the

global coupling p arameter in equation (29) until thecritical point c is reached. However, it is clear fromequation (29) that an equivalent way to induce such aninstability is to keep xed and increase the slope 1 ofthe neural output function . The latter could be achievedby keeping a non-zero uniform input h(r, , t) h0 inequation (1) so that the homogeneous state is non-zero,a(r, , t) a0 6 0 with 1 0(a0). Then variation of theinput h0 and consequently 1, corresponds to changingthe eective neural threshold and hence the level ofnetwork excitability. Indeed, this is thought to be one ofthe p ossible eects of hallucinogens. In summary, themathematically convenient choice of as the bifurcationparameter can be reinterpreted in terms of biologicallymeaningful p arameter variations. It is also possible thathallucinogens act indirectly on the relative levels of inhi-bition and this could also play a role in determining thetyp e of patterns that emergea p articular example isdiscussed below.



1 2 3 4 5

0.9

0.92

0.94

0.96

0.98

1.0

m

odd

even

qc

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

q

q

-q

q+

Alat

(a)

(b)

Figure 16. (a) Plot of marginal stability curves (q) for g(s)given by th e dierence of Gaussians (equation (6)) with

lat 1, lat 3, Alat 1 and - 0:4W1. Also set/1W1 1. The critical wavenumber for spontaneousp attern formation is qc. The marginally stable eigenmodes are

odd functions of . (b) Plot of critical wavenumber q formarginal stability of even ( + ) and odd (7) patterns as afunction of the str ength of inhibitory coup ling Alat . If theinhibition is too weak then there is a bulk instability with

respect to the spatial domain.

m

q

qc

1 2 3 4 5

0.94

0.96

0.98

0.92

even

odd

0.90

Figure 17. Same as gure 16 except that W(q) satises

equation (36) with 0 /3 rather than equation (34). It canbe seen that the marginally stable eigenmodes are now even

functions of .


14/32

(d) The Ermentrout^Cowan model revisited

The marginally stable eigenmodes (equation (33))identied 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 4 andthe corresponding hallucination patterns in the visualeld (obtained by applying an inverse retinocortical map)will be constructed in 5. It turns out that the resultingpatterns account for some of the more complicated form

constants where contours are prominent, includingcobwebs, honeycombs and lattices (gure 4). However,other form constants such as chequer-boards, funnels andspirals (gures 6 and 7) comprise contrasting regions oflight and dark. One possibility is that these hallucinationsare a result of higher-level processes lling 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 fashion

to the cortical retina of Ermentrout & Cowan (1979),that is, can cortical orientation tuning somehow beswitched o ? One possible mechanism is the following:suppose that the relative level of local inhibition, which isspecied by the parameter Aloc in equation (15), isreduced (e.g. by the possible (indirect) action of halluci-nogens). Then W0 maxfWn, n 2 Z g rather than W1,and 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 1

W0 -l (1) -2l(2) . . . , (37)

and

An zn,0 -A(1)n -2A(2)n . . . . (38)

Substituting t hese expansions into the matrix eigenvalueequation (26) and solving the resulting equations tosuccessive orders in - leads to the following results:

l 1

W0 -W0(q) -2m40

Wm(q)2

W0 Wm O(-3)

G0(q)

(39)

and

u() 1 -m40

u0m(q) cos(2m O(-2), (40)

with

u0m(q) Wm(q)

W0 Wm. (41)

Substituting equation (40) into equation (33) shows thatthe marginally stable states are now only weakly depen-dent on the orientation , and to lowest order in - simplycorrespond to the spatially periodic patterns of theErmentrout^Cowan model. The length-scale of thesepatterns is determined by the marginal stability curve0(q) /1 G0(q), an example of which is shown ingure 18.

The occurrence of a bulk instability in orientationmeans that for suciently small - the resulting corticalpatterns will be more like contrasting regions of light anddark rather than a lattice of oriented contours (see 4).However, if the strength of lateral connections - wereincreased, then the eigenfunctions (40) would develop asignicant dependence on the orientation . This could

then provide an alternative mechanism for the generationof even contoured patternsrecall from 3(c) that onlyodd contoured p atterns emerge in the case of a tunedinstability with respect to orientation, unless there issignicant angular spread in the lateral connections.

4. DOUBLY PERIODIC PLANFORMS

As we found in 3 (c) and 3 (d), rotation symmetryimplies that the space of marginally stable eigenfunctionsof the linearized Wilson^Cowan equation is innite-dimensional, that is, if u()eikr is a solution then so is

u( )eiRk

r

. However, translation symmetry suggeststhat we can restrict the space of solutions of the nonlinearWilson^Cowan equation (11) to that of doubly p eriodicfunctions. This restriction is standard in many treatmentsof spontaneous pattern formation, but as yet it has noformal justication. There is, however, a wealth ofevidence from experiments on convecting uids andchemical reaction^diusion systems (Walgraef 1997), andsimulations of neural nets (Von der Malsburg & 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 nite-dimensional. A nite set of specic eigen-functions can then be identied as candidate planforms,in the sense that they approximate time-independentsolutions of equation (11) suciently close to the criticalpoint where the homogeneous state loses stability. In thissection we construct such planforms.



1 2 3 4 5

0.9

0.92

0.94

0.96

0.98

1.0

m

q

qc

Figure 18. Plot of marginal stability curve 0(q) for a bulkinstability with respect to orientation and g(s) given by the

dierence of Gaussians (equation (6)) with lat 1, lat 3,Alat 1:0, - 0:4W0 and /1W0 1. The critical wave-number for spontaneous pattern formation is q

c

.


15/32

(a) Restriction to doubly p eriodic solutions

Let be a planar lattice ; that is, choose two linearlyindependent vectors 1 and 2 and let

f2m1 1 2m2 2 : m1, m2 2 Zg:Note that is a subgroup of the group of planar trans-lations. A function f:R2 S1 ! R is doubly periodicwith respect to if

f(x , ) f(x, ),

for every 2 . Let be the angle between the two basisvectors 1 and 2. We can then distinguish three types oflattice according to the value of : square lattice ( /2),rhombic lattice (055/2, 6 /3) and hexagonal( =3). After rotation, the generators of the p lanarlattices 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, H _ T2, whereH is the holohedry of the lattice, the subgroup of O2)that preserves the lattice, and T2 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 nite-dimensionalwecan only rotate eigenfunctions through a nite set ofangles (for example, multiples of /2 for the square latticeand multiples of /3 for the hexagonal lattice).

It remains to determine the space K of marginallystable eigenfunctions and the action of on this space.In 3 we showed that eigenfunctions either reside in Vk(the even case) or Vk (the odd case) where the length ofk is equal to the critical wavenumber qc . In particular,

the eigenfunctions have the form u( )eik

r

where u iseither an odd or even eigenfunction. We now choose thesize of the lattice so that eikr is doubly periodic withrespect to that lattice, i.e. k is a dual wave vector for thelattice. In fact, there are innitely many choices for thelattice size that satisfy this constraintwe select the onefor which qc is the shortest length of a dual wave vector.The generators for the dual lattices are also given intable 1 with qc 1. The eigenfunctions corresponding todual wave vectors of unit length are given in table 2. Itfollows that KL can be identied with the m-dimensionalcomplex vector space spanned by the vectors(c

1

, : : :, cm

)2

Cm with m

2 for square or rhombiclattices and m 3 for hexagonal lattices. It can be shownthat these form -irreducible representations. Theactions of the group 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) Planforms

We now use an important result from bifurcationtheory in the p resence 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 under the action(9)a subgroup S is axial if the dimension of thespace of vectors that are xed by S is equal to unity. Theaxial subgroups are calculated from the actions presentedin Appendix A(b) (see Bresslo et al. (2000b) for details)and lead to the even planforms listed in table 3 and theodd planforms listed in table 4. The generic p lanformscan 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 -p eriodic function u().In p articular, the perturbation analysis of 3(c) and 3(d) shows that (in the case of weak lateral interactions)

u() can take one of three possible forms:(i) even contoured planforms (equation (30)) with

u() cos(2),(ii) odd contoured planforms (equation (31)) with

u() sin(2),(iii) even non-contoured planforms (equation (40)) with

u() 1.Each planform is an approximate steady-state solution

a(r, ) of the continuum model (equation (11)) dened onthe unbounded domain R2 S1. To determine how thesesolutions generate hallucinations in the visual eld, we

rst need to interpret the planforms in terms of activitypatterns in a bounded domain of V1, which we denote byM R. Once this has been achieved, the resultingpatterns in the visual eld can be obtained by applyingthe inverse retinocortical map as described in 5(a).

The interpretation of non-contoured planforms isrelatively straightforward, since to lowest order in - thesolutions are -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 square,triangular, or rhombic cells that tile

Mas illustrated in

gures 19 and 20.The case of contoured planforms is more subtle. At a

given location r in V1 we have a sum of two or threesinusoids with dierent phases and amplitudes (see tables 3and 4), which can be written as a(r, ) A(r)cos2 20(r). The phase 0(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(1, 3p )

r homb ic ( 1,

cot ) (0, cosec ) (1, 0) (cos , sin )

Table 2. Eigenfunctions corresponding to shortest dual wavevectors

lattice a(r, )

square c1u()eik1 r c2u( /2)eik2 r c:c:

hexagonal c1u()eik1 r c2u( 2/3)eik3 r

c3u(

2/3)eik3 r

c:c:

rhombic c1u()eik1 r c2 u( )eik2 r c:c:


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orientation tuning curve at r (see gure 14). Hence thecontoured solutions generally consist of iso-orientationregions or p atches over which 0(r) is constant, but theamplitude A(r) varies. As in the non-contoured case, these

patches are either square, triangular or rhombic in shape.However, 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 i n gures 21 and 22 forthe square latttice, in gures 23 and 24 for the rhombiclatttice and in gures 25 and 26 for the hexagonal lattice.Note 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 `star shape indicating the

presence of multiple orientations at a given p oint, seegure 26b. Note that this p lanform contains the well-known pi nwheelsdescribed by Blasdel (1992).

5. FROM CORTICAL PATTERNS TO VISUAL

HALLUCINATIONS

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

and the visual eld is that the periodicity of the angularretinal coordinate R implies that the y-coordinate in V1satises cylindrical periodic boundary conditions (seegure 5). This boundary condition should be commensu-

rate with the square, rhombic or hexagonal latticeassociated with the doubly periodic planforms.

(a) The double retinocortical map

An 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 eldin eect totreat them as a vector eld. Let R be the orientation ofsuch a local contour, and its image in V1. What is theappropriate map from R to that must be added to themap zR ! z described earlier? We note that a line in V1of constant slope tan is a level curve of the equation

f(x, y) y cos x sin ,

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

v cos @@x

sin @@y

:

The pre-image of such a line in the visual eld, assumingthe retinocortical map generated by the complexlogarithm is obtained by changing from cortical to retinal

coordinates via the complex exponential, is

f(x, y) ! ~f(rR, R) R cos log rR sin ,

the level curves of which are the logarithmic spirals

rR(R) A exp(cot ()R ).



Table 3. Even planforms wit h u( ) u()(The hexagon solutions (0) and ( ) h ave the same isotrop y subgroup , but they are not conjugate solutions.)

lattice name planform eigenfunction

square even square u()cos x u( /2) cosyeven roll u()cos x

rhombic even rhombic u() cos(k1 r) u( ) cos(k2 r)even roll u() cos(k1 r)hexagonal even hexagon (0) u() cos(k1 r) u( /3) cos(k2 r) u( /3) cos(k3 r)

even hexagon () u() cos(k1 r) u( /3) cos(k2 r) u( /3) cos(k3 r)even roll u() cos(k1 r)

Table 4. Odd planforms with u( ) u()

lattice name planform eigenfunction

square odd square u()cos x u( /2) cosyodd roll u()cos x

rhombic odd rhombic u() cos(k1 r) u( ) cos(k2 r)odd roll u() cos(k1 r)

hexagonal odd hexagon u() cos(k1 r) u( /3) cos(k2 r) u( /3) cos(k3 r)triangle u() sin(k1 r) u( /3) sin(k2 r) u( /3) sin(k3 r)patchwork quilt u( /3) cos(k2 r) u( /3) cos(k3 r)odd roll u() cos(k1 r)


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(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 square

lattice: (a) square, (b) roll.

(b)

(a)

Figure 20. Non-contoured axial eigenfunctions on rhombic

and hexagonal lattices: (a) rhombic, (b) hexagonal.

(b)

(a)

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


18/32



(a)

(b)

Figure 25. Contours of even axial eigenfunctions on the

hexagonal lattice: (a) -hexagonal, (b) 0-hexagonal.

(a)

(b)

Figure 26. Contours of odd axial eigenfunctions on the

hexagonal 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 the

rhombic lattice: (a) rhombic, (b) roll.


19/32

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

~v

rR cos(

R )

@

@xR rR sin(

R)

@

@yR:

Thus the retinal vector eld induced by a constant vectoreld in V1 twists with the retinal angle R and stretcheswith the retinal radius rR. It follows that if R is theorientation of a l ine in the visual eld, then

R R, (42)i.e. local orientation in V1 is relative to the angular coor-dinate of visual eld position. The geometry of the abovesetup is shown in gure 27.

The resulting double map fzR , Rg ! fz, g has veryinteresting properties. As previously noted, the mapzR ! z takes circles, rays and logarithmic spirals intovertical, horizontal and oblique lines, resp ectively. Whatabout the extended map ? Because the tangent to a circle ata given point is perpendicular to the radius at that point,for circles, R R /2, so that /2. Similarly, forrays, R R, so 0. For logarithmic spirals we canwrite either R a ln rR or rR expbR . In retinal coor-dinates we nd the somewhat cumbersome formula

tan R brR sin R ebR cos RbrR cos R ebR sin R

:

However, this can be rewritten as tan (R R) a, sothat in V1 coordinates, tan a. Thus we see that thelocal orientations of circles, rays and logarithmic spirals,measured in relative terms, all lie along the corticalimages of such forms. Figure 28 shows the details.

(b) Planforms in the visual eld

In order to generate a visual eld pattern we split ourmodel V1 domain M into two p ieces, each running72 mm along the x direction and 48 mm along the ydirection, representing the right and left hemields in thevisual eld (see gure 5). Because the y coordinate corre-sponds to a change from

/2 to /2 in 72 mm, which

meets up again smoothly with the representation in theopposite hemield, we must only choose scalings androtations of our planforms that satisfy cylindrical periodicboundary conditions in the y direction. In the x direction,corresponding 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 p eriodic withrespect to a spatial lattice generated by two lattice vectors

1, 2. The cylindrical periodicity is thus equivalent torequiring that there be an integral combination of latticevectors that spans Y 96 mm in the y direction with no

change in the x direction:

096

2m1 1 2m2 2. (43)

If the acute angle of the lattice 0 is specied, then thewavevectors ki are determined by the requirement

ki j 1, i j

0, i 6 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 the

lattice vectors j 1j j 2 j : j j:

j j 96m21 2m1m2 cos(0) m22

. (45)

The commonly reported hallucination p atterns usuallyhave 30^40 repetitions of the pattern around a circumfer-ence of the visual eld, corresponding to length-scalesranging from 2.4 3.2 mm. Therefore, we would expectthe critical wavelength 2/qc for bifurcations to be in thisrange (see 3(c)). Note that when we rotate the planformto match the cylindrical boundary conditions we rotatek

1and hence the maximal amplitude orientations

0(r)

by

cos1m2j j

Ysin(0) 0

2:

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

Similarly, the odd and even contoured planformsobtained by applying the double retinocortical map areshown in gures 31 and 32 for the square lattice, in

gures 33 and 34 for the rhombic lattice, and ingures 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 compare,for example, the pressure phosphenes shown in gure 1with the planform shown in gure 35a, and the cobwebof gure 4 with that of gure 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 p arameter 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

xR

Rq

f

Rf

Rr

Figure 27. The geometry of orientation tuning.


20/32

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 eectively describe the dynamics on a nite-dimensional centre manifold, then determine the selec-tion and stability of patterns (at least suciently close to

the bifurcation point). The symmetries of the systemseverely restrict the allowed forms (Golubitsky et al.1988); however, the coecients 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 on

non-contoured square planforms: (a) square, (b) roll.

(a)

(b)

Figure 30. Action of the single inverse retinocortical map

on non-contoured rhombic and hexagonal p lanforms:(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 m ap s on logarithmic

sp irals. Dashed lines show thelocal tangents to a logarithmic

sp iral contour in the visual eld,and the resulting image in V1.

Since circle and ray contours inthe visual eld are just special

cases of logarithmic spirals, the

same result holds also for suchcontours. (a) Visual eld; (b)

striate cortex, (i) single map,

(ii) double map.


21/32

investigate the selection and stability of both odd patternssatisfying u( ) u() and even patterns satisfyingu( ) u(). A more complete discussion of stabilityand selection based on symmetrical bifurcation theory,which takes into account the p ossible eects of higher-order contributions to the amplitude equation, will bepresented elsewhere (Bressloet al. 2000b).

(a) The cubic amplitude equation

Assume that suciently close to the bifurcation point at

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

a 1a 2 a2 3a3 : : : ,

where 1 00, 2 000/2, 3 0000/3!, etc. Thenwe perform a perturbation expansion of equation (11)with respect to e by writing

a ea1 e2a2 : : : ,

and introducing a slow time-scale t 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 0, c and jkj qc so that

a1(r, , t) N

j 1

cj(t)eikjru( j) c:c:, (46)

with kj qc( cos j, sin j). 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 lattice

First, consider planforms (equation (46)) correspondingto a bimodal structure of the square or rhombic type(N 2). That is, take k1 qc(1, 0) and k2 qc( cos(),sin()), with /2 for the square lattice and 055/2, 6 /3 for a rhombic lattice. The amplitudes evolveaccording to ap air of equations of the form

dc1

dt c1L 0 jc1j2 2 jc2 j2,

dc2

dt c2L

0

jc2

j2

2

jc1

j2

,

(47)

where L c measures the deviation from thecritical point, and

3j3 j

1(3)(), (48)



(a)

(b)

Figure 31. Action of the double inverse retinocortical map on

even square p lanforms: (a) square, (b) roll.

(a)

(b)


odd square p lanforms: (a) square, (b) roll.


22/32



(a)

(b)


even rhombic planforms: (a) rhombic, (b) roll.

(a)

(b)


odd rhombic planforms: (a) rhombic, (b) roll.

(a)

(b)


even hexagonal planforms: (a) -hexagonal, (b) 0-hexagonal.

(b)

(a)


odd hexagonal planforms: (a) triangular, (b) 0-hexagonal.


23/32

for all 045 with

(3)()

0

u( )2u()2 d

. (49)

(ii) Hexagonal lattice

Next consider p lanforms on a hexagonal lattice with

N 3, 1 0, 2 2/3, 3 2/3. The cubic ampli-tude equations take the form

dcj

dt cjL 0 jcjj2 22=3(jcj 1 j2 jcj1 j2) cj1 cj 1,

(50)

where j 1, 2, 3 mod 3, 2=3 is given by equation (49)for 2/3, and

21 1W1

p (2) , (51)

with

(2)

0

u()u( 2/3)u( 2/3) d

. (52)

In deriving equation (50) we have assumed that theneurons are operating close to threshold such that2 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 coecients and are model-dependent and haveto be calculated explicitly. Moreover, these coecients are

dierent for odd and even p atterns because they havedistinct eigenfunctions u(). Note also that because ofsymmetry the quadratic term in equation (50) mustvanish identically in the case of odd patterns.

(iii) Even contoured planforms

Substituting the p erturbation expansion of the eigen-function (equation (30)) for even contoured p lanformsinto equations (52) and (49) gives

(2) 34

-u2 (qc) u0 (qc) OO(-2) (53)

(3)

() 18 2 cos(4) 4-u3 (qc) cos(4) OO(-

2

)(54)

with the coecients un dened by equation (32).

(iv) Odd contoured planforms

Substituting the p erturbation expansion of the eigen-function (equation (31)) for odd contoured planforms intoequations (52) and (49) gives, respectively,

(2) 0, (55)

and

(3)() 182 cos(4) 4-u3 (qc) cos(4) OO(-2),

(56)

with the coecients un dened by equation (32). Notethat the quadratic term in equation (50) vanishes identi-cally in the case of odd patterns.

(v) Even non-contoured planforms

Substituting the p erturbation expansion of the eigen-function (equation (40)) for even non-contoured plan-forms into equations (52) and (49) gives, respectively,

(2) 1 32

-2

m40

u0m(qc)2 cos(2m/3) OO(-3), (57)

and(3)() 1 -2

m40

u0m(qc)21 2 cos(2m) OO(-3),

(58)

with the coecients u0n dened by equation (41).

(b) Even and odd p atterns on square or rhombic

lattices

We now use equation (47) to investigate the selectionand stability of odd or even patterns on square orrhombic lattices. Assuming that 40 and L40, the

following three types of steady state are possible for arbi-trary phases 1, 2.

(i) the homogeneous state: c1 c2 0.(ii) rolls: c1 L/0

pei1 , c2 0 or c1 0, c2 L=0ei2.

(iii) squares or rhombics: c1 L/0 2ei1 ,c2 L/0 2ei2 .

The non-trivial solutions correspond to the axial plan-forms listed in tables 3 and 4. A standard linear stabilityanalysis shows that if 240, then rolls are stablewhereas the square or rhombic patterns are unstable. Theopposite holds if 250. These stability properties

persist when higher-order terms in the amplitudeequation are included (Bressloet al. 2000b).

Using equations (48), (54), (56) and (58) with 3j3j/1 1, we deduce that for non-contoured patterns2 0 1 OO(-),and for (odd or even) contoured patterns

2 0 1 2 cos(4)

8 OO(-).

Hence, in the case of a square or rhombic lattice we havethe following results concerning patterns bifurcating from

the homogeneous state close to the p oint of marginalstability (in the limit of weak lateral interactions):

(i) For non-contoured p atterns 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 6 /2, rollsare stable if cos(4)41/2 whereas -rhombics arestable i f cos(4)51/2, that is, if/655/3.

It should b e noted that this result diers from that obtainedby Ermentrout & Cowan (1979) in which stable squareswere shown to occur for certain parameter ranges (see alsoErmentrout (1991)). We attribute this dierence to theanisotropy of the lateral connections incorporated into thecurrent model and the consequent shift^twist symmetry ofthe Euclidean group action. The eects of this anisotropypersist even in the limit of weak lateral connections, andpreclude the existence of stable s quare patterns.




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(c) Even patterns on a hexagonal lattice

Next we use equations (50) and (51) to analyse thestability of even p lanforms on a hexagonal lattice. Ondecomposing ci Cieii , it is a simple matter to show thattwo of the phases i are arbitrary while the sum 3i1 i and the real amplitudes Ci evolve accordingto the equations

dCidt

LCi Ci 1Ci1 cos 0C3i 22=3(C2i 1 C2i1)Ci(59)

and

d

dt 3

i1

Ci

1 Ci

1

Ci sin , (60)

with i,j 1, 2, 3 mod 3. It immediately follows fromequation (60) that the stable steady-state solution willhave a phase 0 if 40 and a phase if 50.

From equations (48), (54) and (58) with 3j3 j/1 1we see that

22=3 0 1 OO(-2),for even non-contoured patterns, and

22=3 0 -u3 (qc) OO

(-2),

for even contoured patterns. In the parameter regimewhere the marginally stable modes are even contouredplanforms (such as in gure 17) we nd that u3 (qc)40.This is illustrated in gure 37.

Therefore, 22=340 for both the contoured and non-contoured cases. Standard analysis then shows that (tocubic order) there exists a stable hexagonal patternCi C for i 1, 2, 3 with amplitude (Busse 1962)

C 120 42=3

jj 2 40 42=3L , (61)

over the parameter range

240 42=3

5L5220 2=30 22=32

.

The maxima of the resulting hexagonal pattern arelocated on an equilateral triangular lattice for 40

(0-hexagons) whereas the maxima are located on anequilateral hexagonal lattice for 50 (-hexagons).Both classes of hexagonal planform have the same D6axial subgroup (up to conjugacy), see table A2 inAppendix A(b). One can also establish that rolls areunstable versus hexagonal structures in the range

05L5

2

0 22=32 . (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 p atterns are the rstto appear (subcritically) beyond the bifurcation p oint.Subsequently, the stable hexagonal branch exchangesstability with an unstable branch of roll patterns, as

shown in gure 38.Techniques from symmetrical bifurcation theory can beused to investigate the eects of higher order terms in theamplitude equation (Buzano & Golubitsky 1983): in thecase of even planforms the results are identical to th

paul c. bressloff et al- geometric visual hallucinations, euclidean symmetry and the functional...

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