Pinwheel stability, pattern selection and the geometry of visual space

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by  Michael Schnabel

Pinwheel stability, pattern selection and the geometry of visual space
Michael Schnabel1,2, Matthias Kaschube1,2,3,4 and Fred Wolf1,2
1Max Planck Institute for Dynamics and Self-Organization, Goettingen,
Germany, 2Bernstein Center for Computational Neuroscience,
University of Goettingen, Goettingen, Germany, 3Lewis-Sigler Institute,
4Physics Department, Princeton University, Princeton, New Jersey, USA
Ithasbeenproposedthatthedynamicalstabilityoftopologicaldefectsinthevisualcortexre(cid:29)ects
the Euclidean symmetry of the visual world. We analyze defect stability and pattern selection in a
generalized Swift-Hohenberg model of visual cortical development symmetric under the Euclidean
group E(2). Euclidean symmetry strongly in(cid:29)uences the geometry and multistability of model
solutions but does not directly impact on defect stability.
8
Whenanobjectinthevisualworldisrotatedaboutthe
0
axisofsighttheorientationsofdistantcontourschangein
0
2 acoordinatedmanner,leavingrelativeorientationsofdis-
tantcontoursinvariant. Inthevisualcortexofthebrain,
n
it has been predicted that this geometrical property of
a
J visual space imposes the so called shift-twist symmetry
on joint representations of contour position and contour
4
2 orientation [1]. This symmetry requires the equivariance
of dynamical models for these representations under the

] Euclidean group E(2). In particular, it has been hy-
C
pothesized that shift-twist symmetry stabilizes topologi-
N
cal pinwheel-defects in models for the emergence of ori-
. entation selectivity during cortical development [2, 3, 4].
o
i Pinwheel defects are singular points in the visual cor-
b
tex around which each orientation is represented exactly
-
q once [5, 6, 7]. They are initially generated in the visual
[ cortexatthetimeofeyeopening[8, 9]. Thisprocesshas

been theoretically explained by spontaneous symmetry
2
breaking [10, 11]. Why in the brain, pinwheels remain
v
2 present at all developmental stages, although they are
3 dynamically unstable in many models of visual cortical
8 development [2, 3, 11, 12, 13, 14] and in analogous phys- Figure1: (a)Planewaveswithwavevectorinhorizontal(top)
3 ical systems [15, 16, 17], remains unclear. and oblique (bottom) direction for variable strength of SSB
1. Previous studies [2, 3, 11, 12, 13, 14] found pin- ((cid:15) = 0, 0.35, 1) (b) PWCs of varying intersection angle α,
0 π/4 ≤ α ≤ π/2. (c) Energy of solutions depends on (cid:15) and
wheelsdynamicallyunstableonlyinmodelsexhibitingan
8 on α. For su(cid:30)ciently large (cid:15) PWCs are energetically favored
E(2)xU(1)symmetry,whichishigherthantheE(2)sym-
0 relativetoplanewaves. (dashed: energyofplanewaves,plain:
: metry of visual perceptual space. In contradistinction, energies of PWCs for α=π/4, π/3, π/2)
v models exhibiting only Euclidean symmetry have been
i
X shown to exhibit stable pinwheels [2, 3, 18] suggesting
that the stabilization of pinwheel defects may be closely
r tude equations for stationary planforms and (cid:28)nd three
a related to the ’reduced’ Euclidean symmetry. There are,
classes of stationary solutions: stripe patterns without
however, also other scenarios that predict the emergence
anypinwheels,pinwheelcrystalswithpinwheelsregularly
ofstablepinwheelswithhigherthanEuclideansymmetry
arranged on a rhombic lattice, and quasi-periodic pat-
[12,14]leavingithardtojudgetoactualroleofEuclidean
ternscontainingalargenumberofirregularlyspacedpin-
symmetry in orientation map development.
wheels. Wecalculatethephasediagramofthesesolutions
HereweanalyzetheimpactofEuclideansymmetryon depending on the strength of SSB, the e(cid:27)ective strength
patternselection,i.e. thequestionofwhetherstablepin- of nonlocal interactions, and the range of nonlocal inter-
wheel arrangement exist and what their geometric orga- actions. With increasing strength of SSB, pinwheel free
nization is. We construct a generalized Swift-Hohenberg patterns are progressively replaced by pinwheel crystals
model [19, 20] symmetric under the Euclidean group in the phase diagram while both pinwheel free patterns
E(2) that allows to study the transition from higher and pinwheel crystals remain stable. Phases of aperiodic
E(2)xU(1)tolowerE(2)symmetrybychangingaparam- pinwheel rich patterns remain basically una(cid:27)ected. A
eterthatcontrolsthestrengthofshiftsymmetrybreaking critical strength of SSB exists above which multistable
(SSB). Using weakly nonlinear analysis we derive ampli- aperiodic patterns collapse into a single aperiodic state.
2
The spatial structure of an OPM can be represented (cid:80)2j=n−01Ajeikjx, kj = kc(cosαj, sinαj) with 2n modes
by a complex (cid:28)eld z(x) where x denotes the 2D position where we require that to each mode also its antiparallel
ofneuronsinthevisualcortex,θ(x)=arg(z(x))/2their mode is in the set. By symmetry, the dynamics of the
preferred stimulus orientation, and the modulus |z(x)| is amplitudes A at threshold has the form
j
ameasureoftheirselectivity[10]. Inthisrepresentation,
2n−1 2n−1
pinwheel centers are zeros of the (cid:28)eld z(x). The sim- A˙ =A +(cid:15)A¯ e4iαj−(cid:88) g |A |2A −(cid:88) f A A A¯
j j j− jk k j jk k k− j−
plest models for the formation of OPMs are de(cid:28)ned by a
k=0 k=0
dynamics
(7)
where j− denotes the index of the mode antiparallel to
∂ z(x,t)=F[z](x,t). (1)
t mode j, k = −k and with real valued and symmet-
j− j
where t denotes time and F[z] is a nonlinear operator. ric matrices gjk and fjk which determine the coupling
We assume the dynamics equivariant under translation and competition between modes. They can be expressed
T z(x) = z(x + y), rotation R z(x) = e2iαz(Ω x) in terms of angle-dependent interaction functions g(α)
y α −α
with rotation matrix Ωφ, and re(cid:29)ection at the corti- andf(α),whichareobtainedfromthenonlinearityN3[z]
cal (1,0) axis Pz(x) = z¯(x¯), thus expressing the fact (cf.[14, 19, 22]). For simplicity we restrict the following
that within cortical layers there are no special locations analysis to the class of permutation symmetric models,
or directions [21]. In addition, if interactions between de(cid:28)ned in [14], for which g(α)=g(α+π). [23]
OPM development and visuotopy are neglected it is also The simplest solution to Eq.(7) is obtained for n =
equivariant under global shifts of orientation preference 1 and consists of plane waves with wavevector k =
Sβz(x) = eiβz(x) (shift symmetry). Rotations R thus kc(cosα,sinα). For |(cid:15)|≤1/2 it is given by
consistofacompositionofphaseshiftsS andcoordinate e2iα √ √
rotationsD,i.e. R =S ◦D withD z(x)=z(Ω x). z(x)= √ [ 1+2(cid:15) cos(kx+φ)+i 1−2(cid:15)sin(kx+φ)]
α 2α α α −α g
We consider the general class of variational models [3, 00
(8)
10, 14] for which F[z] has the form
with arbitrary phase φ. Hence with SSB orientation an-
gles are no longer equally represented. For (cid:15)>0 cortical
F[z]=Lz+(cid:15)Mz¯+N [z]. (2) areafororientationsαandα+π/2isrecruitedattheex-
3
penseofα+π/4andα+3π/4(andviceversafor(cid:15)<0).
Here L is a linear, translation invariant and self-adjoint Beyond a critical strength of SSB, (cid:15) = 1/2, patterns
∗
operator,thataccountsfora(cid:28)nitewavelengthinstability. only contain two orientations, z(x)=e2iαN cos(kx+φ)
N3 is a cubic nonlinearity which stabilizes the dynamics. for (cid:15) > (cid:15)∗ and z(x) = ie2iαN sin(kx+φ) for (cid:15) < −(cid:15)∗
The second term involves a complex conjugation Cz = with N =(cid:112)4(1+|(cid:15)|)/3g (Fig.1a).
00
z¯ and thus manifestly breaks shift symmetry when (cid:15) (cid:54)=
Another class of solutions, rhombic pinwheel crystals
0. M is assumed to be linear, translation invariant and (PWCs), exist for n = 2 and consist of two pairs of
bounded. Equivariance under rotations, [MC, Rα] = 0, antiparallel modes forming an angle 0 < α ≤ π/2 which
requires are characterized by |A | = |A | = a = |A | = |A |.
0 0− 1 1−
D MD−1 =S M (3) We consider w.l.o.g. the case α0 = −α/2 and α1 = α/2
α α −4α (Fig.1b). With A0,1 = aeiµ0,1, A0−,1− = aeiν0,1 and
and equivariance under parity [M, P]=0. Σ0,1 := µ0,1 + ν0,1 the stationary state is given by
As a concrete example we will consider the model Σ1 = −Σ0 and a2 = (1+(cid:15)cos(Σ0 +2α))/ζ where ζ =
3g +2g +2f cos2Σ . ThephaseΣ isthesolutionto
00 01 01 0 0
L = r−(k2+∇2)2 (4) 0=sin2Σ +(cid:15)[sin(Σ −2α)−(2+3g /g )sin(Σ +2α)]
c 0 0 00 01 0
2−g which bifurcates from Σ =±π/2 for (cid:15)=0. The energy
N3[z] = (1−g)|z(x)|2z(x)− 2πσ2 (5) is given by EPWC = −40a12(1+|(cid:15)|cos(Σ0+2α))+2a4ζ.
(cid:90) (cid:18) 1 (cid:19) For the model Eqs.(4-6) the (cid:15) and α dependence of the
× d2y |z(y)|2z(x)+ z(y)2z¯(x) e−|y−x|2/2σ2 energyisshowninFig.1c. Thesolutionthenreadsz(x)=
2
M = r(∂x+i∂y)4(∂xx+∂yy)−2 (6) 2wait(cid:2)heiaΣr0b/i2tcraorsy(k∆0x0+an∆d0∆/21).+e−iΣ0/2sin(k1x+∆1/2)(cid:3)
A large set of quasiperiodic solutions originates
where L is the Swift-Hohenberg operator [19, 20] with
from the essentially complex planforms (ECP) z(x) =
critical wavenumber k and instability parameter r. N
is adopted from [14], wchere σ sets the range of the non3- (cid:80)nj=−01A+j eiljkjx that solve Eq. (7) for (cid:15) = 0 [14]. Here,
local interactions and g determines whether the local wave vectors kj = kc(cosNπj, sinNπj) (j = 0,..., n−1)
(g > 1) or the nonlocal term (g < 1) stabilizes the dy- are distributed equidistantly on the upper half of the
namics. M is the simplest di(cid:27)erential operator which critical circle and binary variables lj = ±1 determine
transforms according to Eq.(3). It is unitary with spec- whetherthemodewithwavevectorkj orwithwavevec-
trum ∝e4iarg(k). tor −kj is active (Fig.2(a) left column).
Weusedweaklynonlinearanalysis[22]tostudypoten- For |(cid:15)| > 0 we (cid:28)nd that ECPs generalize to z(x) =
tial solutions of Eq.(2). We consider planforms z(x) = (cid:80)n−1(cid:2)A+eiljkjx+A−e−iljkjx(cid:3). For each n there exists
j=0 j j
3
Figure 3: (a)-(c) Pinwheel densities for all realizations of
ECPswith3≤n≤17anddi(cid:27)erentdegreesofshiftsymmetry
breaking (cid:15). (d) Pinwheel densities for n=17 (dots) and for
n→∞ in the Gaussian approximation (gray region).
if(cid:15)<0. Thisimpliesthatallorientationsarerepresented
in patterns with n≥3. The solution can be written
Figure 2: Attractors of amplitude equations (Eq.7) with full, n−1
partially broken and completely broken shift symmetry. (a) z(x)=(cid:112)2γ(cid:88)[(cid:112)1+qze(x,φ )+(cid:112)1−qzo(x,φ )]
j j j j
ECPs. Preferred orientations are color coded [see bars in
j=0
(b)]. Arrangement of active modes on the critical circle and
correspondingOPMs. Forn=3and8thereare2and15dif- with zje(x,φj) = ei2nπjcos(ljkjx+φj) and zjo(x,φj) =
ferent classes of ECPs, respectively. Complete (partial, no)
suppression of opposite modes for full (weakly broken, max- iei2nπjsin(ljkjx+φj)andarbitraryphasesφj. Forn=1
imally broken) shift symmetry (left, middle, right column). this is in agreement with Eq.(8) where (cid:15)∗ = 1/2. Re-
(b) OPM in tree shrew V1 (data: L.E.White, Duke Univ., (cid:29)ectionattheaxisparalleltokj actsonthefunctionszje
USA). Arrows pinwheel centers. Scale bar 1 mm. (c) With andzo as+1and−1,respectively. Thusze andzo corre-
j j j
increasing degree of symmetry breaking (cid:15) amplitudes of an- spond to the even and odd eigenvectors of the nullspace
tiparallelmodesA−growandeventually(at(cid:15)=(cid:15)∗)reachthe of L+(cid:15)MC (cf.[18]). For (cid:15) > 0 ((cid:15) < 0) the even (odd)
same absolute value as active modes A+. part dominates the solution.
ThedynamicsEq.(7)exhibitsapotentiallyexceedingly
high number of multistable solutions. The energy of n-
a critical value (cid:15)∗ := γ[g00 +2(cid:80)nj=−01fijcos4nπj], where ECPs is given by En = −nγ[1+(cid:15)2/(cid:15)∗] for |(cid:15)| ≤ (cid:15)∗ and
γ = [2(cid:80)nj=−01gij]−1. When |(cid:15)| ≤ |(cid:15)∗| stationary ampli- En =−nγ(1+|(cid:15)|)2/[1+(cid:15)∗]for|(cid:15)|≥(cid:15)∗,respectively, and
tudes a± = |A±| ful(cid:28)ll a2 = γ(1±(cid:112)1−(cid:15)2/(cid:15)2). When doesnotdependonthevariableslj whichidentifyapar-
j ± ∗
|(cid:15)|≥|(cid:15) | the amplitude of antiparallel and active modes ticular n-ECP. Due to the growth of antiparallel modes
are equ∗al a2 = γ[1 + |(cid:15)|]/[1 + (cid:15) ] (Fig.2c). A simple with increasing |(cid:15)| patterns for all di(cid:27)erent realizations
measure of t±he degree to which SS∗B a(cid:27)ect n-ECPs is lj with phases φj := ljΦj + 14(1−sign((cid:15)))(1−lj)π (Φj
arbitrary but (cid:28)xed) eventually collapse in a single state
q := (cid:80)jA+j A−j e−i4nπj +c.c.. z(x)∝(cid:80)nj=−01zje/o(x,φj) (Fig.2a).
(cid:80) (cid:12)(cid:12)A+(cid:12)(cid:12)2+(cid:12)(cid:12)A−(cid:12)(cid:12)2 This collapse manifests itself in the pinwheel densi-
j j j ties ρ of various ECPs shown in Fig.3a-c for di(cid:27)erent
n
strength of SSB. The pinwheel density of an n-ECP in
Fanodr aq =stastigionn(cid:15)arify|(cid:15)n|->EC|(cid:15)P∗|.we (cid:28)nd q = (cid:15)/|(cid:15)∗| if |(cid:15)| ≤ |(cid:15)∗| the large n limit is ρ(χ) = π(cid:113)1− π82(1−(cid:15)2/(cid:15)2∗)ζ2 and
Stationary phases φ±j = argA±j ful(cid:28)ll the condition depends through ζ = 41n|(cid:80)nj=−01ljkj| ≤ 1 on the con-
φ++φ− =(4π/n)j if (cid:15)>0 and φ++φ− =(4π/n)j+π (cid:28)guration of active modes. Fig.3d shows that pinwheel
j j j j
4
Figure 5: Stability regions of ECPs with n=20 for di(cid:27)erent
(cid:15). Dashed line denotes the critical line (cid:15)=(cid:15)∗(N,g,σ), above
which q = 1. Shaded region denotes region in the g−σ/Λ
plane for which that planform is a stable solution of the dy-
namicsandcoexistswithplanformsofnearbyvaluesofn,e.g.
n = 18, 19, 21, 22. In the inner region (marked by the two
inner lines) this solution minimizes energy.
space and on (cid:15), a particular angle α minimizing the en-
ergy (c.f. Fig.1c). Large n−ECPs are selected when the
dynamics is stabilized by long-range interactions (g <1,
σ > Λ). In this parameter regime plane waves and pin-
wheelcrystalsareunstable. ThedegreeofSSBqmanifest
in a given n−ECP attractor depends on (cid:15) and on the lo-
cationinthephasediagram. Aboveacriticallinede(cid:28)ned
by |(cid:15) (N,g,σ)|=|(cid:15)| antiparallel modes are maximal and
∗
|q|=1 (gray area), below that line |q|≤1. Figs. 4 and 5
show the high sensitivity of the dynamics to even small
amounts of SSB, a substantial area in phase is occupied
by ECPs with |q|=1 even for (cid:15)=0.02.
Our analysis of pattern selection in visual cortical de-
velopmentdemonstratesthatdynamicalmodelsoforien-
tationmapdevelopmentareverysensitivetothepresence
of interactions imposed by Euclidean E(2) symmetry. It
alsorevealsthattheimpactoftheEuclideansymmetryof
perceptualvisualspaceisnotanallornonephenomenon.
ForweakSSBourEuclideanmodelcloselymimicsthebe-
havior of models possessing the higher E(2)xU(1) sym-
metry. The only qualitative change that we found in the
transition from higher to lower symmetry was the col-
lapse of multistable solutions. This collapse, however,
Figure 4: Phase diagrams of the model, Eqs.(4-6), near crit- only happens when a (cid:28)nite critical strength of SSB is
icality for variable SSB (cid:15). The graph shows the regions of reached. Uptothisthresholdstrength,modelsexhibiting
theg−σ/Λplaneinwhichn-ECPsandPWCshaveminimal E(2) and E(2)xU(1) symmetry seem to be topologically
energy (n=1−25, n>25 dots). Regions of maximally bro- conjugate to one another.
ken shift symmetry [(cid:15)≥(cid:15)∗(N,g,σ)] shaded in gray. Regions Our analysis also reveals that the impact of Euclidean
whereα-PWCsprevailisshadedinblue,intensitylevelcodes
symmetry di(cid:27)ers qualitatively for aperiodic and peri-
for the relative angle α. (light blue: π/4 ≤ α ≤ π/2 :dark
odic patterns. For aperiodic patterns, (i) the parameter
blue)
regime in which they possess minimal energy is virtually
una(cid:27)ected by the strength of SSB, (ii) sets of di(cid:27)erent
multistable solutions become progressively more similar
densities (cid:28)ll a band of values. With increasing degree of and (cid:28)nally merge forming a single unique ground state,
SSBthisbandnarrowsandpinwheeldensitieseventually and (iii) Euclidean symmetry geometrically manifests it-
equal π at the critical value |(cid:15)|=(cid:15)∗. self in speci(cid:28)c two point correlations that are absent in
To reveal how SSB a(cid:27)ects pattern selection we calcu- the E(2)xU(1) symmetric limit. For periodic patterns
latedthephasediagramforthemodelspeci(cid:28)edinEqs.(4- such as pinwheel crystals and pinwheel free states, (i)
6) for various values of (cid:15). Fig.4 shows the con(cid:28)gura- the parameter regime in which these solutions possess
tions of n-ECPs and PWCs minimizing the energy [24]. minimal energy depends on the strength of SSB, and (ii)
Plane waves are progressively replaced by α-PWCs with Euclidean symmetry geometrically manifests itself in a
increasing SSB. Depending on the location in parameter speci(cid:28)cally selected tilt angle of rhombic pinwheel crys-
5
tals and a wave vector dependent underrepresentation of of a model of higher E(2)xU(1) symmetry. Our analy-
particular orientations for pinwheel free states. sis predicts that in this regime, two point correlations
The Swift-Hohenberg model of Euclidean symmetry provide a sensitive measure of the strength of SSB. Such
consideredherepredictsthataperiodicpinwheelrichpat- correlationsarethereforeapromisingtoolforprobingthe
terns resembling the architecture of the primary visual impact of Euclidean symmetry in visual cortex develop-
cortexareonlystablewhenlong-rangeinteractionsdom- ment.
inate pattern selection, con(cid:28)rming previous predictions
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g + (2 − g)e(α) and f(α) = 1g(α) with e(α) =
2
2exp(−σ2k2)cosh(σ2k2cosα) (cf.[14] for details). The
c c
couplingcoe(cid:30)cientsaregivenbygjk =(1−21δjk)g(|αk−
αj|) and fjk =(1−δjk−δjk−)f(|αk−αj|).
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tional E = (cid:80)j|Aj|21 + (cid:15)(cid:80)j(A¯jA¯j−e4inπj + c.c.) −
12(cid:80)jkgjk|Aj|2|Ak|2− 21(cid:80)jkfjkAkAk−A¯jA¯j−.