Paul C. Bressloff and Jack D. Cowan- SO(3) Symmetry Breaking Mechanism for Orientation and Spatial Frequency Tuning in the Visual Cortex

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VOLUME 88, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 18 FEBRUARY 2002

approximately orthogonal so that they generate a localcurvilinear coordinate system. Such results provide func-tional evidence that hypercolumns implement localizedtwo-dimensional spatial frequency analysis.

In this Letter we construct a minimal dynamical modelof a hypercolumn that (a) includes both orientation and

spatial frequency preferences, (b) incorporates the orien-tation pinwheels, and (c) exhibits sharply tuned responsesin the presence of recurrent interactions and weakly biasedLGN inputs via a symmetry breaking mechanism.

For simplicity, we restrict ourselves to a single oculardominance column, and assume that the hypercolumn isparametrized by two cortical labels, which represent theorientation preference f [ 0,p and spatial frequencypreference p [ 0,` of a local patch or column of cells.Given that p is not a periodic variable within a hyper-column, one cannot extend the ring model of orientationtuning by including a second ring so that the networktopology becomes a torus S1 3 S1. An important clueon how to proceed is provided by the fact that each hy-

percolumn (when restricted to a single ocular dominancecolumn) contains two orientation singularities. These areassumed to correspond, respectively, to the two extremesof spatial frequency within the hypercolumn. This sug-gests the network topology of a sphere S2, with the twosingularities identified as the north and south poles, respec-tively [see Fig. 1(b)]. Introducing spherical polar coordi-nates r, u, 2f with r 1, u [ 0,p, and f [ 0,p,we set

u Qp p1 1 p0pb (1)

with p0, b fixed. The compressive nonlinearity is requiredin order to represent the semi-infinite range of spatial fre-

quencies within the bounded domain of a hypercolumn.The gain parameter b determines the effective spatial fre-quency bandwidth of the hypercolumn. The latter is typi-cally around four octaves, that is, pmax 2

npmin withn 4, which corresponds to a gain of around b 1.5.

Let au,f, t denote the activity of a population of cellson the sphere. Our spherical model of a hypercolumn isthen defined according to the evolution equation

au,f, t

t 2 au,f, t 1 hu,f

1ZS2wu, f j u0, f0

3 gau0,f0, tD u0,f0 , (2)

where D u, f 2 sinu du df. Here w represents thedistribution of recurrent interactions within the hypercol-umn, hu, f is a weakly biased input from the LGN, andga is the smooth nonlinear function ga gmax1 1e2ha2ath for constant gain h and threshold ath. Equa-tion (2) is the natural extension of the ring model to the

sphere. It is important to emphasize that the sphere de-scribes the network topology of the local weight distribu-tion expressed in terms of the internal labels for orientationand spatial frequency. It is not required to match the actualspatial arrangement of cells within a hypercolumn.

In order to extend the symmetry breaking mechanism of

the ring model [6] to the spherical model (2), we first haveto construct a weight distribution that is invariant with re-spect to the symmetry group of the sphere, namely, SO(3).This requires the use of spherical harmonics. Any suf-ficiently smooth function fu, f on the sphere can beexpanded in a uniformly convergent double series of spheri-cal harmonicsfu,f

P`n0

Pnm2n fnmY

mn u,f. The

functions Ymn u, f constitute the angular part of the solu-tions of Laplaces equation in three dimensions, and thusform a complete orthonormal set. The orthogonality rela-tion isZ

S2Ym1n1 u, fY

m2n2

u, fD u,f dn1,n2dm1,m2 . (3)

The spherical harmonics are given explicitly by

Ymn u,f 21m

s2n 1 1

4p

n 2 m!

n 1 m!Pmn cosue

2imf

(4)

for n $ 0 and 2n # m # n, where Pmn cosu is anassociated Legendre function. The action of SO(3) onYmn u, f involves 2n 1 1 3 2n 1 1 unitary matricesassociated with irreducible representations of SU(2) [18].From the unitarity of these representations, one canconstruct an SO(3) invariant weight distribution of thegeneral form

wu,f j u0, f0 m`

Xn0Wn

n

Xm2nYmn u

0, f0Ymn u,f

(5)

with Wn real and m a coupling parameter. As an illus-tration, consider the SO(3) distribution joining neuronswith the same spatial frequency (same latitude on thesphere) for the particular case W0 , 0, W1 . 0, Wn 0for n $ 2. One finds that away from the pinwheels (polesof the sphere), cells with similar orientation excite eachother, whereas those with differing orientation inhibit eachother. This is the standard interaction assumption of thering model. On the other hand, around the pinwheels,all orientations uniformly excite, which is consistent withthe experimental observation that local interactions dependon cortical separation [19]. That is, although the cellsaround a pinwheel can differ greatly in their orientationpreference, they are physically close together within thehypercolumn.

We now show how sharp orientation and spatial fre-quency tuning can occur through spontaneous symmetrybreaking of SO(3). First, substitute the distribution (5)into Eq. (2) and assume, for the moment, that there isconstant external drive from the LGN [with hu,f h]

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such that Eq. (2) has au, f a as a fixed point solution.Linearizing about the fixed point and setting au, f, t a 1 eltuu, f leads to an eigenvalue equation for the lin-ear eigenmodes uu, f:

luu, f 2uu,f 1 mZS2`

Xn0n

Xm2nWn3 Ymn u

0, f0Ymn u,fuu

0,f0D u 0, f0

(6)[with a factor g0a absorbed into m]. The orthogonal-ity relation (3) shows that the linear eigenmodes arespherical harmonics with l ln 21 1 mWn foruu, f Ymn u, f, 2n # m # n. Thus the nth eigen-value is 2n 1 1-fold degenerate.

Now suppose that W1 . Wn for all n fi 1. The fixedpoint solution a a then destabilizes at a critical valueof the coupling mc 1W1 due to excitation of the first-order spherical harmonics. Sufficiently close to the bifur-cation point, the resulting activity profile can be written inthe form

au, f a 1X

m0,6

cmfmu, f (7)

for real coefficients c0, c6 and f0u,f cosu,f1u,f sinu cos2f, f2u, f sinu sin2f. Am-plitude equations for these coefficients can be obtainedby carrying out a perturbation expansion of Eq. (2)with respect to the small parameter m 2 mc usingthe method of multiple scales [20]. This leads to theStuart-Landau equations [8,21]

dck

dt ck

m 2 mc 1 L

Xm0,6

c2m

!, k 0, 6 ,

(8)

where

L 3g3

51

2g22

3

mcW01 2 mcW0

18g

22

15

mcW21 2 mcW2

. (9)

Here g2,g3 are coefficients in the Taylor expansion of thefiring rate function, ga 2 gag0a a 2 a 1g2a 2 a

2 1 g3a 2 a3 1 . . .. It is clear that the am-

plitude equations are equivariant with respect to the actionof the orthogonal group SO(3) on c0, c1, c2, which re-flects the underlying spherical symmetry. Moreover, defin-ing R

Pm0,6 c

2m we see that

dR

dt 2Rm 2 mc 1 LR , (10)

which has a stable fixed point at R0 m 2 mcjLj, pro-

vided that L , 0. This corresponds to an SO(3)-invariantsubmanifold of marginally stable states.

Equation (7) represents a tuning surface for orientationand spatial frequency with a solitary peak whose locationis determined by the values of the coefficients c0, c1,c2.Such a solution spontaneously breaks the underlying SO(3)symmetry. However, full spherical symmetry is recovered

by noting that rotation of the solution corresponds to anorthogonal transformation of the coefficients c0, c6. Thusthe action of SO(3) is to shift the location of the peak ofthe activity profile on the sphere, that is, to change theparticular orientation and spatial frequency selected by thetuning surface.

Now suppose that there exists a weakly biased, time-independent input from the LGN of the form

hu,f h 1X

m0,6

hmfmu, f , (11)

where h0 G cosQ, h1 G sinQ cos 2F, h2 G 3sinQ sin2F where G is the effective contrast of the inputstimulus. Equation (11) describes a unimodal functionon the sphere with a single peak at Q, F, which cor-responds to an input orientation fS F and an inputspatial frequency pS Q

21Q with Q defined accord-ing to Eq. (1). Here we ignore higher-order sphericalharmonic contributions to the LGN input, since the patternforming instability amplifies only the first-order harmonic

componentsit is these components that couple to thecubic amplitude equation. If G O e32 then theinput (11) generates an additional contribution hk tothe right-hand side of the cubic amplitude equation thatexplicitly breaks the hidden SO(3) symmetry and fixes thepeak of the tuning surface at Q,F. Since hk O

32,whereas ck O

12, we see that the cortical model actsas an amplifier for the first spherical harmonic componentsof the weakly biased input from the LGN.

In order to confirm the above analytical results, wesolved Eq. (2) numerically, using the discretization schemeconsidered by Varea et al. [22] in their study of patternformation for a reaction-diffusion system on a sphere. InFig. 2 we plot the relative firing rate gagmax in response

to a weakly biased input from the LGN, Eq. (11), withQ p2 (corresponding to an intermediate input fre-quency pS 2cdeg) and F p2. Figure 2(a) shows asurface plot in the p, f plane for G 0.1. It can be seenthat the hypercolumn exhibits a tuning surface that is lo-calized with respect to two-dimensional spatial frequencyand its peak is locked to the LGN input at p 2cdeg,f p2. In Fig. 2(b) we plot the response as a functionof spatial frequency at the optimal orientation for variousinput amplitudes G. The height of the spatial frequencytuning curves increases with the input amplitude G, butthe width at half-height is approximately the same (as canbe checked by rescaling the tuning curves to the sameheight). Since G increases with the contrast of a stimu-lus, this shows that the network naturally exhibits contrastinvariance. Corresponding orientation tuning curves areshown in Fig. 2(c), and are also found to exhibit contrastinvariance.

Note that projecting the spherical tuning surface ontothe p, f plane breaks the underlying SO3 symmetryof the sphere. Consequently, the shape of the planar tun-ing surface is not invariant under shifts in the location of

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050

100150

0.51

24

80

0.2

0.4

0.6

spatialfrequencyp(c/deg.)

orientat

ion

firingrate

0

0.2

0.4

0.6firingrate

= 0.1

90 18045 1350 orientation

= 0.01= 0.05

(c)

0.5 1 2 4 8 16

0

0.2

0.4

0.6 = 0.1= 0.05

= 0.01

spatial frequency p

(b)

(a)

FIG. 2. Plot of normalized firing rate gagmax in response toa weakly biased input from the LGN with Q p2, F p2,and G 1. The firing rate function has a gain h 5, athreshold ath 0.6, and we take a 0. The weight coeffi-cients are W0 22, W1 1, Wn 0, n . 1, and gmaxm 5.(a) Tuning surface in the p, f plane. (b) and (c) Spatial fre-quency and orientation tuning curves.

the peak of the tuning surface. Such distortions generatebehavior that is consistent with recent experimental obser-vations. First, at low and high spatial frequencies (i.e.,towards the pinwheels) there is a broadening of the tunedresponse to orientation, as found in [13]. Second, there is

a systematic shift in the peak of spatial frequency tuningcurves at nonoptimal orientations that is towards the closestpinwheel. Interestingly, there is some suggestion of spatialfrequency shifts in recent optical imaging data [17].

In conclusion, representing the topology of a hyper-column (with respect to orientation and spatial frequencylabels) as a sphere is a natural way to accommodate theorientation pinwheels while providing a recurrent mecha-nism for generating two-dimensional spatial frequency tun-ing. One issue that we have not addressed here is how theprojection of the LGN input on to its first spherical har-monic components encodes information regarding proper-ties of a visual stimulus. Intriguingly, examination of theresponse to a sinusoidal grating stimulus indicates that ifsuch a stimulus is first filtered by the action of the feedfor-

ward pathway from retina to cortex, before the recurrentdynamics amplifies its first-order spherical harmonic com-ponents, then the representation of spatial frequency is notfaithful. That is, there is a mismatch between the spatialfrequency at the peak of the tuning surface and the stimulusfrequency. We expect a similar conclusion to hold for any

recurrent mechanism that amplifies harmonic componentsof two-dimensional stimuli. One possible mechanism forcorrecting this mismatch is via the massive feedback path-ways from V1 back to LGN [21].

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Paul C. Bressloff and Jack D. Cowan- SO(3) Symmetry Breaking Mechanism for Orientation and Spatial Frequency Tuning in the Visual Cortex