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Paul C. Bressloff- Spontaneous symmetry breaking in self–organizing neural fields

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  • 8/3/2019 Paul C. Bressloff- Spontaneous symmetry breaking in selforganizing neural fields


    Biol Cybern (2005) 93: 256274DOI 10.1007/s00422-005-0002-3

    O R I G I NA L PA P E R

    Paul C. Bressloff

    Spontaneous symmetry breaking in selforganizing neural fields

    Received: 1 December 2004 / Accepted: 6 June 2005 / Published online: 29 September 2005 Springer-Verlag 2005

    Abstract We extend the theory of self-organizing neural fieldsin order to analyze the joint emergence of topography andfeature selectivity in primary visual cortex through spontane-ous symmetry breaking. We first show how a binocular one-dimensional topographic map can undergo a pattern forminginstability that breaks the underlying symmetry between leftand right eyes. This leads to the spatial segregation of eyespecific activity bumps consistent with the emergence of oc-ular dominance columns. We then show how a 2-dimensionalisotropic topographic map can undergo a pattern forminginstability that breaks the underlying rotation symmetry. Thisleads to the formation of elongated activity bumps consistentwith the emergence of orientation preference columns.A par-

    ticularly interesting property of the latter symmetry breakingmechanism is that the linear equations describing the growthof the orientation columns exhibits a rotational shift-twistsymmetry, in which there is a coupling between orientationand topography. Such coupling has been found in experimen-tally generated orientation preference maps.

    1 Introduction

    One of the striking features of the visual system is that thevisual world is mapped on to the cortical surface in a topo-graphic manner. This means that neighboring points in a vi-

    sual image evoke activity in neighboring regions of visualcortex. Superimposed upon this topographic map are addi-tional maps reflecting the fact that neurons respond preferen-tially to stimuli with particular features. Neurons in the retina,lateral geniculate nucleus (LGN) of the thalamus, and pri-mary visual cortex (V1) respond to light stimuli in restrictedregions of the visual field called their classical receptive fields(RFs). Patterns of illumination outside the RF of a givenneuron cannot generate a response directly, although they

    P.C. BressloffDepartment of Mathematics,University of Utah,Salt Lake CityUtah

    can significantly modulate responses to stimuli within theRF via longrange cortical interactions (Fitzpatrick 2000).The RF is divided into distinct ON and OFF regions. In anON (OFF) region illumination that is higher (lower) thanthe background light intensity enhances firing. The spatialarrangement of these regions determines the selectivity ofthe neuron to different stimuli. For example, one finds thatthe RFs of most V1 cells are elongated so that the cells re-spond preferentially to stimuli with certain preferred orien-tations (Hubel andWiesel 1962). The RFs of retinal ganglionneurons and LGN neurons, on the other hand, are circularlysymmetric and hence these neurons do not exhibit any stim-ulus orientation preference. Neurons in both the LGN and in-

    put layers of V1 are also segregated according to whether ornot they respond preferentially to left-eye or right-eye stimuli(ocular dominance) (Hubel and Wiesel 1977). Neurons in V1with similar feature preferences tend to arrange themselvesin vertical columns so that to a first approximation the lay-ered structure of cortex can be ignored (LeVay and Nelson1991).The corresponding feature maps then describe the spa-tial distribution of these columns as one moves tangentiallyover the surface of cortex. In recent years much informationhas accumulated regarding the twodimensional distributionof both orientation preferenceand ocular dominance columnsusing optical imaging techniques (Blasdel and Salama 1986;Bonhoeffer and Grinvald 1991). These experimental studies

    indicate that there is an underlying periodicity in the micro-structure of V1 with a period of approximately 1mm (in catsand primates). The fundamental domain of this tiling of thecortical plane is the hypercolumn, which contains two sets oforientation preferences [0, ) per eye, organized arounda pair of orientation singularities or pinwheels (Obermayerand Blasdel 1993).

    It is generally accepted thatthe preference of cortical neu-rons for particular stimulus features such as orientation andocular dominance arises primarily from the spatial arrange-ment of convergent feedforward afferents from the LGN (orfrom other layers of cortex). The experimental observationthat stimulus deprivation can modify ocular dominance col-umns during a critical period of postnatal development in

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    Spontaneous symmetry breaking in selforganizing neural fields 257

    cats and primates provides strong evidence that the formationof these columns is activitydependent (Hubel et al. 1977;LeVay et al. 1978; Stryker and Harris 1986). On the otherhand, since orientation and OD columns are already present

    in newly born primates, and the segregation of OD columnsoccurs as early as one week after LGN axons enter layer 4of V1 in ferrets (Crowley and Katz 2000), it has been sug-gested that activityindependent molecular cues could playa major role in the initial formation of columns. However,specific molecules have not yet been found. Moreover, it ispossible that spontaneous retinal waves (Wong et al. 1993) orendogenous activity in the cortico-geniculate feedback loopcould support an activitydependent mechanism in the earlystages of columnar development (Penn and Shatz 1999). Amuch more likely role for molecular cues is in the initialdevelopment of the topographic map, where geniculate ax-ons are guided to targets in input layer 4 of V1 after reach-ing the cortical subplate (Ghosh and Shatz 1992). However,the resulting map is rather crude and some form of activityappears to be necessary for the subsequent refinement of thetopographic map through the pruning of initially exuberantaxonal arborizations (Catalano and Shatz 1998).

    A large number of models have been proposed that de-scribe activitydependent development as a selforganizingHebbian process (see the review of Swindale 1996). In thecase of correlationbased developmental models (Linsker1986; Miller et al. 1989; Miller 1994; Erwin and Miller1998), the statistical structure of input correlations providesa mechanism for spontaneously breaking some underlyingsymmetry of theneuronalreceptive fields leadingto theemer-gence of feature selectivity. When such correlations are com-bined with intracortical interactions, there is a simultaneousbreaking of translation symmetry across cortex leading tothe formation of a spatially periodic cortical feature map.Correlationbased models are essentially linear, so that con-siderable insight into the developmental process can be ob-tained by solving an associated eigenvalue problem (Mackayand Miller 1990; Miller and MacKay 1994; Wimbauer et al.1998). One of the possible limitations of this class of modelis that a regular topographic map is assumed already to ex-ist before featurebased columns begin to develop. In orderto model the joint development of topography and corticalfeature maps, it appears necessary to introduce some formof nonlinear competition for activation (Willshaw and von

    der Malsburg 1976; Kohonen 1982; Goodhill 1993; Piepen-brockand Obermayer 1999), neurotrophic factors (Elliott andShadbolt 1999) or a combination of the two (Whitelaw andCowan 1981).

    An alternative mathematical formulation of topographicmap formation has been developed byAmari using the theoryof selforganizing neural fields (Takeuchi and Amari 1979;Amari 1980, 1983, 1989). The basic network model involvesa form of noncompetitive Hebbian learning in the presenceof hard threshold, nonlinear firing rate functions. It is foundnumerically that starting from a crude topographic map, thesystem evolves to a more refined continuous map that isdynamically stable. In the simpler onedimensional case,

    conditions for the existence and stability of such a map canbe derived analytically. Moreover, it can be shown that undercertaincircumstances the continuoustopographic map under-goes a pattern forming instability that spontaneously breaks

    continuous translation symmetry, and the map becomes par-titioned into discretized blocks; it has been suggested thatthese blocks could be a precursor for the columnar micro-structure of cortex (Takeuchi and Amari 1979; Amari 1989).Given that cortical columns tend to be associated with stim-ulus features such as ocular dominance and orientation, thisraises the interesting question whether or not such featurescould also emerge through the spontaneous symmetry break-ing of selforganizing neural fields. Some recent numericalstudies support such a possibility (Woodbury et al. 2002;Fellenz and Taylor 2002). In this paper, we explore this issuefrom a mathematical perspective by extending Amaris orig-inal analysis to networks with distinct lefteye and righteyeafferents and to twodimensional networks. Throughout thepaper we emphasize the important role of symmetry.

    2 Neural field theory

    We begin by reviewing Amaris neural field theory for topo-graphic map formation (Takeuchi and Amari 1979; Amari1980, 1983, 1989), and introduce the basic notation that willbe used throughout the paper. We also present an alternativederivation of the linear stability conditions for onedimen-sional topographic maps, which is more easily extended tonetworks with distinct left/right eye afferents (Sect. 3) and totwodimensional networks (Sect. 4).

    2.1 Network model

    A schematic diagram of the basic network model is shown inFig. 1. The lateral geniculate nucleus (LGN) and the primary





    r ILGN








    Fig. 1 Basic network architecture illustrating how a localized inputI centered at position r in the LGN layer induces a correspondingresponse u in the cortical layer. (The global inhibition is not shown)

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    258 P.C. Bressloff

    visual cortex are treated as twodimensional continuous neu-ral sheets. Let r1 = (x1, y1) 1 denote a point in the LGNlayer and r2 = (x2, y2) 2 a point in the cortical layer. Thestrength of feedforward excitatory afferents connecting these

    two points is denoted by s(r2, r1). Suppose, for the moment,that these feedforward afferents are fixed and that there existspresynaptic input activity I (r1|r) centered about the pointr in the LGN. This is supplemented by a global inhibitoryinput I0 with associated feedforward synaptic density s0(r2)(Takeuchi and Amari 1979; Amari 1980). The total weightedinput to a point r2 in cortex is then given by

    v(r2|r) =


    s(r2, r1)I (r1|r)dr1 s0(r2)I0. (2.1)

    Cortical neurons also receive synaptic inputs from recurrentconnections within the layer, which are taken to be homoge-neous and isotropic. Thus, the synaptic density from neurons

    at r2 to neurons at r2 is of the form w(|r2 r2|) for someprescribed function w. Given these two sources of input, theactivity u(r2, t|r) of neurons at r2 at time t in response toa stimulus centered at r satisfies the neural field equation(Takeuchi and Amari 1979; Amari 1980)


    t= u(r2, t|r) +


    w(|r2 r2|)

    H(u(r2, t|r))dr2 + v(r2|r) h, (2.2)where is a membrane time constant, h determines thebackground level of activity in the absence of stimuli, andH(u) denotes the output firing rate function, which is takento be a Heaviside function: H(u)

    =1 ifu > 0 and H(u)


    otherwise. It is assumed that each stimulus is presented tothe network for a sufficiently long time, so the activity u con-verges to a stable equilibrium solution of the integral equation

    u(r2|r) =


    w(|r2 r2|)H(u(r2|r))dr2+v(r2|r) h. (2.3)

    Now suppose that modifications in the strength of thefeedforward afferents s, s0 occur on a much slower time scalethan both the relaxation time of the activity u and the timeinterval over which each input is sampled. This adiabaticcondition implies that the equilibrium activity u is slaved to

    the slowly changing synaptic weights s, s0. A Hebbian ruleis assumed for the dynamics of the feedforward connectionssuch that during the presentation of a single input centeredat r,


    = s(r2, r1, ) + cH(u(r2, |r))I(r1|r) (2.4)



    = s0(r2, ) + cH(u(r2, |r))I0, (2.5)

    where = t for 0 < 1 and c, c are constants. Notethat there is a separation of timescales in which u(r2, |r) =limt u(r2, t , |r) is a stable equilibrium solution of Eqs.

    2.2 and 2.1 with s = s(r2, r1, ), s0 = s0(r2, ) for fixedslow time variable .

    The final step in the formulation of neural field theoryis to assume that the the center r( ) of an input at time

    is generated at random from some probability density (r)(Takeuchi and Amari 1979; Amari 1980). This implies thatEqs. 2.4 and 2.5 become a set of stochastic differential equa-tions. Given the above adiabatic condition, it is then possibleto take an ensemble average over the distribution of inputs toobtain the deterministic equations


    =s(r2, r1, ) + c

    H(u(r2, |r))I(r1|r)




    = s0(r2, ) + c

    H(u(r2, |r))

    I0, (2.7)


    denotes the ensemble average over r. These aver-

    aged equations involve the approximation that u depends ons, s0 rather than s, s0. For ease of notation, the averagess, s0 are then simply denoted by s, s0. The validityof suchan approximation has been established analytically elsewhere(Geman 1979).

    It is convenient to determine the slow variation in theweighted input v induced by changes in the feeedforwardafferents for a fixed input centered at r. Differentiating Eq.2.1 with respect to gives




    s(r2, r1, )

    I (r1|r)dr1


    (r2, ) I 0.

    Using Eqs. 2.6 and 2.7, this reduces to


    = v(r2, |r)



    (r)g(r|r)H(u(r2, |r))dr, (2.8)


    g(r|r) = c


    I (r1|r)I (r1|r)dr1 cI20 . (2.9)

    Further simplification can be achieved by assuming that the

    inputs are homogeneous and isotropic, I (r1|r) = I (|r1 r|),so that the input kernel g(r|r) = g(|rr|). This is onlyvalidif we ignore boundary effects either by setting 1,2 = R2 orby using periodic boundary conditions. For example, taking

    the inputs to be Gaussians, I (|r|) = Aer2 /22 , then Eq. 2.9ensures that the input kernel g is also a Gaussian:

    g(|r|) = c2 A2er2/42 cI20 . (2.10)If we also take (r) to be a uniform distribution then weobtain the homogeneous equations

    u(r2, |r) =


    w(|r2 r2|)H(u(r2, |r))dr2

    +v(r2, |r) h (2.11)

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    Spontaneous symmetry breaking in selforganizing neural fields 259



    = v(r2, |r)

    + 1 g(|r r|)H(u(r2, |r))dr. (2.12)(Note that the normalization factor for the uniform distri-bution can be absorbed into the coefficients c, c). Equations2.11 and 2.12 are the basic neural field equations for topo-graphic map formation (Takeuchi and Amari 1979; Amari1980). Rescaling the LGN and cortical coordinates appropri-ately, that is, ignoring the effects of cortical magnification,one can then look for homogeneous steadystate solutions ofthe form u(r2|r) = U (|r2 r|) and v(r2|r) = V (|r2 r|),where U is a unimodal function satisfying the fixed pointequation

    U (




    = 2 w(|r2 r2|)H(U(|r2 r|))dr2+


    g(|r r|)

    H(U(|r2 r|))dr h, (2.13)and

    V (|r2 r|) =


    g(|r r|)H(U(|r2 r|))dr. (2.14)

    Such a solution represents a continuous topographic map inwhich the center of LGN input activity at r 1 is mappedto the center of cortical output activity at the correspondingpoint r 2. We now discuss the existence and stabilityof such solutions in the simpler onedimensional case orig-inally analyzed by Amari (Takeuchi and Amari 1979; Amari1980, 1989).

    2.2 One-dimensional topographic map

    Onedimensional versions of Eqs. 2.11 and 2.12 take theform

    u(x2, |x) =

    w(x2 x2)H(u(x2, |x))dx2

    +v(x2, |x) h (2.15)and


    = v(x2, |x)


    g(x x )H(u(x2, |x))dx (2.16)


    g(x x) = c

    I (x1 x)I(x1 x)dx1 cI20 . (2.17)

    We will assume that g(x) is a monotonically decreasing func-tion ofx. This will hold, for example, if the inputs I(x) are

    Gaussians I(x) = Aex2/22 and hence

    g(x) = c A2



    0 . (2.18)

    Let us consider an equilibrium solution of the form

    u0(x2|x) = U (x2 x), v0(x2|x) = V (x2 x) (2.19)with U, V satisfying the onedimensional version of the fixed

    point Eqs. 2.13 and 2.14:

    U (x2 x) =

    w(x2 x 2)H(U(x 2 x))dx2


    g(x x)H(U(x2 x))dx h,


    V (x2 x) =

    g(x x )H(U(x2 x))dx.

    In particular, we seek a unimodal solution U with

    U(x) > 0, |x| < a, U (x) = 0, |x| = a,U(x) < 0, |x| > a, (2.20)where 2a is the width of the excited region (activity bump)in cortex. Then

    U(x) = W (x + a) + W (x a) + G(x + a)+G(x a) h (2.21)



    = x


    w(x )dx, G(x)=



    g(x )dx. (2.22)

    The corresponding width of the activity bump is determinedfrom the threshold conditions U (a) = 0, which yields theimplicit equation

    W (2a) + G(2a) = h. (2.23)The stability of the bump with respect to fluctuations on thefast timescale t can be determined by linearizing the equa-tion


    t= U(x,t) +

    w(x x)H(U(x ,t))dx


    g(x x)H(U(x,t))dx h, (2.24)

    about the bump solution, and this leads to the stability con-dition (Amari 1977)

    W(2a) + G(2a) w(2a) + g(2a) < 0. (2.25)As shown elsewhere (Amari 1977; Takeuchi andAmari 1979),ifw consists of shortrange excitation and longrange inhi-bition (the socalled Mexican hat profile) and g is a mono-tonically decreasing function then there exists a unique stablebump solution U for a range of threshold values h. We will

    assume that this holds in the following analysis.

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    260 P.C. Bressloff

    Translation symmetry

    The onedimensionalneural fieldEqs. 2.15 and2.16 areequi-variant withrespect to theproduct groupTTof translationsacting on the space R R according toTs,s (x2, x) = (x2 + s, x2 + s), Ts,s T T.The corresponding group action on the neural fields u, v is

    Ts,s (u(x2|x),v(x2|x)) = (u(x2 s|x s ),v(x2 s|x s )).

    Equivariance means that if (u, v) is a solution of the neuralfield equations then so is Ts,s (u,v). This is a more formalway of expressing the fact that the homogeneous system hasan underlying translation symmetry. It is also important tonote that the homogeneous equilibrium solution u0(x2|x) =U (x2 x),v0(x2|x) = V (x2 x) explicitly breaks the sym-metry group from T


    T with resulting group action

    Ts (u0(x2|x),v0(x2|x)) = (u0(x2 s|x s),v0(x2 s|x s)), Ts T

    that is, Ts = Ts,s . We will show below that the homoge-neous equilibrium solution can undergo a pattern forminginstability that spontaneously breaks the remaining transla-tion symmetry.

    2.3 Linear stability analysis

    In order to investigate the stability of the topographic mapsolution with respect to fluctuations on the slow timescale

    , we linearize Eqs. 2.15 and 2.16 by introducing small per-turbations of the form

    u(x2, |x) = U (x2 x) + p(x2, |x),v(x2, |x) = V (x2 x) + q(x2, |x) (2.26)and expanding to first order in p, q. This leads to the equa-tions (on setting = 1)q

    = q(x2, |x)


    g(x x)H(U(x2 x))p(x2, |x)dx


    p(x2, |x) =

    w(x2 x2)H(U(x2 x))

    p(x 2, |x)dx2 + q(x2, |x).Using the result

    H(U(x)) = 1 [(x a) + (x + a)] , (2.27)where = |U(a)| and (x) is the Dirac delta function, weobtain the pair of linear equations


    = q(x2, |x) + 1 [g(x x2 + a)

    p(x2, |x2 a)+g(x x2 a)p(x2, |x2 + a)] (2.28)


    p(x2, |x) = q(x2, |x)+1 [w(x2 x + a)p(x a, |x)


    2 x



    |x)] .


    Equations 2.28 and 2.29 involve nonlocal terms located atthe boundaries x2 = x a of the unperturbed activity bump.This indicates why it is possible to analyze the stability ofthe topographic map by restricting attention to the effectsof perturbations at the boundaries of the activity bump asoriginally formulated by Amari (Amari 1977; Takeuchi andAmari 1979; Amari 1989). In particular, if u(x2, |x) = 0 atx2 = x a + ( x , ), then0 = U (a + (x,)) + p(x a + (x,),|x)

    = U (a) + U(a)(x,) + p(x a, |x)+O(



    that is,

    ( x , ) = 1p(x a, |x)since U (a) = 0 and U(a) = . Two particular exam-ples of boundary perturbations are illustrated in Fig. 2: auniform expansion of the bump for which p(x + a, |x) =p(x a, |x) and a shift in the center of the bump for whichp(x + a, |x) = p(x a, |x). In this paper we chooseto work directly with the linear Eqs. 2.28 and 2.29, sincethese are more simply extended to the case of twodimen-sional networks. Moreover, they take into account perturba-tions outside the boundary domain of the bump. However, the

    resulting stability conditions are equivalent to those derivedfollowing the boundary approach of Amari (Takeuchi andAmari 1979; Amari 1989): we show this explicitly in thecase of onedimensional topographic maps. Note that a sim-ilar approach to the one adopted here has previously beenused to study the stability of activity bumps in singlelayernetworks with nonadapting synapses (Pinto and Ermentrout2001; Folias and Bressloff 2004).


    p( x , ) = p(x a, |x),q( x , ) = q(x a, |x) (2.30)and setting x2


    a in Eq. 2.29 gives the pair of equations

    p+( x , ) = q+( x , ) + 1[w(2a)p( x , )+w(0)p+(x,)], (2.31)

    p( x , ) = q( x , ) + 1[w(0)p( x , )+w(2a)p+( x , )]. (2.32)

    Similarly, setting x2 = x a in Eq. 2.28 shows thatq+

    = q+( x , ) + 1

    g(0)p+( x , )

    +g(2a)p(x + 2a , )

    , (2.33)


    = q( x , ) + 1

    g(2a)p+(x 2a , )

    +g(0)p( x , ) . (2.34)

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    Spontaneous symmetry breaking in selforganizing neural fields 261













    expansion shift

    a b

    Fig. 2 Perturbations p(x) = p(x a|x) at the boundaries of a homogeneous bump solution centered about x2 = x and having width 2a. Onlythe superthreshold part of the bump is shown. a Expansion of the bump such that p(x) = p+(x). b Shift in the position of the bump such thatp(x) = p+(x)

    We have used the fact that w(x) and g(x) are even functions.Equations 2.312.34 have eigensolutions of the form


    ( x , )

    =eeikx P


    q( x , ) = eeikx Q(k) (2.35)with the eigenvalue and eigenvectors P = (P+, P)T,Q = (Q+, Q)T determined from the matrix equationsQ(k) = Q(k) + 1G(k)P(k),P(k) = Q(k) + 1WP(k), (2.36)where

    W =

    w0 w2w2 w0

    , G(k) =

    g0 g2e


    g2e2ika g0


    with w0 = w(0), w2 = w(2a),g0 = g(0), g2 = g(2a). Itfollows that( + 1)Q(k) = M(k)Q(k), (2.38)where

    M(k) = G(k) [1 W]1 (2.39)= 1

    ( w0)g0 + w2g2e2ika ( w0)g2e2ika + w2g0

    ( w0)g2e2ika + w2g0 ( w0)g0 + w2g2e2ika


    = ( w0)2 w22 . (2.40)Thus





    (k), (2.41)

    where (k) are the eigenvalues of the matrix M(k),

    (k) =1


    (k)2 2(g20 g22 )



    (k) = ( w0)g0 + w2g2 cos(2ka). (2.43)Note that we canexpress in terms of thecoefficients g0,2, w0,2by differentiating Eq. 2.21 with respect to x,

    |U(a)| = g0 g2 + w0 w2. (2.44)First consider the case w2 < 0. Eqs. 2.40 and 2.44 then

    imply that

    = ( w0 + w2)( w0 w2)= (g0 g2)(g0 g2 2w2) > 0,

    since g(x) is a monotonically decreasing function with g0 >g2, see Eq. 2.18. Define

    min = ( w0)g0 |w2g2|,max = ( w0)g0 + |w2g2|such that 0 < min (k) max for all k. It follows fromEqs. 2.42 to 2.44 that

    2min 2[g20 g22 ]= [|w2|g0 ( w0)|g2|]2 0

    and, hence, (k) are real for all k. Combining this with theinequality > 0 shows that

    max maxk

    (k) = 1 + 1 ++

    2+ 2[g20 g22 ]

    = 1 + 1(g0 + |g2|)( w0 + |w2|)

    = 1( w0 + |w2|)(g2 + |g2|),where we have used Eqs. 2.40 and 2.44. Finally, noting that > 0 and w0 + |w2| > 0, we obtain the followingstability conditions:

    1. Ifg2 < 0 then max = +(0) = 0 and the topographicmap is stable2. If g2 > 0 then max = +(/2a) > 0 and the topo-

    graphic map is unstable. Moreover the fastest growingmode has a wavelength equal to 4a, which is twice thewidth of an activity bump, and has vector componentsP(/2a) = (1, 1). That is, the (realvalued) excitedmode is of the form

    p(x) = cos

    (x x)2a


    where x is an arbitrary shift, reflecting hidden translationsymmetry, and the amplitude is arbitrary within the

    linear approximation.

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    262 P.C. Bressloff






    a b

    Fig. 3 a Homogeneous topographic map for g(2a) < 0. b Spatially periodic topographic map for g(2a) > 0 with a blocklike microstructure.The shadedregions indicate where u(x2|x) > 0

    Note that the existence of a zero eigenvalue, +(0) = 0,reflects the underlying translation symmetry of the homoge-neous solution under simultaneous shifts x


    +, x2

    x2 + . In the above analysis we assumed that w2 < 0. Ifw2 > 0 then we require g2 < 0 in order to satisfy the stabilitycondition 2.25. Since the maximum eigenvalue is positive forw2 > 0 and g2 < 0,

    max = 1 + 1(g0 + |g2|)( w0 + w2)= g0 g2

    g0 g2 2w2> 0,

    it follows that the topographic map is unstable. Therefore,g2 < 0 and w2 < 0 are necessary and sufficient conditionsfor the stability of the onedimensional topographic map,as previously shown by Takeuchi and Amari (Takeuchi and

    Amari 1979).The above analysis establishes that the homogeneous equi-

    librium solution u0(x2|x) = U (x2 x) undergoes a patternforming instability as g2 changes from a negative to a posi-tive value induced, for example, by a reduction in the back-ground inhibition cI20 or by an increase in the spread of theGaussian inputs, see Eq. 2.18. Such an instability spontane-ously breaks continuous translation symmetry, leading to thepartitioning of the topographic map into discretized blocks(Takeuchi and Amari 1979). This is illustrated schematicallyin Fig. 3. The fact that the resulting pattern has a blocklikestructure can be understood from the observation that thedominant excited mode satisfies p+( x , )

    = p( x , ) and

    hence +( x , ) = ( x , ). Thus the instability generates aleftward or rightward shift in an activity bump, depending onthelocation of thecenterx of itsassociated receptive field (seeFig. 2). It has been suggested that the blocks could be a pre-cursor for the columnar microstructure of cortex (Takeuchiand Amari 1979; Amari 1989).As we mentioned in the intro-duction, cortical columns tend to be associated with a varietyof stimulus features such as ocular dominance and orienta-tion, which form spatially distributed feature maps that aresuperimposed upon the underlying topographic map (Swin-dale 1996). In the following sections we extend the stabilityanalysis of onedimensional topographic maps in order toinvestigate how such features could also emerge through the

    spontaneous symmetry breaking of selforganizing neural

    fields. First, in Sect. 3 we consider a onedimensional net-work consisting of separate left and right eye afferents fromthe LGN. This introduces an additional Z2 symmetry that canbe spontaneously broken, resulting in the spatial segregationof eye specific activity bumps consistent with the emergenceof ocular dominance columns. Second, in Sect. 4 we consideran isotropic twodimensional network whose rotational sym-metry can be spontaneously broken, leading to the formationof elongated activity bumps consistent with the emergenceof orientation preference columns.

    One final comment regarding Amaris model of topo-graphic map formation is in order before proceeding with ouranalysis. This concerns the inclusion of feedforward inhibi-tory synapses that can also undergo Hebbian learning. Suchinhibition is necessary in order to stabilize the smooth topo-graphic map. However, as far as we are aware, there is noconclusive experimental support for the existence of Heb-bianlike inhibitory synapses. On the other hand, most devel-opmental models involving the Hebbianlike modification ofexcitatory synapses require additional constraints to ensurethat an appropriate form of competition between synapsesoccurs and that a stable distribution of synaptic weights isgenerated (Miller and MacKay 1994). The constraints typi-cally limit the sum of synaptic strengths received by a cell,or the mean activity of the cell. Although the constraints arenot usually biophysically realistic, they are motivated by theidea that there exists some form of global intracellular sig-nal controlling the synaptic weights. The modifiable inhib-itory synaspses in Amaris model play an analogous role to

    these constraints. For example, in the binocular extension ofAmaris model (see Sect. 3), feedforward inhibition ensuresthat the topographic map is stable (unstable) with respect toperturbations that are symmetric (anti-symmetric) under theexchange of left/right eye inputs. This should be comparedwith the use of subtractive normalizationin correlationbasedHebbian models (Miller and MacKay 1994).

    3 Spontaneous symmetry breaking in a binocular onedimensional network

    Our first extension of the theory of selforganizing neural

    fields is to consider a onedimensional network with distinct

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    Spontaneous symmetry breaking in selforganizing neural fields 263

    left and right eye afferents. We derive conditions for the exis-tence of a binocular topographic map, in which the responseto a stimulus is independent of whether it is presented to theleft or right eye. The resulting homogeneous solution is thus

    symmetric with respect to a discrete Z2 left/right exchangesymmetry. We then generalize the linear stability analysispresented in Sec. 2, and show how the binocular state can un-dergo a pattern forming instability that spontaneously breaksthe underlying Z2 symmetry. This leads to the spatial seg-regation of eye specific activity bumps consistent with theemergence of ocular dominance columns.

    3.1 Binocular equilibrium state

    Consider a onedimensional version of the network modelshown in Fig. 1, in whichthere are separate afferents from theleft and right eye denoted by sL(x2, x1, ) and sR(x2, x1, ),

    respectively. The total input to cortical neurons at x2 nowbecomes

    v(x2|x , ) =

    sL(x2, x1)IL(x1|x , )dx1

    +sR(x2, x1)IR(x1|x , )dx1

    s0(x2), (3.1)where is an additional stimulus label that takes intoaccountdifferences in the statistical correlations between same eyeand opposite eye inputs.We have also set the inhibitory inputI0 = 1. For concreteness, we choose Gaussian inputs of theform

    IL(x1|x , ) = A(1 + )e(x




    IR(x1|x , ) = A(1 )e(xx1)2/22 , (3.2)where istakentobeabinaryrandomvariablewithProb ( =0) = Prob( = 0) = 1/2 for some constant 0, 0 0 and g

    S2 < 0.

    This leads to the conditions

    0 < g(2a) < c. (3.35)

    The first inequality is always satisfied, since Eq. 3.9 impliesthat g is a positive function. The dominant excited mode isgiven by (modulo an arbitrary phase)

    p(x,0) = cos(x/2a),p(x, 0) = cos(x/2a), (3.36)which represents a state for which the center of the responseto a left dominated input (+0) is shifted in the oppositedirection to the center of the response to a right dominatedinput (0), see Fig. 4. Moreover, the directions of the shiftsperiodically alternate across space according to the sign ofcos(x/2a). The form of the fastest growing mode suggests

    that ocular dominance columns will form, at least within the

    given linear approximation. Note that the joint developmentof a topographic map and ocular dominance columns has re-cently been demonstrated numerically using selforganizingneural fields with linear threshold nonlinearities (Woodburyet al. 2002).

    The above analysis shows that a certain level of feed-forward inhibition is needed in order to stabilize the topo-graphic map with respect to perturbations that are symmetricunder the exchange of left/right eye inputs. Indeed, if therewere no inhibitory contribution (c = 0), then the symmetriceigenmode would grow faster than the anti-symmetric mode(since 20 < 1) and no OD columns would form. As we com-mented at the end of Sect. 2, feedforward inhibition plays ananalogous role to subtractive normalization in correlationbased Hebbian models (Miller and MacKay 1994).Althoughneither mechanism for stabilizing the symmetric eigenmodemay be biophysically realistic, it is clear that some form ofnormalization is needed if the cortical development of oculardominance columns occurs via Hebbianlike learning. Sucha normalization will depend on properties of the inputs. In thecase of the neural field model with feedforward inhibition,this is expressed by Eqs. 3.9 and 3.35, which show that theminimum level of inhibition c depends on the width andamplitiude A of the Gaussian inputs.

    4 Spontaneous symmetry breaking in an isotropic twodimensional network

    In this section, we extend the analysis presented in Sect. 2to the case of twodimensional topographic maps. We show

    how a dynamical instability of thetopographicmap canoccur,in which there is a spontaneous breaking of continuous rota-tion symmetry, leading to the formation of elongated activitybumps; these are consistent with the emergence of orienta-tion preference columns. Our analysis is based on a directlinearization of the neural field Eqs. 2.11 and 2.12 about aradially symmetric homogeneous solution.

    4.1 Twodimensional topographic map

    Consider a radially symmetric, homogeneous equilibrium

    solution of Eq. 2.13 such that

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    266 P.C. Bressloff

    U(r) > 0, 0 < r < a, U (r ) = 0, r = a,U(r) < 0, r > a, (4.1)

    where a is the radius of the twodimensional activity bumpin cortex. Substituting into Eq. 2.13 gives

    U(r) = F(a,r) h, (4.2)where

    F(a,r) =2



    f (|r r|)r dr d (4.3)

    and we have defined f(r) = w(r) + g(r). The radius of thebump is determined from the threshold condition U(a) = 0,which yields

    F(a,a) = h. (4.4)As in the onedimensional case, suppose that w(r) is a Mex-ican hat function and the input I(r) is a Gaussian so that g(r)

    is a monotonically decreasing function of r , see Eq. 2.10. Aunique stable bump solution then exists for a range of thresh-olds h. (The issue of stability will be addressed below). How-ever, as has been pointed out elsewhere (Werner and Richter2001), certain care has to be taken with regards the existenceof twodimensional bumps in the presence of shortrangeexcitation and longrange inhibition. That is, in contrast tothe onedimensional case, the thresholdcondition maynot besufficient for existence, since the activity u could dip belowthresholdwithinthe interior of thedisc r < a. We will assumein the following that the stable bump solution is superthresh-old for r < a.

    It is possible to simplify the double integral in Eq. 4.3 us-

    ing a Fourier transform, which for radially symmetric func-tions reduces to a Hankel transform (Folias and Bressloff2004). To see this, consider the two-dimensional Fouriertransform of the radially symmetric function f, expressedin polar coordinates,

    f(r) = 12


    eirkf(k)dk= 1




    eir k cos()f(k)d kdk,where

    f denotes the Fourier transform of f and k = (k,).

    Using the integral representation




    eir k cos()d = J0(rk),

    where J (z) is the Besselfunction of the first kind, we expressf in terms of its Hankel transform of order zero,

    f(r) =


    f (k)J0(rk)k dk (4.5)which, when substituted into Eq. 4.3, gives



    0 f(k)



    J0(k|r r|)r drd

    k dk.


    In polar coordinates,20


    J0(k|r r|)r drd

    = 20


    J0kr 2 + r 2 2rr cos( ) r drd

    To separate variables, we use the addition theorem



    r 2 + r 2 2rr cos



    ) cos m

    where 0 = 1 and n = 2 for n 1. Since2

    0cos m

    d = 0 for m 1, it follows that



    J0(k|r r|)r drd

    = 2 J0(kr)a


    J0(kr)r dr

    = 2 ak


    Hence, F(a,r) has the integral representation

    F(a,r) = 2 a


    f (k)J0(rk)J1(ak)dk. (4.7)Stability of twodimensional bumps

    The stability of a twodimensional bump with respect to fluc-tuations on the fast timescale t can be determined from lin-earizing the equation


    t= U (r, t )


    R2f (|r r|)H(U(r, t ) )dr h (4.8)

    about the radially symmetric equilibrium solution. This par-ticular problem has previously been studied in the restrictedcase of radially symmetric perturbations by Taylor (Taylor

    1999). However, as recently shown by Folias and Bressl-off (Folias and Bressloff 2004), it is also necessary to takeinto account nonradially symmetric perturbations in orderto fully determine the stability of a twodimensional activitybump. It is useful to review this latter analysis here beforeconsidering the stability of the associated topographic map.Consider the time-dependent perturbation U (r, t ) = U(r)+p(r, t ) and expand to first order in p. This leads to the line-arized equation


    t= p(r, t )

    + R2 f (|r r|)H(U(r ) )p(r, t ) ) dr (4.9)

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    Spontaneous symmetry breaking in selforganizing neural fields 267

    which has solutions of the form p(r, t ) = p(r)et. Introduc-ing polar coordinates r = (r, ) and using the result

    H(U(r)) = (U(r)) = (r a)


    |we obtain the eigenvalue equation

    ( + 1)p(r) = a|U(a)|2


    f (|r a|)p(a, ) d ,(4.10)

    where a = (a, ).If the eigenfunction p satisfies the condition2


    f (|r a|)p(a, ) d = 0

    for all r then the associated eigenvalue is = 1. This ispart of the essential spectrum and does not cause instability.

    Ifp does not satisfy the above condition, then we must studythe solutions of the integral equation

    p(r, ) = a2


    F(a,r; )p(a, ) d ,

    where ( + 1)|U(a)| = andF(a,r; )) = f

    r 2 + a2 2ra cos


    It follows that p(r,) is determined completely by the restric-tion p(a,). Hence we need only consider r = a, yieldingthe integral equation

    p(a, )=

    a 2




    ) d. (4.11)

    The solutions of this equation are exponential functions ein

    where n Z. Thus the integral operator with kernel F has adiscrete spectrum given by

    n = a2


    F(a,a; )ein d

    = a2



    a2 + a2 2a2 cos

    ein d

    = a2


    f (2a sin (/2)) ein d

    (after rescaling ). Note that n

    is real since

    Im{n(a)} = a2


    f (2a sin(/2)) sin(n)d = 0,

    i.e. the integrand is odd-symmetric about . Hence,

    n(a) = Re{n(a)}

    = a2


    f (2a sin(/2)) cos(n)d (4.12)

    with the integrand even-symmetric about .We conclude from the above analysis that an activity

    bump of radius a (assuming that it exists) will be stableprovided that n(a) |U(a)| for all n Z. This en-sures that the corresponding eigenvalues are nonnegative,

    n = 1+|U(a)|1n(a) 0 for all n Z. DifferentiatingEq. 4.3 with respect to r shows that

    U(a) = r





    a2 + r 2 2r a cos a2 + r 2 2r a cos

    (a r cos )r drd




    cos f

    r + sin

    r f


    r drd




    2a2 2a2 cos

    cos d

    = 1(a).The final step in the above derivation involves integrating

    by parts the term r cos f/r with respect to r and theterm sin f/ with respect to . It follows that1 = 1 + |U(a)|11(a) = 0. The existence of a zeroeigenvalue reflects the underlying translation symmetry ofthe system, which implies that the activity bump is margin-ally stable with respect to uniform shifts in space (see alsoFig. 5 below). It follows that the bump will be stable if thezero eigenvalue is simple and all other eigenvalues are neg-ative, that is, n(a) < |U(a)| for all n = 1. From Eqs. 4.2and 4.3 we have

    0(a) |U(a)| = a





    = dda

    F (a, a). (4.13)

    Hence, a necessary condition for stability is dF(a,a)/ da 1 (4.33)Combining Eqs. 4.24, 4.25 and 4.32 yields a vector equa-

    tion of the form

    b(k)(b(k) P(k)) = (1 + )P(k), (4.34)where denotes complex conjugate, andPn(k) = nPn(k),bn(k) =



    in . (4.35)

    There are two classes of solution to Eq. 4.34. If b P = 0then = 1 and the topographic map is stable with respectto excitation of the corresponding eigenmodes. On the other

    hand, if b P = 0 then P = b (up to a constant multipli-cative factor). Substituting into the Fourier series (4.23), theresulting eigenmode is of the form P (k) = P ( k , ) withk = (k,),

    P ( k , ) =


    0+ 2


    (1)n J2n(ka) 2n

    cos(2n )


    (1)n J2n1(ka) 2n1

    sin((2n 1) )



    where is an arbitrary amplitude. The corresponding eigen-

    value is = (k) with

    (k) = 1 + |b|2 = 1 +nZ



    2. (4.37)

    The Bessel functions Jn for n = 0, 1, 2 are plotted in Fig. 6.For the sake of illustration, suppose that n < for all

    n Z. This is plausible given Eq. 4.33 and the conditions onn. Equation (4.37) implies that if g(0) < 0 then the topo-graphic map is stable since (k) < 0 for all k. On the otherhand, ifg(0) > 0 such that (kc) = maxk (k) > 0 then thetopographic map is unstable and the fastest growing eigen-modes have the critical wavenumber kc. Recall from Sect.

    4.1 that = 1. It then follows from Eq. 4.33 that 1

    and the dominant contribution to the sum in Eq. 4.37 willarise from the n = 1 term, at least distance from the zerosof J1(ka). Hence, kc is approximately given by the point atwhich the first order Bessel function attains its global max-

    imum, that is, |J1(akc)| = maxk |J1(ak)|. Fig. 6 shows thatkc 3/a. One of the major differences between the lineartheory of onedimensional and two dimensional topographicmaps, is that in the latter case the eigenvalues (k), k = 0,have an infinite degeneracy that reflects the additional rota-tion symmetry of the system. That is, all eigenmodes P (k)with |k| = k have the same eigenvalue. It follows that thepattern forming instability will be dominated by some lin-ear combination of eigenmodes lying on the critical circle|k| = kc:

    p (r) =N

    i=1 zi e

    iki r + zi eiki r

    P (kc, i ), (4.38)

    where ki = (kc, i ) and zi is a complex amplitude. Supposethat each eigenmode can be approximated by the first threeterms of Eq. 4.36 so that

    P (kc, )


    0+ 2J1(kca)

    1sin( )

    2J2(kca) 2

    cos(2 )

    , (4.39)

    The first term generates an expansion of the bump, the secondterm generates a uniform shift of the bump and the third termgenerates an elongation of the bump (see Fig. 5). In gen-eral, we expect the eigenmode (Eq. 4.39) to be dominated

    by the first harmonic term sin( ), since 1 . However, if2 as well, then there could alsobe a significant contribu-tion from the term cos(2 ). Thus the spontaneous symmetrybreaking mechanism has the potential for generating elon-gated receptive fields that are consistent with the formationof orientation columns. Moreover, since each eigenmode inthe sum (Eq. 4.38) then represents an elongation in the direc-tion i or /2 + i (depending on the sign of its associatedcoefficient z(r) = zi eiki r + zi eiki r), it follows that there issome complicated variation in the preferred orientation as rvaries across the cortex. We note that the emergence of orien-tation selectivity in selforganizing neural fields has recentlybeen demonstrated numerically (Fellenz and Taylor 2002).

    However, whether or not such a model can reproduce thedetailed structure of orientation maps found experimentallyremains to be established. For example, it might be neces-sary to develop a more detailed model that takes into accountseparate ON and OFF pathways as previously considered byMiller using correlationbased methods (Miller 1994).

    Calculation of eigenmodes: narrow inputs

    As in the onedimensional case, if the excitatory inputs be-come sufficiently narrow then the topographic map is stablein the presence of feedforward inhibition. In the absence of

    such inhibition (c = 0), it is possible to find an approximate

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    Spontaneous symmetry breaking in selforganizing neural fields 271

    0 2 4 6 8 10 12 14 16 18 20-0.5






    n = 0

    n = 1

    n = 2

    Fig. 6 Bessel functions Jn for n = 0, 1, 2





    Fig. 7 Action of a rotation by : p (r) p (r) where (r, ) = (R r, + ). Here r represents the position of the center of a twodimensionalbump and p represents the perturbation of steadystate activity at a point on the boundary of the bump

    solution for the eigenmodes in the limit of narrow inputs( a) to determine the dominant eigenmode. For sim-plicity, we take the excitatory input kernel g to be a narrowstep function rather than a Gaussian such that g(



    if |r| < and zero otherwise. Under this approximation,g(2a sin([ ]/2)) = g0 for /a and is zerootherwise. Substitution into Eq. 4.26 shows that

    Gnn (k) g0a2





    eia k[cos()cos(+)]dd


    = g0a2




    (+) (1

    +ia k sin( + ) +O






    = 2g0asin(n/a)



    n,n1ei n,n+1ei 1in




    = g0a


    an,n +O([/a]3)

    . (4.40)

    Note in particular that the O([/a ]2) term is zero. Substitu-tion into Eq. 4.24 and using Eq. 4.25 implies that to lowestorder in /a, the eigenmodes are kindependent and of theform ein with corresponding eigenvalues





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    272 P.C. Bressloff

    Given that n = n +O(/a) (see Eq. 4.28) and 1 = , itfollows that the dominant eigenmode is going to be n = 1,which represents a uniform shift in the topographic map. Weconclude that even in the absence of feedforward inihibition,

    destabilization of the topographic map cannot generate elon-gated receptive fields nor an associated orientation map if theexcitatory inputs are too narrow.

    Euclidean shifttwist symmetry

    The basic structure of the eigenmodes p (k) can be under-stood from a more general group theoretic perspective bynoting that the linear equations 4.18 and 4.19 are equivari-ant with respect to the socalled shifttwist action of theEuclidean group E(2) on the space R2 S1 (Bressloff et al.2001a,b):


    (r, )


    +s, ) ,

    R (r, ) = (R r, + ),R (r, ) = (R r, ),


    where R (x,y) = (x, y). The corresponding action on thefields p (r) and q (r) is

    Ts (p (r), q (r)) = (p (r s), q (r s)),

    R (p (r), q (r)) = (p (R r), q (R r)),

    R (p (r), q (r)) = (p (R r), q (R r)).It can be seen that the rotation operation comprises a trans-lation or shift of the angle to + , together with a rota-tion or twist of the position vector r by the angle . This is

    illustrated in Fig. 7. One of the consequences of the under-lying Euclidean symmetry is that the associated eigenfunc-tions form irreduciblerepresentations of the shifttwist groupaction (Bressloff et al. 2001a,b).This explains why the eigen-modes P (k) have the basic structure given by Eq. 4.36, withthe angular variable coupled to the direction of the wave-vector k. Interestingly, there is growing evidence that thereis a coupling between orientation and topography consistentwith an underlying rotational shifttwist symmetry (Boskinget al. 1997; Lee et al. 2003), as highlighted in the discussionbelow.

    5 Discussion

    In this paper we have extended the theory of selforganiz-ing neural fields in order to investigate from a mathemati-cal perspective the possible joint emergence of topographyand feature selectivity through spontaneous symmetry break-ing. We first showed how a binocular onedimensional topo-graphic map can undergo a pattern forming instability thatbreaks the underlying Z2 symmetry between left and righteyes.This leads to thespatial segregation of eye specific activ-ity bumpsconsistent with the emergence of ocular dominancecolumns. We then showed how a twodimensional isotropictopographic map can undergo a pattern forming instability

    that breaks the underlying rotation symmetry. This leads tothe formation of elongated activity bumps consistent withthe emergence of orientation preference columns. A partic-ularly interesting property of the latter symmetry breaking

    mechanism is that the linear equations describing the growthof the orientation columns exhibits a rotational shifttwistsymmetry, in which there is a coupling between orientationand topography. A recent statistical analysis of orientationpreference maps in primates indicates that there are correla-tions between the direction of the topographic axis joiningpairs of columns with similar orientation preferences andtheir common orientation (Lee et al. 2003). Thus the orienta-tion preference map exhibits a form of rotational shifttwistsymmetry as predicted from our analysis of twodimensionaltopographic maps. Numerical simulations of a featurebaseddynamical spin mode has led to the suggestion that such asymmetry could help to stabilize the emerging orientationpreference map with its associated set of pinwheels (Lee etal. 2003).As previously shown by Wolf and Geisel (Wolf andGeisel 1998),in the absence of such a coupling,the pinwheelstypically annihilate in pairs. Hence, in order to maintain pin-wheels, either development has to be stopped or one has tointroduce inhomogeneities that trap the pinwheels. (Note thatThomas and Cowan (Thomas and Cowan2004)have recentlyanalyzeda spin model with a different form of rotational cou-pling between orientation and topography, and shown howdislocations in the topographic map can occur).

    Another aspect of cortical structure that appears to exhibitshifttwist symmetry is the distribution of patchy horizon-tal connections found in superficial layers of cortex. Opticalimaging combined with labeling techniques has establishedthat these connections tend to link cells with similar featurepreferences (Malach et al. 1993;Yoshioka et al. 1996). More-over, in tree shrew and cat there is a pronounced anisotropy inthe distribution of patchy connections, with isoorientationpatches preferentially connecting to neighboring patches insuch a way as to form continuous contours along the topo-graphic axis(Bosking et al. 1997).Thereis alsoa clear anisot-ropy in the patchy connections of owl (Sincich and Blasdel2001) and macaque (Angelucci et al. 2002) monkeys. How-ever, in these cases most of the anisotropy can be accountedfor by the fact that V1 is expanded in the direction orthog-onal to ocular dominance columns. It is possible that whenthis expansion is factored out, there remains a weak anisot-

    ropy correlated with orientation selectivity but this remainsto be confirmed experimentally. Interestingly, the recentlyobserved patchyfeedback connectionsfrom extrastriateareasin macaque tend to be more strongly anisotropic (Angelucciet al. 2002); it is likely that the patchiness again signifiesthat feedback correlates cells with similar feature preferences(Shmuel et al. 1998). It has been shown elsewhere that theshifttwist symmetry of anisotropic horizontal connectionshas a nontrivial affect on the dynamics of neural activity invisual cortex (Bressloff et al. 2001a,b, 2002). It would beinteresting to extend the analysis of this paper in order todetermine how such connections selforganize through Heb-bian learning. This will require treating both feedforward

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    Spontaneous symmetry breaking in selforganizing neural fields 273

    and intracortical connections as adaptive (Bartsch and vanHemmen 2001), rather than keeping the latter fixed.

    Acknowledgements This work was partially supported by NSF grantDMS 0515725.


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