{"title": "Development and Spatial Structure of Cortical Feature Maps: A Model Study", "book": "Advances in Neural Information Processing Systems", "page_first": 11, "page_last": 17, "abstract": null, "full_text": "Development and Spatial Structure of Cortical \n\nFeature Maps: A Model Study \n\nK. 0 berulayer \nBeckman-Institute \nUniversity of Illinois \nUrbana, IL 61801 \n\nH. Ritter \n\nTechnische Fakultiit \nU niversitiit Bielefeld \n\nD-4800 Bielefeld \n\nAbstract \n\nK. Schulten \nBeckman -Insti t u te \nUniversity of Illinois \nUrbana, IL 61801 \n\nFeature selective cells in the primary visual cortex of several species are or(cid:173)\nganized in hierarchical topographic maps of stimulus features like \"position \nin visual space\", \"orientation\" and\" ocular dominance\". In order to un(cid:173)\nderstand and describe their spatial structure and their development, we in(cid:173)\nvestigate a self-organizing neural network model based on the feature map \nalgorithm. The model explains map formation as a dimension-reducing \nmapping from a high-dimensional feature space onto a two-dimensional \nlattice, such that \"similarity\" between features (or feature combinations) \nis translated into \"spatial proximity\" between the corresponding feature \nselective cells. The model is able to reproduce several aspects of the spatial \nstructure of cortical maps in the visual cortex. \n\n1 \n\nIntroduction \n\nCortical maps are functionally defined structures of the cortex, which are charac(cid:173)\nterized by an ordered spatial distribution of functionally specialized cells along the \ncortical surface. In the primary visual area( s) the response properties of these cells \nmust be described by several independent features, and there is a strong tendency to \nmap combinations of these features onto the cortical surface in a way that translates \n\"similarity\" into \"spatial proximity\" of the corresponding feature selective cells (see \ne.g. [1-6]). A neighborhood preserving mapping between a high-dimensional fea(cid:173)\nture space and the two dimensional cortical surface, however, cannot be achieved, so \nthe spatial structure of these maps is a compromise, preserving some neighborhood \nrelations at the expense of others. \n\nThe compromise realized in the primary visual area(s) is a hierarchical represen(cid:173)\ntation of features. The variation of the secondary features \"preferred orientation\", \n\n11 \n\n\f12 \n\nObermayer, Ritter, and Schulten \n\n\"orientation specifity\" and \"ocular dominance\" is highly repetitive across the pri(cid:173)\nmary map of retinal location, giving rise to a large number of small maps, each \ncontaining a complete representation of the full range of the seconda.ry features. If \nthe neighborhood relations in feature space are to be preserved and maps must be \ncontinuous, the spatial distributions of the secondary features \"orientation prefer(cid:173)\nence\", \"orientation specifity\" and \"ocular dominance\" can no longer be independent. \nInterestingly, there is experimental evidence in the macaque that this is the case, \nnamely, that regions with smooth change in one feature (e.g. \"ocular dominance\") \ncorrelate with regions of rapid change in another feature (e.g. \"orientation\") [7,8]. \nPreliminary results [9] indicate that these correlations may be a natural consequence \nof a dimension reducing mapping which preserves neighborhood relations. \n\nIn a previous study, we investigated a model for the joint formation of a retino(cid:173)\ntopic projection and an orientation column system (10], which is based on the \nself-organizing feature map algorithm [11,12]. This algorithm generates a repre(cid:173)\nsentation of a given manifold in feature space on a neural network with prespecified \ntopology (in our case a two-dimensional sheet), such that the mapping is continous, \nsmooth and neighborhood relations are preserved to a large extent.! The model \nhas the advantage that its rules can be derived from biologically plausible devel(cid:173)\nopmental principles [15,16]. Therefore, it can be interpreted not only as a pattern \nmodel, which generates a representation of feature combinations subject to a set of \nconstraints, but also as a pattern formation model, which describes an input driven \ndevelopmental process. In this contribution we will extend our previous work by the \naddition of another secondary feature, \"ocular dominance\" and we will concentrate \non the hierarchical mapping of feature combinations as a function of the set of input \npatterns. \n\n2 Description of the Model \nIn our model the cortical surface is divided into N x N small patches, units r, which \nare arranged 011 a two-dimensional lattice ( network layer) with periodic boundary \nconditions (to avoid edge effects). The functional properties of neurons located in \neach patch are characterized by a feature vector Wr , which is associated with each \nunit r and whose components (wrh are interpreted as receptive field properties of \nthese neurons. The feature vectors, Wr, as a function of unit locations r, describe the \nspatial distribution of feature selective cells over the cortical layer, i.e. the cortical \nmap. \n\nTo generate a representation of features along the network layer, we use the self(cid:173)\norganizing feature map algorithm [1,2]. This algorithm follows an iterative proce(cid:173)\ndure. At each step an input vector V, which is of the same dimensionality as Wr, \nis chosen at random according to a probability distribution P( V). Then the unit \ns, whose feature vector w; is . closest to the input pattern V, is selected and the \ncomponents (wr h of it's feature vector are changed according to the feature map \nlearning rule: \n\n1 For ot.h~r mod~lling nppron.ch~fl along t.h~fl~ lin~fl fl~e [13,14]. \n\n(1) \n\n\fDevelopment and Spatial Structure of Cortical ftature Maps: A Model Study \n\n13 \n\nwhere hU;', s, t), the neighborhood junction, is given by: \n\n(2) \n\n3 Coding of Receptive Field Properties \n\nIn the following we describe the receptive field properties by the feature vector Wi~ \ngiven by W,~ = (XI~' YI~' q,~cos(2\u00a2,~), q,~sin(2\u00a2,~), ZI~) where (xr, YI~) denotes the \nposition of the receptive field centers in visual space, (\u00a2r) the preferred orientation, \nand (ql~)' (zr) two quantities, which qualitatively can be interpreted as orientation \nspecificity (see e.g. [17]) and ocular dominance (see e.g . [lS]). If qr is zero, then the \nunits are unspecific for orientation; the larger q,~ becomes, the sharper the units are \ntuned. \"Binocular\" units are characterized by Zr = 0, \"monocular\" units by a large \npositive or negative value of Zi~' \"Similarity\" between receptive field properties is \nthen given by the euclidean distance between the corresponding feature vectors. \nThe components WI~ of the input vector v= (x, y, qcos(2\u00a2), qsin(2\u00a2), z) describe \nstimulus features which should be represented by the cells in the cortical map. \nThey denote position in the visual field (x, V), orientation \u00a2, and two quantities q \nand Z qualitatively describing pattern eccentricity and the distribution of activity \nbetween both eyes, respectively. Round stimuli are characterized by q = 0 and the \nmore eliptic a pattern is the larger is the value of q. A \"binocular\" stimulus is \ncharacterized by Z = 0, while a \"monocular\" stimulus is characterized by a large \npositive or negative value of Z for \"right eye\" or \"left eye\" preferred, respectively. \n\nInput vectors were chosen with equal probability from the manifold \n\ni.e. all feature combinations characterized by a fixed value of q and Izi were selected \nequally often. If the model is interpreted from a developmental point of view, the \nmanifold V describes properties of (subcortical) activity patterns, which drive map \nformation. The quantities d, qpat and Zpat determine the feature combinations to \nbe represented by the map. As we will see below, their values crucially influence \nthe spatial structure of the feature map. \n\n(3) \n\n4 Hierarchical Maps \n\nIf qpat and Zpat are smaller than a certa.in threshold then \"orientation preference\" , \n\"orientation selectivity\" and \"ocular dominance\" are not represented in the map (i.e. \nqr = Z,~ = 0) but fluctuate around a stationary sta.te of eq. (1), which corresponds \nto a perfect topographic representation of visual space. In this parameter regime, \nthe requirement of a continous dimension-reducing map leads to the suppression of \nthe additional features \"orientation\" and \"ocular dominance\". \nLet us consider an ensemble of networks, each characterized by a set {wr } of feature \nvectors, and denote the time-dependent distribution function of this ensemble by \n\n\f14 \n\nObermayer, Ritter, and Schulten \n\nS( w, t). Following a method derived in [19], we can describe the time-development \nof S( W, t) near the stationary state by the Fokker-Planck equation \n\n({ .... }) I: 8 \nU,~' t = \n\n-8.... Bpmqnuqn S Ur , t + -2 \nu-\n\n.... \n\n({ .... }) f I: \n\n18 \n-\nf \n\nt S \n\npmqn \n\npm \n\nDpmqn 8.... \nu-\n\n8 2S({ur},t) () \n4 \n\n8.... \nu-\n\nqn \n\npm \n\npmqn \n\nwhere the origin of S(.,t) was shifted to the stationary state {Uir }, using now the \nnew argument variable ur = wr - Uir. The eigenvalues of B determine the stability \nof the stationary state, the topographic representation, while Band D together \ngovern size and time development of fluctuations < UpiUqj >. \nLet us define the Fourier modes uk of the equilibrium deviations ur by uk = \nl/N Ereikrur. For small values of qpat and Zpat the eigenvalues of B are all neg(cid:173)\native, hence the topographic stationary state is stable. If qpat and Zpat are larger \nthan2 \n\nqthrel = ~ ~ min(O\"hl,O\"h2), \n\nZthre$ = ~Ve ~ min(O\"hl,O\"h2), \n\n(5) \n\nhowever, the eigenvalues corresponding to the set of modes uk which are perpen(cid:173)\ndicular to the (x, y)-plane and whose wave-vectors k are given by \n\n\u00b12/O\"hl } \na \n\n. \n'l/ O\"hl < O\"h2, \n\n(6) \n\nbecome positive. For larger values of qpat and Zpat then, the topographic state \nbecomes unstable and a \"column system\" forms. \nFor an isotropic neighborhood function (O\"hl = O\"h2 = O\"h), the matrices B(k) and \nb( k) can be diagonalized simultaneously and the mean square amplitude of the \nfluctuations around the stationary state can be given in explicit form: \n\n2 (k .... ) \n\n< U II \n\n> - 7r - 0\" -\n\ng 2 d2 (0\"~k2/4+ 1/12)exp(-0\"~k2/4) \n~':\"\"'-~-;::---::-':--:--'---\"-'--:~:--:-'----'-\n2 h N2 \n\nexp(0\"~k2/4) - 1 + 0\"~k2/2 \n\n-\n\n2 \n\ng 2 d2 exp( -0\"~k2 /4) \n< u.L(k) >= 7r 240\"h N2 exp(0\"~k2/4)-1 \n\n.... \n\n2 \n\n.... \n\n2.... \n\ng 2 2 \n\nexp(-0\"~k2/4) \n\n< u y l(k) >=< uy2 (k) >= 7r40\"hqPatexp(0\"~k2/4) _ (N2q;atk2)/(2d2) \n\n2 .... \n\ng 2 2 \n\nexp( -0\"~k2 /4) \n\n< uz(k) >= 7r20\"hZpatexp(0\"~k2/4) _ (N2q;atk2)/d2 \n\n(7) \n\n(8) \n\n(9) \n\n(10) \n\n2Tn t.he derivat.ion of t.he following formulas several approximat.ions have t.o he made. A \ncomparison wit.h numerical simulat.ions, however, demonst.rat.e t.hat. t.hese approximat.ions are \nvalid except. if t.he value qpat or Zpat is wit.hin a few percent. of qthre .. or Zthre.\" respectively. \nDet.ails of t.hese calculat.ions will be published elsewhere \n\n\fDevelopment and Spatial Structure of Cortical Feature Maps: A Model Study \n\n15 \n\nFigure 1: \"Orientation preference\" (a, left), \"ocular dominance\" (h, center) and \nlocations of receptive field centers (c, right) as a function of unit loaction. Figure \nla displays an enlarged section of the \"orientation map\" only. Parameters of the \nsimulation were: N = 256, d = 256, qpat = 12, Zpat = 12, Uh = 5, e = 0.02 \n\nwhere ulI' U.L denote the amplitude of fluctuations parallel and orthogonal to k \nin the (x, y)-plane, Uyl' U y 2 parallel to the orientation feature dimension and U z \nparallel to the ocular dominance feature dimension, respectively. \n\nThus, for qpat ~ qthres or Zpat ~ Zthres the mean square amplitudes of fluctuations \ndiverge for the modes which become unstable at the threshold (the denominator of \neqs. (9,10) approaches zero) and the relaxation time of these fluctuations goes to \ninfinity (not shown). The fact that either a ring or two groups of modes become \nunstable is reflected in the spatial structure of the maps above threshold. \n\nFor larger values of qpat and Zpat orientation and ocular dominance are represented \nby the network layer, i.e. feature values fluctuate around a stationary state which \nis characterized by a certain distribution of feature-selective cells. Figure 1 displays \norientation preference \u00a2r (Fig. la), ocular dominance Z,~ (Fig. 1b) and the locations \n(xr' Yr) of receptive field centers in visual space (Fig. 1c) as a function of unit \nlocation r. Each pixel of the images in Figs. 1a,b corresponds to a network unit \nr. Feature values are indicated by gray values: black ~ white corresponds to an \nangle of 00 ~ 1800 (Fig. 1a) and to an ocular dominance value of a ~ max (Fig. \n1b). White dots in Fig. 1a mark regions where units still completely unspecific for \norientation are located (\"foci\"). In Fig. 1c the receptive field center of every unit \nis marked by a dot. The centers of units which are neighbors in the network layer \nwere connected by lines, which gives rise to the net-like structure. \n\nThe overall preservation of the lattice topology, and the absence of any larger dis(cid:173)\ncontinuities in Fig. 1c, demonstrate that \"position\" plays the role of the primary \nstimulus variable and varies in a topographic fashion across the network layer. On \na smaller length scale, however, numerous distortions are visible which are caused \nby the representation of the other features, \"orientation\" and \"ocular dominance\". \nThe variation of these secondary features is highly repetitive and patterns strongly \nresembling orientation columns (Fig. 1b) and ocular dominance stripes (Fig. 1c) \nhave formed. Note that regions unspecific for orientation as well as \"binocular\" \nregions exist in the final map, although these feature combinations were not present \nin the set of input patterns (3). They are correlated with regions of high magnitude \nof the \"orientation\" and \"ocular dominance\"-gradients, respectively (not shown). \nThese structures are a consequence of the neighborhood preserving and dimension \n\n\f16 \n\nObermayer, Ritter, and Schulten \n\n' .:.: \n\n\u00b7*, :~{AJ . ::;::}. \n\n:.:~:;:~;.:.:::::. \n. .... :;::::::~::: ... \n: .. :::::\\;~ \n:':':':~:: . . \n\"\n< \u00b7:::{:\u00b7r~if as a function of the distance s between cells \nin the network layer. The origin of the s-plane is located in the center of the image \nand the brightness indicates a positive (white), zero (medium gray) or negative \n(black) value of 5 33 . The autocorrelation functions have a Mexican-hat form. The \n(negative) minimum is located at half the wavelength A associated with the the \nwave number If I of the modes with high amplitude in Fig. 2a. At this distance \nthe response properties of the units are anticorrelated to some extent. If cells are \nseparated by a distance larger than A, the response properties are uncorrelated. \nIf qpat and Zpat are large enough, the feature hierarchy observed in Figs. 1,2 breaks \ndown and \"preferred orientation\" or \"ocular dominance\" plays the role of the pri(cid:173)\nmary stimulus variable. Figure 3 displays orientation preference \u00a2r (Fig. 3a) and \nocular dominance Zr (Fig. 3b) as a function of unit location 1-:. There is only one \ncontinous region for each interval of \"preferred orientation\" and for each eye, but \neach of these regions now contains a representation of a large part of visual space. \nConsequently the position map shows multiple representations of visual space. \n\nHierarchical maps are generated by the feature map algorithm whenever there is a \nhierarchy in the variances of the set of patterns along the various feature dimensions \n\n:'lIn t.he cort.ex, however, cellfl unflpecific for orient.ation fleem t.o he import.nnt for viflual \nprocf'Bsing. 1'0 improve t.he deflcript.ion oft.he spat.ial st.ruct.ure of cort.ical maps, it is neCf'Bflary \nt.o include t.hese feat,lIre comhinnt.iOlls into t.he flet V' of inpnt, patternfl (see [0]). \n\n\fDevelopment and Spatial Structure of Cortical ~ature Maps: A Model Study \n\n17 \n\ni :. \n\nFigure 3: \"Orientation preference\" ( a, left) and \"ocular dominance\" (h, center) as \na function of unit loaction for a map generated using a large value of qpat and Zpat. \nParameters were: N = 128, d = 128, qpat = 2500, Zpat = 2500, Uh = 5, c; = 0.1 \n(In our example a hierarchy in the magnitudes of d, qpat and Zpat). The features \nwith the largest variance become the primary feature; the other features become \nsecondary features, which are represented multiple times on the network layer. \n\nAcknow ledgelnents \n\nThe authors would like to thank the Boehringer-Ingelheim Fonds for financial sup(cid:173)\nport by a scholarship to K. O. This research has been supported by the National \nScience Foundation (grant number 9017051). Computer t.ime on the Connection \nMachine CM-2 has been made available by the National Center for Supercomputer \nApplications at Urbana-Champaign and the Pittsburgh Supercomputing Center \nboth supported by the National Science Foundation. \n\nReferences \n\n[1] Hubel D.H. and Wiesel T.N. (1974), J. Compo Neurol. 158, 267-294 \n[2] Blasdel G.G. and Salama G. (1986), Nature 321, 579-585 \n[3] Grinvald A. et aI. (1986), Nature 324, 361-364 \n[4] Swindale N.V. et al. (1987), J. Neurosci. 7,1414-1427 \n[5] Lowel S. et al. 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Cybern. 60,59-71 \n\n\f", "award": [], "sourceid": 299, "authors": [{"given_name": "Klaus", "family_name": "Obermayer", "institution": null}, {"given_name": "Helge", "family_name": "Ritter", "institution": null}, {"given_name": "Klaus", "family_name": "Schulten", "institution": null}]}