{"title": "A comparison between a neural network model for the formation of brain maps and experimental data", "book": "Advances in Neural Information Processing Systems", "page_first": 83, "page_last": 90, "abstract": "", "full_text": "A comparison between a neural network model for \nthe formation of brain maps and experimental data \n\nK. Obermayer \nBeckman-Institute \nUniversity of Illinois \nUrbana, IL 61801 \n\nK. Schulten \n\nBeckman-Institute \nUniversity of Illinois \n\nUrbana, IL 61801 \n\nG.G. Blasdel \nHarvard Medical School \nHarvard University \nBoston, MA 02115 \n\nAbstract \n\nRecently, high resolution images of the simultaneous representation of \norientation preference, orientation selectivity and ocular dominance have \nbeen obtained for large areas in monkey striate cortex by optical imaging \n[1-3]. These data allow for the first time a \"local\" as well as \"global\" \ndescription of the spatial patterns and provide strong evidence for corre(cid:173)\nlations between orientation selectivity and ocular dominance. \nA quantitative analysis reveals that these correlations arise when a five(cid:173)\ndimensional feature space (two dimensions for retinotopic space, one each \nfor orientation preference, orientation specificity, and ocular dominance) is \nmapped into the two available dimensions of cortex while locally preserving \ntopology. These results provide strong evidence for the concept of topology \npreserving maps which have been suggested as a basic design principle of \nstriate cortex [4-7]. \n\nMonkey striate cortex contains a retinotopic map in which are embedded the highly \nrepetitive patterns of orientation selectivity and ocular dominance. The retinotopic \nprojection establishes a \"global\" order, while maps of variables describing other \nstimulus features, in particular line orientation and ocularity, dominate cortical \norganization locally. A large number of pattern models [8-12] as well as models \nof development [6,7,13-21] have been proposed to describe the spatial structure of \nthese patterns and their development during ontogenesis. However, most models \nhave not been compared with experimental data in detail. There are two reasons \nfor this: (i) many model-studies were not elaborated enough to be experimentally \ntestable and (ii) a sufficient amount of experimental data obtained from large areas \nof striate cortex was not available. \n83 \n\n\f84 \n\nObermayer, Schulten, and Blasdel \n\nFigure 1: Spatial pattern of orientation preference and ocular dominance in mon(cid:173)\nkey striate cortex (left) compared with predictions of the SOFM-model (right). Iso(cid:173)\norientation lines (gray) are drawn in intervals of 11.25\u00b0 (left) and IS.00 (right), re(cid:173)\nspectively. Black lines indicate the borders (ws(rj = 0) of ocular dominance bands. \nThe areas enclosed by black rectangles mark corresponding elements of organization \nin monkey striate cortex and in the simulation result (see text). Left: Data obtained \nfrom a 3.1mm x 4.2mm patch of the striate cortex of an adult macaque (macaca \nnemestrina) by optical imaging [1-3]. The region is located near the border with \narea IS, close to midline. Right: Model-map generated by the SOFM-algorithm. \nThe figure displays a small section of a network of size N = d = 512. The param(cid:173)\neters of the simulation were: \u20ac = 0.02, tTh = 5, vr,:x = 20.48, vrax = 15.36, 9 . 101 \niterations, with retinotopic initial conditions and periodic boundary conditions. \n\n1 Orientation and ocular dominance columns in monkey \n\nstriate cortex \n\nRecent advances in optical imaging [1-3,22,23] now make it possible to obtain high \nresolution images of the spatial pattern of orientation selectivity and ocular domi(cid:173)\nnance from large cortical areas. Prima vista analysis of data from monkey striate \ncortex reveals that the spatial pattern of orientation preference and ocular domi(cid:173)\nnance is continuous and highly repetitive across cortex. On a global scale orienta(cid:173)\ntion preferences repeat along every direction of cortex with similar periods. Locally, \norientation preferences are organized as parallel slabs (arrow 1, Fig. 1a) in linear \nzones, which start and end at singularities (arrow 2, Fig. la), point-like disconti(cid:173)\nnuities, around which orientation preferences change by \u00b1IS00 in a pinwheel-like \nfashion. Both types of singularities appear in equal numbers (359:354 for maps \nobtained from four adult macaques) with a density of 5.5/mm2 (for regions close to \n\n\fA Neural Network Model for the Formation of Brain Maps Compared with Experimental Data \n\n85 \n\nFourier transforms \n\nWj (k) \n\ncorrelation functions Cij (P) \n\nfeature gradients \n\nI V;rWj (r) I \n\nGabor transforms \n\n9j (k, r) \n\nE;r exp{ikr) Wj{r) \n< Wier) Wj{r + p) >;r \n({Wj{rl + 1, r2) - Wj{rl' r2\u00bb2 \n+ (Wj(rl, r2 + 1) - Wj{rl' r2\u00bb2}i/2 \n(211'0';)-t f d2r'wj{r') \n\nexp{-\n\n(;r ;t'r~ \n;(1~ + ik{r' - !r)} \n\n~ \n\nTable 1: Quantitative measures used to characterize cortical maps. \n\nthe midline). Figure la reveals that the iso-orientation lines cross ocular dominance \nbands at nearly right angles most of the time (region number 2) and that singular(cid:173)\nities tend to align with the centers of the ocular dominance bands (region number \n1). Where orientation preferences are organized as parallel slabs (region number \n2), the iso-orientation contours are often equally spaced and orientation preferences \nchange linearly with distance. \nThese results are confirmed by a quantitative analysis (see Table 1). For the \nfollowing we denote cortical location by a two-dimensional vector r. At each lo(cid:173)\ncation we denote the (average) position of receptive field centroids in visual space \nby (Wl(r), tlJ2{r). Orientation selectivity is described by a two-dimensional vector \n( W3( r), W4 ( r), whose length and direction code for orientation tuning strength and \npreferred orientation, respectively [1,10]. Ocular dominance is described by a real(cid:173)\nvalued function ws{r), which denotes the difference in response to stimuli presented \nto the left and right eye. Data acquisition and postprocessing are described in detail \nin [1-3]. \nA Fourier transform of the map of orientation preferences reveals a spectrum which \nis a nearly circular band (Fig. 2a), showing that orientation preferences repeat \nwith similar periods in every direction in cortex. Neglecting the slight anisotropy \nin the experimental datal, a power spectrum can be approximated by averaging \namplitudes over all directions of the wave-vector (Fig. 2b, dots~. The location of \nthe peak corresponds to an average period Ao = 710pm \u00b1 50pm and it's width to \na coherence length of 820pm \u00b1 130pm. The coherence length indicates the typical \ndistance over which orientation preferences can change linearly and corresponds \nto the average size of linear zones in Fig. la. The corresponding autocorrelation \nfunctions (Fig. 2c) have a Mexican hat shape. The minimum occurs near 300pm, \nwhich indicates that orientation preferences in regions separated by this distance \ntend to be orthogonal. In summary, the spatial pattern of orientation preference is \ncharacterized by local correlation and global \"disorder\" . \n\n, ,\\ long axes parallel t.o t.he ocular dominance fllabfl, orient.ation preferencM repeat on \naverage every 660/Jm \u00b1 40pm; perpendicular t.o the fltripes every 840/Jm \u00b1 40/Jm. The fllight. \nhorizontal elongation reflects t.he fact that. iR(H)rientation fllabR tend to r.onned t.he centerR \nof ocul:\" dominance bandR. \n\n2,\\11 quantities regarding experimental data are averages over four animalR, nml-nm4, \n\nunleAA Rtated ot.herwifle. F.rror marginfl indicat.e fltandard deviations. \n\n\f. . ::\\~. >: ;:':.: \n\nii.i;\u00b7\u00b7':i~l;'~;::\u00b7\u00b7;i:>\u00b7~ \u2022..\u2022.. ~;, \n\nC44 \n\nc) \n\n.-nm2 \n_ \ntheory \n\na \nspatial frequency \n\n2 \n\n1 \n\n3 \n\n(nonnalized) \n\n0.5 \n\n0.0 \ndistance (nonnalized) \n\n1.0 \n\n1.5 \n\n86 \n\nObermayer, Schulten. and Blasdel \na) \n\n:.:. \n\n:. :\u00a5 .. : ::::-:':0: ::::~;:::-;:::::\n\n. \n\n.. \"' \n\nX: :: :::.- .... \n\n~'j': :. .. .. :~~;~ .: : : .. ', . \n\niI;ip;:lr:, \n\n1.0 \n\n\",-.... 1.5 \n13 \n-\n.~ \ne \nC':S \n\n0 \nc:: \n'-' \n~ \n& 0.0 \n\n0.5 \n\nb) \n\n.. nm2 \n\n-theory \n\n.-~\",-.... \n6 \n0\"0 \nc:: ~ \ncE .- 0.5 \n.9 e \nc::~ \n~ 0 \n-d \n~'-' \n8 \n\n-0.5 \n\n0.0 \n\nFigure 2: Fourier analysis and correlation functions of the orientation map in \nmonkey striate cortex (animal nm2) compared with the predictions of the SOFM(cid:173)\nmodel. Simulation results were taken from the data set described in Fig. 1, right. \n(a) Fourier spectra of nm2 (left) and simulation results (right). Each pixel repre(cid:173)\nsents one mode; location and gray value of the pixel indicate wave-vector and energy, \nrespectively. (b) Approximate power spectrum (normalized) obtained by averaging \nthe Fourier-spectra in (a) over all directions of the wave-vector. Peak frequency \nof 1.0 corresponds to 1.4/mm for nm2. (c) Correlation functions (normalized). A \ndistance of 1.0 corresponds to 725IJm for nm2. \n\nLocal properties of the spatial patterns, as well as correlations between orientation \npreference and ocular dominance, can be quantitatively characterized using Gabor(cid:173)\nHelstrom-transforms (see Table 1). If the radius ug of the Gaussian function in the \nGabor-filter is smaller than the coherence length the Gabor-transform of any of the \nquantities W3Cr'), W4(r') and ws(r') typically consists of two localized regions of high \nenergy located on opposite sides of the origin. The length Ikd of the vectors ki' \ni E [3,4,5], which corresponds to the centroids of these regions, fluctuates around \nthe characteristic wave-number 27r/>'o of this pattern, and its direction gives the \nnormal to the ocular dominance bands and iso-orientation slabs at the location r, \nwhere the Gabor-transform was performed. \n\n\fA Neural Network Model for the Fonnation of Brain Maps Compared with Experimental Data \n\n87 \n\nnml-nm4 \n\ntheory \n\n10 \n\n.:::1 \n\n\u00a7 \n-\n8 \n'C) \n& \nrP PIBllel alalll S \n/SI \nc:: \n2 \n~ 0 \n8. \n\nainguJaritiel \n\n6 \n\n4 \n\n.. \n\n8 \n\ncP \nU \n\n.. \n\nU) \n\n-\n\n6 10 \n.:::1 \n~ \n8 \n8 \n6 \n'-\n0 \n& 4 \nS \n2 \n~ \n8. \n\n0 \n\nfP \nII \n\n~ \n\n90\u00b0 \n\n.J. \n\nFigure 3: Gabor-analysis of cortical maps. The percentage of map locations is \nplotted against the parameters 81 and 82 (see text) for 3,421 locations randomly \nselected from the cortical maps of four monkeys, nml-nm4, (left) and for 1,755 \nlocations randomly selected from simulation results (right). Error bars indicate \nstandard deviations. Simulation results were taken from the data set described in \nFig. 1. CTg was 150l'm for the experimental data and 28 pixels for the SOFM-map. \n\nResults of this analysis are shown in Fig. 3 (left) for 3,434 samples selected randomly \nfrom data of four animals. The angle between k3 and k4 is represented along the 81 \naxis. Histograms at the back, where 81 = 0\u00b0, represent regions where iso-orientation \nlines are parallel. Histograms in the front, where 81 = 90\u00b0, represent regions con(cid:173)\ntaining singularities. The intersection angle of iso-orientation slabs and ocular dom(cid:173)\ninance bands is represented along the 82 axis. The proportion of sampled regions \nincreases steadily with decreasing 81. As 81 approaches zero, values accumulate \nat the right, where orientation and ocular dominance bands are orthogonal. Thus \nlinear zones and singularities are important elements of cortical organization but \nlinear zones (back rows) are the most prominent features in monkey striate cortex3 . \nWhere iso-orientation regions are organized as parallel slabs, orientation slabs in(cid:173)\ntersect ocular dominance bands at nearly right angles (back and right corner of \ndiagrams). \n\n2 Topology preserving maps \n\nRecently, topology preserving maps have been suggested as a basic design principle \nunderlying these patterns and its was proposed that these maps are generated by \nsimple and biologically plausible pattern formation processes [4,6,7]. In the following \nwe will test these models against the recent experimental data. \nWe consider a five-dimensional feature 8pace V which is spanned by quantities de(cid:173)\nscribing the most prominent receptive field properties of cortical cells: position \nof a receptive field in retinotopic space (VI, V2), orientation preference and tuning \nstrength (V3, V4), and ocular dominance (V5). If all combinations of these properties \n\n!iDBtB from BreB 17 of the eBt indkBte t.hat in thifl flpeciefl, Blthough hot,h elementfl are \n\nprPJlent, flingulBritiefi Bre more import.ant [23] \n\n\f88 \n\nObermayer, Schulten, and Blasdel \n\nare represented in striate cortex, each point in this five-dimensional feature space \nis mapped onto one point on the two-dimensional cortical surface A. \nIn order to generate these maps we employ the feature map (SOFM-) algorithm \nof Kohonen [15,16] which is known to generate topology preserving maps between \nspaces of different dimensionality [4,5]4. The algorithm describes the development \nof these patterns as unsupervised learning, i.e. the features of the input patterns \ndetermine the features to be represented in the network [4]. Mathematically, the \nalgorithm assignes feature vectors w(r), which are points in the feature space, to \ncortical units r, which are points on the cortical surface. In our model the surface is \ndivided into N x N small patches, units r, which are arranged on a two-dimensional \nlattice (network layer) with periodic boundary conditions (to avoid edge effects). \nThe average receptive field properties of neurons located in each patch are char(cid:173)\nacterized by the feature vector w( r) whose components (Wj (r) are interpreted as \nreceptive field properties of these neurons. The algorithm follows an iterative pro(cid:173)\ncedure. At each step an input vector V, which is of the same dimensionality as w(;;') \nis chosen at random according to a probability distribution P( V). Then the unit \ni whose feature vector w(S) is closest to the input pattern v is selected and the \ncomponents (Wj(r) of its feature vector are changed according to the feature map \nlearning rule [15,16]' \n\nP( V) was chosen to be constant within a cylindrical manifold in feature space, \n\nwhere vg::x and vrax are some real constants, and zero elsewhere. \n\nFigure 4 shows a typical map, a surface in feature space, generated by the SOFM(cid:173)\nalgorithm. For the sake of illustration the five-dimensional feature space is projected \nonto a three-dimensional subspace spanned by the coordinate-axes corresponding \nto retinotopic location (VI and V2) and ocular dominance (vs). The locations of \nfeature vectors assigned to the cortical units are indicated by the intersections of \na grid in feature space. Preservation of topology requires that the feature vectors \nassigned to neighboring cortical units must locally have equal distance and must be \narranged on a planar square lattice in feature space. Consequently, large changes \nin one feature, e.g. ocular dominance vs, along a given direction on the network \ncorrelate with small changes of the other features, e.g. retinotopic location VI and \nV2, along the same direction (crests and troughs of the waves in Fig. 4) and vice \nversa. Other correlations arise at points where the map exhibits maximal changes \nin two features. For example for retinotopic location (VI) and ocular dominance \n(vs) to vary at a maximal rate, the surface in Fig. 4 must be parallel to the (VI, vs)(cid:173)\nplane. Obviously, at such points the directions of maximal change of retinotopic \nlocation and ocular dominance are orthogonal on the surface. \nIn order to compare model predictions with experimental data the surface in the five(cid:173)\ndimensional feature space has to be projected into the three-dimensional subspace \n\n\"'fhp. p.xBd form of t.hp. Bigorit.hm iA not P.A.'lP.ntiBI, howp.vp.r. A Igorit.hmA hR.'JP.