{"title": "The Role of Lateral Cortical Competition in Ocular Dominance Development", "book": "Advances in Neural Information Processing Systems", "page_first": 139, "page_last": 145, "abstract": null, "full_text": "The Role of Lateral Cortical Competition \n\nin Ocular Dominance Development \n\nChristian Piepenbrock and Klaus Obermayer \n\nDept. of Computer Science, Technical University of Berlin \n\nFR 2-1; Franklinstr. 28-29; 10587 Berlin, Germany' \n\n{piep,oby}@cs.tu-berlin.de; http://www.ni.cs.tu-berlin.de \n\nAbstract \n\nLateral competition within a layer of neurons sharpens and localizes the \nresponse to an input stimulus. Here, we investigate a model for the ac(cid:173)\ntivity dependent development of ocular dominance maps which allows \nto vary the degree of lateral competition. For weak competition, it re(cid:173)\nsembles a correlation-based learning model and for strong competition, \nit becomes a self-organizing map. Thus, in the regime of weak compe(cid:173)\ntition the receptive fields are shaped by the second order statistics of the \ninput patterns, whereas in the regime of strong competition, the higher \nmoments and \"features\" of the individual patterns become important. \nWhen correlated localized stimuli from two eyes drive the cortical de(cid:173)\nvelopment we find (i) that a topographic map and binocular, localized \nreceptive fields emerge when the degree of competition exceeds a critical \nvalue and (ii) that receptive fields exhibit eye dominance beyond a sec(cid:173)\nond critical value. For anti-correlated activity between the eyes, the sec(cid:173)\nond order statistics drive the system to develop ocular dominance even \nfor weak competition, but no topography emerges. Topography is estab(cid:173)\nlished only beyond a critical degree of competition. \n\n1 Introduction \n\nSeveral models have been proposed in the past to explain the activity depending develop(cid:173)\nment of ocular dominance (00) in the visual cortex. Some models make the ansatz of \nlinear interactions between cortical model neurons [2, 7], other approaches assume com(cid:173)\npetitive winner-take-all dynamics with intracortical interactions [3, 5]. The mechanisms \nthat lead to ocular dominance critically depend on this choice. In linear activity models, \nsecond order correlations of the input patterns determine the receptive fields. Nonlinear \ncompetitive models like the self-organizing map. however, use higher order statistics of the \ninput stimuli and map their features. In this contribution. we introduce a general nonlinear \n\n\f140 \n\nOil x \n\npLIl \n\n\u2022 pRIl \u2022 \n\nC. Pie pen brock and K. Obermayer \n\nFigure I: Model for OD development: the in(cid:173)\n~~~~~~mQQQ!~\u00a7QJ Cortex put patterns p/!J. and P!RJ-L in the LGN drive \nthe Hebbian modification of the cortical affer-\nent synaptic weights S~i and S~. Cortical neu-\n~:=:::=:::. left-eye rons are in competition and interact with effec(cid:173)\nright-eye tive strengths f xy. Locations in the LGN are in-\nRl \ndexed i or j, cortical locations are labeled x or y. \n\nLGN \n\nR \nYJ \n\nJ \n\nHebbian development rule which interpolates the degree of lateral competition and allows \nus to systematically study the role of non-linearity in the lateral interactions on pattern for(cid:173)\nmation and the transition between two classes of models. \n\n2 Ocular Dominance Map Development by Hebbian Learning \n\nFigure I shows our basic model framework for ocular dominance development. We con(cid:173)\nsider two input layers in the lateral geniculate nucleus (LGN). The input patterns f1 = \n1, ... , U on these layers originate from the two eyes and completely characterize the in(cid:173)\nput statistics (the mean activity P is identical for all input neurons). The afferent synaptic \nconnection strengths of cortical cells develop according to a generalized Hebbian learning \nrule with learning rate \",. \n\nASLJ-L - ~ f 0-J-LpLJ-L \nL). xi \n\n'\" ~ xy \n\ny \n\ni \n\n-\n\n( I ) \n\ny \n\nAn analogous rule is used for the connections from the right eyes S~. We use v = 2 in the \nfollowing and rescale the length of each neurons receptive field weight vector to a constant \nlength after a learning step. The model includes effective cortical interactions fry for the \ndevelopment of smooth cortical maps that spread the output activities 6~ in the neighbor(cid:173)\nhood of neuron x (with a mean j = ~ Lx fry for N output neurons). The cortical output \nsignals are connectionist neurons with a nonlinear activation function g(.), \n\n_ \nOt = g(H~) = Lz exp(;3~r) with H~ = L(S~jPjLJ-L + s~pt!J.) , \n\nexp({3HJ-L) \n\nj \n\n(2) \n\nwhich models the effect of cortical response sharpening and competition for an input stim(cid:173)\nulus. The degree of competition is determined by the parameter {3. Such dynamics may re(cid:173)\nsult as an effect of local excitation and long range inhibition within the cortical layer [6, I], \nand in the limits of weak and strong competition, we recover two known types of develop(cid:173)\nmental models-the correlation based learning model and the self-organizing map. \n\n2.1 From Linear Neurons to Winner-take-all Networks \n\nIn the limit ;3 ---+ 0 of weak cortical competition. the output 6~ becomes a linear function \nof the input. A Taylor series expansion around 13 = 0 yields a correlation-based-learning \n(CBL) rule in the average over all patterns \n\n~st \n\nT];3L ~(l:rz - j)(S~jctL + S~CflL) +const .. \n\nz .j \n\nwhere CfiL = b LJ-L ptJ-L p/J-L is the correlation function of the input patterns. Ocular \ndominance development under this rule requires correlated activity between inputs from \n\n\fRole of Lateral Cortical Competition in Ocular Dominance Development \n\n141 \n\nCBL limit \n\n(3 = 2.5 \n\n(3 = 32 \n\nSOM limit \n\nFigure 2: The network response for different degrees of cortical competition: the plots \nshow the activity rates Ly 1 xy 6~ for a network of cortical output neurons (the plots are \nscaled to have equal maxima). Each gridpoint represents the activity of one neuron on a \n16 x 16 grid. The interactions Ixy are Gaussian (variance 2.25 grid points) and all neu(cid:173)\nrons are stimulated with the same Gaussian stimulus (variance 2.25). The neurons have \nGaussian receptive fields (variance (J'2 = 4.5) in a topographic map with additive noise \n(uniformly distributed with amplitude 10 times the maximum weight value). \n\nwithin one eye and anti-correlated activity (or uncorrelated activity with synaptic competi(cid:173)\ntion) between the two eyes [2,4]. It is important to note, however, that CBL models cannot \nexplain the emergence of a topographic projection. The topography has to be hard-wired \nfrom the outset of the development process which is usually implemented by an \"arbor \nfunction\" that forces all non-topographic synaptic weights to zero. \n\nStrong competition with (3 --t 00, on the other hand, leads to a self-organizing map [3, 5], \n\n~sf: = TJlxq(Jl)P/Jl with q(ll) = argmaXy I)S~jPf!l + s~ptJl) . \n\nj \n\nModels of this type use the higher order statistics of the input patterns and map the impor(cid:173)\ntant features of the input. In the SOM limit, the output activity pattern is identical in shape \nfor all input stimuli. The input influences only the location of the activity on the output \nlayer but does not affect its shape. \n\nFor intermediate values of (3, the shape of the output activity patterns depends on the input. \nThe activity of neurons with receptive fields that match the input stimulus better than oth(cid:173)\ners is amplified, whereas the activity of poorly responding neurons is further suppressed as \nshown in figure 2. On the one hand, the resulting output activity profiles for intermediate (3 \nmay be biologically more realistic than the winner-take-alllimit case. On the other hand, \nthe difference between the linear response case (low (3) and the nonlinear competition (in(cid:173)\ntermediate (3) is important in the Hebbian development process-it yields qualitatively dif(cid:173)\nferent results as we show in the next section. \n\n2.2 Simulations of Ocular Dominance Development \n\nIn the following, we study the transition from linear CBL models to winner-take-all SOM \nnetworks for intermediate values of 13. We consider input patterns that are localized and \nshow ocular dominance \n\np.LJl = 0.5 + eyeL (11) exp (_ (i -IOC(f-L))2) with eyeL (f-L) = -eyeR (f-L) \n\n(3) \n\nl \n\n2rr(J'2 \n\n2(J'2 \n\nEach stimulus 11 is of Gaussian shape centered on a random position loc(ll) within the input \nlayer and the neuron index i is interpreted as a two-dimensional location vector in the input \nlayer. The parameter eye (f-L) sets the eye dominance for each stimulus. eye = 0 produces \nbinocular stimuli and eye = \u00b1 ~ results in uncorrelated left and right eye activities. \nWe have simulated the development of receptive fields and cortical maps according to \nequations 1 and 2 (see figure 3) for square grids of model neurons with periodic bound(cid:173)\nary conditions, Gaussian cortical interactions. and 00 stimuli (equation 3). The learning \n\n\f142 \n\n5 \n\n4 \n\n(!) \nN \n.