{"title": "Models of Ocular Dominance Column Formation: Analytical and Computational Results", "book": "Advances in Neural Information Processing Systems", "page_first": 375, "page_last": 383, "abstract": null, "full_text": "LINEAR LEARNING: LANDSCAPES AND ALGORITHMS \n\n65 \n\nPierre Baldi \n\nJet Propulsion Laboratory \n\nCalifornia Institute of Technology \n\nPasadena, CA 91109 \n\nWhat follows extends some of our results of [1] on learning from ex(cid:173)\namples in layered feed-forward networks of linear units. In particu(cid:173)\nlar we examine what happens when the ntunber of layers is large or \nwhen the connectivity between layers is local and investigate some \nof the properties of an autoassociative algorithm. Notation will be \nas in [1] where additional motivations and references can be found. \nIt is usual to criticize linear networks because \"linear functions do \nnot compute\" and because several layers can always be reduced to \none by the proper multiplication of matrices. However this is not the \npoint of view adopted here. It is assumed that the architecture of the \nnetwork is given (and could perhaps depend on external constraints) \nand the purpose is to understand what happens during the learning \nphase, what strategies are adopted by a synaptic weights modifying \nalgorithm, ... [see also Cottrell et al. (1988) for an example of an ap(cid:173)\nplication and the work of Linsker (1988) on the emergence of feature \ndetecting units in linear networks}. \n\nConsider first a two layer network with n input units, n output units \nand p hidden units (p < n). Let (Xl, YI), ... , (XT, YT) be the set of \ncentered input-output training patterns. The problem is then to find \ntwo matrices of weights A and B minimizing the error function E: \n\nE(A, B) = L IIYt - ABXtIl2. \n\nl \nIf I = {i t , ... ,ip}(l < it < ... < ip < n) is any or(cid:173)\n... > An. \ndered p-index set, let Uz = [Ui t , \u2022\u2022\u2022 , Uip ] denote the matrix formed \nby the orthononnal eigenvectors of ~ associated with the eigenvalues \nAil' ... , Aip \u2022 Then two full rank matrices A and B define a critical \npoint of E if and only if there exist an ordered p-index set I and an \ninvertible p x p matrix C such that \n\nA=UzC \n\nFor such a critical point we have \n\nE(A,B) = tr(~yy) - L Ai. \n\niEZ \n\n(8) \n\n(9) \n\n(10) \n\n(11 ) \n\nTherefore a critical point of W of rank p is always the product of the \nordinary least squares regression matrix followed by an orthogonal \nprojection onto the subspace spanned by p eigenvectors of~. The map \nW associated with the index set {I, 2, ... ,p} is the unique local and \nglobal minimum of E. The remaining (;) -1 p-index sets correspond \nto saddle points. All additional critical points defined by matrices \nA and B which are not of full rank are also saddle points and can \nbe characerized in terms of orthogonal projections onto subspaces \nspanned by q eigenvectors with q < p. \n\n\f68 \n\nBaldi \n\nDeep Networks \n\nConsider now the case of a deep network with a first layer of n input \nunits, an (m + 1 )-th layer of n output units and m - 1 hidden layers \nwith an error function given by \n\nE(AI, ... ,An)= L IIYt-AIAl ... AmXtll2. \n\nl Xd J1( k) ~ +00. Therefore the algorithm can converge \nonly for a < /-leO) < Xd. When the learning rate is too large, i. e. \nwhen 7],\\ > 1/2 then even if /-leO) is in the interval (0, Xd) one can see \nthat the algorithm does not converge and may even exhibit complex \noscillatory behavior. However when 7],\\ < 1/2, if 0 < J1(0) < Xa then \nJ1( k) ~ 1, if /-leO) = Xa then /-l( k) = a and if Xa < J1(0) < Xd then \n/-l(k) ~ 1. \n\nIn conclusion, we see that if the algorithm is to be tested, the \nlearning rate should be chosen so that it does not exceed 1/2,\\, where \n,\\ is the largest eigenvalue of ~xx. Even more so than back propaga(cid:173)\ntion, it can encounter problems in the proximity of saddle points. \nOnce a non-principal eigenvector of ~xx is learnt, the algorithm \nrapidly incorporates a projection along that direction which cannot \nbe escaped at later stages. Simulations are required to examine the \neffects of \"noisy gradients\" (computed after the presentation of only \na few training examples), multiple starting points, variable learning \nrates, momentum terms, and so forth. \n\n\f72 \n\nBaldi \n\nAknowledgement \n\nWork supported by NSF grant DMS-8800323 and in part by ONR \ncontract 411P006-01. \n\nReferences \n\n(1) Baldi, P. and Hornik, K. (1988) Neural Networks and Principal \nComponent Analysis: Learning from Examples without Local Min(cid:173)\nima. Neural Networks, Vol. 2, No. 1. \n(2) Chauvin, Y. (1989) Another Neural Model as a Principal Compo(cid:173)\nnent Analyzer. Submitted for publication. \n(3) Cottrell, G. W., Munro, P. W. and Zipser, D. (1988) Image Com(cid:173)\npression by Back Propagation: a Demonstration of Extensional Pro(cid:173)\ngramming. In: Advances in Cognitive Science, Vol. 2, Sharkey, N. E. \ned., Norwood, NJ Abbex. \n(4) Linsker, R. (1988) Self-Organization in a Perceptual Network. \nComputer 21 (3), 105-117. \n( 5) Willi ams, R. J. (1985) Feature Discovery Through Error-Correction \nLearning. ICS Report 8501, University of California., San Diego. \n\n\f375 \n\nMODELS OF OCULAR DOMINANCE \n\nCOLUMN FORMATION: ANALYTICAL AND \n\nCOMPUTATIONAL RESULTS \n\nKenneth D. Miller \n\nUCSF Dept. of Physiology \n\nJoseph B. Keller \nSF, CA 94143-0444 \nMathematics Dept., Stanford ken@phyb.ucsf.edu \n\nMichael P. Stryker \nPhysiology Dept., UCSF \n\nABSTRACT \n\nWe have previously developed a simple mathemati(cid:173)\n\ncal model for formation of ocular dominance columns in \nmammalian visual cortex. The model provides a com(cid:173)\nmon framework in which a variety of activity-dependent \nbiological machanisms can be studied. Analytic and com(cid:173)\nputational results together now reveal the following: \nif \ninputs specific to each eye are locally correlated in their \nfiring, and are not anticorrelated within an arbor radius, \nmonocular cells will robustly form and be organized by \nintra-cortical interactions into columns. Broader corre(cid:173)\nlations withln each eye, or anti-correlations between the \neyes, create a more purely monocular cortex; positive cor(cid:173)\nrelation over an arbor radius yields an almost perfectly \nmonocular cortex. Most features of the model can be un(cid:173)\nderstood analytically through decomposition into eigen(cid:173)\nfunctions and linear stability analysis. This allows predic(cid:173)\ntion of the widths of the columns and other features from \nmeasurable biological parameters. \n\nINTRODUCTION \n\nIn the developing visual system in many mammalian species, there is initially a uni(cid:173)\nform, overlapping innervation of layer 4 of the visual cortex by inputs representing \nthe two eyes. Subsequently, these inputs segregate into patches or stripes that are \nlargely or exclusively innervated by inputs serving a single eye, known as ocular \ndominance patches. The ocular dominance patches are on a small scale compared \nto the map of the visual world, so that the initially continuous map becomes two \ninterdigitated maps, one representing each eye. These patches, together with the \nlayers of cortex above and below layer 4, whose responses are dominated by the \neye innervating the corresponding layer 4 patch, are known as ocular dominance \ncolumns. \n\n\f376 \n\nMiller, Keller and Stryker \n\nThe discoveries of this system of ocular dominance and many of the basic features \nof its development were made by Hubel and Wiesel in a series of pioneering studies \nin the 1960s and 1970s (e.g. Wiesel and Hubel (1965), Hubel, Wiesel and LeVay \n(1977)). A recent brief review is in Miller and Stryker (1989). \n\nThe segregation of patches depends on local correlations of neural activity that are \nvery much greater within neighboring cells in each eye than between the two eyes. \nForcing the eyes to fire synchronously prevents segregation, while forcing them to \nfire more asynchronously than normally causes a more complete segregation than \nnormal. The segregation also depends on the activity of cortical cells. Normally, if \none eye is closed in a young kitten during a critical period for developmental plas(cid:173)\nticity (\"monocular deprivation\"), the more active, open eye comes to dominantly or \nexclusively drive most cortical cells, and the inputs and influence of the closed eye \nbecome largely confined to small islands of cortex. However, when cortical cells are \ninhibited from firing, the opposite is the case: the less active eye becomes dominant, \nsuggesting that it is the correlation between pre- and post-synaptic activation that \nis critical to synaptic strengthening. \n\nWe have developed and analyzed a simple mathematical model for formation of \nocular dominance patches in mammalian visual cortex, which we briefly review \nhere (Miller, Keller, and Stryker, 1986). The model provides a common framework \nin which a variety of activity-dependent biological models, including Hebb synapses \nand activity-dependent release and uptake of trophic factors, can be studied. The \nequations are similar to those developed by Linsker (1986) to study the development \nof orientation selectivity in visual cortex. We have now extended our analysis and \nalso undertaken extensive simulations to achieve a more complete understanding of \nthe model. Many results have appeared, or will appear, in more detail elsewhere \n(Miller, Keller and Stryker, 1989; Miller and Stryker, 1989; Miller, 1989). \n\nEQUATIONS \n\nConsider inputs carrying information from two eyes and co-innervating a single cor(cid:173)\ntical sheet. Let SL(x, 5, t) and SR(x, 5, t), respectively, be the left eye and right \neye synaptic weight from eye-coordinate 5 to cortical coordinate x at time t. Con(cid:173)\nsideration of simple activity-dependent models of synaptic plasticity, such as Hebb \nsynapses or activity-dependent release and uptake of trophic or modification factors, \nleads to equations for the time development of SL and SR: \n\n8t S J (x,5,t) = AA(x-5) L I(x-y)OJK(5-P)SK(y, p,t)_-ySK(x, 5,t)-e. (1) \n\nf/,{l,K \n\nJ, K are variables which each may take on the values L, R. A(x-5) is a connectivity \nfunction, giving the number of synapses from 5 to x (we assume an identity mapping \nfrom eye coordinates to cortical coordinates). OJ K (5 - P) measures the correlation \nin firing of inputs from eyes J and K when the inputs are separated by the distance \n5 - p. I(x - y) is a model-dependent spread of influence across cortex, telling how \ntwo synapses which fire synchronously, separated by the distance x-y, will influence \n\n\fModels of Ocular Dominance Column Formation \n\n377 \n\none another's growth. This influence incorporates lateral synaptic interconnections \nin the case of Hebb synapses (for linear activation, 1= (1- B)-l, where 1 is \nthe identity matrix and B is the matrix of cortico-cortical synaptic weights), and \nincorporates the effects of diffusion of trophic or modification factors in models \ninvolving such factors. .A, \"1 and \u20ac are constants. Constraints to conserve or limit \nthe total synaptic strength supported by a single cell, and nonlinearities to keep left(cid:173)\nand right-eye synaptic weights positive and less than some maximum, are added. \nSubtracting the equation for SR from that for SL yields a model equation for the \ndifference, SD(x, 0, t) = SL(x, 0, t) - SR(x, 0, t): \n\n8t SD(x, 0, t) = .AA(x - 0) L I(x - y)eD(o - p)SD(y, p, t) - \"1SD(x, 0, t). \n\n\".Il \n\n(2) \n\nHere, CD = eSameEye _ eOppEye, where eSameEye = eLL = eRR, eOppEye = \ne LR = e RL , and we have assumed statistical equality of the two eyes. \n\nSIMULATIONS \n\nThe development of equation (1) was studied in simulations using three 25 x 25 \ngrids for the two input layers, one representing each eye, and a single cortical layer. \nEach input cell connects to a 7 x 7 square arbor of cortical cells centered on the \ncorresponding grid point (A(x) = 1 on the square of \u00b13 grid points, 0 otherwise). \nInitial synaptic weights are randomly assigned from a uniform distribution between \n0.8 and 1.2. Synapses are allowed to decrease to 0 or increase to a weight of 8. \nSynaptic strength over each cortical cell is conserved by subtracting after each \niteration from each active synapse the average change in synaptic strength on that \ncortical cell. Periodic boundary conditions on the three grids are used. \n\nA typical time development of the cortical pattern of ocular dominance is shown \nin figure 1. For this simulation, correlations within each eye decrease with distance \nto zero over 4-5 grid points (circularly symmetric gaussian with parameter 2.8 \ngrid points). There are no opposite-eye correlations. The cortical interaction func(cid:173)\ntion is a \"Mexican hat\" function excitatory between nearest neighbors and weakly \ninhibitory more distantly (I(x) = exp((-1;1)2) - ~exp((;l:1)2), .A1 = 0.93.) Indi(cid:173)\nvidual cortical cell receptive fields refine in size and become monocular (innervated \nexclusively by a single eye), while individual input arbors refine in size and become \nconfined to alternating cortical stripes (not shown). \n\nDependence of these results on the correlation function is shown in figure 2. Wider \nranging correlations within each eye, or addition of opposite-eye anticorrelations, \nincrease the monocularity of cortex. Same-eye anticorrelations decrease monocu(cid:173)\nlarity, and if significant within an arbor radius (i.e. within \u00b13 grid points for the \n7 x 7 square arbors) tend to destory the monocular organization, as seen at the \nlower right. Other simulations (not shown) indicate that same-eye correlation over \nnearest neighbors is sufficient to give the periodic organization of ocular dominance, \nwhile correlation over an arbor radius gives an essentially fully monocular cortex. \n\n\f378 \n\nMiller, Keller and Stryker \n\nT=O \n\nT=10 \n\nT=20 \n\nR \n\nT=30 \n\nT=40 \n\nT=80 \n\nL \nFigure 1. Time development of cortical ocular dominance. Results shown after 0, \n10, 20,30, 40, 80 iterations. Each pixel represents ocular dominance (E a SD(x, a)) \nof a single cortical cell. 40 X 40 pixels are shown, repeating 15 columns and rows of \nthe cortical grid, to reveal the pattern across the periodic boundary conditions. \n\nSimulation of time development with varying cortical interaction and arbor func(cid:173)\ntions shows complete agreement with the analytical results presented below. The \nmodel also reproduces the experimental effects of monocular deprivation, including \nthe presence of a critical developmental period for this effect. \n\nEIGENFUNCTION ANALYSIS \n\nConsider an initial condition for which SD ~ 0, and near which equation (2) \nlinearizes some more complex, nonlinear biological reality. SD = 0 is a time(cid:173)\nindependent solution of equation (2). Is this solution stable to small perturbations, \nso that equality of the two eyes will be restored after perturbation, or is it unstable, \nso that a pattern of ocular dominance will grow? If it is unstable, which pattern \nwill initially grow the fastest? These are inherently linear questions: they depend \nonly on the behavior of the equations when SD is small, so that nonlinear terms \nare negligible. \n\nTo solve this problem, we find the characteristic, independently growing modes of \nequation (2). These are the eigenfunctions of the operator on the right side of that \nequation. Each mode grows exponentially with growth rate given by its eigenvalue. \nIf any eigenvalue is positive (they are real), the corresponding mode will grow. Then \n\nthe SD = 0 solution is unstable to perturbation. \n\n\fModels of Ocular Dominance Column Formation \n\n379 \n\nSAME-EYE \n\nCORRELATIONS \n\n+ OPP-EYE \nANTI-CORR \n\n+ SAME-EYE \nANTI-CORR \n\n2.8 \n\n1.4 \n\nFigure f. Cortical ocular dominance after fOO iterations for 6 choices of correlation \nfunctions. Top left is simulation of figure 1. Top and bottom rows correspond to \ngaussian same-eye correlations with parameter f.B and 1.4 grid points, respectively. \nMiddle column shows the effect of adding weak, broadly ranging anticorrelations \nbetween the two eyes (gaussian with parameter 9 times larger than, and amplitude \n- ~ that oj, the same-eye correlations). Right column shows the effect of instead \nadding the anticorrelation to the same-eye correlation function. \n\nANALYTICAL CHARACTERIZATION OF EIGENFUNC(cid:173)\nTIONS \n\nChange variables in equation (2) from cortex and inputs, (x, a), to cortex and re(cid:173)\nceptive field, (x, r) with r = x-a. Then equation 2 becomes a convolution in the \ncortical variable. The result (assume a continuum; results on a grid are similar) \nis that eigenfunctions are of the form S~,e(x, a, t) = eimoz RFm,e(r). RFm,e is a \ncharacteristic receptive field, representing the variation of the eigenfunction as r \nvaries while cortical location x is fixed. m is a pair of real numbers specifying a two \ndimensional wavenumber of cortical oscillation, and e is an additional index enumer(cid:173)\nating RFs for a given m. The eigenfunctions can also be written eimoa ARBm..,{r) \nwhere ARBm..,(r) = eimor RFm..,(r). ARB is a characteristic arbor, representing \nthe variation of the eigenfunction as r varies while input location a is fixed. While \nthese functions are complex, one can construct real eigenfunctions from them with \nsimilar properties (Miller and Stryker, 1989). A monocular (real) eigenfunction is \nillustrated in figure 3. \n\n\f380 \n\nMiller, Keller and Stryker \n\nCHARACTERISTIC RECEPTIVE FIELD \n\nI \n\nvvv \n\nCHARACTERISTIC ARBOR \n\nFigure 9. One of the set (identical but for rotations and reflections) of fastest(cid:173)\ngrowing eigenfunctions for the functions used in figure 1. The monocular receptive \nfields of synaptic differences SD at different cortical locations, the oscillation across \ncortez, and the corresponding arbors are illustrated. \n\nModes with RFs dominated by one eye C~::II RFm.dY) -:F 0) will oscillate in domi(cid:173)\nnance with wavelength ~ across cortex. A monocular mode is one for which RF \ndoes not change sign. The oscillation of monocular fields, between domination by \none eye and domination by the other, yields ocular dominance columns. The fastest \ngrowing mode in the linear regime will dominate the final pattern: if its receptive \nfield is monocular, its wavelength will determine the width of the final columns. \n\nOne can characterize the eigenfunctions analytically in various limiting cases. The \ngeneral conclusion is as follows. The fastest growing mode's receptive field RF \nis largely determined by the correlation function CD. If the peak of the fourier \ntransform of CD corresponds to a wavelength much larger than an arbor diameter, \nthe mode will be monocular; if it corresponds to a wavelength smaller than an arbor \ndiameter, the mode will be binocular. If CD selects a monocular mode, a broader \nCD (more sharply peaked fourier spectrum about wavenumber 0) will increase the \ndominance in growth rate of the monocular mode over other modes; in the limit \n\n\fModels of Ocular Dominance Column Formation \n\n381 \n\nin which CD is constant with distance, only the monocular modes grow and all \nother modes decay. If the mode is monocular, the peak of the fourier transform of \nthe cortical interaction function selects the wavelength of the cortical oscillation, \nand thus selects the wavelength of ocular dominance organization. In the limit in \nwhich correlations are broad with respect to an arbor, one can calculate that the \ngrowth rate of monocular modes as a function of wavenumber of oscillation m is \nproportional to E, i(m -1)6(1)..42 (1) (where X is the fourier transform of X). In \nthis limit, only l's which are close to 0 can contribute to the sum, so the peak will \noccur at or near the m which maximizes i(m). \n\nThere is an exception to the above results if constraints conserve, or limit the change \nin, the total synaptic strength over the arbor of an input cell. Then monocular \nmodes with wavelength longer than an arbor diameter are suppressed in growth \nrate, since individual inputs would have to gain or lose strength throughout their \narborization. Given a correlation function that leads to monocular cells, a purely \nexcitatory cortical interaction function would lead a single eye to take over all \nof cortex; however, if constraints conserve synaptic strength over an input arbor, \nthe wavelength will instead be about an arbor diameter, the largest wavelength \nwhose growth rate is not suppressed. Thus, ocular dominance segregation can occur \nwith a purely excitatory cortical interaction function, though this is a less robust \nphenomenon. Analytically, a constraint conserving strength over afferent arbors, \nimplemented by subtracting the average change in strength over an arbor at each \niteration from each synapse in the arbor, transforms the previous expression for the \ngrowth rates to E, i(m -1)0(1)..42(1)(1- A~~!?)). \n\nCOMPUTATION OF EIGENFUNCTIONS \n\nEigenfunctions are computed on a grid, ~nd the resulting growthrates as a function \nof wavelength are compared to the analytical expression above, in the absence of \nconstraints on afferents. The results, for the parameters used in figure (2), are \nshown in figure (4). The grey level indicates monocularity of the modes, defined as \nEr RF(r) normalized on a scale between 0 and 1 (described in Miller and Stryker \n(1989)). The analytical expression for the growth rate, 1rhose peak coincides in \nevery case with the peak of i(m), accurately predicts the growth rate of monocular \nmodes, even far from the limiting case in which the expression was derived. Broader \ncorrelations or opposite-eye anticorrelations enhance the monocularity of modes \nand the growth rate of monocular modes, while same-eye anticorrelations have the \nopposite effects. When same-eye anticorrelations are short range compared to an \narbor radius, the fastest growing modes are binocular. \n\nResults obtained for calculations in the presence of constraints on afferents are \nalso as predicted. With an excitatory cortical interaction function, the spectrum \nis radically changed by constraints, selecting a mode with a wavelength equal to \nan arbor diameter rather than one with a wavelength as wide as cortex. With the \nMexican hat cortical interaction function used in the simulations, the constraints \nsuppress the growth of long-wavelength monocular modes but do not alter the basic \n\n\f382 \n\nMiller, Keller and Stryker \n\nSAME-EYE \n\nCORRELATIONS \n\n+ OPP-EYE \nANTI-CORR \n\n+ SAME-EYE \nANTI-CO~~ \n\n35.0 \n\n19.7 \n\n27.3 . : \n\n\u00b7 \u00b7 '.-.. \u00b7 \n\n2.8 \n\n8.98 \n\n13.8 \n\n5.92 \n\n1.4 \n\nFigure 4. Growth rate (vertical axis) as a function of inverse wavelength (horizontal \naxis) for the six sets of functions used in figure 2, computed on the same grids. Grey \nlevel codes maximum monocularity of modes with the given wavelength and growth \nrate, from fully monocular ( white) to fully binocular (black). The black curve \nindicates the prediction for relative growth rates of monocular modes given in the \nl~\u00b7mit of broad correlations, as described in the text. \n\nstructure or peak of the spectrum. \n\nCONNECTIONS TO OTHER MODELS \n\nThe model of Swindale (1980) for ocular dominance segregation emerges as a lim(cid:173)\niting case of this model when correlations are constant over a bit more than an ar(cid:173)\nbor diameter. Swindale's model assumed an effective interaction between synapses \ndepending only on eye of origin and distance across cortex. Our model gives a \nbiological underpinning to this effective interaction in the limiting case, allows con(cid:173)\nsideration of more general correlation functions, and allows examination of the \ndevelopment of individual arbors and receptive fields and their relationships as well \nas of overall ocular dominance. \n\nEquation 2 is very similar to equations studied by others (Linsker, 1986, 1988; \nSanger, this volume). There are several important differences in our results. First, \nin this model synapses are constrained to remain positive. Biological synapses are \n\n\fModels of Ocular Dominance Column Fonnation \n\n383 \n\neither exclusively positive or exclusively negative, and in particular the projection of \nvisual input to visual cortex is purely excitatory. Even if one is modelling a system \nin which there are both excitatory and inhibitory inputs, these two populations will \nalmost certainly be statistically distinct in their activities and hence not treatable as \na single population whose strengths may be either positive or negative. S D, on the \nother hand, is a biological variable which starts near 0 and may be either positive \nor negative. This allows for a linear analysis whose results will remain accurate in \nthe presence of nonlinearities, which is crucial for biology. \n\nSecond, we analyze the effect of intracortical synaptic interactions. These have two \nimpacts on the modes: first, they introduce a phase variation or oscillation across \ncortex. Second, they typically enhance the growth rate of monocular modes relative \nto modes whose sign varies across the receptive field. \n\nAcknowledgements \n\nSupported by an NSF predoctoral fellowship and by grants from the McKnight \nFoundation and the System Development Foundation. Simulations were performed \nat the San Diego Supercomputer Center. \n\nReferences \n\nHubel, D.H., T.N. Wiesel and S. LeVay, 1977. Plasticity of ocular dominance \ncolumns in monkey striate cortex, Phil. Trans. R. Soc. Lond. B. 278:377-409. \nLinsker, R., 1986. From basic network principles to neural architecture, Proc. Nat!. \n\nAcad. Sci. USA 83:7508-7512, 8390-8394, 8779-8783. \n\nLinsker, R., 1988. Self-Organization in a Perceptual Network. IEEE Computer \n\n21:105-117. \n\nMiller, K.D., 1989. Correlation-based models of neural development, to appear in \n\nNeuroscience and Connectionist Theory (M.A. Gluck & D.E. Rumel(cid:173)\nhart, Eds.), Hillsdale, NJ: Lawrence Erlbaum Associates. \n\nMiller, K.D., J.B. Keller & M.P. Stryker, 1986. Models for the formation of ocular \ndominance columns solved by linear stability analysis, Soc. N eurosc. Abst. \n12:1373. \n\nMiller, K.D., J.B. Keller & M.P. Stryker, 1989. Ocular dominance column develop(cid:173)\n\nment: analysis and simulation. Submitted for publication. \n\nMiller, K.D. & M.P. Stryker, 1989. The development of ocular dominance columns: \nmechanisms and models, to appear in Connectionist Modeling and Brain \nFunction: The Developing Interface (S. J. Hanson & C. R. Olson, Eds.), \nMIT Press/Bradford. \n\nSanger, T.D., 1989. An optimality principle for unsupervised learning, this volume. \nSwindale, N. V., 1980. A model for the formation of ocular dominance stripes, Proc. \n\nR. Soc. Lond. B. 208:265-307. \n\nWiesel, T.N. & D.H. Hubel, 1965. Comparison of the effects of unilateral and \nbilateral eye closure on cortical unit responses in kittens, J. Neurophysiol. \n28:, 1029-1040. \n\n\f", "award": [], "sourceid": 124, "authors": [{"given_name": "Kenneth", "family_name": "Miller", "institution": null}, {"given_name": "Joseph", "family_name": "Keller", "institution": null}, {"given_name": "Michael", "family_name": "Stryker", "institution": null}]}