{"title": "Competition and Arbors in Ocular Dominance", "book": "Advances in Neural Information Processing Systems", "page_first": 203, "page_last": 209, "abstract": null, "full_text": "Competition and Arbors in Ocular Dominance \n\nPeter Dayan \n\nGatsby Computational Neuroscience Unit, UCL \n17 Queen Square, London, England, WCIN 3AR. \n\nd a y a n @gat sby.u c l.a c .uk \n\nAbstract \n\nHebbian and competitive Hebbian algorithms are almost ubiquitous in \nmodeling pattern formation in cortical development. We analyse in the(cid:173)\noretical detail a particular model (adapted from Piepenbrock & Ober(cid:173)\nmayer, 1999) for the development of Id stripe-like patterns, which places \ncompetitive and interactive cortical influences, and free and restricted ini(cid:173)\ntial arborisation onto a common footing. \n\n1 Introduction \n\nCats, many species of monkeys, and humans exibit ocular dominance stripes, which are \nalternating areas of primary visual cortex devoted to input from (the thalamic relay associ(cid:173)\nated with) just one or the other eye (see Erwin et aI, 1995; Miller, 1996; Swindale, 1996 \nfor reviews of theory and data). These well-known fingerprint patterns have been a seduc(cid:173)\ntive target for models of cortical pattern formation because of the mix of competition and \ncooperation they suggest. A wealth of synaptic adaptation algorithms has been suggested \nto account for them (and also the concomitant refinement of the topography of the map \nbetween the eyes and the cortex), many of which are based on forms of Hebbian learning. \nCritical issues for the models are the degree of correlation between inputs from the eyes, \nthe nature of the initial arborisation of the axonal inputs, the degree and form of cortical \ncompetition, and the nature of synaptic saturation (preventing weights from changing sign \nor getting too large) and normalisation (allowing cortical and/or thalamic cells to support \nonly a certain total synaptic weight). Different models show different effects of these pa(cid:173)\nrameters as to whether ocular dominance should form at all, and, if it does, then what \ndetermines the widths of the stripes, which is the main experimental observable. \nAlthough particular classes of models excite fervid criticism from the experimental com(cid:173)\nmunity, it is to be hoped that the general principles of competitive and cooperative pattern \nformation that underlie them will remain relevant. To this end we seek models in which we \ncan understand the interactions amongst the various issues above. Piepenbrock & Ober(cid:173)\nmayer (1999) suggested an interesting model in which varying a single parameter spans \na spectrum from cortical competition to cooperation. However, the nature of competition \nin their model makes it hard to predict the outcome of adaptation completely, except in \nsome special cases. In this paper, we suggest a slightly different model of competition \nwhich makes the analysis tractable, and simultaneously generalise the model to consider \nan additional spectrum between flat and peaked arborisation. \n\n2 The Model \n\nFigure 1 depicts our model. It is based on the competitive model of Piepenbrock & Ober(cid:173)\nmayer (1999), who developed it in order to explore a continuum between competitive and \nlinear cortical interactions. We use a slightly different competition mechanism and also \n\n\fA \n\nveal \n\ncortex.-- competitive \ninteraction \n\nW'(a,b)A (a,b)~ \n\nW '(a,b) A (a,b) \n\nB \n\na \n\nL A R \n\nc \n\nL W R \n\nD ocularity \n\nw -\n\no \nu'(b) \n\n60000 0 \n\nleft \n\nthalamus \n\n0000 \nu'(b) \n\nright \n\nb \n\nFigure 1: Competitive ocular dominance model. A) Left (L) and right (R) input units (with activi(cid:173)\nties uL (b) and uR(b) at the same location b in input space) project through weights WL(a, b) and \nWR(a, b) and a restricted topography arbor function A(a, b) (B) to an output layer, which is subject \nto lateral competitive interactions. C) Stable weight patterns W(a , b) showing ocular dominance. D) \n(left) difference in the connections W- = W R - W L from right and left eye; (right) sum difference \nacross b showing the net ocularity for each a. Here, O\"A = 0.2, 0\"[ = 0.08, O\"u = 0.075 , f3 = 10, \nI = 0.95, n = 3. There are N = 100 units in each input layer and the output layer. Circular \n(toroidal) boundary conditions are used with bE [0, 1). \n\nextend the model with an arbor function (as in Miller et aI, 1989). The model has two \ninput layers (representing input from the thalamus from left 'L' and right 'R' eyes), each \ncontaining N units, laid out in a single spatial dimension. These connect to an output layer \n(layer IV of area VI) with N units too, which is also laid out in a single spatial dimension. \nWe use a continuum approximation, so labeling weights W L ( a, b) and W R ( a, b) . An ar(cid:173)\nbor function, A(a, b), represents the multiplicity of each such connection (an example is \ngiven in figure IB). The total strengths of the connections from b to a are the products \nWL(a,b)A(a, b) and WR(a,b)A(a, b). \nFour characteristics define the model: the arbor function, the statistics of the input; the map(cid:173)\nping from input to output; and the rule by which the weights change. The arbor function \nA(a, b) specifies the basic topography of the map at the time that the pattern of synaptic \ngrowth is being established. We consider A(a, b) ()( e-(a-b)2 /20-1 , where O\"A is a parameter \nspecifies its width (figure IB). The two ends of the spectrum for the arbor are fiat, when \nA(a, b) = 0: is constant (O\"A = 00), and rigid or punctate, when A(a, b) ()( c5(a - b) (O\"A = 0) \nand so input cells are mapped only to their topographically matched cells in the cortex. \nThe second component of the model is the input. Since the model is non-linear, pattern \nformation is a function of aspects of the input in addition to the two-point correlations \nbetween input units that drive development of standard, non-competitive, Hebbian models. \nWe follow Piepenbrock & Obermayer (1999) and consider highly spatially simplified input \nactivities at location b in the left (uL (b) and right (uR (b) projections, refiecting just a \nsingle Gaussian bump (of width oV) which is stronger to the tune of I in (a randomly \nchosen) one of the input projections than the other \n\nuL(b) = 0.5(1 + zl)e-(b-e)2/20-~ \n\nuR(b) = 0.5(1- zl)e-(b-e) 2 /20-~ \n\n(1) \n\nwhere ~ E [0,1) is the randomly chosen input location, z is -lor 1 (with probability 0.5 \neach), and determines whether the input is more from the right or left projection. 0::::: I ::::: 1 \ngoverns the weakness of correlations between the projections . \nThe third component of the model is the way that input activities and the weights conspire \nto form output activities. This happens in linear (I), competitive (c) and interactive (i) steps: \n\nI: \n\nv(a) = JdbA(a,b) (WL(a,b)uL(b) + WR(a,b)uR(b)) , \n\n(2) \n\nc : v~a) = (v(a))/3 / Jda' (v(a'))/3 \n\n(3) \nWeights, arbor and input and output activities are all positive. In equation 3c, f3 ~ 1 is a \nparameter governing the strength of competition between the cortical cells. As f3 -+ 00, the \nactivation process becomes more strongly competitive, ultimately having a winner-takes-all \neffect as in the standard self-organising map. This form of competition makes it possible \n\ni : vi(a) = Jda' I(a, a')v~a) \n\n\fto perform analyses of pattern formation that are hard for the model of Piepenbrock & \nObermayer (1999). A natural form for the cortical interactions of equation 3i is the purely \npositive Gaussian I(a, at) = e-(a-a')2/ 2o} . \nThe fourth component of the model is the weight adaptation rule which involves the \nHebbian correlation between input and output activities, averaged over input patterns ez. \nThe weights are constrained W(a, b) E [0,1], and also multiplicatively normalised so \nfdbA(a, b)(WL(a, b) + WR(a, b)) = n, for all a. \n\nWL(a, b) -+ WL(a, b) + E( (vi(a)uL(b))~z - A(a)WL(a, b)) . \n\n(4) \n\n(similarly for WR) where A(a) = A(a)(WL, WR) is chosen to enforce normalisation. \nThe initial values for the weights are WL,R = we-(a-b)2/20'~ +1]8WL,R, where w is cho(cid:173)\nsen to satisfy the normalisation constraints, 1] is small, and 8WL(a, b) and 8WR(a, b) are \nrandom perturbations constrained so that normalisation is still satisfied. Values of u~ < 00 \ncan emerge as equilibrium values of the weights if there is sufficient competition (suffi(cid:173)\nciently large (3) or a restricted arbor (ul < 00). \n\n3 Pattern Formation \n\nWe analyse pattern formation in the standard manner, finding the equilibrium points (which \nrequires solving a non-linear equation), linearising about them and finding which linear \nmode grows the fastest. By symmetry, the system separates into two modes, one involving \nthe sum of the weight perturbations 8W+ =8WR+8WL, which governs the precision of \nthe topography of the final mapping, and one involving the difference 8W+ = 8WR-;5WL, \nwhich governs ocular dominance. The development of ocular dominance requires that a \nmode of 8W- (a, b) # 0 grows, for which each output cell has weights of only one sign \n(either positive or negative). The stripe width is determined by changes in this sign across \nthe output layer. Figure 1 C;D show the sort of patterns for which we would like to account. \n\nEquilibrium solution \n\nThe equilibrium values of the weights can be found by solving \n\n(5) \nfor the A+ determined such that the normalisation constraint fdb W L (a, b) + W R ( a, b) = \nn is satisfied for all a. v(a) is a non-linear function of the weights; however, the sim(cid:173)\nple form of the inputs means that at least one set of equilibrium values of WL(a, b) and \nWR(a, b) are the same, WL(a, b) = we-(a-b)2 /20'~ for a particular width Uw that de(cid:173)\npends on I = 1/ ul, A = 1/ ul, U = 1/ ub and (3 according to a simple quadratic equation. \nWe assume that w < 1, so the weights do not reach their upper saturating limit, and this im-\nplies thatw = 2~J(A + W)/1l'. \nThe quadratic equation governing the equilibrium width can be derived by postulating \nGaussian weights, and finding the values successively of v(a), v\"(a) and vi(a) of equa(cid:173)\ntions 2 and 3, calculating ((vi(a)uL (b)) ~z and finding a consistency condition that W must \nsatisfy in orderfor W L (a, b) -+ W L (a, b) in equation 4. The result is \n\n(((3 + I)I + (3U)W2 + (A(((3 + I)I + (3U) - ((3 - I)UI)W - (3AIU = 0 \n\n(6) \nFigure 2 shows how the resulting physically realisable (W > 0) equilibrium value of Uw \ndepends on (3, UA and UI, varying each in turn about a single set of values in figure 1. \nFigure 2A shows that the width rapidly asymptotes as (3 grows, and it only gets large as the \narbor function gets large for (3 near 1. Figure 2B shows this in another way. For (3 = 1 (the \ndotted line), which quite closely parallels the non-competitive case of Miller et al (1989), \n\n\fA \n\n0.5. - - - - - - , \n\nB \n\n0.3 ':.. ITA = 2.0 \nOW~ \n0.1 \n\n-\".:~:.\"\"!0:\"':!.0~00~1-~ \n\n10\u00b0 (3 \n\n101 \n\nfJ = 1.0 .. \u00b7\u00b7\u00b7 \n\"~.~ . ., .\". \n, .. _' _.IJ\u00b7;'~_ = 1.25 \n\nfJ \n\n10 \n\nC 0 0.5 \n\n0.3 \n\n0.1 \n\nFigure 2: Log-log plots of the equilibrium values of ow in the case of multiplicative normalisation. \nSolid lines based on parameters as in figure 1 (aA = 0.2, a[ = 0.08, au = 0.075, fl = 10). A) aw \nas a function of fl for aA = 0.2 (solid), aA = 2.0 (dotted) and aA = 0.0001 (dashed). B) aw as a \nfunction of aA for fl = 10 (solid), fl = 1.25 (dashed) and fl = 1.0 (dotted). C) aw as a function of \na[. Other parameters as for the solid lines. \n\naw grows roughly like the square root of aA as the arborisation gets flatter. For any (3 > 1, \none equilibrium value of aw has a finite asymptote with UA. For absolutely flat topography \n(UA = 00) and (3 > 1, there are actually two equilibrium values for uw, one with Uw = 00, \nie flat weights; the other with Uw taking values such as the asymptotic values for the dotted \nand solid lines in figure 2B. \n\nThe sum mode \nThe update equation for (normalised) perturbations to the sum mode is 8W+ (a, b) -t \n(1 - f.A+)oW+(a, b) + f~ II daldb l O(a, b, al, bdoW+(al' bl ) - f.A'(a)W+(a, b) \n\nwhere the operator 0 = 0 1 - 0 2 is defined by averaging over ~ with z = 1, 'Y = 1 \n01 (a, b, aI, bl ) = (I da2I(a, a2)v\"(a2) 6~t:t) A(al' bl)uR(bl)uR(b)) \n02(a,b,al,bl ) = (I da2I(a,a2)v\"(a2)~t:SA(al,bt)uR(bl)uR(b)) , \n\n(7) \n\n(8) \n\n(9) \n\nwhere, for convenience, we have hidden the dependence of v(a) and v\"(a) on ~ and z. \nHere, the values of A+ and \n\nA'Ca) = (3 III dbdaldbl A(a, b)O(a, b, aI, bl )8W+(al, bl )/2f2 \n\n(10) \n\ncome from the normalisation condition. The value of A+ is determined by W+(a, b) and \nnot by 8W+(al,bl ). Except in the special case that UA = 00, the term f.A'(a)W+(a,b) \ngenerally keeps stable the equilibrium solution. \nWe consider the full eigenfunctions ofO(a, b, aI, bl ) below. However, the case that Piepen(cid:173)\nbrock & Obermayer (1999) studied of a flat arbor function (u A = 00) turns out to be spe(cid:173)\ncial, admitting two equilibrium solutions, one flat, one with topography, whose stability \ndepends on (3. For UA < 00, the only Gaussian equilibrium solution for the weights has \na refined topography (as one might expect), and this is stable. This width depends on the \nparameters in a way shown in equation 6 and figure 2, in particular, reaching a non-zero \nasymptote even as (3 gets very large. \n\nThe difference mode \nThe sum mode controls the refinement of topography, whereas the difference mode controls \nthe development and nature of ocular dominance. The equilibrium value of W- (a, b) is \nalways 0, by symmetry, and the linearised difference equation for the mode is \n\noW- (a , b) -t (l-f.A+)oW-(a, b) + fflt II daldbl O(a, b, al, bl)OW- (al' bd \n\n(11) \n\n\fn= 0 \n\n10.86 \n\n0.81 \n\nk= \n\n2 \n0 .06 \n\no \n10.86 \n\n0 .00 \n\n2 \n0 .00 \n\n0.81 \n\n0.81 \n\no \n\n2 \n\n3 \n\nFigure 3: Eigenfunctions and eigenvalues of 0 1 (left block), 0 2 (centre block), and and the theoret(cid:173)\nical and empirical approximations to 0 (right columns). Here, as in equation 12, k is the frequency \nof alternation of ocularity across the output (which is integral for a finite system); n is the order of \nthe Hermite polynomial. The numbers on top of each eigenfunction is the associated eigenvalue. \nParameters are as in figure 1 with I = 1. \n\nwhich is almost the same as equation 7 (with the same operator 0), except that the mul(cid:173)\ntiplier for the integral is (3\"(2 /2 rather than (3/2. Since \"( < 1, the eigenvalues for the \ndifference mode are therefore all less than those for the sum mode, and by the same frac(cid:173)\ntion. The multiplicative decay term EA+JW- (a, b) uses the same A+ as equation 7, whose \nvalue is determined exclusively by properties of W+ (a, b); but the non-multiplicative term \nEA'(a)W+(a, b) is absent. Note that the equilibrium values of the weights (controlled by \now) affect the operator 0, and hence its eigenfunctions and eigenvalues. \nProvided that the arbor and the initial values of the weights are not both flat (aA =j:. 00 or \naw =j:. 00), the principal eigenfunctions of 0 1 and 0 2 have the general form \n\n(12) \n\nwhere Pn(r, k) is a polynomial (related to a Hermite polynomial) of degree n in r whose \ncoefficients depend on k. Here k controls the periodicity in the projective field of each \ninput cell b to the output cells, and ultimately the periodicity of any ocular dominance \nstripes that might form. The remaining terms control the receptive fields of the output cells. \nOperator 0 2 has zero eigenvalues for the polynomials of degree n > 0. The expressions \nfor the coefficients of the polynomials and the non-zero eigenvalues of 0 1 and 0 2 are \nrather complicated. Figure 3 shows an example of this analysis. The left 4 x 3 block \nshows eigenfunctions and eigenvalues of 0 1 for k = 0 ... 5 and n = 0, 1, 2; the middle \n4 x 3 block, the equivalent eigenfunctions and eigenvalues of 0 2 . The eigenvalues come \nessentially from a Gaussian, whose standard deviation is smaller for 0 2 . To a crude first \napproximation, therefore, the eigenvalues of 0 resemble the difference of two Gaussians in \nk, and so have a peak at a non-zero value of k, ie a finite ocular dominance periodicity. \nHowever, this approximation is too crude. Although the eigenfunctions of 0 1 and 0 2 \nshown in figure 3 look almost identical, they are, in fact, subtly different, since 0 1 and 0 2 \ndo not commute (except for flat or rigid topography). The similarity between the eigenfunc(cid:173)\ntions makes it possible to approximate the eigenfunctions of 0 very closely by expanding \nthose of 0 2 in terms of 0 1 (or vice-versa). This only requires knowing the overlap between \nthe eigenfunctions, which can be calculated analytically from their form in equation 12. Ex(cid:173)\npanding for n ~ 2 leads to the approximate eigenfunctions and eigenvalues for 0 shown in \nthe penultimate column on the right of figure 3. The difference, for instance, between the \n\n\fA \n\nB \n\n':E: \n\n10-' \n\n10-3 \n\n10-2 \n(II \n\nFigure 4: A) The constraint term >'+(0./ N) (dotted line) and the ocular dominance eigenvalues \ne(k)(Q/N) (solid line 7 = 1; dotted line 7 = 0.5) of /3720/2 as a function of C>[ , where k is the \nstripe frequency associated with the maximum eigenvalue. For C>[ too large, the ocular dominance \neigenfunction no longer dominates. The star and hexagon show the maximum values of C>r such that \nocular dominance can form in each case. The scale in (A) is essentially arbitrary. B) Stripe frequency \nk associated with the largest eigenvalue as a function of C>r. The star and hexagon are the same as in \n(A), showing that the critical preferred stripe frequency is greater for higher correlations between the \ninputs (lower 7). Only integer values are considered, hence the apparent aliasing. \n\neigenfunction of 0 for k = 3 and those for 0 1 and 0 2 is striking, considering the simi(cid:173)\nlarity between the latter two. For comparison, the farthest right column shows empirically \ncalculated eigenfunctions and eigenvalues of 0 (using a 50 x 50 grid). \nPutting 8W- back in terms of ocular dominance, we require that eigenmodes of 0 resem(cid:173)\nbling the modes with n = 0 should grow more strongly than the normalisation makes them \nshrink; and then the value of k associated with the largest eigenvalue will be the stripe fre(cid:173)\nquency that should be expected to dominate. For the parameters of figure 3, the case with \nk = 3 has the largest eigenvalue, and exactly this leads to the outcome of figure IC;D. \n\n4 Results \n\nWe can now predict the outcome of development for any set of parameters. First, the \nanalysis of the behavior of the sum mode (including, if necessary, the point about multiple \nequilibria for flat initial topography) allows a prediction of the equilibrium value of c>w, \nwhich indicates the degree of topographic refinement. Second, this value of C>w can be used \nto calculate the value of the normalisation parameter ).+ that affects the growth of 8W+ \nand 8W-. There is then a barrier of 2),+ / f3'''-? that the eigenvalues of 0 must surmount \nfor a solution that is not completely binocular to develop. Third, if the peak eigenvalue of \no is indeed sufficiently large that ocular dominance develops, then the favored periodicity \nis set by the value of k associated with this eigenvalue. Of course, if many eigenfunctions \nhave similarly large eigenvalues, then slightly different stripe periodicities may be observed \ndepending on the initial conditions. \nThe solid line in figure 4A shows the largest eigenvalue of f37 2 0/2 as a function of the \nwidth of the cortical interactions C>[, for 7 = 1, the value of C>w specified through the \nequilibrium analysis, and values of the other parameters as in figure 1. The dashed line \nshows ).+, which comes from the normalisation. The largest value of C>[ for which ocular \ndominance still forms is indicated by the star. For 7 = 0.5, the eigenvalues are reduced by \na factor of 7 2 = 0.25, and so the critical value of C>[ (shown by the hexagram) is reduced. \nFigure 4B shows the frequency of the stripes associated with the largest eigenvalue. The \nsmaller C>[ , the greater the frequency of the stripes. This line is jagged because only integers \nare acceptable as stripe frequencies. \nFigure 5 shows the consequences of such relationships slightly differently. Some models \nconsider the possibility that C>[ might change during development from a large to a small \nvalue. If the frequency of the stripes is most strongly determined by the frequency that \ngrows fastest when C>[ is first sufficiently small that stripes grow, we can analyse plots such \nas those in figure 4 to determine the outcome of development. The figures in the top row \n\n\fIi = 1.5 \n\nIi = 10 \n\nIi = 100 \n\n~;:D/ ~ !~lLj \n\n...... / ~ \n'--=~~='----:! k '~ 1 \n\n1 \n\n0.5 \n\n1 \n\n1 \n\n0.5 \n\n1 \n\n0 5 \n\n> \u00b7' 1 0 \n\nO'A=2.0 / Joe G: ~' \n\nIi = 100 \n\nIi = 1.5 \n\nIi = 10 \n\nl \n1 :;~ .... , \n\n1 '~:~-:-'-.:-, 0 \n\nFigure 5: First three figures : maximal values of fr[ for which ocular dominance will develop as a \nfunction of /. All other parameters as in figure 1, except that frA = 0.2 (solid), frA = 2.0 (dashed); \nfrA = 0.0001 (dotted). Last three figures: value of stripe frequency k associated with the maximal \neigenvalue for parameters as in the left three plots at the critical value of fr[. \n\nshow the largest values of fr[ for which ocular dominance can develop; the bottom plots \nshow the stripe frequencies associated with these critical values of fr[ (like the stars and \nhexagons in figure 4), in both cases as a function of /. The columns are for successively \nlarger values of fJ; within each plot there are three lines, for frA =0.0001 (dotted); frA =0.2 \n(solid), and frA = 2.0 (dashed). Where no value of fr[ permits ocular dominance to form, \nno line is shown. From the plots, we can see that the more similar the inputs, (the smaller \n'Y) or the less the competition (the smaller fJ), the harder it is for ocular dominance to form. \nHowever, if ocular dominance does form, then the width of the stripes depends only weakJy \non the degree of competition, and slightly more strongly on the width of the arbors. The \nnarrower the arbor, the larger the frequency of the stripes. For rigid topography, as frA -t 0, \nthe critical value of fr[ depends roughly linearly on 'Y. We analyse this case in more detail \nbelow. Note that the stripe width predicted by the linear analysis does not depend on the \ncorrelation between the input projections unless other parameters (such as a[) change, \nalthough ocular dominance might not develop for some values of the parameters. \n\n5 Discussion \n\nThe analytical tractability of the model makes it possible to understand in depth the inter(cid:173)\naction between cooperation, competition, correlation and arborisation. Further exploration \nof this complex space of interactions is obviously required. Simulations across a range of \nparameters have shown that the analysis makes correct predictions, although we have only \nanalysed linear pattern formation. Non-linear stability turns out to playa highly signifi(cid:173)\ncant role in higher dimensions (such as the 2d ocular dominance stripe pattern) where a \ncontinuum of eigenmodes share the same eigenvalues (Bressloff & Cowan, personal com(cid:173)\nmunication), and also in Id models involving very strong competition (fJ -t 00) like the \nself-organising map (Kohonen, 1995). \n\nAcknowledgements \nFunded by the Gatsby Charitable Foundation. I am very grateful to Larry Abbott, Ed Erwin, \nGeoff Goodhill, John Hertz, Ken Miller, Klaus Obermayer, Read Montague, Nick Swin(cid:173)\ndale, Peter Wiesing and David Willshaw for discussions and to Zhaoping Li for making \nthis paper possible. \n\nReferences \nErwin, E, Obermayer, K & Schulten, K (1995) Neural Computation 7:425-468. \nKohonen, T (1995) Self-Organizing Maps . Berlin, New York:Springer-Verlag. \nMiller, KD (1996) In E Domany, JL van Hemmen & K Schulten, eds, Models of Neural Networks, Ill. New \nYork:Springer-Verlag, 55-78. \nMiller, KD, Keller, JB & Stryker, MP (1989) Science 245:605-615. \nPiepenbrock, C & Obermayer, K (1999). In MS Keams, SA SoBa & DA Cohn, eds, Advances in Neuralfnforma(cid:173)\nlion Processing Systems, fl. Cambridge, MA: MIT Press. \nSwindale, NV (1996) Network: Computation in Neural Systems 7: 161-247. \n\n\f", "award": [], "sourceid": 1904, "authors": [{"given_name": "Peter", "family_name": "Dayan", "institution": null}]}