Competition and Arbors in Ocular Dominance 
Peter Dayan 
Gatsby Computational Neuroscience Unit, UCL 
17 Queen Square, London, England, WC1N 3AR. 
dayan@ gatsby.ucl. ac.uk 
Abstract 
Hebbian and competitive Hebbian algorithms are almost ubiquitous in 
modeling pattern formation in cortical development. We analyse in the- 
oretical detail a particular model (adapted from Piepenbrock & Ober- 
mayer, 1999) for the development of 1 d stripe-like patterns, which places 
competitive and interactive conical influences, and free and restricted ini- 
tial arborisation onto a common footing. 
1 Introduction 
Cats, many species of monkeys, and humans exibit ocular dominance stripes, which are 
alternating areas of primary visual cortex devoted to input from (the thalamic relay associ- 
ated with) just one or the other eye (see Erwin et al, 1995; Miller, 1996; Swindale, 1996 
for reviews of theory and data). These well-known fingerprint patterns have been a seduc- 
tive target for models of cortical pattern formation because of the mix of competition and 
cooperation they suggest. A wealth of synaptic adaptation algorithms has been suggested 
to account for them (and also the concomitant refinement of the topography of the map 
between the eyes and the cortex), many of which are based on forms of Hebbian learning. 
Critical issues for the models are the degree of correlation between inputs from the eyes, 
the nature of the initial arborisation of the axonal inputs, the degree and form of conical 
competition, and the nature of synaptic saturation (preventing weights from changing sign 
or getting too large) and normalisation (allowing cortical and/or thalamic cells to support 
only a certain total synaptic weight). Different models show different effects of these pa- 
rameters as to whether ocular dominance should form at all, and, if it does, then what 
determines the widths of the stripes, which is the main experimental observable. 
Although particular classes of models excite fervid criticism from the experimental com- 
munity, it is to be hoped that the general principles of competitive and cooperative pattern 
formation that underlie them will remain relevant. To this end we seek models in which we 
can understand the interactions amongst the various issues above. Piepenbrock & Ober- 
mayer (1999) suggested an interesting model in which varying a single parameter spans 
a spectrum from conical competition to cooperation. However, the nature of competition 
in their model makes it hard to predict the outcome of adaptation completely, except in 
some special cases. In this paper, we suggest a slightly different model of competition 
which makes the analysis tractable, and simultaneously generalise the model to consider 
an additional spectrum between flat and peaked arborisation. 
2 The Model 
Figure 1 depicts our model. It is based on the competitive model of Piepenbrock & Ober- 
mayer (1999), who developed it in order to explore a continuum between competitive and 
linear cortical interactions. We use a slightly different competition mechanism and also 
v(.) cortex  competitive 
 interaction 
0000000000 0000000000 
left thalamus right 
B C 
L A- R L  R 
b b b b 
D ocularity 
b L R 
Figure 1: Competitive ocular dominance model. A) Left (L) and right (R) input units (with activi- 
ties u z' (b) and u R (b) at the same location b in input space) project through weights W z' (a, b) and 
W R (a, b) and a restricted topography arbor function A(a, b) (B) to an output layer, which is subject 
to lateral competitive interactions. C) Stable weight patterns W(a, b) showing ocular dominance. D) 
(left) difference in the connections W- = W  - W z' from right and left eye; (right) sum difference 
across b showing the net ocularity for each a. Here, o-.4 = 0.2, o- = 0.08, o'u = 0.075,/ = 10, 
3' = 0.95,  = 3. There are N = 100 units in each input layer and the output layer. Circular 
(toroidal) boundary conditions are used with b E [0, 1). 
extend the model with an arbor function (as in Miller et al, 1989). The model has two 
input layers (representing input from the thalamus from left 'L' and right 'R' eyes), each 
containing N units, laid out in a single spatial dimension. These connect to an output layer 
(layer IV of area V1) with N units too, which is also laid out in a single spatial dimension. 
We use a continuum approximation, so labeling weights W z' (a, b) and WR(a, b). An ar- 
bor function, A(a, b), represents the multiplicity of each such connection (an example is 
given in figure lB). The total strengths of the connections from b to a are the products 
WZ'(a,b)A(a,b) and W](a,b)A(a,b). 
Four characteristics define the model: the arbor function, the statistics of the input; the map- 
ping from input to output; and the rule by which the weights change. The arbor function 
A(a, b) specifies the basic topography of the map at the time that the pattern of synaptic 
growth is being established. We consider A(a, b) cre -(a-t)2/2 , where or,4 is a parameter 
specifies its width (figure lB). The two ends of the spectrum for the arbor are flat, when 
A(a, b) = o is constant (O'A = OO), and rigid or punctate, when A(a,b)orS(a - b) (aA =0) 
and so input cells are mapped only to their topographically matched cells in the cortex. 
The second component of the model is the input. Since the model is non-linear, pattern 
formation is a function of aspects of the input in addition to the two-point correlations 
between input units that drive development of standard, non-competitive, Hebbian models. 
We follow Piepenbrock & Obermayer (1999) and consider highly spatially simplified input 
activities at location b in the left (uZ'(b)) and right (u](b)) projections, reflecting just a 
single Gaussian bump (of width cry) which is stronger to the tune of 7 in (a randomly 
chosen) one of the input projections than the other 
uZ'(b) = 0.5(1 + z"/)e -(b-()2/2 u](b) = 0.5(1 - z"/)e -(-)/2 (1) 
where ( 6 [0, 1) is the randomly chosen input location, z is -1 or I (with probability 0.5 
each), and determines whether the input is more from the right or left projection. 0 _< "/_< 1 
governs the weakness of correlations between the projections. 
The third component of the model is the way that input activities and the weights conspire 
to form output activities. This happens in linear (1), competitive (c) and interactive (i) steps: 
l' v(a) = fA(a,)(W(a,)() + W(a,)()) , (2) 
c' vC(a)=(v(a))Z/fda'(v(a'))  i'vi(a)= fda'I(a,a')vC(a) (3) 
Weights, arbor and input and output activities are all positive. In equation 3c,/ _> I is a 
parameter governing the strength of competition between the cortical cells. As/ --> oo, the 
activation process becomes more strongly competitive, ultimately having a winner-takes-all 
effect as in the standard self-organising map. This form of competition makes it possible 
to perform analyses of pattern formation that are hard for the model of Piepenbrock & 
Obermayer (1999). A natural form for the cortical interactions of equation 3i is the purely 
positive Gaussian I(a, a ) = e 
The fourth component of the model is the weight adaptation rule which involves the 
Hebbian correlation between input and output activities, averaged over input patterns z. 
The weights are constrained W(a,b) E [0, 1], and also multiplicatively normalised so 
fdb A(a,b)(WZ'(a,b) + WR(a, b)) = , for all a. 
WZ'(a, b) --> WZ'(a, b) + e((vi(a)uZ'(b))z - ,(a)WZ'(a, b)). (4) 
(similarly for W R) where ,(a) = ,()(W z', W ) is chosen to enforce normalisation. 
The initial values for the weights are W z', = coe -(a-b)2/2rv q-ll(W L'R, where co is cho- 
sen to satisfy the normalisation constraints, r/is small, and 6WZ'(a, b) and 6W(a, b) are 
random perturbations constrained so that normalisation is still satisfied. Values of crv < 
can emerge as equilibrium values of the weights if there is sufficient competition (suffi- 
ciently large fi) or a restricted arbor (cr < 
3 Pattern Formation 
We analyse pattern formation in the standard manner, finding the equilibrium points (which 
requires solving a non-linear equation), linearising about them and finding which linear 
mode grows the fastest. By symmetry, the system separates into two modes, one involving 
the sum of the weight perturbations 6W + = 6W R +6W z', which governs the precision of 
the topography of the final mapping, and one involving the difference 6W + = 6Wu-dW z', 
which governs ocular dominance. The development of ocular dominance requires that a 
mode of 6W- (a, b)  0 grows, for which each output cell has weights of only one sign 
(either positive or negative). The stripe width is determined by changes in this sign across 
the output layer. Figure 1C;D show the sort of patterns for which we would like to account. 
Equilibrium solution 
The equilibrium values of the weights can be found by solving 
((a),L(b)) = +WL,b) ((),(b)) = +W,b) 
(5) 
for the ,+ determined such that the normalisation constraint fdb W z' (a, b) + WU(a, b) = 
Q is satisfied for all a. v(a) is a non-linear function of the weights; however, the sim- 
ple form of the inputs means that at least one set of equilibrium values of W z' (a, b) and 
WR(a, b) are the same, W(a, b) = coe -("-)/2v for a particular width crw that de- 
pends on I = 1/cry, A = 1/cr], U = 1/cr and fi/according to a simple quadratic equation. 
We assume that co < 1, so the weights do not reach their upper saturating limit, and this im- 
plies that co = -9--yx/(A + W)/r. 
The quadratic equation governing the equilibrium width can be derived b.y postulating 
Gaussian weights, and finding the values successively of v(a), v(a) and v(a) of equa- 
tions 2 and 3, calculating ((vi(a)u z' (b))z and finding a consistency condition that W must 
satisfy in order for WZ'(a, b) --> WZ'(a, b) in equation 4. The result is 
((fi/+ 1)I + filU)W 2 + (A((fil + 1)I + filU) - (ill- 1)UI)W - filAIU = 0 (6) 
Figure 2 shows how the resulting physically realisable (W > 0) equilibrium value of crw 
depends on fi/, an and cri, varying each in turn about a single set of values in figure 1. 
Figure 2A shows that the width rapidly asymptotes as  grows, and it only gets large as the 
arbor function gets large for fi/near 1. Figure 2B shows this in another way. For fi/= 1 (the 
dotted line), which quite closely parallels the non-competitive case of Miller et al (1989), 
A B 
0.3 .0:4 = 2.0 ' = ' 
"" ,,,,.-.,.,-"-"fl = 1.2 
0.1 001  = 10 l 
10 o ,8 101 10 2 10 -20'A 10 -1 10 o 
c 
10 -3 o'er 1 0 -2 
, 0.5 
0.3 
0.1 
10 -1 
Figure 2: Log-log plots of the equilibrium values of o'w in the case of multiplicative normalisation. 
Solid lines based on parameters as in figure 1 (o-4 = 0.2, o-t = 0.08, o'v = 0.075,/ = 10). A) o'w 
as a function of/ for o-4 = 0.2 (solid), o-4 = 2.0 (dotted) and o-4 = 0.0001 (dashed). B) o'w as a 
function of o-4 for/ = 10 (solid),/ = 1.25 (dashed) and/ = 1.0 (dotted). C) o'w as a function of 
o-t. Other parameters as for the solid lines. 
crw grows roughly like the square root of cry4 as the arborisation gets flatter. For any  > 1, 
one equilibrium value of crw has a finite asymptote with cry4. For absolutely flat topography 
(cry4 = ec) and  > 1, there are actually two equilibrium values for crw, one with crw = , 
ie flat weights; the other with crw taking values such as the asymptotic values for the dotted 
and solid lines in figure 2B. 
The sum mode 
The update equation for (normalised) perturbations to the sum mode is 6W + (a, b) 
(1 - eA+)6w+(a, 6) + eq ff daldt, 10(a,t,,al,t, 1)6W+(al,t, 1) - eA't)w+(a, 6) (7) 
where the operator 0 = 0  - 0 2 is defined by averaging over  with z = 1, 7 = 1 
( 6() A(a,b)u(b)u(b)) (8) 
0(a,b,a,b) = f da2l(a, a2)vC(a2) 
O2(a,b,a,b) = (f da2I(a, a2)vC(a2)A(a,b)u(b)u(b)) , (9) 
where, convenience, we have hidden the dependence v(a) and vC(a) on 
Here, the values of + and 
() =  fff dbdadb A(a,b)O(a,b,a,b)SW+(a,b)/2Q (10) 
come from the noalisation condition. The value of + is deteined by W + (a, b) and 
not by 5W+(a, b). Except in the special case that a = , the te e()W+(a, b) 
generally keeps stable the equilibrium solution. 
We consider the full eigenfunctions of 0 (a, b, a, b) below. However, the case that Piepen- 
brock & Obermayer (1999) studied of a fiat tabor function (a = ) tums out to be spe- 
cial, admitting two equilibrium solutions, one fiat, one with topography, whose stability 
depends on . For a < , the only Gaussian equilibrium solution for the weights has 
a refined topography (as one might expect), and this is stable. This width depends on the 
parameters in a way shown in equation 6 and figure 2, in particular, reaching a non-zero 
asymptote even as  gets vew large. 
The difference mode 
The sum mode controls the refinement of topography, whereas the difference mode controls 
the development and nature of ocular dominance. The equilibrium value of W- (a, b) is 
always 0, by symmetry, and the linearised difference equation for the mode is 
/,.2 bl)W- (1, bl) 
5W-(a,b) --> (1-eA+)SW-(a,b) + -- ff dldbl O(,b,l, 
O 1 
n---- 0 1 2 
10.86 0.81 0.06 
10.03 0.75 0.06 
7.92 0.59 0.04 
5.35 0.40 0.03 
0 2 
0 1 2 
10.86 0.00 0.00 
9.81 0.00 0.00 
7.23 0.00 0.00 
4.34 0.00 0.00 
o 
0.81 
0.98 
1.29 
1.38 
0.81 
0.98 
1.29 
1.38 
Figure 3: Eigenfunctions and eigenvalues of O  (left block), O 2 (centre block), and and the theoret- 
ical and empirical approximations to O (right columns). Here, as in equation 12, k is the frequency 
of alternation of ocularity across the output (which is integral for a finite system); r, is the order of 
the Hermite polynomial. The numbers on top of each eigenfunction is the associated eigenvalue. 
Parameters are as in figure 1 with 7 = 1. 
which is almost the same as equation 7 (with the same operator 0), except that the mul- 
tiplier for the integral is /"/2/2 rather than fi/2. Since 7 < 1, the eigenvalues for the 
difference mode are therefore all less than those for the sum mode, and by the same frac- 
tion. The multiplicative decay term e+6W- (a, b) uses the same + as equation 7, whose 
value is determined exclusively by properties of W + (a, b); but the non-multiplicative term 
(a)W + (a, b) is absent. Note that the equilibrium values of the weights (controlled by 
o'w) affect the operator O, and hence its eigenfunctions and eigenvalues. 
Provided that the arbor and the initial values of the weights are not both flat (or,4  ec or 
o'w  ec), the principal eigenfunctions of 0  and 0 2have the general form 
W-(a, b) = e2-t' e-C?(t)(b-')2 +2-t(t)(b-') pn(b -- a, k) (12) 
where p (r, k) is a polynomial (related to a Hermite polynomial) of degree n in r whose 
coefficients depend on k. Here k controls the periodicity in the projective field of each 
input cell b to the output cells, and ultimately the periodicity of any ocular dominance 
stripes that might form. The remaining terms control the receptive fields of the output cells. 
Operator 0 2has zero eigenvalues for the polynomials of degree n > 0. The expressions 
for the coefficients of the polynomials and the non-zero eigenvalues of 0  and 0 2are 
rather complicated. Figure 3 shows an example of this analysis. The left 4 x 3 block 
shows eigenfunctions and eigenvalues of 0  for k -- 0... 5 and n = 0, 1, 2; the middle 
4 x 3 block, the equivalent eigenfunctions and eigenvalues of 0 2 The eigenvalues come 
essentially from a Gaussian, whose standard deviation is smaller for 0 2 To a crude first 
approximation, therefore, the eigenvalues of 0 resemble the difference of two Gaussians in 
k, and so have a peak at a non-zero value of k, ie a finite ocular dominance periodicity. 
However, this approximation is too crude. Although the eigenfunctions of 0  and 0 2 shown in figure 3 look almost identical, they are, in fact, subtly different, since 0  and 0 2 do not commute (except for flat or rigid topography). The similarity between the eigenfunc- 
tions makes it possible to approximate the eigenfunctions of 0 very closely by expanding 
those of 0 2in terms of 0  (or vice-versa). This only requires knowing the overlap between 
the eigenfunctions, which can be calculated analytically from their form in equation 12. Ex- 
panding for n _< 2 leads to the approximate eigenfunctions and eigenvalues for 0 shown in 
the penultimate column on the right of figure 3. The difference, for instance, between the 
A 
10 -3 10 -2 10 - 
B 
1 
1"0 -3 10 -2 10 - 
o'i 
Figure 4: A) The constraint term ,k+(I/N) (dotted line) and the ocular dominance eigenvalues 
e(k)(i2/N) (solid line 7 = 1; dotted line 7 = 0.5) of 720/2 as a function of o-r, where k is the 
stripe frequency associated with the maximum eigenvalue. For o-r too large, the ocular dominance 
eigenfunction no longer dominates. The star and hexagon show the maximum values of o-r such that 
ocular dominance can form in each case. The scale in (A) is essentially arbitrary. B) Stripe frequency 
k associated with the largest eigenvalue as a function of o-r. The star and hexagon are the same as in 
(A), showing that the critical preferred stripe frequency is greater for higher correlations between the 
inputs (lower 7)- Only integer values are considered, hence the apparent aliasing. 
eigenfunction of 0 for k = 3 and those for 0  and 02 is striking, considering the simi- 
larity between the latter two. For comparison, the farthest right column shows empiric ally 
calculated eigenfunctions and eigenvalues of 0 (using a 50 x 50 grid). 
Putting 6W- back in terms of ocular dominance, we require that eigenmodes of 0 resem- 
bling the modes with r = 0 should grow more strongly than the normalisation makes them 
shrink; and then the value of k associated with the largest eigenvalue will be the stripe fre- 
quency that should be expected to dominate. For the parameters of figure 3, the case with 
k = 3 has the largest eigenvalue, and exactly this leads to the outcome of figure 1C;D. 
4 Results 
We can now predict the outcome of development for any set of parameters. First, the 
analysis of the behavior of the sum mode (including, if necessary, the point about multiple 
equilibria for flat initial topography) allows a prediction of the equilibrium value of crw, 
which indicates the degree of topographic refinement. Second, this value of crw can be used 
to calculate the value of the normalisation parameter ,+ that affects the growth of 6W + 
and 6W-. There is then a barrier of 2,+/7 2 that the eigenvalues of 0 must surmount 
for a solution that is not completely binocular to develop. Third, if the peak eigenvalue of 
0 is indeed sufficiently large that ocular dominance develops, then the favored periodicity 
is set by the value of k associated with this eigenvalue. Of course, if many eigenfunctions 
have similarly large eigenvalues, then slightly different stripe periodicities may be observed 
depending on the initial conditions. 
The solid line in figure 4A shows the largest eigenvalue of 720/2 as a function of the 
width of the cortical interactions cri, for 7 = 1, the value of crw specified through the 
equilibrium analysis, and values of the other parameters as in figure 1. The dashed line 
shows ,+, which comes from the normalisation. The largest value of cri for which ocular 
dominance still forms is indicated by the star. For 7 = 0.5, the eigenvalues are reduced by 
a factor of 3 ,2 = 0.25, and so the critical value of cri (shown by the hexagram) is reduced. 
Figure 4B shows the frequency of the stripes associated with the largest eigenvalue. The 
smaller cri, the greater the frequency of the stripes. This line is jagged because only integers 
are acceptable as stripe frequencies. 
Figure 5 shows the consequences of such relationships slightly differently. Some models 
consider the possibility that cri might change during development from a large to a small 
value. If the frequency of the stripes is most strongly determined by the frequency that 
grows fastest when o'i is first sufficiently small that stripes grow, we can analyse plots such 
as those in figure 4 to determine the outcome of development. The figures in the top row 
1.5  = 10 
015 
Ol 0 
? 0.5 
/ 
0.5 1 
= lOO j = 1.5 
k 
]o 
0.5 1 o '7 05 
=10 
1 0 '7 05 1 '7 05 1 
Figure 5: First three figures: maximal values of trr for which ocular dominance will develop as a 
function of 7- All other parameters as in figure 1, except that o-n = 0.2 (solid), o-n = 2.0 (dashed); 
o-n = 0.0001 (dotted). Last three figures: value of stripe frequency k associated with the maximal 
eigenvalue for parameters as in the left three plots at the critical value of o-r. 
show the largest values of cri for which ocular dominance can develop; the bottom plots 
show the stripe frequencies associated with these critical values of cri (like the stars and 
hexagons in figure 4), in both cases as a function of 3'. The columns are for successively 
larger values of/3; within each plot there are three lines, for an = 0.0001 (dotted); an = 0.2 
(solid), and O' A ---- 2.0 (dashed). Where no value of O' I permits ocular dominance to form, 
no line is shown. From the plots, we can see that the more similar the inputs, (the smaller 
3') or the less the competition (the smaller/3), the harder it is for ocular dominance to form. 
However, if ocular dominance does form, then the width of the stripes depends only weakly 
on the degree of competition, and slightly more strongly on the width of the arbors. The 
narrower the arbor, the larger the frequency of the stripes. For rigid topography, as an --> 0, 
the critical value of cri depends roughly linearly on 3'. We analyse this case in more detail 
below. Note that the stripe width predicted by the linear analysis does not depend on the 
correlation between the input projections unless other parameters (such as cri) change, 
although ocular dominance might not develop for some values of the parameters. 
5 Discussion 
The analytical tractability of the model makes it possible to understand in depth the inter- 
action between cooperation, competition, correlation and arborisation. Further exploration 
of this complex space of interactions is obviously required. Simulations across a range of 
parameters have shown that the analysis makes correct predictions, although we have only 
analysed linear pattern formation. Non-linear stability turns out to play a highly signifi- 
cant role in higher dimensions (such as the 2d ocular dominance stripe pattern) where a 
continuum of eigenmodes share the same eigenvalues (Bressloff & Cowan, personal com- 
munication), and also in ld models involving very strong competition (/3 --> co) like the 
self-organising map (Kohonen, 1995). 
Acknowledgements 
Funded by the Gatsby Charitable Foundation. I am very grateful to Larry Abbott, Ed Erwin, 
Geoff Goodhill, John Hertz, Ken Miller, Klaus Obermayer, Read Montague, Nick Swin- 
dale, Peter Wiesing and David Willshaw for discussions and to Zhaoping Li for making 
this paper possible. 
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