375 
MODELS OF OCULAR DOMINANCE 
COLUMN FORMATION: ANALYTICAL AND 
COMPUTATIONAL RESULTS 
Kenneth D. Miller 
UCSF Dept. of Physiology 
Joseph B. Keller SF, CA 94143-0444 
Mathematics Dept., Stanford kenQphyb. ucsf. edu 
Michael P. Stryker 
Physiology Dept., UCSF 
ABSTRACT 
We have previously developed a simple mathemati- 
cal model for formation of ocular dominance columns in 
msmmalinn visual cortex. The model provides a com- 
mon framework in which a variety of activity-dependent 
biological machnnisms can be studied. Analytic and com- 
putational results together now reveal the following: if 
inputs specific to each eye are locally correlated in their 
firing, and are not anticorrelated within an arbor radius, 
monocular cells will robustly form and be organized by 
intra-cortical interactions into column. Broader corre- 
lations within each eye, or anti-correlations between the 
eyes, create a more purely monocular cortex; positive cor- 
relation over an arbor radius yields an almost perfectly 
monocular cortex. Most features of the model can be un- 
derstood analytically through decomposition into eigen- 
functions and linear stability analysis. This allows predic- 
tion of the widths of the coh,mns and other features from 
measurable biological parameters. 
INTRODUCTION 
In the developing visual system in many mammalian species, there is initially a uni- 
form, overlapping innervation of layer 4 of the visual cortex by inputs representing 
the two eyes. Subsequently, these inputs segregate into patches or stripes that are 
largely or exclusively innervated by inputs serving a single eye, known as ocular 
dominance patches. The ocular dominance patches are on a small scale compared 
to the map of the visual world, so that the initially continuous map becomes two 
interdigitated maps, one representing each eye. These patches, together with the 
layers of cortex above and below layer 4, whose responses are dominated by the 
eye innervating the corresponding layer 4 patch, are known as ocular dominance 
columns. 
376 Miller, Keller and Stryker 
The discoveries of this system of ocular dominance and many of the basic features 
of its development were made by Hubel and Wiesel in a series of pioneering studies 
in the 1960s and 1970s (e.g. Wiesel and Hubel (1965), Hubel, Wiesel and LeVay 
(1977)). A recent brief review is in Miller and Stryker (1989). 
The segregation of patches depends on local correlations of neural activity that are 
very much greater within neighboring cells in each eye than between the two eyes. 
Forcing the eyes to fire synchronously prevents segregation, while forcing them to 
fire more asynchronously than normally causes a more complete segregation than 
normal. The segregation also depends on the activity of cortical cells. Normally, ff 
one eye is closed in a young kitten during a critical period for developmental plas- 
ticity ("monocular deprivationS), the more active, open eye comes to dominantly or 
exclusively drive most cortical cells, and the inputs and influence of the closed eye 
become largely confined to small islands of cortex. However, when cortical cells are 
inhibited from firing, the opposite is the case: the less active eye becomes dominant, 
suggesting that it is the correlation between pre- and post-synaptic activation that 
is critical to synaptic strengthening. 
We have developed and analyzed a simple mathematical model for formation of 
ocular dominance patches in mammalian visual cortex, which we briefly review 
here (Miller, Keller, and Stryker, 1986). The model provides a common framework 
in which a variety of activity-dependent biological models, including Hebb synapses 
and activity-dependent release and uptake of trophic factors, can be studied. The 
equations are similar to those developed by Linsker (1986) to study the development 
of orientation selectivity in visual cortex. We have now extended our analysis and 
also undertaken extensive simulations to achieve a more complete understanding of 
the model Many results have appeared, or will appear, in more detail elsewhere 
(Miller, Keller and Stryker, 1989; Miller and Stryker, 1989; Miller, 1989). 
EQUATIONS 
Consider inputs carrying information from two eyes and co-innervating a single cor- 
tical sheet. Let SL(z, 5, t) and SR(z, 5, t), respectively, be the left eye and right 
eye synaptic weight from eye-coordinate 5 to cortical coordinate x at time t. Con- 
sideration of simple activity-dependent models of synaptic plasticity, such as Hebb 
synapses or activity-dependent release and uptake of trophic or modification factors, 
leads to equations for the time development of S L and SR: 
a, = (1) 
y, fi,K 
J, K are variables which each may take on the values L, R. A(z-5) is a connectivity 
function, giving the number of synapses from 5 to x (we assume an identity mapping 
from eye coordinates to cortical coordinates). C 'K (5- f) measures the correlation 
in firing of inputs from eyes J and K when the inputs are separated by the distance 
 - f. I(x - y) is a model-dependent spread of influence across cortex, telling how 
two synapses which fire synchronously, separated by the distance z- y, will influence 
Models of Ocular Dominance Column Formation 377 
one another's growth. This influence incorporates lateral synaptic interconnections 
in the case of Hebb synapses (for linear activation, I = (1 - B) -z, where 1 is 
the identity matrix and B is the matrix of cortico-cortical synaptic weights), and 
incorporates the effects of diffusion of trophic or modification factors in models 
involving such factors. ),  and e are constants. Constraints to conserve or limit 
the total synaptic strength supported by a single cell, and nonlinearities to keep left- 
and right-eye synaptic weights positive and less than some maximum, are added. 
Subtracting the equation for $R from that for $L yields a model equation for the 
difference, $D(z, 6, t) ---- $L(z, 6, t) -- $t(z, 6,): 
= z(:- as(:, 
(2) 
Here, C D = C SameEye - CPPEY% where C SameEye  C LL - C RIl, C OppEye - 
C 'R = C 'R, and we have assumed statistical equality of the two eyes. 
SIMULATIONS 
The development of equation (1) was studied in simulations using three 25 x 25 
grids for the two input layers, one representing each eye, and a single cortical layer. 
Each input cell connects to a 7 x 7 square arbor of cortical cells centered on the 
corresponding grid point (A(x) = 1 on the square of +3 grid points, 0 otherwise). 
Initial synaptic weights are randomly assigned from a uniform distribution between 
0.8 and 1.2. Synapses are allowed to decrease to 0 or increase to a weight of 8. 
Synaptic strength over each cortical cell is conserved by subtracting after each 
iteration from each active synapse the average change in synaptic strength on that 
cortical cell. Periodic boundary conditions on the three grids are used. 
A typical time development of the cortical pattern of ocular dominance is shown 
in figure 1. For this simulation, correlations within each eye decrease with distance 
to zero over 4-5 grid points (circularly symmetric gaussian with parameter 2.8 
grid points). There are no opposite-eye correlations. The cortical interaction func- 
tion is a "Mexican hat  function excitatory between nearest neighbors and weakly 
inhibitory more distantly ([(x) = exp((L) 2) - exp(I---12, )I 0.93.) Indi- 
vidual cortical cell receptive fields refine in size and become monocular (innervated 
exclusively by a single eye), while individual input arbors refine in size and become 
confined to alternating cortical stripes (not shown). 
Dependence of these results on the correlation function is shown in figure 2. Wider 
ranging correlations within each eye, or addition of opposite-eye anticorrelations, 
increase the monocularity of cortex. Same-eye anticorrelations decrease monocu- 
larity, and if significant within an arbor radius (i.e. within +3 grid points for the 
7 x 7 square arbors) tend to destory the monocular organization, as seen at the 
lower right. Other simulations (not shown) indicate that same-eye correlation over 
nearest neighbors is sufficient to give the periodic organization of ocular dominance, 
while correlation over an arbor radius gives an essentially fully monocular cortex. 
378 Miller, Keller and Stryker 
T--0 T--10 T20 
R 
T=30 T=40 T=80 
L 
Figure i. Time development of cortical ocular dominance. Results shown after O, 
10, 0, 30, 40, 80 iterations. Each pizel represents ocular dominance (a SD( 
of a single cortical cell. 40 x 40 pizels are shown, repeating I5 columns and rows of 
the cortical grid, to reveal the pattern across the periodic boundat&t conditions. 
Simulation of time development with varying cortical interaction and arbor func- 
tions shows complete agreement with the analytical results presented below. The 
model also reproduces the experimental effects of monocular deprivation, including 
the presence of a critical developmental period for this effect. 
EI(gENFUNCTION ANALYSIS 
Consider an initial condition for which S   0, and near which equation (2) 
linearises some more complex, nonlinear biological reality. S z --- 0 is a time- 
independent solution of equation (2). Is this solution stable to small perturbations, 
so that equality of the two eyes will be restored after perturbation, or is it unstable, 
so that a pattern of ocular dominance will grow? If it is unstable, which pattern 
will initially grow the fastest? These are inherently linear questions: they depend 
only on the behavior of the equations when S  is small, so that nonlinear terms 
are negligible. 
To solve this problem, we find the characteristic, independently growing modes of 
equation (2). These are the eigenfunctions of the operator on the right side of that 
equation. Each mode grows exponentially with growth rate given by its eigenvalue. 
If any eigenvalue is positive (they are real), the corresponding mode will grow. Then 
the S  -- 0 solution is unstable to perturbation. 
Models of Ocular Dominance Column Formation 379 
SAME-EYE 
CORRELATIONS 
+ OPP-EYE 
ANTI-CORR 
+ SAME-EYE 
ANTI-CORR 
2.8 
1.4 
Figure . Cortical ocular dominance after 00 iterations for 6 choices of correlation 
functions. Top left is simulation of figure 1. Top and bottom rows correspond to 
gaussian same-eye correlations with parameter .8 and 1.J grid points, respectively. 
Middle column shows the effect of adding weak, broadly ranging anticorrelations 
between the two eyes (gaussian with parameter $ times larger than, and amplitude 
-- that of, the same-eye correlations). Right column shows the effect of instead 
9 
adding the anticorrelation to the same-eye correlation function. 
ANALYTICAL CHARACTERIZATION OF EIGENFUNC- 
TIONS 
Change variables in equation (2) from cortex and inputs, (x, c), to cortex and re- 
ceptive field, (z, r) with r  z - c. Then equation 2 becomes a convolution in the 
cortical variable. The result (assume a continuum; results on a grid are similar) 
is that eigenfunctions are of the form D t) ei"'-"RF.,(r). RF., is a 
a, = 
chactertic receptive field, representing the variation of the eigenfunction as r 
varies while corticM location z  fixed. m is a pa of reM numbers specyg a two 
dimensionM wavenumber of corticM oscillation, and  is an additionM index enumer- 
atg RFs for a given m. The eigenfunctions can Mso be written eim'aARBm(r) 
where ARBor(r) = ei'RF(r). ARB   chacteristic arbor, representg 
the viation of the eigenfunction  r varies while input location a  fixed. While 
these functions e complex, one can construct reM eigenfunctions from them with 
similar properties (Miller and Stoker, 1989). A monocul (real) eigenfunction is 
illustrated in figure 3. 
380 Miller, Keller and Stryker 
CHARACTERISTIC RECEPTIVE FIELD 
! 
CHARACTERISTIC ARBOR 
Figure $. One of the set (identical but for rotations and reflections} of fastest- 
growing eigenfunetions for the functions used in figure 1. The monocular receptive 
fields of stnaptie differences 8  at different cortical locations, the oscillation across 
cortez, and the corresponding arbors are illustrated. 
Modes with RFs dominated by one eye (Yy RFm,($t)  O) will oscillate in domi- 
nance with wavelength - across cortex. A monocular mode is one for which RF 
does not change sign. The oscillation of monocular fields, between domination by 
one eye and domination by the other, yields ocular dominance columns. The fastest 
growing mode in the linear regime will dominate the final pattern: if its receptive 
field is monocular, its wavelength will determine the width of the final columns. 
One can characterize the eigenfunctions analytically in various limiting cases. The 
general conclusion is as follows. The fastest growing mode's receptive field RF 
is largely determined by the correlation function C 'D. If the peak of the fourier 
transform of C 'D corresponds to a wavelength much larger than an arbor diameter, 
the mode will be monocular; if it corresponds to a wavelength smaller than an arbor 
diameter, the mode will be binocular. If C 'D selects a monocular mode, a broader 
C 'D (more sharply peaked fourier spectrum about wavenumber 0) will increase the 
dominance in growth rate of the monocular mode over other modes; in the limit 
Models of Ocular Dominance Column Formation 381 
in which C 'D is constant with distance, only the monocular modes grow and all 
other modes decay. If the mode is monocular, the peak of the fourier transform of 
the cortical interaction function selects the wavelength of the cortical oscillation, 
and thus selects the wavelength of ocular dominance organization. In the limit in 
which correlations are broad with respect to an arbor, one can calculate that the 
growth rate of monocular modes as a function of wavenumber of oscillation m is 
proportional to , i(m-/)(l)2(l) (where ? is the fourier transform of X). In 
this limit, only l's which are close to 0 can contribute to the sum, so the peak will 
occur at or near the m which maximizes (m). 
There is an exception to the above results if constraints conserve, or limit the change 
in, the total synaptic strength over the arbor of an input cell. Then monocular 
modes with wavelength longer than an arbor diameter are suppressed in growth 
rate, since individual inputs would have to gain or lose strength throughout their 
arborization. Given a correlation function that leads to monocular cells, a purely 
excitatory cortical interaction function would lead a single eye to take over all 
of cortex; however, if constraints conserve synaptic strength over an input arbor, 
the wavelength will instead be about an arbor diameter, the largest wavelength 
whose growth rate is not suppressed. Thus, ocular dominance segregation can occur 
with a purely excitatory cortical interaction function, though this is a less robust 
phenomenon. Analytically, a constraint conserving strength over afferent arbors, 
implemented by subtracting the average change in strength over an arbor at each 
iteration from each synapse in the arbor, transforms the previous expression for the 
growth rates to 
(o) )' 
COMPUTATION OF EIGENFUNCTIONS 
Eigenfunctions are computed on a grid, and the resulting growthrates as a function 
of wavelength are compared to the analytical expression above, in the absence of 
constraints on afferents. The results, for the parameters used in figure (2), are 
shown in figure (4). The grey level indicates monocularity of the modes, defined as 
,. RF(r) normalized on a scale between 0 and i (described in Miller and Stryker 
(1989)). The analytical expression for the growth rate, 'whose peak coincides in 
every case with the peak of (rn), accurately predicts the growth rate of monocular 
modes, even far from the limiting case in which the expression was derived. Broader 
correlations or opposite-eye anticorrelations enhance the monocularity of modes 
and the growth rate of monocular modes, while same-eye anticorrelations have the 
opposite effects. When same-eye anticorrelations are short range compared to an 
arbor radius, the fastest growing modes are binocular. 
Results obtained for calculations in the presence of constraints on afferents are 
also as predicted. With an excitatory cortical interaction function, the spectrum 
is radically changed by constraints, selecting a mode with a wavelength equal to 
an arbor diameter rather than one with a wavelength as wide as cortex. With the 
Mexican hat cortical interaction function used in the simulations, the constraints 
suppress the growth of long-wavelength monocular modes but do not alter the basic 
382 Miller, Keller and Stryker 
SAME-EYE + OPP-EYE + SAME-EYE 
CORRELATIONS ANTI-CORR AN'I'I-CORR 
1.4 
azis} for the siz sets of functions used in fure , computed on the same 9rids. Grey 
level codes mazimum monocularitt of modes with the 9iven wavelength and 9rowth 
rate, from fully monocular ( white] to futt binocular (black]. The black curve 
indicates the prediction for relative 9rowth rates of monocular modes 9iven in the 
limit of broad correlations, as described in the tezt. 
structure or peak of the spectrum. 
CONNECTIONS TO OTHER MODELS 
The model of Swindale (1980) for ocular dominance segregation emerges as a lim- 
iting case of this model when correlations are constant over a bit more than an ar- 
bor diameter. Swindale's model assumed an effective interaction between synapses 
depending only on eye of origin and distance across cortex. Our model gives a 
biological underpinning to this effective interaction in the limiting case, allows con- 
sideration of more general correlation functions, and allows examination of the 
development of individual arbors and receptive fields and their relationships as well 
as of overall ocular dominance. 
Equation 2 is very similar to equations studied by others (Linsker, 1986, 1988; 
Sanger, this volume). There are several important differences in our results. First, 
in this model synapses are constrained to remain positive. Biological synapses are 
Models of Ocular Dominance Column Formation 383 
either exclusively positive or exclusively negative, and in particular the projection of 
visual input to visual cortex is purely excitatory. Even if one is modelling a system 
in which there are both excitatory and inhibitory inputs, these two populations will 
almost certainly be statistically distinct in their activities and hence not treatable as 
a single population whose strengths may be either positive or negative. S D, on the 
other hand, is a biological variable which starts near 0 and may be either positive 
or negative. This allows for a linear analysis whose results will remain accurate in 
the presence of nonlinearities, which is crucial for biology. 
Second, we analyze the effect of intracortical synaptic interactions. These have two 
impacts on the modes: first, they introduce a phase variation or oscillation across 
cortex. Second, they typically enhance the growth rate of monocular modes relative 
to modes whose sign varies across the receptive field. 
Acknowledgements 
Supported by an NSF predoctoral fellowship and by grants from the McKnight 
Foundation and the System Development Foundation. Simulations were performed 
at the San Diego Supercomputer Center. 
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