Topography and Ocular Dominance 
with Positive Correlations 
Geoffrey J. Goodhill 
University of Edinburgh 
Centre for Cognitive Science 
2 Buccleuch Place 
Edinburgh EH8 9LW 
SCOTLAND 
gj gcns. ed. ac. uk 
Abstract 
A new computational model that addresses the formation of both topog- 
raphy and ocular dominance is presented. This is motivated by exper- 
imental evidence that these phenomena may be subserved by the same 
mechanisms. An important aspect of this model is that ocular domi- 
nance segregation can occur when input activity is both distributed, and 
positively correlated between the eyes. This allows investigation of the 
dependence of the pattern of ocular dominance stripes on the degree of 
correlation between the eyes: it is found that increasing correlation leads 
to narrower stripes. Experiments are suggested to test whether such be- 
haviour occurs in the natural system. 
1 INTRODUCTION 
The development of topographic and interdigitated mappings in the nervous sys- 
tem has been much studied experimentally, especially in the visual system (e.g. 
[8, 15]). Here, each eye projects in a topographic manner to more central brain 
structures: i.e. neighbouring points in the eye map to neighbouring points in the 
brain. In addition, when fibres from the two eyes invade the same target struc- 
985 
986 Goodhill 
ture, a competitive interaction often appears to take place such that eventually 
postsynaptic cells receive inputs from only one eye or the other, in a pattern of 
interdigitating "ocular dominance" stripes. 
These phenomena have received a great deal of theoretical attention: several mod- 
els have been proposed, each based on a different variant of a Hebb-type rule (e.g. 
[16, 17, 14, 13]). However, there are two aspects of the experimental data which 
previous models have not satisfactorily accounted for. 
Firstly, experimental manipulations in the frog and goldfish have shown that when 
fibres from a second eye invade a region of brain which is normally innervated 
by only one eye, ocular dominance stripes can be formed (e.g. [2]). This suggests 
that ocular dominance may be a byproduct of the expression of the rules for topo- 
graphic map formation, and does not require additional mechanisms [1]. However, 
previous models of topography have required additional implausible assumptions 
to account for ocular dominance (e.g. [16, 11], [17, 10]), while previous models of 
ocular dominance (e.g. [14]) have not simultaneously addressed the development 
of topography. 
Secondl the simulation results presented for most previous models of ocular dom- 
inance have used only localized rather than distributed patterns of input activity 
(e.g. [11]), or zero or negative correlations in activity between the two eyes (e.g. 
[3]). It is clear that, in realit between-eye correlations have a minimum value of 
zero (which might be achieved for instance in the case of strabismus), and in gen- 
eral these correlations will be positive after eye-opening. In the cat for instance, the 
majority of ocular dominance segregation occurs anatomically three to six weeks 
after birth, whereas eye opening occurs at postnatal day 7-10. 
Here I present a new model that accounts for (a) both topography and ocular 
dominance with the same mechanisms, and (b) ocular dominance segregation 
and receptive field refinement for input patterns which are both distributed, and 
positively correlated between the eyes. 
2 OUTLINE OF THE MODEL 
The model is formulated at a general enough level to be applicable to both the 
retinocortical and the retinotectal systems. It consists of two two dimensional 
sheets of input units (indexed by r) connected to one two-dimensional sheet of 
output units (indexed by c) by fibres with variable synaptic weights Wc. It is 
assumed in the retinocortical case that the topography of the retina is essentially 
unchanged by the lateral geniculate nucleus (LGN) on its way to the cortex, and 
in addition, the effects of retinal and LGN processing are taken together. Thus for 
simplicity we refer to the input layers of the model as being retinae and the output 
layer as being the cortex. An earlier version of the model appeared in [4], and a 
fuller description can be found in [5]. 
Both retina and cortex are arranged in square arrays. All weights and unit activities 
are positive. Lateral interactions exist in the cortical sheet of a circular center- 
surround excitation/inhibition form, although these are not modeled explicitly. 
Initially there is total connectivity of random strengths between retinal and cortical 
Topography and Ocular Dominance with Positive Correlations 987 
units, apart from a small bias that specifies an orientation for the map. At each time 
step, a pattern of activity is presented by setting the activities ct of retinal units. 
Each cortical unit c calculates its total input Xc according to a linear summation 
rule: 
X c  -WcrO. r 
We use assumptions similar to [9] concerning the effect of inhibitory lateral con- 
nections, to obtain the learning rule: 
= + ocas(c, 0) 
oc is a small positive constant, g is the cortical unit with the largest input xg, and 
s is the function that specifies how the activities of units c near to g decrease with 
distance from . We assume s to be a gaussian function of the Euclidean distance 
between units in the cortical sheet, with standard deviation tyc. 
Inputs to the model are random dot patterns with short range spatial correlation 
introduced by convolution with a blurring function. Locally correlated patterns of 
activity were generated by assigning the value 0 or 1 to each pixel in each eye with 
a probability of 50%, and then convolving each eye with a gaussian function of 
standard deviation try. Between-eye correlations were produced in the following 
way. Once each retina has been convolved individually with a gaussian function, 
activity ct i of each unit j in each retina is replaced with Ixct i + (l - Ix)cg i, where ct' i 
is the activity of the corresponding unit to j in the other eye, and lx specifies the 
similarity between the two eyes. Thus by varying IX it is possible to vary the degree 
of correlation between the eyes: if IX = 0 they are uncorrelated, and if IX = 0.5 they 
are perfectly correlated (i.e. the pattern of activity is identical in the two eyes). 
The correlations existing in the biological system will clearly be more complicated 
than this. However, the simple correlational structure described above aims to 
capture the key features of the biological system: on average, cells in each retina are 
correlated to an extent that decreases with distance between cells, and (after eye 
opening) corresponding positions in the two eyes are also on the average somewhat 
correlated. 
The sum of the weights for each postsynaptic unit is maintained at a constant fixed 
value. However, whereas this constraint is most usually enforced by dividing each 
weight by the sum of the weights for that postsynaptic unit ("divisive" normal- 
ization), it is enforced in this model by subtracting a constant amount from each 
weight ("subtractive" normalization), as in [13]. 
3 RESULTS 
Typical results for the case of two positively correlated eyes are shown in figure 
1. Gradually receptive fields refine over the course of development, and cortical 
units eventually lose connections from one or the other eye (figure l(a-c)). After 
a large number of input patterns have been presented, cortical units are almost 
entirely monocular, and units dominant for the left and right eyes are laid out in 
a pattern of alternating stripes (figure l(c)). In addition, maps from the two eyes 
are in register and topographic (figure l(d-f)). The map of cortical receptive fields 
988 Goodhill 
(a) (b) (c) 
(d) (e) (f) 
Figure 1: Typical results for two eyes. (a-c) show the ocular dominance of cortical 
units after 0, 50,000, and 350,000 iterations respectively. Each cortical unit is repre- 
sented by a square with colour showing the eye for which it is dominant (black for 
right eye, white for left eye), and size showing the degree to which it is dominant. 
(d-f) represent cortical topography. Here the centre of mass of weights for each 
cortical unit is averaged over both eyes, imagining the retinae to be lying atop 
one another, and neighbouring units are connected by lines to form a grid. This 
type of picture reveals where the map is folded to take into account that the cortex 
must represent both eyes. It can be seen that discontinuities in terms of folds tend 
to follow stripe boundaries: first particular positions in one eye are represented, 
and then the cortex "doubles back"as its ocularity changes in order to represent 
corresponding positions in the other eye. 
Topography and Ocular Dominance with Positive Correlations 989 
_-=l -_-'l 
Figure 2: The receptive fields of cortical units, showing topography and eye pref- 
erence. Units are coloured white if they are strongly dominant for the left eye, 
black if they are strongly dominant for the right eye, and grey if they are primarily 
binocular. "Strongly dominant"is taken to mean that at least 80% of the total 
weight available to a cortical unit is concentrated in one eye. Within each unit is a 
representation of its receptive field: there is a 16 by 16 grid within each cortical unit 
with each grid point representing a retinal unit, and the size of the box at each grid 
point encodes the strength of the connection between each retinal unit and the cor- 
tical unit. For binocular (grey) units, the larger of the two corresponding weights in 
the two eyes is drawn at each position, coloured white or black according to which 
eye that weight belongs. It can be seen that neighbouring positions in each eye 
tend to be represented by neighbouring cortical units, apart from discontinuities 
across stripe boundaries. For instance, the bottom right corner of the right retina is 
represented by the bottom right cortical unit, but the bottom left corner of the right 
retina is represented by cortical unit (3,3) (counting along and up from the bottom 
left corner of the cortex), since unit (1,1) represents the left retina. 
990 Goodhill 
(a) 
Frequency 
(b) 
(c) 
Figure 3: Effect on stripe width of the degree of correlation between the two eyes. 
Shown from left to right are the cortical topography averaged over both eyes, the 
stripe pattern, and the power spectrum of the fourier transform for each case. (a) 
1 = 0.0 (b) tr = 0.1 (c) tr = 0.2. Note that stripe width tends to decrease as 1 
increases, and also that the topography becomes smoother. 
Topography and Ocular Dominance with Positive Correlations 991 
(figure 2) confirms that, as described for the natural system [8], there is a smooth 
progression of retinal position represented across a stripe, followed by a doubling 
back at stripe boundaries for the cortex to "pick up where it left off" in the other 
eye. 
An important aspect of the model is that the effect on stripe width of the strength 
of correlation between the two eyes can be investigated, which has not been done 
in previous models. Figure 3 shows a series of results for the model, from which 
it can be seen that stronger between-eye correlations lead to narrower stripes. It 
is interesting to note that a similar relationship is seen in the elastic net model of 
topography and ocular dominance [6], even though this is formulated on a rather 
different mathematical basis to the model presented here. 
4 DISCUSSION 
It has sometimes been argued that it is not necessary to also consider the develop- 
ment of topography in models for ocular dominance, since in the cat for instance, 
topography develops first, and is established before ocular dominance segrega- 
tion occurs. However, non-simultaneity of development does not imply different 
mechanisms. As a theoretical example, in the elastic net model [6], minimisation 
by gradient descent of a particular objective function produces two clear stages of 
development: first topography formation, then ocular dominance segregation. A 
similar (though less marked) effect can be seen with the present model in figure 1: 
the rough form of the map is established before ocular dominance segregation is 
complete. 
Interest in the contribution to map formation and eye-specific segregation of pre- 
visual activity in the retina has been re-awakened recently by the finding that 
spontaneous retinal activity takes the form of waves sweeping across the retina 
in a random direction [12]. Although this finding has in turn generated a wave 
of theoretical activity, it is important to note that the theoretical principles of how 
correlated activity can guide map formation have been fairly well worked out since 
the 1970's [16, 11]. Discovery of the precise form that these correlations take does 
not invalidate earlier modelling studies. 
Finally, the results for the model presented in figure 3 raise the question of whether 
stronger between-eye correlations lead to narrower stripes in the natural system. 
Perhaps the simplest way to test this experimentally would be to look for changes in 
stripe width in the cat after artificially induced strabismus, which severely reduces 
the correlations between the two eyes. Although the effect of strabismus on the 
degree of monocularity of cortical cells has been extensively investigated (e.g. 
[7]), the effect on stripe width has not been examined. Such experiments would 
shed light on the extent to which the periodicity of ocular dominance stripes is 
determined by environmental as opposed to innate factors. 
Acknowledgements 
This work was funded by an SERC postgraduate studentship, and an MRC/JCI 
postdoctoral training fellowship. I thank Harry Barrow for advice relating to this 
992 Goodhill 
work, and David Willshaw and David Price for helpful comments on an earlier 
draft of this paper. 
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