Ocular Dominance and Patterned Lateral 
Connections in a Self-Organizing Model of the 
Primary Visual Cortex 
Joseph Sirosh and Risto Miikkulainen 
Department of Computer Sciences 
University of Texas at Austin, Austin, TX 78712 
em a i I: si rosh, risto(cs. utexas.ed u 
Abstract 
A neural network model for the self-organization of ocular dominance and 
lateral connections from binocular input is presented. The self-organizing 
process results in a network where (1) afferent weights of each neuron or- 
ganize into smooth hill-shaped receptive fields primarily on one of the reti- 
nas, (2) neurons with common eye preference form connected, intertwined 
patches, and (3) lateral connections primarily link regions of the same eye 
preference. Similar self-organization of cortical structures has been ob- 
served experimentally in strabismic kittens. The model shows how pat- 
terned lateral connections in the cortex may develop based on correlated 
activity and explains why lateral connection patterns follow receptive field 
properties such as ocular dominance. 
1 Introduction 
Lateral connections in the primary visual cortex have a patterned structure that closely 
matches the response properties of cortical cells (Gilbert and Wiese11989; Malach et al. 1993). 
For example, in the normal visual cortex, long-range lateral connections link areas with sim- 
ilar orientation preference (Gilbert and Wiesel 1989). Like cortical response properties, the 
connectivity pattern is highly plastic in early development and can be altered by experience 
(Katz and Callaway 1992). In a cat that is brought up squint-eyed from birth, the lateral con- 
nections link areas with the same ocular dominance instead of orientation (L6wel and Singer 
1992). Such patterned lateral connections develop at the same time as the orientation selectiv- 
ity and ocular dominance itself (Burkhalter et al. 1993; Katz and Callaway 1992). Together, 
110 Joseph Sirosh, Risto Miikkulainen 
these observations suggest that the same experience-dependent process drives the develop- 
ment of both cortical response properties and lateral connectivity. 
Several computational models have been built to demonstrate how orientation preference, 
ocular dominance, and retinotopy can emerge from simple self-organizing processes (e.g. 
Goodhill 1993; Miller 1994; Obermayer et al. 1992; von der Malsburg 1973). These models 
assume that the neuronal response properties are primarily determined by the afferent con- 
nections, and concentrate only on the self-organization of the afferent synapses to the cor- 
tex. Lateral interactions between neurons are abstracted into simple mathematical functions 
(e.g. Gaussians) and assumed to be uniform throughout the network; lateral connectivity is not 
explicitly taken into account. Such models do not explicitly replicate the activity dynamics 
of the visual cortex, and therefore can make only limited predictions about cortical function. 
We have previously shown how Kohonen's self-organizing feature maps (Kohonen 1982) 
can be generalized to include self-organizing lateral connections and recurrent activity dy- 
namics (the Laterally Interconnected S ynergetically Self-Organizing Map (LIS SOM); Sirosh 
and Miikkulainen 1993, 1994a), and how the algorithm can model the development of ocu- 
lar dominance columns and patterned lateral connectivity with abstractions of visual input. 
LIS SOM is a low-dimensional abstraction of cortical self-organizing processes and models a 
small region of the cortex where all neurons receive the same input vector. This paper shows 
how realistic, high-dimensional receptive fields develop as part of the self-organization, and 
scales up the LISSOM approach to large areas of the cortex where different parts of the corti- 
cal network receive inputs from different parts of the receptor surface. The new model shows 
how (1) afferent receptive fields and ocular dominance columns develop from simple reti- 
nal images, (2) input correlations affect the wavelength of the ocular dominance columns and 
(3) lateral connections self-organize cooperatively and simultaneously with ocular dominance 
properties. The model suggests new computational roles for lateral connections in the cortex, 
and suggests that the visual cortex maybe maintained in a continuously adapting equilibrium 
with the visual input by coadapting lateral and afferent connections. 
2 The LISSOM Model of Receptive Fields and Ocular Dominance 
The LISSOM network is a sheet of interconnected neurons (figure 1). Through afferent con- 
nections, each neuron receives input from two "retinas". In addition, each neuron has recip- 
rocal excitatory and inhibitory lateral connections with other neurons. Lateral excitatory con- 
nections are short-range, connecting only close neighbors. Lateral inhibitory connections mn 
for long distances, and may even implement full connectivity between neurons in the network. 
Neurons receive afferent connections from broad overlapping patches on the retina called 
anatomical receptive fields, or RFs. The N x N network is projected on to each retina of 
/ x/ receptors, and each neuron is connected to receptors in a square area of side s around 
the projections. Thus, neurons receive afferents from corresponding regions of each retina. 
Depending on the location of the projection, the number of afferents to a neuron from each 
  (at the comers) to s x s (at the center). 
retina could vary from s x s 
The external and lateral weights are organized through an unsupervised learning process. At 
each training step, neurons start out with zero activity. The initial response r]ij of neuron (i, j) 
Ocular Dominance and Patterned Lateral Connections 111 
Inhlbllmy 
nn, 
$hrt-range 
Figure 1' The Receptive-Field LISSOM architecture. The afferent and lateral connectionsof a single 
neuron in the LISSOM network are shown. All connection weights are positive. 
is based on the scalar product 
ij -- r (E ab/.tij,ab-J- E cd/.tij,cd) , (1) 
 a,b c,d 
where b and c are the activations of retinal receptors (a, b) and (c, d) within the receptive 
fields of the neuron in each retina, izij,.b and izij,ca are the corresponding afferent weights, 
and r is a piecewise linear approximation of the familiar sigmoid activation function. The 
response evolves over time through lateral interaction. At each time step, the neuron com- 
bines the above afferent activation '. /z with lateral excitation and inhibition: 
riij(t) -- r (E iz + 7e E Eij,k,rik,(t --1) -- 7i E I, j,k,rIk,(t --1)) , (2) 
k,l k,l 
where Eij,kl is the excitatory lateral connection weight on the connection from neuron (k, l) 
to neuron (i, j), Iij,kt is the inhibitory connection weight, and r/l (t - 1) is the activity of 
neuron (k, l) during the previous time step. The constants % and 7i determine the relative 
strengths of excitatory and inhibitory lateral interactions. The activity pattern starts out dif- 
fuse and spread over a substantial part of the map, and converges iteratively into stable focused 
patches of activity, or activity bubbles. After the-activity has settled, typically in a few iter- 
ations of equation 2, the connection weights of each neuron are modified. Both afferent and 
lateral weights adapt according to the same mechanism: the Hebb role, normalized so that the 
sum of the weights is constant: 
wij,m(t) + (3) 
where ij stands for the activity of neuron (i, j) in the final activity bubble, wij,mn is the affer- 
ent or lateral connection weight (/z, E or I), a is the learning rate for each type of connection 
(as for afferent weights, az for excitatory, and ar for inhibitory) and X,, is the presynaptic 
activity ( for afferent, V for lateral). 
112 Joseph Sirosh, Risto Miikkulainen 
_ 2O 
(a) Random Initial Weights 
(b) Monocular RF 
(c) Binocular RF 
Figure 2: Serf-organization of the afferent input weights into receptive fields. The afferent weights 
of a neuron at position (42, 39) in a 60 x 60 network are shown before (a) and after self-organization 
(b). This particular neuron becomes monocular with strong connections to the right eye, and weak con- 
nections to the left. A neuron at position (38, 23) becomes binocular with appoximately equal weights 
to both eyes (c). 
Both excitatory and inhibitory lateral connections follow the same Hebbian learning pro- 
cess and strengthen by correlated activity. The short-range excitation keeps the activity of 
neighboring neurons correlated, and as self-organization progresses, excitation and inhibi- 
tion strengthen in the vicinity of each neuron. At longer distances, very few neurons have 
correlated activity and therefore most long-range connections become weak. Such weak con- 
nections are eliminated, and through weight normalization, inhibition concentrates in a closer 
neighborhood of each neuron. As a result, activity bubbles become more focused and local, 
weights change in smaller neighborhoods, and receptive fields become better tuned to local 
areas of each retina. 
The input to the model consists of gaussian spots of "light" on each retina: 
(z - zi) 2 + (y- yi)2) (4) 
r,y : exp(- u2 
where ,y is the activation of receptor (x, y), u 2 is a const determining the width of the 
spot, and (xi,Yi): 0 _ xi, Yi < R its center. At each input presentation, one spot is randomly 
placed at (xi,Yi) in the left retina, and a second spot within a radius of p x RN of (xi, Yi) 
in the right retina. The parameter p E [0, 1] specifies the spatial correlations between spots 
in the two retinas, and can be adjusted to simulate different degrees of correlations between 
images in the two eyes. 
3 Simulation results 
To see how correlation between the input from the two eyes affects the columnar structures 
that develop, several simulations were run with different values of p. The afferent weights of 
all neurons were initially random (as shown in figure 2a), with the total strength to both eyes 
being equal. 
Figures 2b, c show the final afferent receptive fields of two typical neurons in a simulation 
with p -- 1. In this case, the inputs were uncorrelated, simulating perfect strabismus. In 
the early stages of such simulation, some of the neurons randomly develop a preference for 
one eye or the other. Nearby neurons will tend to share the same preference because lateral 
Ocular Dominance and Patterned Lateral Connections 113 
(a) Connections of a Monocular Neuron 
(b) Connections of a Binocular Neuron 
Figure 3: Ocular dominance and lateral connection patterns. The ocular dominance of a neuron is 
measured as the difference in total afferent synaptic weight from each eye to the neuron. Each neuron 
is labeled with a grey-scale value (black  white) that represents continuously changing eye prefer- 
ence from exclusive left through binocular to exclusive right. Small white dots indicate the lateral input 
connections to the neuron marked with a big white dot. (a) The surviving lateral connections of a left 
monocular neuron predominantly link areas of the same ocular dominance. (b) The lateral connections 
of a binocular neuron come from both eye regions. 
excitation keeps neural activity partially correlated over short distances. As self-organization 
progresses, such preferences are amplified, and groups of neurons develop strong weights to 
one eye. Figure 2b shows the afferent weights of a typical monocular neuron. 
The extent of activity correlations on the network determines the size of the monocular neu- 
ronal groups. Farther on the map, where the activations are anticorrelated due to lateral in- 
hibition, neurons will develop eye preferences to the opposite eye. As a result, alternating 
ocular dominance patches develop over the map, as shown in figure 3. In areas between oc- 
ular dominance patches, neurons will develop approximately equal strengths to both eyes and 
become binocular, like the one shown in figure 2c. 
The width and number of ocular dominance columns in the network (and therefore, the wave- 
length of ocular dominance) depends on the input correlations (figure 4). When inputs in the 
two eyes become more correlated (p < 1), the activations produced by the two inputs in the 
network overlap closely and activity correlations become shorter range. By Hebbian adapta- 
tion, lateral inhibition concentrates in the neighborhood of each neuron, and the distance at 
which activations becomes anticorrelated decreases. Therefore, smaller monocular patches 
develop, and the ocular dominance wavelength decreases. Similar dependence was very re- 
cently observed in the cat primary visual cortex (L6wel 1994). The LISSOM model demon- 
strates that the adapting lateral interactions and recurrent activity dynamics regulate the wave- 
length, and suggests how these processes help the cortex develop feature detectors at a scale 
 For a thorough treatment of the mathematical principles underlying the development of ocular dom- 
inance columns, see (Goodhill 1993; Miller et al. 1989; yon der Malsburg and Singer 1988). 
114 Joseph Sirosh, Risto Miikkulainen 
(a) Strabismic case 
(b) Normal case 
Figure 4: Ocular dominance wavelength in strabismic and normal models. In the strabismic case, 
there are no between-eye correlations (p = 1), and broad ocular dominance columns are produced (a). 
With normal, partial between-eye correlations (p -- 0.45 in this example), narrower stripes are formed 
(b). As a result, there are more ocular dominance columns in the normal case and the ocular dominance 
wavelength is smaller. 
that matches the input correlations. 
As eye preferences develop, left or right eye input tends to cause activity only in the left or 
right ocular dominance patches. Activity patterns in areas of the network with the same oc- 
ular dominance tend to be highly correlated because they are caused by the same input spot. 
Therefore, the long-range lateral connections between similar eye preference areas become 
stronger, and those between opposite areas weaker. After the weak lateral connections are 
eliminated, the initially wide-ranging connections are pruned, and eventually only connect 
areas of similar ocular dominance as shown in figure 3. Binocular neurons between ocular 
dominance patches will see some correlated activity in both the neighboring areas, and main- 
tain connections to both ocular dominance columns (figure 3b). 
The lateral connection patterns shown above closely match observations in the primary vi- 
sual cortex. Ltwel and Singer (1992) observed that when between-eye correlations are abol- 
ished in kittens by surgically induced strabismus, long-range lateral connections primarily 
link areas of the same ocular dominance. However, binocular neurons, located between ocu- 
lar dominance columns, retained connections to both eye regions. The receptive field model 
confirms that such patterned lateral connections develop based on correlated neuronal activity, 
and demonstrates that they can self-organize simultaneously with ocular dominance columns. 
The model also predicts that the long-range connections have an inhibitory function. 
4 Discussion 
In LISSOM, evolving lateral interactions and dynamic activity patterns are explicitly mod- 
eled. Therefore, LISSOM has several novel properties that set it apart from other self- 
organizing models of the cortex. 
Previous models (e.g. Goodhill 1993; Miller et al. 1989; Obermayer et al. 1992; von der Mals- 
burg 1973) have concentrated only on forming ordered topographic maps where clusters of 
adjacent neurons assume similar response properties such as ocular dominance or orientation 
preference. The lateral connections in LISSOM, in addition, adapt to encode correlations be- 
Ocular Dominance and Patterned Lateral Connections 115 
tween the responses? This property can be potentially very useful in models of cortical func- 
tion. While afferent connections learn to detect the significant features in the input space (such 
as ocularity or orientation), the lateral connections can learn correlations between these fea- 
tures (such as Gestalt principles), and thereby form a basis for feature grouping. 
As an illustration, consider a single spot of light presented to the left eye. The spot causes dis- 
joint activity patterns in the left-eye-dominant patches. How can these multiple activity pat- 
terns be recognized as representing the same spatially coherent entity? As proposed by Singer 
et al. (1990), the long-range lateral connections between similar ocular dominance columns 
could synchronize cortical activity, and form a coherently firing assembly of neurons. The 
spatial coherence of the spot will then be represented by temporal coherence of neural activ- 
ity. LISSOM can be potentially extended to model such feature binding. 
Even after the network has self-organized, the lateral and afferent connections remain plastic 
and in a continuously-adapting dynamic equilibrium with the input. Therefore, the receptive 
field properties of neurons can dynamically readapt when the activity correlations in the net- 
work are forced to change. For example, when a small area of the cortex is set inactive (or 
lesioned), the sharply-tuned afferent weight profiles of the neurons surrounding that region 
expand in size, and neurons begin to respond to the stimuli that previously activated only the 
lesioned area (Sirosh and Miikkulainen 1994b, 1994c). This expansion of receptive fields is 
reversible, and when the lesion is repaired, neurons return to their original tuning. Similar 
changes occur in response to retinal lesions as well. Such dynamic expansions of receptive 
fields have been observed in the visual cortex (Pettet and Gilbert 1992). The LISSOM model 
demonstrates that such plasticity is a consequence of the same self-organizing mechanisms 
that drive the development of cortical maps. 
5 Conclusion 
The LISSOM model shows how a single local and unsupervised self-organizing process can 
be responsible for the development of both afferent and lateral connection structures in the pri- 
mary visual cortex. It suggests that this same developmental mechanism also encodes higher- 
order visual information such as feature correlations into the lateral connections. The model 
forms a framework for future computational study of cortical reorganization and plasticity, as 
well as dynamic perceptual processes such as feature grouping and binding. 
Acknowledgments 
This research was supported in part by National Science Foundation under grant #IRI- 
9309273. Computer time for the simulations was provided by the Pittsburgh S upercomputing 
Center under grants IRI930005P and TRA940029P. 
References 
Burkhalter, A., Bernardo, K. L., and Charles, V. (1993). Development of local circuits in 
human visual cortex. Journal of Neuroscience, 13:1916-1931. 
Gilbert, C. D., and Wiesel, T. N. (1989). Columnar specificity of intrinsic horizontal and 
corticocortical connections in cat visual cortex. Journal of Neuroscience, 9:2432-2442. 
2 The idea was conceived by yon der Malsburg and Singer (1988), but not modeled. 
116 Joseph Sirosh, Risto MiikkuIainen 
Goodhill, G. (1993). Topography and ocular dominance: a model exploring positive correla- 
tions. Biological Cybernetics, 69:109-118. 
Katz, L. C., and Callaway, E. M. (1992). Development of local circuits in mammalian visual 
cortex. Annual Review of Neuroscience, 15:31-56. 
Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biolog- 
ical Cybernetics, 43:59-69. 
L6wel, S. (1994). Ocular dominance column development: Strabismus changes the spacing 
of adjacent columns in cat visual cortex. Journal of Neuroscience, 14(12):7451-7468. 
L6wel, S., and Singer, W. (1992). Selection of intrinsic horizontal connections in the visual 
cortex by correlated neuronal activity. Science, 255:209-212. 
Malach, R., Amir, Y., Harel, M., and Grinvald, A. (1993). Relationship between intrinsic 
connections and functional architecture revealed by optical imaging and in vivo targeted 
biocytin injections in the primate striate cortex. Proceedings of the National Academy 
of Sciences, USA, 90:10469-10473. 
Miller, K. D. (1994). A model for the development of simple cell receptive fields and the 
ordered arrangement of orientation columns through activity-dependent competition be- 
tween on- and off-center inputs. Journal of Neuroscience, 14:409-441. 
Miller, K. D., Keller, J. B., and Stryker, M.P. (1989). Ocular dominance column development: 
Analysis and simulation. Science, 245:605-615. 
Obermayer, K., Blasdel, G. G., and Schulten, K. J. (1992). Statistical-mechanical analysis of 
self-organization and pattern formation during the development of visual maps. Physical 
Review A, 45:7568-7589. 
Pettet, M. W., and Gilbert, C. D. (1992). Dynamic changes in receptive-field size in cat pri- 
mary visual cortex. Proceedings of the National Academy of Sciences, USA, 89:8366- 
8370. 
Singer, W., Gray, C., Engel, A., K6nig, P., Artola, A., and Br6cher, S. (1990). Formation of 
cortical cell assemblies. In CoM Spring Harbor Symposia on Quantitative Biology, Vol. 
LV, 939-952. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory. 
Sirosh, J., and Miikkulainen, R. (1993). How lateral interaction develops in a self-organizing 
feature map. In Proceedings of the IEEE International Conference on Neural Networks 
(San Francisco, CA), 1360-1365. Piscataway, NJ: IEEE. 
Sirosh, J., and Miikkulainen, R. (1994a). Cooperative self-organization of afferent and lateral 
connections in cortical maps. Biological Cybernetics, 71 (1):66-78. 
Sirosh, J., and Miikkulainen, R. (1994b). Modeling cortical plasticity based on adapting lat- 
eral interaction. In The Neurobiology of Computation: Proceedings of the Annual Com- 
putationaINeuroscience Meeting. Dordrecht; Boston: Kluwer. In Press. 
Sirosh, J., and Miikkulainen, R. (1994c). A neural network model of topographic reorganiza- 
tion following cortical lesions. In Proceedings of the Worm Congress on Computational 
Medicine, Public Health and Biotechnology (Austin, TX). World Scientific. In Press. 
von der Malsburg, C. (1973). Self-organization of orientation-sensitive cells in the striate 
cortex. Kybernetik, 15:85-100. 
von der Malsburg, C., and Singer, W. (1988). Principles of cortical network organization. In 
Rakic, P., and Singer, W., editors, Neurobiology of Neocortex, 69-99. New York: Wiley. 
