Simple Spin Models 
for the Development of Ocular Dominance 
Columns and Iso-Orientation Patches 
J.D. Cowan & A.E. Friedman 
Department of Mathematics, Committee on 
Neurobiology, and Brain Research Institute, 
The University of Chicago, 5734 S. Univ. Ave., 
Chicago, Illinois 60637 
Abstract 
Simple classical spin models well-known to physicists as the ANNNI 
and Heisenberg XY Models, in which long-range interactions occur in 
a pattern given by the Mexican Hat operator, can generate many of the 
structural properties characteristic of the ocular dominance columns 
and iso-orientation patches seen in cat and primate visual cortex. 
1 INTRODUCTION 
In recent years numerous models for the formation of ocular dominance columns 
(Malsburg, 1979; Swindale, 1980; Miller, Keller, & Stryker, 1989) and of iso-orientation 
patches (Malsburg 1973; Swindale 1982 & Linsker 1986)have been published. Here we 
show that simple spin models can reproduce many of the observed features. Our work is 
similar to, but independent of a recent study employing spin models (Tanaka, 1990). 
26 
Simple Spin Models 27 
1.1 OCULAR DOMINANCE COLUMNS 
We use a one-dimensional classical spin HamiltonJan on a two-dimensional lattice with 
long-range interactions. Let ci be a spin vector restricted to the orientations ? and , in 
the lattice space, and let the spin Hamiltonian be: 
HOD =- E E wij m' cj, (1) 
1 ji 
where wij is the well-known "Mexican Hat" distribution of weights: 
2 
wij = a+ exp(- li-jl2/c+)- a_ exp(- li-j12/o 5 (2) 
with c+ < c_ and a+ / a_ = c2/c+ 2. Evidently ci  cj = + I ci I Ijl = +1, so that 
s o 
HOD=-ZZ wij-ZZ wij 
i ji i ji 
s o 
where wij = wij if ci = cj, and wij = o wij if 6i  6j. 
(3) 
Figure 1. Pattern of Ocular Dominance which 
results from simulated annealing of the energy 
function HOD. Light and dark shadings correspond 
respectively to the two eyes. 
Let s denote retinal fibers from the same eye and o fibers from the opposite eye. Then 
HOD represents the "energy" of interactions between fibers from the two eyes. It is 
relatively easy to find a configuration of spins which minimizes HOD by simulated 
annealing (Kirkpatrick, Gelatt& Vecchi 1983). The result is shown in figure 1. It will 
be seen that the resulting pattern of right and left eye spins c R and c L is disordered, but 
at a constant wavelength determined in large part by the space constants c+ and c_. 
28 Cowan and Friedman 
Breaking the symmetry of the initial conditions (or letting the lattivce grow 
systematically) results in ordered patterns. 
If HOD is considered to be the energy function of a network of spins exhibiting gradient 
dynamics (Hirsch & Smale, 1974), then one can write equations for the evolution of spin 
pauerns in the form: 
Z 
t (i -- -((l HOD = j:g.i 1 j J 
= wi'joi j:i aJ 1 = wijoi ' wijoi 13' 
where ot = R or L, 3 = L or R respectively. 
proposed by Swindale in 1979. 
(4) 
Equation (4) will be recognized as that 
1. 2 ISO-ORIENTATION PATCHES 
Now let 6i represent avector in the plane of the lattice which runs continuously from 
to , without reference to eye class. It follows that 
ci  cj = Ici I lcfii cos (0i- 0j) (5) 
where 0i is the orientation of the ith spin vector. The appropriate classical spin 
Hamiltonian is: 
HIO= -,Z wijci cj = -ZZ wij Icil Icilcos(Oi-oj). 
i j,q i ji 
(6) 
Physicists will recognize HOD as a form of the Ising Lattice Hamiltonian with long-range 
alternating next nearest neighbor interactions, a type of ANNNI model (Binder, 1986) 
and HIO as a similar form of the Heisenberg XY Model for antiferromagnetic materials 
(Binder 1986). 
Again one can find a spin configuration that minimizes HIO by simulated annealing. The 
result is shown in figure 2 in which six differing orientations are depicted, corresponding 
to 30  increments (note that 0 + = is equivalent to 0). It will be seen that there are long 
stretches of continuously changing spin vector orientations, with intercalated 
discontinuities and both clockwise and counter-clockwise singular regions around which 
the orientations rotate. A one-dimensional slice shows some of these features, and is 
shown in figure 3. 
Simple Spin Models 29 
Figure 2. Pattern of orientation patches obtained by 
simulated annealing of the energy function HIO. Six 
differing orientations varying from 0  to 180  are 
represented by the different shadings. 
180 
i O0 
0 10 20 30 40 50 
Cell Number 
Figure 3. Details of a one-dimensional slice through 
the orientation map. Long stretches of smoothly 
changing orientations are evident. 
The length of ci is also correlated with these details. Figure 4 shows that Ici I is large in 
smoothly changing regions and smallest in the neighborhood of a singularity. In fact this 
model reproduces most of the details of iso-orientation patches found by Blasdel and 
Salama (1986) 
30 Cowan and Friedman 
10 
0 10 20 30 40 50 
Cell Number 
Figure 4. Variation of Ici I along the same one-dim. 
slice through the orientation map shown in figure 3. 
The amplitude drops only near singular regions. 
For example, the change in orientation per unit length, Igrad0il is shown in figure 5. It 
will be seen that the lattice is "tiled", just as in the data from visual cortex, with max 
Igrad0il located at singularities. 
Figure 5. Plot of Igrad0il corresponding to the 
orientation map of figure 2. Regions of maximum 
rate of change of 0i are shown as shaded. These 
correspond with the singular regions of figure 2. 
Simple Spin Models 31 
Once again, if HIO is taken to be the energy of a gradient dynamical system, there results 
the equation: 
d 
(7) 
3i 
which is exactly that equation introduced by Swindale in 1981 as a model for the 
structure of iso-orientation patches. There is an obvious relationship between such 
equations, and recent similar treatments (Durbin & Mitchison 1990; Schulten, K. 1990 
(Preprint); Cherjnavsky & Moody, 1990). 
2 CONCLUSIONS 
Simple classical spin models well-known to physicists as the ANNNI and Heisenberg 
XY Models, in which long-range interactions occur in a pattern given by the Mexican 
Hat operator, can generate many of the structural properties characteristic of the ocular 
dominance columns and iso-orientation patches seen in cat and primate visual cortex. 
Acknowledgements 
This work is based on lectures given at the Institute for Theoretical Physics (Santa 
Barbara) Workshop on Neural Networks and Spin Glasses, in 1986. We thank the 
Institute and The University of Chicago Brain Research Foundation for partial support of 
this work. 
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