M. Golubitsky, L-J. Shiau and A. Torok
Bifurcation on the visual cortex with weakly anisotropic lateral coupling
SIAM J. Appl. Dynam. Sys. 2 (2003) 97-143.
Ermentrout and Cowan used the Wilson-Cowan equations to model the evolution of an activity variable a(x) that represents, for example, the voltage potential a of the neuron located at point x in V1. Bressloff, Cowan, Golubitsky, Thomas, and Wiener generalize this class of models to include the orientation tuning of neurons in V1 and the Hubel and Wiesel hypercolumns. In these models a(x,phi) represents the voltage potential a of the neuron in the hypercolumn located at x and tuned to direction phi. The Bressloff et al. work assumes that lateral connections between hypercolumns are anisotropic; that is, neurons in neighboring hypercolumns are connected only if they are tuned to the same orientation and then only if the neurons are oriented in the cortex along the direction of their cells' preference. In this work, we first assume that lateral connections are isotropic: neurons in neighboring hypercolumns are connected whenever they have the same orientation tuning. Wolf and Giesel use such a model to study development of the visual cortex. Then we consider the effect of perturbing the lateral couplings to be weakly anisotropic.
There are two common features in these models: the models are continuum models (neurons and hypercolumns are idealized as points and circles) and the models all have planar Euclidean E(2) symmetry (when cortical lateral boundaries are ignored). The approach to pattern formation is also common. It is assumed that solutions are spatially periodic with respect to a fixed planar lattice. There are two common features in these models: the models are continuum models (neurons and hypercolumns are idealized as points and circles) and the models all have planar Euclidean E(2) symmetry (when cortical lateral boundaries are ignored). The approach to pattern formation is also common. It is assumed that solutions are spatially periodic with respect to a fixed planar lattice and that patterns are formed by symmetry-breaking bifurcations (corresponding to wave vectors of shortest length) from a spatially uniform state. There are also substantial differences. In the Ermentrout-Cowan model E(2) acts in its standard representation on R^2, whereas in the Bressloff et al. model E(2) acts on R^2xS^1 via the shift-twist action. In our model isotropic coupling introduces an additional S^1 symmetry. Weak anisotropy is then thought of as a small forced symmetry-breaking from E(2)dot{+}S^1 to E(2) in its shift-twist action.
The bifurcation analyses in each of these theories proceeds along similar lines, but each produce different hallucinatory images --- many of which have been reported in the psychophysics literature. The Ermentrout and Cowan model produces spirals and funnels, whereas the Bressloff et al. model produces in addition thin line images including honeycombs and cobwebs. Finally, our model produces two types of time-periodic states: rotating structures such as spirals and states that appear to rush into (or out from) a tunnel with its hole in the center of the visual field. Although it is known that branches of time-periodic states can emanate from steady-state bifurcations in systems with symmetry, this model provides the first examples of this phenomena in a specific class of models.