Bifurcation on the Visual Cortex with Weakly Anisotropic Lateral Coupling

Mathematical studies of drug induced geometric visual hallucinations include three components: a model (or class of models) that abstracts the structure of the primary visual cortex V1; a mathematical procedure for finding geometric patterns as solutions to the cortical models; and a method for interpreting these patterns as visual hallucinations. 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, φ) represents the voltage potential a of the neuron in the hypercolumn located at x and tuned to direction φ. The work of Bressloff et al. 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 Geisel 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 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, whereas in the Bressloff et al. model, E(2) acts on R × S via the shift-twist action. In our model, isotropic coupling introduces an additional S-symmetry. Weak anisotropy is then thought of as a small forced symmetry-breaking from E(2)+̇S to E(2) in its shift-twist action. The bifurcation analyses in each of these theories proceed along similar lines, but each produces different hallucinatory images—many of which have been reported in the psychophysics literature. The Ermentrout–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 three types of time-periodic states: rotating structures such as spirals, states that appear to rush ∗Received by the editors June 18, 2002; accepted for publication (in revised form) by G. Kriegsmann December 12, 2002; published electronically May 8, 2003. This work was supported in part by NSF grant DMS-0071735, ARP grant 003652-0032-2001, and the Faculty Research Support Fund of UHCL (LJS). http://www.siam.org/journals/siads/2-2/40988.html †Department of Mathematics, University of Houston, Houston, TX 77204-3008 ([email protected], [email protected]). ‡Department of Mathematics, University of Houston Clear Lake, Houston, TX 77058 ([email protected]). 97 SIAM J. APPLIED DYNAMICAL SYSTEMS c © 2003 Society for Industrial and Applied Mathematics Vol. 2, No. 2, pp. 97–143 into (or out from) a tunnel with its hole in the center of the visual field, and pulsating images. 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.