Symmetry and Pattern Formation on the Visual Cortex

Mathematical studies of drug induced geometric visual hallucinations include three components: a model 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. In this note we survey the symmetry based ways in which geometric visual hallucinations have been modelled. Ermentrout and Cowan model the activity of neurons in the primary visual cortex. Bressloff, Cowan, Golubitsky, Thomas, and Wiener include the orientation tuning of neurons in V1 and assume that lateral connections in V1 are anisotropic. Golubitsky, Shiau, and Török assume that lateral connections are isotropic and then consider the effect of perturbing the lateral couplings to be weakly anisotropic. These models all have planar Euclidean E(2) symmetry. Solutions are assumed to be spatially periodic and patterns are formed by symmetrybreaking bifurcations from a spatially uniform state. In the ErmentroutCowan 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. Isotropic coupling introduces an additional S symmetry, and weak anisotropy is then thought of as forced symmetry-breaking from E(2)+̇S to E(2) in its shift-twist action. We outline the way symmetry appears in bifurcations in these different models.