M. Golubitsky, L-J. Shiau and A. Torok
Symmetry and pattern formation on the visual cortex
In: Dynamics and Bifurcation of Patterns in Dissipative Systems. (G. Danglmayer and J. Opera, eds.) Series on Nonlinear Science 12 World Scientific Publishing Co., Singapore, 2004, 3-19.
These models all have planar Euclidean E(2) symmetry. Solutions are assumed to be spatially periodic and patterns are formed by symmetry-breaking bifurcations from a spatially uniform state. 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^2 X S^1 via the shift-twist action. Isotropic coupling introduces an additional S^1 symmetry, and weak anisotropy is then thought of as forced symmetry-breaking from E(2)+S^1 to E(2) in its shift-twist action. We outline the way symmetry appears in bifurcations in these different models.