Spatiotemporal Dynamics of Neural Fields on Product Spaces

Motivated by the functional architecture of the primary visual cortex, we analyze solutions to a neural field equation defined on the product space R × S, where the circle S represents the orientation preferences of neurons. We show how standard solutions such as orientation bumps in S and traveling wavefronts in R can destabilize in the presence of this product structure. Using bifurcation theory, we derive amplitude equations describing the spatiotemporal evolution of these instabilities. In the case of destabilization of an orientation bump, we find that synaptic weight kernels representing the patchiness of horizontal cortical connections yield new stable pattern forming solutions. For traveling wavefronts, we find that cross-orientation inhibition induces the formation of a stable propagating orientation bump at the location of the wavefront.