Motivated by the functional architecture of the primary visual cortex, we analyze solutions to a
neural field equation defined on the product space R × S1, where the circle S1 represents the
orientation preferences of neurons. We show how standard solutions such as orientation bumps in
S1 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.