Hopf Bifurcations of Two Population Neural Fields on the Sphere with Diffusion and Distributed Delays

A common model for studying pattern formation in large groups of neurons is the neural field. We investigate a neural field with excitatory and inhibitory neurons, like [H. R. Wilson and J. D. Cowan (1972), Biophys. J., 12, pp. 1-24], with transmission delays and gap junctions. We build on the work of [S. Visser, R. Nicks, O. Faugeras, and S. Coombes (2017), Phys. D, 349, pp. 27-45] by investigating pattern formation in these models on the sphere. Specifically, we investigate how gap junctions, modelled by a diffusion term, influence the behavior of the neural field. We look in detail at the periodic orbits that are generated by Hopf bifurcations in the presence of spherical symmetry. To this end, we derive general formulas to compute the normal form coefficients for these bifurcations up to third order and predict the stability of the resulting branches. A novel numerical method to solve delay equations with diffusion on the sphere is formulated and applied in simulations.