A natural model to study pattern formation in large groups of neurons is the neural field. We investigate a neural field model on a sphere, with excitatory and inhibitory neurons, with space-dependent delays and gap junctions. This work is an extension of [1] in the following directions: we add a diffusion term to the model to simulate gap junctions. Moreover, we consider two distinct populations of excitatory and inhibitory neurons in a Wilson-Cowan type model, instead of an Amari type model. The main focus is on the investigation of pattern formation in these systems on the sphere. Specifically, we look in detail at the periodic and quasi periodic orbits which are generated by Hopf bifurcation in the presence of spherical symmetry. We derive formulas to compute the normal form coefficients of these bifurcations and predict the stability of the resulting branches. All these results are used to study the effect of the gap junctions on the resulting patterns of the neural field.
Predictions of the emerging spatio-temporal patterns are found to be in excellent agreement with the results from our direct numerical simulations. The numerical method which we develop is an extension of the method used in [1] for solving integro-differential equations with delays on large meshes. We show the advantage of using cubic Hermite splines for interpolating the history of delay differential equation. On an almost regular triangulation of the two-sphere, derived from the icosahedron, we discretize the surface Laplacian using finite difference formulas. Finally, for time integration we employ an implicit-explicit scheme, where the linear diffusion term is evaluated implicitly and the nonlinear synaptic term is evaluated explicitly.

[1] S. Visser, R. Nicks, O. Faugeras, and S. Coombes, Standing and travelling waves in a spherical brain model: The Nunez model revisited, Physica D, 349 (2017), pp. 27-45