Stability and Bifurcations in Neural Fields with Finite Propagation Speed and General Connectivity

A stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the connectivity kernel. In the case of nonstationary bifurcations, bounds are determined on the frequencies of the resulting oscillatory solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillations, and traveling waves. For high propagation speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible systems with combinations of Gaussian excitatory and inhibitory connections.