Neural Fields with Distributed Transmission Speeds and Long-Range Feedback Delays

We introduce distributed axonal transmission speeds and a long-range constant feedback loop into the standard neural field model. We analyze the stability of spatially homogeneous equilibrium solutions for general connectivity kernels. By studying reduced models based on the assumption of small delays, we determine the effects of the delays on the stability and bifurcations. We show in a reduced model that delayed excitatory feedback generally facilitates stationary bifurcations and Turing patterns, while suppressing the bifurcation of periodic solutions and traveling waves. The reverse conclusion holds for inhibitory feedback. In case of oscillatory bifurcations, the variance of the distributed propagation and feedback delays affect the frequency of periodic solutions and the phase speed of traveling waves. Moreover, we give a nonlinear analysis of traveling fronts, and find that distributed transmission speeds can maximize the front speed.