Propagation dynamics for neural field equations in time-space periodic media.

Abstract

Neural field equations model population dynamics of large-scale networks of neurons. To investigate multiple effects of spatiotemporal heterogeneity on wave propagation, we propose a neural field equation with monostable nonlinearity in time-space periodic media. We first establish the existence of a positive, globally attractive, time-space periodic solution under appropriate conditions. For exponentially bounded kernels, we determine the spreading speed and demonstrate its equivalence to the minimal speed of time-space periodic traveling wave solutions. We also provide a variational characterization of this spreading speed via principal eigenvalues. Furthermore, employing the monotone iteration method and partial metric theory, we obtain an attractive traveling wave solution at noncritical speeds. In contrast, for exponentially unbounded kernels, we find the occurrence of accelerated spreading. Leveraging properties of subexponential kernels, we precisely determine the rate of acceleration. Our results comprehensively address the problem posed by Fang and Faye (Math. Models Methods Appl. Sci., 2016) in the absence of synaptic delay.