A Hamilton-Jacobi approach to neural field equations

Abstract

This paper explores the long time/large space dynamics of the neural field equation with an exponentially decaying initial data. By establishing a Harnack type inequality, we derive the Hamilton-Jacobi equation corresponding to the neural field equation due to the elegant theory developed by Freidlin [Ann. Probab. (1985)], Evans and Souganidis [Indiana Univ. Math. J. (1989)]. In addition, we obtain the exact formula for the motion of the interface by constructing the explicit viscosity solutions for the underlying Hamilton-Jacobi equation. It is then shown that the propagation speed of the interface is determined by the decay rate of the initial value. As an intriguing implication, we find that the propagation speed of interface is related to the speed of traveling waves. Finally, we study the spreading speed of the corresponding Cauchy problem. To the best of our knowledge, it is the first time that the Hamilton-Jacobi approach is used in the study of dynamics of neural field equations.