A space-time finite element method for neural field equations with transmission delays

We present and analyze a new space-time finite element method for the solution of neural field equations with transmission delays. The numerical treatment of these systems is rare in the literature and currently has several restrictions on the spatial domain and the functions involved, such as connectivity and delay functions. The use of a space-time discretization, with basis functions that are discontinuous in time and continuous in space (dGcG-FEM), is a natural way to deal with space-dependent delays, which is important for many neural field applications. In this article we provide a detailed description of a space-time dGcG-FEM algorithm for neural delay equations, including an a-priori error analysis. We demonstrate the application of the dGcG-FEM algorithm on several neural field models, including problems with an inhomogeneous kernel.