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

## TL;DR

This paper introduces a novel space-time finite element method using discontinuous Galerkin in time for neural field equations with transmission delays, enabling more flexible and accurate simulations of neural dynamics.

## Contribution

It develops and analyzes a new dGcG-FEM algorithm for neural delay equations, addressing limitations of existing methods and including an a-priori error analysis.

## Key findings

- Effective handling of space-dependent delays in neural models
- Successful application to models with inhomogeneous kernels
- Provides a detailed algorithm and error analysis

## Abstract

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.