Numerical Solution of the Neural Field Equation in the Two-dimensional Case

TL;DR

This paper introduces a high-accuracy, efficient numerical method for solving two-dimensional Neural Field Equations with space-dependent delays, addressing computational challenges and demonstrating its effectiveness through numerical examples.

Contribution

It presents a novel numerical scheme with improved accuracy and reduced complexity for two-dimensional Neural Field Equations, including delays, which enhances computational efficiency.

Findings

The proposed method achieves higher accuracy in time discretisation.

It reduces computational complexity compared to existing algorithms.

Numerical examples confirm the method's effectiveness and convergence.

Abstract

We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in Neuroscience and Robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretisation. Since computational efficiency is a key issue in this type of calculations, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples is presented, which illustrate the performance of the method.