The simulation of spatio-temporal neural field equations with delay depending on the position of neural fibers using the Galerkin method based on moving least squares

One of the most significant domains in neurodynamics revolves around models founded on neural field equations (NFEs), commonly referred to as Amari’s equations. These equations intricately depict neural activity within individual neural fields and networks of such fields. The current study investigates a numerical approach to solve the time-spatio Hammerstein integro-differential equations including space-dependent delay on 2D irregular domains derived from NFEs. The presented scheme utilizes a modified discrete Galerkin technique, involving the moving least squares (MLS) approximation. This method starts by discretizing the temporal direction using the MLS method and proceeds to obtain the solution within the domain space employing the MLS scheme. The MLS method incorporates locally weighted functions into its framework and utilizes scattered data, eliminating the need for a background mesh, and is known as a meshless method. The offered method not only achieves high accuracy in approximation but also offers a simple algorithm, inheriting the advantageous features of the MLS technique such as, independence from the geometry of the domain. Furthermore, the error bound and the convergence rate of the proposed scheme have been estimated. The effectiveness of the approach is further verified through various numerical simulations, illustrating its performance and consistency with the theoretical expectations derived.