Collocation Methods for Solving Two-Dimensional Neural Field Models on Complex Triangulated Domains

Neural field models are commonly used to describe the average activity of large populations of cortical neurons, treating the spatial domain as being continuous. Here we present an approach for solving neural field equations that enables us to consider more physiologically realistic scenarios, including complicated domains obtained from MRI data, and more general connectivity functions that incorporate, for example, cortical geometry. To illustrate our methods, we solve a popular 2D neural field model over a square domain, which we triangulate, first uniformly and then randomly. Our approach involves solving the integral form of the partial integro-differential equation directly using collocation techniques, which we compare to the commonly used method of Fast Fourier transforms. Our goal is to apply and extend our methods to analyse more physiologically realistic neural field models, which are less restrictive in terms of the types of geometries and/or connectivity functions they can treat, when compared to Fourier based methods.