PDE Methods for Two-Dimensional Neural Fields

We consider neural field models in both one and two spatial dimensions and show how for some coupling functions they can be transformed into equivalent partial differential equations (PDEs). In one dimension we find snaking families of spatially-localised solutions, very similar to those found in reversible Reversible fourth-order ordinary differential equations. In two dimensions we analyse spatially-localised bump and ring Rings solutions and show how they can be unstable with respect to perturbations which break rotational symmetry, thus leading to the formation of complex patterns. Finally, we consider spiral waves in a system with purely positive coupling and a second slow variable. These waves are solutions of a PDE in two spatial dimensions, and by numerically following these solutions as parameters are varied, we can determine regions of parameter space in which stable spiral waves exist.