Numerical Bifurcation Theory for High-Dimensional Neural Models

Abstract

Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of interest satisfies a parametrised set of algebraic equations, and branches of solutions are followed as the parameter is varied. An effective way to do this is with pseudo-arclength continuation. We give an introduction to pseudo-arclength continuation and then demonstrate its use in investigating the behaviour of a number of models from the field of computational neuroscience. The models we consider are high dimensional, as they result from the discretisation of neural field models—nonlocal differential equations used to model macroscopic pattern formation in the cortex. We consider both stationary and moving patterns in one spatial dimension, and then translating patterns in two spatial dimensions. A variety of results from the literature are discussed, and a number of extensions of the technique are given.