7. Spots: Breathing, Drifting and Scattering in a Neural Field Model

Two dimensional neural field models with short range excitation and long range inhibition can exhibit localised solutions in the form of spots. Moreover, with the inclusion of a spike frequency adaptation current, these models can also support breathers and travelling spots. In this chapter we show how to analyse the properties of spots in a neural field model with linear spike frequency adaptation. For a Heaviside firing rate function we use an interface description to derive a set of four nonlinear ordinary differential equations to describe the width of a spot, and show how a stationary solution can undergo a Hopf instability leading to a branch of periodic solutions (breathers). For smooth firing rate functions we develop numerical codes for the evolution of the full space-time model and perform a numerical bifurcation analysis of radially symmetric solutions. An amplitude equation for analysing breathing behaviour in the vicinity of the bifurcation point is determined. The condition for a drift instability is also derived and a center manifold reduction is used to describe a slowly moving spot in the vicinity of this bifurcation. This analysis is extended to cover the case of two slowly moving spots, and establishes that these will reflect from each other in a head-on collision.