Numerical continuation of travelling waves and pulses in neural fields

We study travelling waves and pulses in neural fields. Neural fields are a macroscopic description of the activity of brain tissue, which mathematically are formulated as integro-differential equations. While linear and weakly nonlinear analysis can describe instabilities and small amplitude patterns, numerical techniques are needed to study the nonlinear properties of travelling waves and pulses. Here we report on progress of such an approach using an excitatory neural field with linear adaptation. First, we find and analyse an anti-pulse, where the whole field is active except for a moving region with lowered activity. Such an anti-pulse may be relevant in modelling spreading depression. Second, we consider dynamics for relatively shallow, smooth activation functions where the neural field behaves as an excitable medium. We compute numerically the dispersion curves for travelling waves, i.e. the wavespeed as a function of the spatial period. This allows a kinematic analysis, see [3], that may be useful for analysing spreading epileptiform activity. Third, we present a numerical continuation method for periodic orbits of integro-differential equations based on fast Fourier transforms. Hence, neural fields with biophysically realistic mechanisms may be analysed beyond linearisation.