Research

Travelling waves in models of neural tissue: from localised structures to periodic waves

⁵ Department of Applied Mathematics, MIRA Institute of Biomedical Technology and Technical Medicine, University of Twente, Postbus 217, 7500 AE, Enschede, The Netherlands
⁶ Center for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, NG7 2RD, Nottingham, UK

^* e-mail: h.g.e.meijer@utwente.nl

Received: 21 November 2013
Accepted: 18 February 2014
Published online: 6 March 2014

Abstract

Abstract

We consider travelling waves (fronts, pulses and periodics) in spatially extended one dimensional neural field models. We demonstrate for an excitatory field with linear adaptation that, in addition to an expected stable pulse solution, a stable anti-pulse can exist. Varying the adaptation strength we unravel the organizing centers of the bifurcation diagram for fronts and pulses, with a mixture of exact analysis for a Heaviside firing rate function and novel numerical schemes otherwise. These schemes, for non-local models with space-dependent delays, further allow for the construction and continuation of periodic waves. We use them to construct the dispersion curve – wave speed as a function of period – and find that they can be oscillatory and multi-valued, suggesting bistability of periodic waves. A kinematic theory predicts the onset of wave instabilities at stationary points in the dispersion curve, leading to period doubling behaviour, and is confirmed with direct numerical simulations. We end with a discussion of how the construction of dispersion curves may allow a useful classification scheme of neural field models for epileptic waves.

PACS codes

Primary 87.19.lj; 87.19.le; 87.19.lq; 87.19.lf