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Spots: breathing, drifting and scattering in a neural field model

Coombes, Stephen; Schmidt, Helmut; Avitabile, Daniele

Authors

Helmut Schmidt

Daniele Avitabile



Contributors

Stephen Coombs
Editor

Peter Beim Graben
Editor

Roland Potthast
Editor

James Wright
Editor

Abstract

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.

Citation

Coombes, S., Schmidt, H., & Avitabile, D. (2014). Spots: breathing, drifting and scattering in a neural field model. In S. Coombs, P. Beim Graben, R. Potthast, & J. Wright (Eds.), Neural fields: theory and applications (187-211). Springer. https://doi.org/10.1007/978-3-642-54593-1_7

Acceptance Date Jan 1, 2014
Publication Date May 12, 2014
Deposit Date Jul 21, 2016
Peer Reviewed Peer Reviewed
Pages 187-211
Book Title Neural fields: theory and applications
ISBN 978-3-642-54592-4
DOI https://doi.org/10.1007/978-3-642-54593-1_7
Public URL https://nottingham-repository.worktribe.com/output/730485
Publisher URL http://www.springer.com/us/book/9783642545924
Contract Date Jul 20, 2016