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Dynamic instabilities in scalar neural field equations with space-dependent delays

Venkov, N.A.; Coombes, S.; Matthews, P.C.

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Authors

N.A. Venkov

P.C. Matthews



Abstract

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These nonlocal models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O (1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but also those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities. © 2007 Elsevier Ltd. All rights reserved.

Journal Article Type Article
Acceptance Date Apr 30, 2007
Online Publication Date May 18, 2007
Publication Date Aug 1, 2007
Deposit Date May 2, 2007
Publicly Available Date Oct 9, 2007
Journal Physica D: Nonlinear Phenomena
Print ISSN 0167-2789
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 232
Issue 1
Pages 1-15
DOI https://doi.org/10.1016/j.physd.2007.04.011
Keywords neuronal networks, integral equations, space dependent delays, dynamic pattern formation, travelling waves, amplitude equations
Public URL https://nottingham-repository.worktribe.com/output/1016861
Publisher URL https://www.sciencedirect.com/science/article/abs/pii/S0167278907001285?via%3Dihub

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