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

Venkov, Nikola Atanasov; Coombes, Stephen; Matthews, Paul C


Nikola Atanasov Venkov

Paul C Matthews


In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These non-local 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 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.


Venkov, N. A., Coombes, S., & Matthews, P. C. (2007). Dynamic instabilities in scalar neural field equations with space-dependent delays

Journal Article Type Article
Publication Date May 1, 2007
Deposit Date May 2, 2007
Publicly Available Date Oct 9, 2007
Peer Reviewed Peer Reviewed
Keywords neuronal networks, integral equations, space dependent delays, dynamic pattern formation, travelling waves, amplitude equations
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