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Evans functions for integral neural field equations with Heaviside firing rate function

Coombes, Stephen; Owen, Markus R.

Authors

Markus R. Owen Markus.Owen@nottingham.ac.uk



Abstract

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

Journal Article Type Article
Journal SIAM Journal on Applied Dynamical Systems
Print ISSN 1536-0040
Electronic ISSN 1536-0040
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 3
Issue 4
Institution Citation Coombes, S., & Owen, M. R. Evans functions for integral neural field equations with Heaviside firing rate function. SIAM Journal on Applied Dynamical Systems, 3(4), doi:10.1137/040605953
DOI https://doi.org/10.1137/040605953
Keywords traveling waves, neural networks, integral equations, Evans functions
Publisher URL http://epubs.siam.org/doi/abs/10.1137/040605953
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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