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

被引:106
作者
Coombes, S [1 ]
Owen, MR [1 ]
机构
[1] Univ Nottingham, Dept Math Sci, Nottingham NG7 2RD, England
来源
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS | 2004年 / 3卷 / 04期
基金
英国工程与自然科学研究理事会;
关键词
traveling waves; neural networks; integral equations; Evans functions;
D O I
10.1137/040605953
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the. ring 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 Zhang [Differential 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 speeds. 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.
引用
收藏
页码:574 / 600
页数:27
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