Inhomogeneity-induced bifurcation of stationary and oscillatory pulses
type: Item
instance of: scholarly article
title: Inhomogeneity-induced bifurcation of stationary and oscillatory pulses
zbMATH Open document ID: 1082.35025
author: Alain Prat
author: Yue-Xian Li
author: Paul C. Bressloff
published in: Physica D
publication date: 2005-06-01T00:00:00Z
review text: This paper discusses the existence and stability of stationary and periodic pulse solutions to two different mathematical models for the electrical potential in one-dimensional neural networks. One of the models is a Fitzhugh-Nagumo reaction-diffusion system whereas in the other one the diffusion term is substituted by a convolution integral. For a special form of the kernel, the integro-differential equation can be reduced to a partial differential equation which has stationary solutions identical to those of the reaction-diffusion model. In both cases two forms of the nonlinearity are considered: a cubic one and a piecewise linear one. Moreover a spatial inhomogeneity is assumed taking the form of an additive term which has the shape of a Gaussian or Mexican-hat function. When there is no inhomogeneity no stable stationary pulse solution can exist, but the introduction of inhomogeneity makes the existence of stable stationary pulse solutions possible which are not Turing patterns. Mainly taking as parameters the size of the inhomogeneity and the time constant of the slow negative feedback component, an exhaustive construction of the bifurcation diagrams is performed both analytically and numerically in the case of the PDE model giving rise to regions of stability of the stationary pulses, to super- and sub-critical Hopf bifurcations (stable and unstable periodic pulses) and to pulse generators (an oscillatory pulse emitting traveling pulses). In the case of the integro-differential model the stability of the stationary pulse solutions is shown to be qualitatively the same as the one of the PDE system both in the case of piecewise linear nonlinearity and in the case of the cubic one.
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 2173558
zbMATH Keywords: inhomogeneous excitable media
zbMATH Keywords: one-dimensional neural networks
zbMATH Keywords: Fitzhugh-Nagumo
zbMATH Keywords: convolution integral
zbMATH Keywords: super- and sub-critical Hopf bifurcations
reviewed by: Ángel Calsina
MaRDI profile type: Publication
full work available at URL: j.physd.2005.02.005
OpenAlex ID: W2037923566
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DOI: 10.1016/J.PHYSD.2005.02.005
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description: scientific article; zbMATH DE number 2173558