zbMATH — the first resource for mathematics

Propagating waves in discrete bistable reaction-diffusion systems. (English) Zbl 0787.92010
Summary: We consider a discrete bistable reaction-diffusion system modeled by \(N\) coupled Nagumo equations. We develop an asymptotic method to describe the phenomenon of propagation failure. The Nagumo model depends on two parameters: the coupling constant \(d\) and the bistability parameter \(a\). We investigate the limit \(a \to 0\) and \(d(a) \to 0\) and construct traveling front solutions. We obtain the critical coupling constant \(d=d^*(a)\) above which propagation is possible and determine the propagation speed \(c=c(d)\) if \(d>d^*\).
We investigate two different cases for the initiation of a propagating front solution. Case 1 considers a uniform steady state distribution. A propagating front appears as the result of a fixed boundary condition. Case 2 also considers a uniform steady state distribution but a propagating front appears as the result of a localized perturbation.

92C20 Neural biology
34B99 Boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX Cite
Full Text: DOI
[1] Keener, J.P., A mathematical model for the initiation of ventricular tachycardia in myocardium, (), 589-608
[2] Keener, J.P., The effects of discrete gap junctions coupling on propagation in myocardium, J. theor. biol., 148, 49-82, (1991)
[3] Bell, J., Excitability behavior of myelineated axon models, (), 95-116
[4] Rinzel, J., Mechanisms for nonuniform propagation along excitable cables, Ann. NY acad. sci., 591, 51-61, (1990)
[5] Laplante, J.P.; Erneux, T., Propagation failure in arrays of coupled bistable chemical reactors, J. phys. chem., 96, 4931-4934, (1992)
[6] Keener, J.P., Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. appl. math., 47, 556-572, (1987) · Zbl 0649.34019
[7] Pauwelussen, J.P., One way of traffic of pulses in a neuron, J. math. biol., 15, 151-171, (1982) · Zbl 0497.92007
[8] Murray, J.D., Mathematical biology, () · Zbl 1284.92009
[9] Defontaines, A.-D.; Pomeau, Y.; Rostand, B., Chain of coupled bistable oscillators: a model, Physica D, 46, 201-216, (1990) · Zbl 0721.34032
[10] Booth, V.; Erneux, T., Propagation failure in the Fitzhugh-Nagumo discrete reaction-diffusion model, (1993), in preparation
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.