A model of a myelinated nerve axon: threshold behaviour and propagation
type: Item
instance of: scholarly article
title: A model of a myelinated nerve axon: threshold behaviour and propagation
zbMATH Open document ID: 0579.92004
DOI: 10.1007/BF00276561
published in: Journal of Mathematical Biology
publication date: 1985-01-01T00:00:00Z
review text: The approach in this paper differs from that of other authors in that the nodes of Ranvier are assumed to be of small but nonzero width. The model considered describing the transmembrane potential u(x,t) consists of the following coupled equations: \[ c_ 1u_ t=u_{xx}-u/r,\quad x\in (X_ i,X_{i+1}),\quad i-even, \] \[ c_ 2u_ t=u_{xx}+u(1-u)(u- a),\quad x\in (X_ i,X_{i+1}),\quad i-odd, \] with continuity conditions at \(X_ i\), \(i=1,2,..\). Several particular cases are considered. For the case \[ X_ i=k+\theta \quad if\quad i=2k+1,\quad and\quad =k\quad if\quad i=2k,\quad k=0,1,...,\quad \theta \in (0,1), \] sufficient conditions are obtained for the uniqueness and global stability of the trivial 1-periodic steady state solution. In the general case it is shown that propagation of a superthreshold solution may be prevented in the absence of nodes. The mathematical theorems are given some biological interpretation. A conjecture is put forward at the end.
Mathematics Subject Classification ID: 92Cxx
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 3928768
zbMATH Keywords: myelinated nerve axons
zbMATH Keywords: FitzHugh-Nagumo dynamics
zbMATH Keywords: threshold
zbMATH Keywords: behaviour
zbMATH Keywords: wave fronts
zbMATH Keywords: nodes of Ranvier
zbMATH Keywords: transmembrane potential
zbMATH Keywords: global stability
zbMATH Keywords: periodic steady state solution
zbMATH Keywords: propagation of a superthreshold solution
Wikidata QID: Q52679074
author: Peter Grindrod
author: B. D. Sleeman
MaRDI profile type: Publication
cites work: Q4089232
cites work: Threshold conditions for a diffusive model of a myelinated axon
cites work: Q4774804
cites work: Geometric theory of semilinear parabolic equations
cites work: Time-periodic solutions to semilinear parabolic equations
description: scientific article; zbMATH DE number 3928768