A model of a myelinated nerve axon: threshold behaviour and propagation
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.
propagation of a superthreshold solution
scientific article; zbMATH DE number 3928768