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Propagation and its failure in coupled systems of discrete excitable cells. (English) Zbl 0649.34019

Consider the continuous model of excitable cells in neuro-physiology and cardiophysiology

$\left(1\right)\phantom{\rule{1.em}{0ex}}\frac{\partial v}{\partial t}=d\frac{{\partial }^{2}v}{\partial {x}^{2}}+f\left(u,v\right),\phantom{\rule{1.em}{0ex}}\frac{\partial u}{\partial t}=g\left(u,v\right),\phantom{\rule{1.em}{0ex}}v\in ℝ,\phantom{\rule{1.em}{0ex}}u\in ℝ\phantom{\rule{1.em}{0ex}}n,$

where v represents the membrane potential of the cell and u comprises additional variables (such as gating variables, chemical concentrations, etc.) necessary to the model. Assuming that the cells are coupled resistively, and currents between cells satisfy Kirkhoff’s laws, one has

$\left(2\right)\phantom{\rule{1.em}{0ex}}\frac{d{v}_{n}}{dt}=d\left({v}_{n+1}-2{v}_{n}+{v}_{n-1}\right)+f\left({u}_{n},{v}_{n}\right),\phantom{\rule{1.em}{0ex}}\frac{d{u}_{n}}{dt}=g\left({u}_{n},{v}_{n}\right),$

where the subscript n indicates the nth cell in a string of cells. This paper shows that the behavior of the systems (1) and (2) can be markedly different. In particular, if (1) has traveling wave solutions for some value of d, it does so for all values of $d>0$; on the other hand, there are values of $d>0$ for which solutions of (2) fail to propagate, regardless of initial data. In this paper, the case of $g\equiv 0$ and u constant independent of n is studied:

$\left(3\right)\phantom{\rule{1.em}{0ex}}\frac{d{v}_{n}}{dt}=d\left({v}_{n+1}-2{v}_{n}+{v}_{n-1}\right)+f\left({v}_{n}\right)·$

The results are complementary to those of J. Bell and C. Cosner, Q. Appl. Math. 42, 1-14 (1984; Zbl 0536.34050)].

Reviewer: Li Bingxi
##### MSC:
 34A35 ODE of infinite order 35K57 Reaction-diffusion equations 92Cxx Physiological, cellular and medical topics