Steady state plane wave propagation speed in excitable media

Abstract

With the view to excitation waves in neuromuscular tissue we study the propagation speed $c$ of steady state plane trigger waves as a function of the shape parameters of the nonlinear source function in the reaction diffusion equation (the slow recovery variable is assumed frozen). The nonlinear eigenvalue problem, which yields $c$ as the eigenvalue and the wave profile $u(\ensuremath{\xi})$ as the eigenfunction, is reformulated to allow us to construct a variety of exactly solvable models, in particular the waves with profiles having central symmetry about their midpoints. Trigger waves with asymmetric profiles are also considered. The propagation speed $c$ is expressed in terms of the shape parameters of the nonlinear source function $i(u).$ We show that among shape parameters of $i(u)$ there are only three essential ones which control the propagation speed. We derive a general expression for the propagation speed, which has the same simple form as in the well known case with the sawtoothwise $i(u)$ but with one parameter appropriately redefined. We also introduce a simple iterative procedure for solving the nonlinear eigenvalue problem. Finally introducing a new exactly solvable model we show that the effect of noninstantaneous activation on the propagation speed can be reduced to renormalization of one of the steady state model's parameter.