Pulselike Exact Solutions for a Model Describing Nerve Fibers

Abstract

A quasi-linear nonhomogeneous first order hyperbolic system describing nerve pulse transmission is considered. By requiring the compatibility of the governing equations with quasi-linear differential constraints, exact solutions to the model in question are determined. Furthermore classes of material response functions amenable to the mathematical approach are characterized. Initial and/or boundary value problems of interest in nerve pulse propagation are solved. It is proved that the governing model admits solutions which describe a "space clamp" situation and the propagation of a localized action potential pulse along the nerve.

A reduction approach for determining generalized simple waves

“…In most cases the qualitative analyses based upon these techniques show that the effects due to the source-like term B produce a time damping in the process of nonlinear distortion of the wave amplitude whose dynamics is ruled by the differential operator involved in the left hand side of system (1) [4]. Exact wave-like solutions for models belonging to the class (1) were obtained by means of different reduction methods [24], [18], [16], [5], [6], [7], [9], [17], [3], [15], [8]. Whithin such a theoretical framework, of relevant interest in modeling nonlinear wave propagation is developing appropriate approaches for obtaining solutions of (1) in a closed form which generalize the standard simple wave solution ( [12], [20]) of the homogeneous and autonomous system…”