On theory of pulse transmission in a nerve fibre

Abstract

The ray method is used to analyse pulse propagation in a nerve fibre. The initial equations are considered to be hyperbolic, i. e. with a finite velocity of propagation, and the corresponding evolution equation is obtained. The evolution equation that describes the dynamical side of the process is an equation of the first order, being the natural result of applying the ray method. This approach avoids the paradox of infinite velocity arising from the initial parabolic system used in the Hodgkin-Huxley model, but does not make the final governing equation more complicated. The biochemical side of the process is described by means of the Hodgkin-Huxley or FitzHugh-Nagumo models. The steady-state solution, if it exists, leads to an equation of the Liénard type, and the correspondence between the equations of the catastrophe theory and the usual nerve pulse equations is established.