On hyperbolicity for nerve pulse propagation in axons.

The classical Hodgkin-Huxley (HH) model describes the propagation of an action potential (AP) in unmyelinated axons. Here, the key difference from the classical HH model is the hypothesis that the AP propagation in axons depends not only on the HH ion mechanism but also on capacitance and inductance. The modified model is hyperbolic (wave-equation type) compared to the classical HH model, which is parabolic (diffusion-equation type). In this paper, we revisit a modification of the classical HH model proposed by Lieberstein for describing propagating AP in an unmyelinated axon, including the possible effect of inductance that might influence velocity, into the governing equation. We briefly revisit and discuss the underlying assumptions about the hypothesis of including the inductivity in the HH model, systematically check the behaviour of the solutions of the modified governing equations to make sure it is correct for describing the AP and finally discuss some aspects which are relevant for the further modifications (like, for example, taking into account the effect of myelination). The numerical simulation using the physical variables demonstrates the changes in the velocity of an AP as well as the changes in its profile. These results match well the known effects from experimental studies and prepare the basis for the further improvement of modelling the propagation of the AP taking into account the effects of the possible phenomenological inductance.