A Single Differential Equation Description of Membrane Properties Underlying the Action Potential and the Axon Electric Field

TL;DR

This paper introduces a novel analytical model of nerve action potentials using a single nonlinear differential equation, simplifying the complex Hodgkin-Huxley framework and linking membrane potential with electric field.

Contribution

It presents a new, simplified differential equation-based description of the nerve action potential that unifies membrane potential and electric field analysis.

Findings

The model provides an analytical solution for the action potential.

It demonstrates the equivalence of membrane potential and electric field descriptions.

The approach simplifies understanding of nerve excitation mechanisms.

Abstract

In a succession of articles published over 65 years ago, Sir Alan Lloyd Hodgkin and Sir Andrew Fielding Huxley established what now forms our physical understanding of excitation in nerve, and how the axon conducts the action potential. They uniquely quantified the movement of ions in the nerve cell during the action potential, and demonstrated that the action potential is the result of a depolarizing event across the cell membrane. They confirmed that a complete depolarization event is followed by an abrupt increase in voltage that propagates longitudinally along the axon, accompanied by considerable increases in membrane conductance. In an elegant theoretical framework, they rigorously described fundamental properties of the Na+ and K+ conductances intrinsic to the action potential. Notwithstanding the elegance of Hodgkin and Huxley's incisive and explicative series of discoveries, their model is mathematically complex, relies on no small number of stochastic factors, and has no analytical solution. Solving for the membrane action potential and the ionic currents requires integrations approximated using numerical methods. In this article I present an analytical formalism of the nerve action potential, Vm and that of the accompanying cell membrane electric field, Em. To conclude, I present a novel description of Vm in terms of a single, nonlinear differential equation. This is an original stand-alone article: the major contribution is the latter, and how this description coincides with the cell membrane electric field. This work has necessitated unifying information from two preceding papers, each being concerned with the development of closed-form descriptions of the nerve action potential.