A model of Na-K exchange during nerve excitation

Abstract

A system of nonlinear equations in the ionic charge densities is derived from the conservation laws of charge and momentum and Poisson's equation for the electrostatic potential. It has been shown previously that a similar set of equations in the chemical potentials has oscillatory solutions with amplitudes and frequencies of the same order of magnitude as those determined from the neural membrane. It is proposed that the action potential is a nonlinear oscillation in the electric potential associated with an increase in the number density of K ions and a corresponding decrease in the Na density within the membrane channels and external Debye layer. However, the large amplitude of this mode requires an initial asymmetry between the fluctuations in the K and Na densities, and this may develop in the presence of Ca at the membrane surface. The model can account for the existence of a threshold potential, the dependence of the amplitude of the action potential on log([Na]ext), and the oppositely directed currents of Na and K. The transient inward current or “early channel” may be identified with the external K-Na mode, whereas the delayed K current or “late channel” is associated with an increase in the K charge density at the external surface of the membrane. When the nonlinear diffusion equations are expressed in terms of the relative charge densities of K and Na, they have a form which is equivalent to the Hodgkin-Huxley equations in the activation variables n and m.