Derivation of an Improved Hodgkin-Huxley Model for Potassium Channel by Means of the Fokker-Planck Equation

Abstract

This paper demonstrates the derivation of Hodgkin-Huxley-like equations from the Fokker-Planck equation. The primary result is that instead of the familiar$g_K = \hat g_K n^4$ equation expressing the potassium conductance as a function of the variablen which obeys a first order differential equation, the expression$g_K = g_o exp[L^2 - (n - L)^2 ]$, whereL = 2.7, is to be used. This form is obtained by solving analytically an approximate solution to a Fokker-Planck partial difference equation. Instead of the Hodgkin-Huxley interpretation as the probability of occupying the conducting state, the parameter n(t) is now interpreted as the position of the “peak” of the population distribution function P(N, t), which changes in time described by the Fokker-Planck equation.

This new approach enables close fitting of the experimental voltage clamp data for potassium conductance. In addition, the Cole-Moore shift paradox can be quantitatively explained in terms of the shift of the distribution function P(N,t) by the initial clamped transmembrane potentialV _i before the final clamped transmembrane potentialV _f is applied, thus increasing the time necessary for the establishment of equilibrium.