Derivation of Hodgkin-Huxley equations for aNa+channel from a master equation for coupled activation and inactivation

Abstract

The Na+ current in nerve and muscle membranes may be described in terms of the activation variable m(t) and the inactivation variable h(t), which are dependent on the transitions of S4 sensors of each of the Na+ channel domains DI to DIV. The time-dependence of the Na+ current and the rate equations satisfied by m(t) and h(t) may be derived from the solution to a master equation which describes the coupling between two or three activation sensors regulating the Na+ channel conductance and a two stage inactivation process. If the inactivation rate from the closed or open states increases as the S4 sensors activate, a more general form for the Hodgkin-Huxley expression for the open state probability may be derived where m(t) is dependent on both activation and inactivation processes. The voltage dependence of the rate functions for inactivation and recovery from inactivation are consistent with the empirically determined expressions, and exhibit saturation for both depolarized and hyperpolarized clamp potentials. 1 INTRODUCTIONThe opening and subsequent inactivation of Na+ channels and the activation of K+ channels generate the action potential in nerve and muscle membranes [1]. The time-dependence of the Na+ current in the squid axon may be described in terms of the expression m(t) 3 h(t) where the activation variable m(t) and inactivation variable h(t) satisfy the rate equations

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

“…whereḡ j is the conductance, V j is the equilibrium potential for each channel j (Na+, K+ and leakage), and i e is the external current. When the fast inactivation transition rates α ik ≪ γ ik , δ ik ≪ β ik , and γ ik + β ik is greater than the activation and deactivation rate functions, for each k, the occupation probabilities of A 1 to A 4 attain quasi-stationary values in a time that is smaller than the relaxation of the membrane potential and the closed, open and inactivated states [24], and Eqs. (4) to (15) may be reduced to an eight state system by expressing the solution as a two-scale asymptotic expansion and eliminating secular terms [17] (see Fig.…”

“…In Eqs. (24), (25) and (29), it is assumed that the transition rates between fast inactivated states with occupation probabilities I 2 , I 3 and I 4 are an order of magnitude larger than inactivation and recovery rates, and activation and deactivation rates between closed and open states, and therefore, by expressing the solution as a two-scale asymptotic expansion and eliminating secular terms [17], it may be shown that Eqs. (18) to (25) may be reduced to a five state kinetic model (see Fig.…”

“…For a moderate hyperpolarization (σ 1 ≫ α I1 , β I1 ),σ 1r ≈σ 1 ≈ β I1 , and therefore, the voltage dependence of α h is approximately exponential [1] (see Fig. 8), but may attain a plateau value for a large hyperpolarization [6,21,24]. If the previous conditions for each stage of reduction are satisfied, the solution of the twelve state kinetic model, Eqs.…”

“…By expressing the solution as a two-scale asymptotic expansion and eliminating secular terms [17], Eqs. ( 61) to (75) may be reduced to an eleven state system when the two-stage inactivation transitions satisfy α ik ≪ γ ik , δ ik ≪ β ik and γ ik + β ik is greater than the activation and deactivation rate functions, for each k [24] (see Fig. 12)…”