Rate equations for a Na+ channel gating master equation during the action potential within a neural membrane

TL;DR

This paper derives rate equations for Na+ channel gating during action potentials, simplifying complex master equations into manageable kinetic models under certain assumptions, linking detailed channel dynamics to Hodgkin-Huxley equations.

Contribution

It introduces a reduced kinetic model for Na+ channel gating based on assumptions about activation sensors and inactivation, connecting detailed master equations to Hodgkin-Huxley rate equations.

Findings

A twelve-state master equation can be reduced to a five-state system.

Inactivation rate depends on activation variables under certain conditions.

Master equation solutions can be approximated by Hodgkin-Huxley equations when m(t) is faster than h(t).

Abstract

The action potential in a neural membrane is generated by Na+ and K+ channel ionic currents that may be calculated from a current equation and the rate equations for activation variables m and n, and the Na+ inactivation variable h. Assuming that a Na+ channel has three activation sensors, and activation and inactivation are cooperative processes, a twelve state master equation that describes channel gating may be reduced to kinetic equations for a five state system when the occupational probability of the first inactivated state is small, and the remaining inactivated states contribute to a total inactivated state. In the case of independent activation sensors, the inactivation rate is, in general, dependent on the activation variable m(t) as well as the forward inactivation transition rates. However, when m(t) has a faster time constant than h(t), the inactivation rate may be approximated by a voltage-dependent function, and therefore, the solution of the master equation during an action potential may be approximated by the solution of Hodgkin-Huxley rate equations for m and h.