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

TL;DR

This paper presents a reduced kinetic model for Na+ channel gating that captures activation, fast, and slow inactivation processes, enabling analysis of neural and cardiac membrane behaviors with simplified equations.

Contribution

The authors derive a simplified six-state model from a complex fifteen-state system using multiple scales, capturing key gating dynamics in neurons and cardiac cells.

Findings

Model reproduces spike frequency adaptation.

Captures bursting oscillations in neural membranes.

Describes cardiac action potential plateau oscillation.

Abstract

A fifteen state kinetic model for Na+ channel gating that describes the coupling between three activation sensors, a two-stage fast inactivation process and slow inactivated states, may be reduced to equations for a six state system by application of the method of multiple scales. By expressing the occupation probabilities for closed states and the open state in terms of activation and fast inactivation variables, and assuming that activation has a faster relaxation than inactivation and that the activation sensors are mutually independent, the kinetic equations may be further reduced to rate equations for activation, and coupled fast and slow inactivation that describe spike frequency adaptation, a repetitive bursting oscillation in the neural membrane, and a cardiac action potential with a plateau oscillation. The fast inactivation rate function is, in general, dependent on the activation variable m(t) but may be approximated by a voltage-dependent function, and the rate function for entry into the slow inactivated state is dependent on the fast inactivation variable.