Fast-slow asymptotics for a Markov chain model of fast sodium current

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

A 15-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 6-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.