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

Abstract

A fifteen state kinetic model for Na+ channel gating that describes the coupling between three activation sensors, a twostage 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. 1 INTRODUCTIONDuring prolonged or repetitive depolarization, in addition to the fast inactivation of Na channels that contributes to repolarization of the membrane [1], a slow inactivation process reduces the number of Na+ channels available for activation. The increase in slow inactivation of Na+ channels during depolarization is associated with a delay to the next spike or a reduction in the firing frequency (spike frequency adaptation) [2] and is the result of a structural rearrangement in the selectivity filter region of the ion channel that generally occurs following the inactivation of the pore [3]. Slow inactivation of the transient and persistent components of the Na+ current in a mesencephalic V neuron is associated with the termination of a bursting oscillation, and the increase in the amplitude of the subthreshold oscillation between bursts occurs during the recovery from slow inactivation [4]. In subicular neurons adjacent to the hippocampus, the transition from bursting to single spiking is influenced by the slow inactivation of Na+ channels, and this may provide a mechanism for enhancing the effect of input signals [5].For Na+ channels with slow inactivation, the Na+ current I N a may be described by the expression m 3 hs(V N a − V ) [2] where V N a is the equilibrium potential, and the activation variable m, the fast inactivation variable h, and the slow inactivation variable s satisfy the equations