Theoretical reconstruction of myotonia and paralysis caused by incomplete inactivation of sodium channels

Muscle fibers from individuals with hyperkalemic periodic paralysis generate repetitive trains of action potentials (myotonia) or large depolarizations and block of spike production (paralysis) when the extracellular K+ is elevated. These pathologic features are thought to arise from mutations of the sodium channel alpha subunit which cause a partial loss of inactivation (steady-state Popen approximately 0.02, compared to < 0.001 in normal channels). We present a model that provides a possible mechanism for how this small persistent sodium current leads to repetitive firing, why the integrity of the T-tubule system is required to produce myotonia, and why paralysis will occur when a slightly larger proportion of channels fails to inactivate. The model consists of a two-compartment system to simulate the surface and T-tubule membranes. When the steady-state sodium channel open probability exceeds 0.0075, trains of repetitive discharges occur in response to constant current injection. At the end of the current injection, the membrane potential may either return to the normal resting value, continue to discharge repetitive spikes, or settle to a new depolarized equilibrium potential. This after-response depends on both the proportion of noninactivating sodium channels and the magnitude of the activity-driven K+ accumulation in the T-tubular space. A reduced form of model is presented in which a two-dimensional phase-plane analysis shows graphically how this diversity of after-responses arises as extracellular [K+] and the proportion of noninactivating sodium channels are varied