Dynamical analysis of a model of skeletal muscles with myotonia or periodic paralysis
AbstractMedical research has indicated that abnormalities of skeletal muscles, myotonia and periodic paralysis, are caused by alteration in the voltage-gated sodium channels. This assumption led to studies of channel behavior based on the dynamics of membrane potentials. To simulate such behavior, the Hodgkin-Huxley model has been modified by reformulation of the sodium current term. A singular perturbation analysis is carried out on the model system consisting of rate equations on the membrane potentials and the probability function connected to the opening of the potassium channels activation gates. The conditions on the system parameters under which the model exhibits dynamic behavior that resembles clinical observations shall be derived. We are able to detect slow-fast limit cycles which generate bursts of action potentials characteristic of the clinical case where active and non-active phases are observed to alternate in a pulsatile fashion, such as that in patients with Hyperkalemic periodic paralysis. Relying on the observation that the state variables possess drastically diversified dynamics, we are able to explain the differences between the action potential dynamics of a normal subject and those of myotonia or periodic paralysis cases.