A model study of stability and oscillations in the myocardial cell membrane

As a step towards an improved understanding of cardiac arrhythmias caused by abnormal automaticity, we perform a stability analysis of a Hodgkin-Huxley model of the myocardial cell membrane (modified Beeler-Reuter, MBR). The bifurcation structure of the model is obtained as a function of three parameters: the intensity of an applied constant current; the potassium equilibrium potential representing the accumulation of K+ ions in the external medium; and the maximum conductance of the slow inward current mimicking the local application of catecholamines on the membrane. For a range of parameter values, the model exhibits either stable automaticity or bistability between two quiescent states or between a quiescent state and an oscillatory state. These transformations of the bifurcation structure are shown to depend on the interrelationship between three elements: the activation of the slow inward current, the region of high slope conductance of the time-independent potassium current functions, and the slow variables controlling the activation of the potassium current and the inactivation of the slow inward current. Reduced two- and three-dimensional models are shown to reproduce the main stability properties of the full MBR model and to facilitate the understanding of its dynamic behavior. The onset of instability and the oscillatory features of the MBR model are in good agreement with relevant experimental results, and possible sources of disagreement on certain points are discussed.