Steady-state solutions in mathematical models of atrial cell electrophysiology and their stability

The steady states of the Fenton-Karma, the Courtemanche and the Nygren cell models were studied by determining the fixed points of the dynamical system describing their cell kinetics. The linear stability of the fixed points was investigated, as well as their response to external stimuli. Symbolic calculations were carried out as far as possible in order to prove the existence of these fixed points. In the Fenton-Karma model, a unique stable fixed point was found, namely the resting state. In contrast, the Courtemanche model had an infinite number of fixed points. A bifurcation diagram was constructed by classifying these fixed points according to a conservation law. Initial conditions were identified, for which the dynamical behavior of the cell was auto-oscillatory. In its original formulation, the Nygren model had no fixed point. After having restored charge conservation, the system was found to have an infinite number of fixed points, resulting in a bifurcation diagram similar to that of the Courtemanche model. The approach proposed in this paper assists in the exploration of the high-dimensional parameter space of the cell models and the identification of the conditions leading to spontaneous pacemaker activity.