Bistabilities and annihilation phenomena in electrophysiological cardiac models

We have investigated the oscillatory behavior of cardiac cellular elements simulated by two electrophysiological models: the van Capelle and Durrer (VCD) model and the sinoatrial node cell model of Yanagihara, Noma, and Irisawa (YNI). The VCD model behavior was examined systematically by using continuation-bifurcation analysis. Bifurcation diagrams were constructed as a function of Qit1, an intrinsic parameter of the model, which sets both maximum diastolic potential and depolarization threshold of the cell. The existence of stable high amplitude oscillations was evidenced between two Hopf bifurcation points (HB). Near each HB, a zone of bistability was detected. Close to the HB that corresponded to high values of Qit1, a high amplitude periodic stable state coexisted with a stable steady state. Close to the other HB, in a narrow range of lower Qit1 values, a relatively high amplitude periodic stable state coexisted with a low amplitude periodic stable state. There was no stable steady state in the latter bistability zone. Through the use of phase-plane representations and the determination of separatrices between the different attractor basins, we could deduce the conditions of timing, polarity, and strength needed for a pulse perturbation to send the system from one state to another and vice versa. The YNI model was analyzed by numerical simulation, and the oscillatory behavior of the sinoatrial node cell was explored while applying a depolarizing bias current of various strengths. Results were similar to those obtained from the VCD model in that there were two bistability regions for two different ranges of applied bias current. Depending on current intensity, annihilation of pacemaker activity could be achieved in both zones. However, the coexistence of two oscillatory stable states was never observed in the YNI model. From the behavioral similarities of these different models, we can conclude that bistabilities and annihilation phenomena can be found in transitional zones between quiescence and rhythmic activity.