Early afterdepolarizations in cardiac action potentials as mixed mode oscillations due to a folded node singularity.

Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials. They are considered as potential precursors to cardiac arrhythmias and have recently gained much attention in the context of preclinical drug safety testing under the Comprehensive in vitro Proarrhythmia Assay (CiPA) paradigm. From the viewpoint of multiple time scales theory, the onset of EADs has previously been studied by means of mathematical action potential models with one slow ion channel gating variable. In this article, we for the first time associate EADs with mixed mode oscillations in dynamical systems with two slow gating variables and present a folded node singularity of the slow flow as a novel mechanism for EADs genesis. We derive regions of the pharmacology parameter space in which EADs occur using both the folded node analysis and a full system bifurcation analysis, and we suggest the normal distance to the boundary of the EADs region as a mechanism-based risk metric to computationally estimate a drug's proarrhythmic liability.

Period doubling cascades of limit cycles in cardiac action potential models as precursors to chaotic early Afterdepolarizations.

BACKGROUND: Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials (APs). EADs are caused by drugs, oxidative stress or ion channel disease, and they are considered as potential precursors to cardiac arrhythmias in recent attempts to redefine the cardiac drug safety paradigm. The irregular behaviour of EADs observed in experiments has been previously attributed to chaotic EAD dynamics under periodic pacing, made possible by a homoclinic bifurcation in the fast subsystem of the deterministic AP system of differential equations. RESULTS: In this article we demonstrate that a homoclinic bifurcation in the fast subsystem of the action potential model is neither a necessary nor a sufficient condition for the genesis of chaotic EADs. We rather argue that a cascade of period doubling (PD) bifurcations of limit cycles in the full AP system paves the way to chaotic EAD dynamics across a variety of models including a) periodically paced and spontaneously active cardiomyocytes, b) periodically paced and non-active cardiomyocytes as well as c) unpaced and spontaneously active cardiomyocytes. Furthermore, our bifurcation analysis reveals that chaotic EAD dynamics may coexist in a stable manner with fully regular AP dynamics, where only the initial conditions decide which type of dynamics is displayed. CONCLUSIONS: EADs are a potential source of cardiac arrhythmias and hence are of relevance both from the viewpoint of drug cardiotoxicity testing and the treatment of cardiomyopathies. The model-independent association of chaotic EADs with period doubling cascades of limit cycles introduced in this article opens novel opportunities to study chaotic EADs by means of bifurcation control theory and inverse bifurcation analysis. Furthermore, our results may shed new light on the synchronization and propagation of chaotic EADs in homogeneous and heterogeneous multicellular and cardiac tissue preparations.

Long time evolution of atherosclerotic plaques.

The evolution of atherosclerosis in general, and the influence of wall shear stress on the growth of atherosclerotic plaques in particular, is an intricate phenomenon which is still only partly understood. We therefore propose a qualitative mathematical model which consists of a number of ordinary differential equations for the concentrations of the most relevant constituents of the atherosclerotic plaque. These equations were studied both for the case that the wall shear stress is a parameter (model A), and for the case in which the plaque evolution is coupled to the blood flow (model B) which results in a time dependent wall shear stress. We find that both models exhibit a class of marginally stable equilibria, all reflecting states in which the plaque only grows for a short period of time after a perturbation. The uncoupled model A, however, shows bi-stability between this class of equilibria and another equilibrium state in which the plaque experiences unlimited growth in time, if the LDL cholesterol intake exceeds a threshold value. In model B the bi-stability vanishes, but we find that there is still a critical value of the LDL cholesterol intake beyond which the lumen radius drastically decreases. We show that this decrease is quite sensitive to the value of the wall shear stress.