Action potential duration alternans in mathematical models of excitable cells

Action potential duration alternans has been associated with the onset of one of the most common forms of abnormal heart rhythm, atrial fibrillation (Cherry et al., 2012; Nattel, 2002). This thesis concerns identifying variables and parameters responsible for inducing action potential duration alternans. In order to achieve this, we apply asymptotic reduction methods to models of cardiac electrophysiology described by a system of ordinary differential equations and derive explicit discrete restitution maps which specify the action potential duration as a function of the preceding diastolic interval. The bifurcations of equilibria of these maps are studied to determine regions in the parameter space of the models where normal response and alternans occur. Furthermore, explicit parametric representations of both the normal and the alternans equilibrium branches of the restitution map are found. We also develop a framework formulated in terms of a boundary value problems for studying cardiac restitution. This framework can be used to derive analytically or compute numerically different branches of the action potential duration restitution map from the full excitable models. Our method is validated by comparing the asymptotic restitution map with the boundary value problem formulated restitution curves. The proposed method is applied to investigate the restitution properties of three excitable models: one generic excitable model and two ionic cardiac models. The first model is the McKean (1970) model which is a simplified version of the classical FitzHugh (1961) model. The other two models are the Caricature version of the Noble (1962) model derived by Biktashev et al. (2008) and an asymptotically reduced version of the Courtemanche et al. (1998) model of the atrial cell, reduced by Suckley (2004). After deriving the action potential duration restitution map for each of the mentioned model, the region of the models parameters in which alternans occurs is determined. We conclude that alternans appears if the dynamics in the diastolic stage of an action potential are faster than the dynamics in the systolic stage. Furthermore, we show that the time scale for the slow gating variable is responsible for inducing alternans. We outline that the oscillation in the slow activation of the K+ current and the slow inactivation of the L-type Ca+2 current can induce or suppress alternans