A new approach to modelling the dynamics of cardiac action potentials
This thesis is concerned with the development of a new approach to the modelling of cardiac action potentials. Electrophysiological models of the heart have become very accurate in recent years giving rise to extremely complicated systems of differential equations. Although describing the behaviour of cardiac cells well, the models are computationally demanding for numerical simulations and are very difficult to analyse from a mathematical (dynamical-systems) viewpoint. Simplified mathematical models that capture the underlying dynamics to a certain extent are therefore frequently used. However, from a physiological viewpoint these equations are unrealistic and often fail to reproduce important quantitative properties of the tissue. In this thesis we introduce a different approach to the mathematical modelling of cardiac action potentials with the aim of gaining a clearer insight into the origin of the dynamics of electrophysiological models. Chapter 1 contains an introduction to the research and outlines the main aims of the work. In Chapter 2 various background material is introduced. This includes some basic electrophysiology, ideas currently used in mathematical modelling of excitable media, and details of models previously developed for the study of cardiac tissue. In Chapter 3, following a detailed analysis of an early physiological model, we develop a mathematical model based on the currents involved. This model reproduces, to good accuracy, action potentials of heart tissue and we discuss the essential ideas behind the dynamics. In Chapter 4 the mathematical model developed in the previous chapter is analysed in more detail and simpler equations using similar ideas are introduced. Various types of action potentials of varying behaviours are studied. In Chapter 5 we investigate some spatial simulations of the new mathematical models. We principally concentrate on one-dimensional studies but towards the end of the chapter we look at some two-dimensional simulations. Finally, in Chapter 6, we discuss our conclusions and some possible ideas for further related work. Details of our methods of numerical simulation are included in Appendix A.