Methods for model reduction in cardiac dynamics

Mathematical models have been crucial for understanding biological systems because they help us organize our knowledge about the system and allow us to not only test new ideas without harming or perturbing expensive in vivo, in vitro, or in situ subjects, but to further test new hypothesis. Cardiac electrophysiology is a field that requires a deep understanding of a wide span of physiological scales. From the single-cell ionic membrane exchanges, to the fiber distribution and geometry of the heart. Naturally, this complexity draws several kinds of modelling proposals, many of which describe, with different degrees of complexity, the process of excitation and propagation of an action potential (AP). In this thesis we will present two model reduction paradigms and the computational tools to use them. First, we introduce a new parsimonious phenomenological model based on the FitzHugh-Nagumo model. We focus on describing its main characteristics and presenting a variety of applications that cover a wide range of subjects. In particular, our model can fit experimental data of several animal species. Moreover, analytical expressions for the restitution and dispersion curves are available. Next, we expand our idea of model reduction by taking advantage of the symmetries of the electrical patterns. We specifically look at translational and rotational invariant solutions. We then present a numerical scheme for symmetry reduction of spiral waves. Afterwards, we tested the method with several models and multiple spiral wave solutions. Finally, we investigated the performance of several parallel programming languages for graphic processing units by comparing the speeds of multiple implementations of a cardiac solver. In this work, we develop the theory and provide the numerical schemes to reproduce our results.Ph.D