(Georgia Institute of Technology, 2022-05-05)
Velasco Perez, Hector Augusto
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.
(Georgia Institute of Technology, 2021-08-23)
Herndon, Conner J.
This thesis explores the electrical dynamics of the heart, and in particular, the mechanisms that underlie cardiac arrhythmia, their variability across and within species, and the impact they may pose as a selective pressure. As a living system, we can study the heart and its electrical activity through perspectives ranging from subcellular processes to the evolutionary trajectories of entire species. With broader consideration of interrelated components and their implications on all scales, we are armed with context. In this thesis, I examine the contextual significance of cardiac arrhythmia. My main contributions fall within three categories: (i) the development and improvement of experimental techniques, (ii) the strength of cardiac arrhythmia as a selective pressure, and (iii) a symbiotic partnership between comparative physiology and nonlinear dynamics.
(Georgia Institute of Technology, 2021-07-29)
Detal, Noah
In this thesis, two problems involving the macroscopic ordering of coupled nonlinear elements are studied. The first problem involves analyzing a synchronizing transition in a population of coupled oscillators. Synchronization of the oscillators in this model corresponds to coherent propagation of solitary waves in a nonlinear Schrödinger equation. The second problem concerns the elimination of chaotic fibrillatory dynamics in excitable cardiac tissue. In the fibrillatory state, reentrant spiral waves of electrical activity entrain the excitable cells and interrupt the healthy heart rhythm. By appealing to the topological structure of excitable dynamics, conditions are derived for the stimulated elimination of spiral waves and associated fibrillation. Insights from this topological framework are then applied to the optimization of a novel low-energy multi-pulse defibrillation scheme.