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ItemMethods for model reduction in cardiac dynamics(Georgia Institute of Technology, 2022-05-05) Velasco Perez, Hector AugustoMathematical 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.
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ItemTrade-offs in cardiac electrophysiology(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.
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ItemDisorder to order in oscillatory and excitable systems(Georgia Institute of Technology, 2021-07-29) Detal, NoahIn 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.
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ItemComing together individuals at different scales working for a common goal(Georgia Institute of Technology, 2019-08-27) Welsh, Andrea J.Patterns in biology, chemistry, physics and mathematics can occur from self-organization and the interaction of constituents. In this thesis defense, I will explore patterns in two very different systems: (i) “chimera states” in a biologically-relevant model of excitable tissue, namely a modified version of the FitzHugh-Nagumo model, and (ii) collective motion of living many-agent systems such as swarms of brine shrimp. The FitzHugh-Nagumo model is a simple dynamical system that adequately describes many phenomena in excitable biological systems, such as firing neurons. The excitability is modeled via cubic terms added to the otherwise linear differential equations that describe the time evolution of two dependent variables that characterize the state of a cell. When many of these cells are then coupled in space, the model results in either a stable fixed point or a stable limit cycle which describes synchronized oscillating cells. However, chimera states in which stable fixed-point and limit-cycle regions coexist are not described within this model, even though they are observed in the heart and the brain. By adding a 5th order term in the membrane potential to this 3rd order system, we can recover chimeras, dependent on only initial conditions of the cells. Chimeras have previously been shown in systems with non-local coupling. Interestingly, however, they appear in this new system with purely local coupling. We study the dynamics of these chimeras in a few situations: in 1-dimensional cables and rings with two different simultaneous dynamics and in 2-dimensional grids representing tissues. Switching gears, I then discuss the patterns that occur in swarming, a self-organization phenomenon exhibited in many biological systems such as flocks of bird and insect, schools of fish, and collections of bacteria. This sort of behavior emerges spontaneously, arising without any sort of centralized control or leadership. Many crustaceans such as brine shrimp produce swarms, in which individuals cluster together rather than spread out uniformly in their environment. The size and distribution of these swarms are governed by local interactions between individuals. We will discuss the three-dimensional patterns that can be observed in brine shrimp swarms, specifically of the Great Salt Lake strain of Artemia franciscana, at high concentration. These patterns can be easily observed with simple tabletop experiments. We experimentally test the effects of certain environmental conditions on the dynamics of the individuals and on the development of these swarms.
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ItemMinimal models in cardiac dynamics(Georgia Institute of Technology, 2019-04-30) Chen, DiandianCardiac Arrhythmia is a leading cause of death in the western industrialized world. To date, multiple pathways to examine and treat this disease exist. In this thesis, I focus on computational modeling and nonlinear analysis of cardiac dynamics. While a variety of cardiac models exist, I examine minimal models that produce phenomenological properties of cardiac dynamics. The usefulness of such models is that they are intuitively easy to understand and manipulated. From our results, I first demonstrate the usefulness of minimal models by using a two variable model to produce a novel technique to predict the onset of instability. By reducing current models to a minimal version, I show through graphical and nonlinear methods that action potential amplitude alternans is of equal importance to action potential duration alternans. By further reduction of the two-variable model through fitting to simple equations, I show that phenomenological models can reproduce results that better fit experimental data. Moreover, not only can my constructed minimal produce common phenomena, they can also demonstrate novel dynamics with the adjustment of a small group of parameters. To further expand the usefulness of minimal models, in the last chapter, I construct a minimal model of not only voltage but also the calcium cycling system. Overall, while mathematically complex models are useful and necessary, in this thesis, we present an alternative perspective to study arrhythmia.
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ItemA numerical study of cardiac arrhythmia and defibrillation validated by experiments(Georgia Institute of Technology, 2018-08-27) Ji, YanyanDefibrillation is termination of arrhythmias by altering the transmembrane voltage through the delivery of electric shocks. Debates on the mechanisms behind defibrillations, however, have never ceased. More recent studies affirmed the contribution of inhomogeneities to depolarization in tissues during a defibrillation shock, stating the heterogeneities, such as vessels and bundles, which have different electrical conductivities than cardiac tissue, serve as virtual electrodes during an electric shock, creating excitations in tissues far away from the anode and cathode. Low energy anti-fibrillation pacing (LEAP) has been suggested as an alternative method to traditional defibrillation method, which applies a strong electric pulse to terminate the arrhythmia. LEAP delivers multiple low amplitude electric shocks through field electrodes close to, or inside the tissue. The main goal of this thesis is to investigate the mechanism of LEAP and to suggest ways to improve it. The main finding is that LEAP works by gradually synchronizing the electric activity to the same frequency through each additional shock. Because the tissue is synchronized to the same frequency, both depolarization and repolarization are synchronized and additional shocks will not restart arrhythmia. Modified Kuramoto phase diagrams showed that, during arrhythmias, phase is relatively evenly distributed, and once LEAP is applied, the phase over the domain is increasingly focused with each shock. To further quantify this synchronicity, we calculated the fraction of tissue excited (FTE) as a function of time. The FTE peak progressively increases to one with each pulse for successful LEAP and its derivative indicates how fast the tissue synchronizes. In contrast, during one-shock defibrillation, the FTE upstroke is much slower compared to LEAP, indicating that all cells are eventually excited but not at the same time. Therefore, the mechanism of one-shock defibrillation is not through synchronization but rather by resetting all cells to an excited state, which requires the use of stronger electrical shocks, as some cells are less excitable than the others due to the repolarization gradients during fibrillation. Numerical simulations in this study suggested some ways to improve LEAP by adjusting the pacing period as well as the shock timings. The success rate is higher when the pacing cycle length is close to the dominant period of the arrhythmia and when the first shock was applied at the downslope of the fraction of tissue excited (FTE) curve.