Model-Based and Machine Learning-Based Control of Biological Oscillators

Nonlinear oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. This dissertation investigates the dynamics of such oscillators arising in biology, and develops several control algorithms to modify their collective behavior. We demonstrate that these control algorithms have potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag. Phase reduction, a classical reduction technique, has been instrumental in understanding such biological oscillators. In this dissertation, we investigate a new reduction technique called augmented phase reduction, and calculate its associated analytical expressions for six dynamically different planar systems: This helps us to understand the dynamical regimes for which the use of augmented phase reduction is advantageous over the standard phase reduction. We further this study by developing a novel optimal control algorithm based on the augmented phase reduction to change the phase of a single oscillator using a minimum energy input. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier of the oscillator is close to unity; in such cases, the previously proposed control algorithm based on the standard phase reduction fails.We then devise a novel framework to control a population of biological oscillators as a whole, and change their collective behavior. Our first two control algorithms are Lyapunov-based, and our third is an optimal control algorithm which minimizes the control energy consumption while achieving the desired collective behavior of an oscillator population. We show that the developed control algorithms can synchronize, desynchronize, cluster, and phase shift the population.We continue this investigation by developing two novel machine learning control algorithms, which have a simple and intelligent structure that makes them effective even with a sparse data set. We show that these algorithms are powerful enough to control a wide variety of dynamical systems and not just biological oscillators. We conclude this study by understanding how the developed machine learning algorithms work in terms of phase reduction.In this dissertation, we have developed all these algorithms with the goal of ease of experimental implementation, for which the model parameters/training data can be measured experimentally. We close the loop on this dissertation by carrying out robustness analysis for the developed algorithms; demonstrating their resilience to noise, and thus their suitability for controlling living biological tissue. They truly hold great potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag

Classifying Mechanisms of Spiral Wave Breakup Underlying Cardiac Fibrillation Using Quantitative Metrics

Cardiovascular diseases are one of the leading causes of death in the world and manifest themselves in several forms, including arrhythmias. These disruptions in the normal rhythm of the heart inhibit the regular transmission of electrical signals that are essential for the heart to contract and pump blood to the rest of the body. During reentrant arrhythmias, spiral or scroll waves of electrical activation are conducted through the cardiac tissue and excite it repeatedly. As these waves propagate through the heart, they can break up in an irregular manner, leading to the onset of fibrillation. There are several mechanisms by which these reentrant waves can destabilize, but they are known mostly from computational studies. Experimentally, it has not been possible so far to distinguish among these mechanisms based on straightforward observations of the heart\u27s voltage during fibrillation. As a preliminary step in this direction, we aim to determine whether quantifying certain observable properties of the system will allow us to identify the mechanism underlying a given fibrillation episode. Toward this end, we propose a number of metrics that could help us classify mechanisms underlying fibrillation, including chaos in the system as assessed by the largest Lyapunov exponent; the amount of information (mutual information) and dependency (spatial correlation) shared by various spatial points in the domain; and reentrant wave properties like the number of reentries, wave birth and death rates, reentrant wave lifetimes, and spiral wave tip speeds. We implement and apply these metrics to simulated data obtained by numerically solving partial differential equations describing electrical wave propagation in the heart. Specifically, we analyze data achieved through six different mechanisms of reentrant wave breakup: steep APD restitution, discordant alternans, bistability, Doppler effect, supernormal conduction velocity and periodic boundary conditions. Our results suggest that of the various reentrant wave properties, the distribution of the number of reentries over time serves to be the most useful metric by providing a visual representation of how the breakup proceeds with time for each mechanism. When the mutual information and spatial correlation are studied in the context of the distribution of reentries over time, they help us gauge any spatial dependencies that may be present in the system. To validate our findings, we carried out a blind test to classify breakup mechanisms in four provided data sets with established breakup mechanisms. Our metrics correctly classified the mechanisms for three of these cases, and we are condent that further optimization could improve the reliability of our approach. Our work forms the basis for future studies that apply these and other metrics towards identifying the mechanisms responsible for fibrillation in experimental settings

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

Subcellular calcium patterns in ventricular myocytes

Understanding the biology and mechanisms as to how the heart contracts has long been a point of interest for biologists and mathematicians alike. Since inconsistent beating of the heart has been linked to multiple pathological conditions, research into this area has been extensive but we still only have some of the answers. One of the key findings over the last century has been the role of calcium in activating the machinery within the heart that drives contraction. Further studies have shown that when calcium is mishandled by the heart's myocytes, it can lead to some of these pathological conditions. Since such discoveries a major point of research into the heart has focused on the possible avenues that calcium mishandling can occur. This thesis explores some of these avenues using a mathematical model of the ventricular myocyte developed by Thul and Coombes in their 2010 paper 'Understanding cardiac alternans: A piecewise linear modeling framework'. The chosen model contains key components involved in the movement of calcium within the myocyte. Moreover, the model used is piecewise linear and the stability of some important behaviours can be studied exactly without the need for approximations and reductions. This is often an issue in many other models used to study the calcium dynamics within a ventricular myocyte. The avenue towards calcium mishandling that this thesis predominantly focuses on is that of intracellular calcium diffusion between the building blocks of ventricular myocytes known as sarcomeres. Our research extends previous research into how strong diffusion between sarcomeres can cause unwanted calcium dynamics. Further to this, we explore how the balance in the strength of different forms of calcium diffusion between sarcomeres can drive a variety of spatial patterns in terms of how the calcium is distributed throughout the cell. Throughout these studies we also investigate the role of other parts of the myocyte, particularly the sarcoplasmic reticulum Calcium-ATPase pumps and sarcoplasmic reticulum release in relation to diffusion driven instabilities. As well as intracellular diffusion of calcium, this thesis considers the role of intercellular diffusion of calcium through gap junctions. This form of diffusion has historically been considered to a lesser extent than intracellular diffusion. As such this thesis introduces new ideas concerning gap junctions. These include a role in driving the mishandling of calcium as well as altering behaviours driven by intracellular diffusion. An important message is that calcium diffusion within the myocyte is far more important in terms of how unwanted behaviours can appear than previous studies suggest

Modelling pathological effects in intracellular calcium dynamics leading to atrial fibrillation

The heart beating is produced by the synchronization of the cardiac cells' contraction. A dysregulation in this mechanism may produce episodes of abnormal heart contraction. The origin of these abnormalities often lies at the subcellular level where calcium is the most important ion that controls the cell contraction. The regulation of calcium concentration is determined by the ryanodine receptors (RyR), the calcium channels that connect the cytosol and the sarcoplasmic reticulum. RyRs open and close stochastically with calcium-dependent rates. The fundamental calcium release event is known as calcium spark, which refers to a local release of calcium through one or more RyRs. Thus, a deep knowledge on both the spatio-temporal characteristics of the calcium patterns and the role of the RyRs is crucial to understand the transition between healthy to unhealthy cells. The aim of this Thesis has been to figure out these changes at the submicron scale, which may induce the transition to Atrial Fibrillation (AF) in advanced stages. To address this issue, I have developed, and validated, a subcellular mathematical model of an atrial myocyte which includes the electro-physiological currents as well as the fundamental intracellular structures. The high resolution of the model has allowed me to study the spatio-temporal calcium features that arise from both the cell stimulation and the resting conditions. Simulations show the relevance of the assembly of RyRs into clusters, leading to the formation of macro-sparks for heterogeneous distributions. These macro-sparks may produce ectopic beats under pathophysiological conditions. The incorporation of RyR-modulators into the model produces a nonuniform spatial distribution of calcium sparks, a situation observed during AF. In this sense, calsequestrin (CSQ) has emerged as a key calcium buffer that modifies the calcium handling. The lack of CSQ produces an increase in the spark frequency and, during calcium overload, it also promotes the appearance of global calcium oscillations. Finally, I have also characterized the effect of detubulation, a common issue in cells with AF and heart failure. Thus, the present work represents a step forward in the understanding of the mechanisms leading to AF, with the development of computational models that, in the future, can be used to complement in vitro or in vivo studies, helping find therapeutic targets for this disease