Mathematical Studies of the Dynamics of Excitable Systems

University Of Utah, Salt Lake City UT

Investigators

Abstract

Keener 0211366 The investigator uses mathematical models and analysis to study the dynamics of several physiological systems in which excitability is a common and fundamental feature. The three broad problem areas to be investigated are 1) The initiation and dynamics of cardiac arrhythmias; 2) The dynamics of defibrillation; 3) The dynamics of autoimmune disease. The study of these problems involves the development and use of sophisticated mathematical models and tools, including advanced numerical methods such as immersed boundary methods, bifurcation theory and dynamical systems theory, and asymptotic analysis. These tools are used to give theoretical understanding to problems of border zone and ischemia related cardiac arrhythmias, APD instabilities and their relationship to the onset and development of reentrant arrhythmias, mechanical alternans deriving from instabilities of the calcium handling system or crossbridge dynamics, the success or failure of defibrillation, and oscillatory autoimmune responses. Serious attention is given to obtaining agreement between theoretical results and experimental findings. To that end collaborations with experimentalists examine gap junction coupling of rabbit myocytes, the effect of hypertrophy on excitation-contraction coupling through the sodium-calcium exchanger, the effect of mutations on the delayed rectifier potassium channel that lead to increased proclivity for fatal arrhythmias, and the role of mimicry in oscillatory dynamics of cystic fibrosis in mice. Excitability is one of the most important features enabling signalling in physiological systems. It is crucial to the operation of nerves, muscle, the heart, the immune system, and blood clotting as well as numerous gene regulatory networks. The specific goal of this project is to gain an improved theoretical understanding of how the normal function of systems that rely on excitability can go awry, exhibiting pathological behaviors, and how these pathological behaviors might be controlled or prevented. The behaviors that are studied here are the result of interwoven interactions of simpler behaviors, resulting in complex behaviors that defy understanding through intuition or simple means. The investigation therefore focuses on the development and analysis of mathematical models of these behaviors in the belief that substantial new and important insights can be gained, and new hypotheses formulated that are not apparent from phenomenological descriptions. While this project has as its main focus several problems relating to abnormalities of the cardiac cycle, it extends far beyond to many areas of physiology where common principles are involved. The project includes a significant training component for students.