Mathematical Studies of the Dynamics of Excitable Systems

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Keener0211366 The investigator uses mathematical models and analysis tostudy the dynamics of several physiological systems in whichexcitability is a common and fundamental feature. The three broadproblem 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 ofsophisticated mathematical models and tools, including advancednumerical methods such as immersed boundary methods, bifurcationtheory and dynamical systems theory, and asymptotic analysis.These tools are used to give theoretical understanding to problemsof border zone and ischemia related cardiac arrhythmias, APDinstabilities and their relationship to the onset and developmentof reentrant arrhythmias, mechanical alternans deriving frominstabilities of the calcium handling system or crossbridgedynamics, the success or failure of defibrillation, andoscillatory autoimmune responses. Serious attention is given toobtaining agreement between theoretical results and experimentalfindings. To that end collaborations with experimentalistsexamine gap junction coupling of rabbit myocytes, the effect ofhypertrophy on excitation-contraction coupling through thesodium-calcium exchanger, the effect of mutations on the delayedrectifier potassium channel that lead to increased proclivityfor fatal arrhythmias, and the role of mimicry in oscillatorydynamics of cystic fibrosis in mice. Excitability is one of the most important features enablingsignalling in physiological systems. It is crucial to theoperation of nerves, muscle, the heart, the immune system, andblood clotting as well as numerous gene regulatory networks. Thespecific goal of this project is to gain an improved theoreticalunderstanding of how the normal function of systems that rely onexcitability can go awry, exhibiting pathological behaviors, andhow these pathological behaviors might be controlled orprevented. The behaviors that are studied here are the result ofinterwoven interactions of simpler behaviors, resulting incomplex behaviors that defy understanding through intuition orsimple means. The investigation therefore focuses on thedevelopment and analysis of mathematical models of thesebehaviors in the belief that substantial new and importantinsights can be gained, and new hypotheses formulated that arenot apparent from phenomenological descriptions. While thisproject has as its main focus several problems relating toabnormalities of the cardiac cycle, it extends far beyond to manyareas of physiology where common principles are involved. Theproject includes a significant training component for students.