Mathematical Sciences: Dynamic Bifurcation Phenomena in Excitable Systems

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The modeling and analysis of dynamic bifurcation phenomena is important in science and engineering because experiments are often performed by dynamically tuning one or more parameters until destabilization causes a transition to a new observed state or behavior. Using numerical and computational methods, this project will investigate accommodation phenomena in spatially-unrestricted axons and myelinated nerve segments. A primary goal is to establish the interrelationship between passive loading (both conductive and capacitive) and accommodation in nerve cells and how it affects signaling. The project requires computational and analytic bifurcation studies of reaction-diffusion equations with slowly varying boundary and interface conditions. Another objective is to classify and explore how different kinds of random fluctuations influence accommodation in nerve models. Stochastic perturbation methods for exit time estimates and numerical simulations will be employed. In neuroscience, threshold and accommodation experiments at the cellular level are performed by slowly depolarizing an excitable membrane by ramping or stepping up an applied current. Such experiments to investigate accommodation phenomena in spatially-unrestricted axons and myelinated nerve segments are important for understanding why and how conduction failures occur in partially demyelinated nerve; debilitating diseases such as Multiple Sclerosis can induce lesions that partially demyelinate the myelin sheath. In addition, many of the tools developed in this project are useful for studying dynamic bifurcation phenomena, which occur in many other areas of science and engineering.