Modeling and Analyzing Pattern Formation Mechanisms Derived from Nonlinear Physiological and Neuronal Systems

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This research is aimed at modeling and analyzing temporal, spatial, and spatio-temporal pattern-formation mechanisms arising from nonlinear systems in cell physiology and neural sciences. Analytical and numerical methods are employed/developed in the study of the nonlinear ordinary, partial, and integral differential equations that describe these systems. Unlike closed physical systems that evolve toward a state of increased disorder, biological systems are typically open and evolving into structures/patterns of increasing complexity. Understanding the mechanisms driving the formation of these patterns is one of the major goals of biological research and is vital in revealing their functions and developing potential ways of controlling them. Due to the nonlinear and dynamical nature of these processes, mathematical methods are often essential in achieving an understanding that is otherwise difficult or even impossible to obtain using only experimental tools. The topics of this proposed research are: (1) single-cell behaviors and synchronized rhythmic secretion of hormones from neuroendocrine cells; (2) general synchronization mechanisms in coupled cells; (3) the wave propagation in excitable single- and multi-cellular media; and (4) effects of noise in the formation of biological patterns. When developing models, we use data and results from most recent experiments. Once a model is developed, we seek in-depth understanding of the mathematics behind each novel behavior. A model describing the synchronization and generation of pulsatile GnRH release in diffusely distributed GnRH neurons is a good example. The novelty and potential impact of the proposed research is evidenced by the close relationship between these projects and the frontier developments in experimental cell physiology. Progress in the mathematical understanding of the equations arising in these models will deepen our understanding of nonlinear differential equations that were poorly explored previously and has the potential of being applied to solving other problems of interest in related areas.