| Oscillatory behavior is observed in ecological (predator-prey model), physiological (neuronal signal propagation), chemical (oxidation of malonic acid), and in many other disciplines of science. Mathematics has been used extensively to study both the qualitative and quantitative aspects of this behavior, such as the Lotka-Volterra equations for predator-prey modeling, Hodgkin-Huxley equations for neuronal signal propagation, and Belousov-Zhabatinskii equations for the oxidation of malonic acid.;This dissertation examines spatial effects on the single-cell analog model for the bursting electrical activity in the pancreatic beta-cells using reaction-diffusion equations. Reaction-diffusion systems are known to possess many types of solutions and exhibit a rich variety of spatial and spatiotemporal patterns. One class of solutions is the traveling wave solution, in which the solution connects two steady states (bistability) of the reaction equation and moves with fixed velocity. Analytic methods are employed to seek traveling wave solutions in the beta-cell model. Numerical methods are used and discovered a further connection between a steady state and a stable periodic solution. We attack this traveling-wave-like solution via reduction of the reaction-diffusion equations into normal form. Analysis is done on the resulting complex Ginzburg-Landau equation. Comparisons are made between the numerical results from the reaction-diffusion equations and the complex Ginzburg-Landau equation. Finally we address the class of coherent structures the complex Ginzburg-Landau equation possesses and how it relates to the dynamics of the beta-cell model.;In the early 70's, scientists began to study the mechanism behind the release of insulin in the pancreas. They found a strong correlation between the rate of insulin release and the membrane potential of the beta-cells in the pancreas: insulin is released when the cells are depolarized and the membrane potential undergoes rapid oscillates, and not released when the cells are hyperpolarized. The first mathematical model for the electrophysiological activity in the beta-cells was proposed in the early 80's and it duplicated results strikingly similar to those found in experiments. The model consisted of two time-scales, which resulted in two oscillatory stages: fast oscillations embedded inside slow oscillations. |
