Temporal and spatio-temporal nonlinear systems
Abstract
This thesis contains two projects that are quite different from each other. Both projects involve analyzing the behavior of nonlinear systems. In the first project a multiple time scale analysis of two four-variable models of the peroxidase-oxidase reaction, the DOP and the Olsen models, is carried out. It is shown that autonomous limit cycle oscillations are exhibited by the fast subsets of these two models, but only in certain regions of parameter space, confirming the prior suggestion that the slow variable, (NADH), is not essential for oscillatory behavior. By treating (NADH) as a driving term, it is found that the slow variable is essential for oscillatory behavior over other ranges of parameter values and is always essential for complex oscillatory and chaotic behavior. Further study of the driven system allows for the identification of a set of rules for applying parametric driving to a bistable system in order to generate a more complete Farey sequence from a truncated Farey sequence. These conclusions are used to compare the very similar DOP and Olsen models to account for the difference in their Farey sequences and routes to chaos. The second project introduces an excitable system of mapping equations that has many nerve-like characteristics and is capable of modeling both inhibitory and excitatory neurons. The mapping system is used in coupled map lattices (CMLs) to simulate "healthy" neural tissue and "unhealthy" neural tissue that contains a focal region (or regions). Focal regions are simulated with a patch of purely excitatory cells. Results from the CML simulations are consistent with the thesis that epileptic seizures may occur when a sufficiently large focal region (or regions) is stimulated adequately while the inhibitory activity of the neural tissue connected to the focal region is adequately reduced. The required reduction in inhibitory activity may only be a small percentage of the inhibitory activity of the healthy neural tissue.
Degree
Ph.D.
Subject Area
Chemistry|Anatomy & physiology|Animals|Neurology
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