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Neurons and endocrine cells display various patterns of electrical activity, including periodic bursting. Bursting oscillations are characterized by the alternation between periods of fast spiking (the active phase) and quiescent periods (the silent phase), and are accompanied by slow variations in one or more slowly changing variables. Bursts are often more efficient than periodic spiking in evoking the release of neurotransmitter or hormone. The technique of two-fast/one-slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the mechanisms for bursting in endocrine cells are just quantitative variations of those for bursting in neurons. The two-fast/one-slow analysis used to make the transition between plateau and pseudo-plateau bursting, and to understand the dynamics of plateau bursting is of limited use for pseudo-plateau bursting. Using a one-fast/two-slow analysis technique, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. Finally we show the relationship between the two-fast/one-slow analysis and one-fast/two-slow analysis techniques.
A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Bibliography Note
Includes bibliographical references.
Advisory Committee
Richard Bertram, Professor Directing Dissertation; Paul Q. Trombley, University Representative; Jöel Tabak, Committee Member; Nick G. Cogan, Committee Member; Xiaoming Wang, Committee Member.
Publisher
Florida State University
Identifier
FSU_migr_etd-5219
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