Date of Award

Spring 2008

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Amitabha Koshal Bose

Second Advisor

Farzan Nadim

Third Advisor

Jorge P. Golowasch

Fourth Advisor

Horacio G. Rotstein

Fifth Advisor

Georgi Medvedev

Abstract

This dissertation studies the roles of gap junctions in the dynamics of neuronal networks in three distinct problems. First, we study the circumstances under which a network of excitable cells coupled by gap junctions exhibits sustained activity. We investigate how network connectivity and refractory length affect the sustainment of activity in an abstract network. Second, we build a mathematical model for gap junctionally coupled cables to understand the voltage response along the cables as a function of cable diameter. For the coupled cables, as cable diameter increases, the electrotonic distance decreases, which cause the voltage to attenuate less, but the input of the second cable decreases, which allows the voltage of the second cable to attenuate more. Thus we show that there exists an optimal diameter for which the voltage amplitude in the second cable is maximized. Third, we investigate the dynamics of two gap-junctionally coupled theta neurons. A single theta neuron model is a canonical form of Type I neural oscillator that yields a very low frequency oscillation. The coupled system also yields a very low frequency oscillation in the sense that the ratio of two cells' spiking frequencies obtains the values from a very small number. Thus the network exhibits several types of solutions including stable suppressed and 1 N spiking solutions. Using phase plane analysis and Denjoy's Theorem, we show the existence of these solutions and investigate some of their properties.

Included in

Mathematics Commons

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