Date of Award

Spring 2005

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

Robert M. Miura

Fourth Advisor

David C. Stickler

Fifth Advisor

G. Miller Jonakait

Abstract

A conditional oscillator is one that requires input to oscillate. An example of such is the gastric mill network of the stomatogastric ganglion of the crab Cancer borealis which requires modulatory input from outside the stomatogastric ganglion and fast input from the pyloric network of the animal in order to become active. This dissertation studies how the frequency of the gastric mill network is determined when it is simultaneously subjected to two different rhythmic inputs whose timing may be mismatched. We derive a mathematical model of the gastric mill network and deduce that the difference in timing between the pyloric and modulatory inputs is crucial in determining what effect it will have on the frequency of the gastric mill network. Over a certain range of the time mismatch, the pyloric input plays no role in determining the network frequency, while in another range of the time mismatch, both inputs work together to determine the frequency. The existence and stability of periodic solutions to the modeling set of equations are obtained analytically using geometric singular perturbation theory and an analytic approximation of the frequency is obtained. The results are validated through numerical simulations of the model and are shown to extend to a detailed Hodgkin-Huxley type compartmental model of the gastric mill network. Comparisons to experiments are also presented.

Included in

Mathematics Commons

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