Date of Award
Dissertation - Restricted
Doctor of Philosophy (PhD)
Mathematics, Statistics and Computer Science
This thesis is intended to explain and reveal some of the underlying common mechanisms that generate lamprey fictive swimming, a periodic neural bursting phenomenon that mimics the oscillatory pattern of controlled swimming. This investigation is accomplished through mathematical modeling. Analysis of these models utilizes techniques from non-linear dynamical systems theory. This topic is motivated by attempting to bridge the gap between Buchanan's proposed lamprey spinal cord neural circuit and Kopell's general mathematical framework for coupled chains of oscillators. The dynamical systems constructed from the proposed neural circuit present a rich set of behaviors, which qualitatively agree with experimental data. Some conclusions and hypotheses can be drawn concerning the biological subjects, such as the rhythmicity of the unit circuit, the influence of certain drugs on the oscillations, the origin of the observed slow modulation and the observed constant phase lag along the length of lamprey spinal cord. The mathematical aspect of this thesis focuses on the subjects in the research of nonlinear dynamical systems, such as the existence, uniqueness and stability of a limit cycle solution, the Hopf bifurcation phenomenon and the construction of a torus containing attracting periodic solutions in space. The dynamical analysis is needed to apply Kopell 's oscillator results.