Date of Award

7-15-2009

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Andrey Shilnikov - Chair

Second Advisor

Gennady Cymbalyuk - Co-Chair

Third Advisor

Robert Clewley - Co-Chair

Fourth Advisor

Michael Stewart

Fifth Advisor

Igor Belykh

Sixth Advisor

Vladimir Bondarenko

Seventh Advisor

Mukesh Dhamala

Abstract

A multifunctional central pattern generator (CPG) can produce bursting polyrhythms that determine locomotive activity in an animal: for example, swimming and crawling in a leech. Each rhythm corresponds to a specific attractor of the CPG. We employ a Hodgkin-Huxley type model of a bursting leech heart interneuron, and connect three such neurons by fast inhibitory synapses to form a ring. This network motif exhibits multistable co-existing bursting rhythms. The problem of determining rhythmic outcomes is reduced to an analysis of fixed points of Poincare mappings and their attractor basins, in a phase plane defined by the interneurons' phase differences along bursting orbits. Using computer assisted analysis, we examine stability, bifurcations of attractors, and transformations of their basins in the phase plane. These structures determine the global bursting rhythms emitted by the CPG. By varying the coupling synaptic strength, we examine the dynamics and patterns produced by inhibitory networks.

Included in

Mathematics Commons

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