Pattern Formation in Coupled Networks with Inhibition and Gap Junctions
In this dissertation we analyze networks of coupled phase oscillators. We consider systems where long range chemical coupling and short range electrical coupling have opposite effects on the synchronization process. We look at the existence and stability of three patterns of activity: synchrony, clustered state and asynchrony. In Chapter 1, we develop a minimal phase model using experimental results for the olfactory system of Limax. We study the synchronous solution as the strength of synaptic coupling increases. We explain the emergence of traveling waves in the system without a frequency gradient. We construct the normal form for the pitchfork bifurcation and compare our analytical results with numerical simulations. In Chapter 2, we study a mean-field coupled network of phase oscillators for which a stable two-cluster solution exists. The addition of nearest neighbor gap junction coupling destroys the stability of the cluster solution. When the gap junction coupling is strong there is a series of traveling wave solutions depending on the size of the network. We see bistability in the system between clustered state, periodic solutions and traveling waves. The bistability properties also change with the network size. We analyze the system numerically and analytically. In Chapter 3, we turn our attention to a very popular model about network synchronization. We represent the Kuramoto model in its original form and calculate the main results using a different technique. We also look at a modified version and study how this effects synchronization. We consider a collection of oscillators organized in m groups. The addition of gap junctions creates a wave like behavior.
Advisor:Nathan Urban; William Troy; Jonathan Rubin; G. Bard Ermentrout
School:University of Pittsburgh
School Location:USA - Pennsylvania
Source Type:Master's Thesis
Date of Publication:06/29/2007