Phase models and clustering in networks of oscillators with delayed coupling

2016/07/19 by Sue Ann Campbell, Zhen Wang, Campbell, Sue Ann +1
Computer Science · Mathematics · Physics and Astronomy · #Neural Networks Stability and Synchronization #Nonlinear Dynamics and Pattern Formation #math.DS #msc:34C15 #nlin.AO #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.1607.05759

arxiv created 2016/07/19 · arxiv updated 2016/07/21

Abstract

We consider a general model for a network of oscillators with time delayed, circulant coupling. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to study the existence and stability of cluster solutions. Cluster solutions are phase locked solutions where the oscillators separate into groups. Oscillators within a group are synchronized while those in different groups are phase-locked. We give model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.

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