The Existence and Classification of Synchrony-Breaking Bifurcations in Regular Homogeneous Networks Using Lattice Structures

Abstract

For regular homogeneous networks with simple eigenvalues (real or complex), all possible explicit forms of lattices of balanced equivalence relations can be constructed by introducing lattice generators and lattice indices [Kamei, 2009]. Balanced equivalence relations in the lattice correspond to clusters of partially synchronized cells in a network. In this paper, we restrict attention to regular homogeneous networks with simple real eigenvalues, and one-dimensional internal dynamics for each cell. We first show that lattice elements with nonzero indices indicate the existence of codimension-one synchrony-breaking steady-state bifurcations, and furthermore, the positions of such lattice elements give the number of partially synchronized clusters. Using four-cell regular homogeneous networks as an example, we then classify a large number of regular homogeneous networks into a small number of lattice structures, in which networks share an equivalent clustering type. Indeed, some of these networks even share the same generic bifurcation structure. This classification leads us to explore how regular homogeneous networks that share synchrony-breaking bifurcation structure are topologically related.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

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Bifurcations and Dynamics Emergent From Lattice and Continuum Models of Bioactive Porous Media

“…This model arises as a mathematical caricature of a model developed in [Krause et al, 2017] representing fluid and cell interactions in a bioactive porous medium. While lattice-dynamical systems have been studied in a variety of different settings [Buzano & Golubitsky, 1983;Winterbottom, 2004;Wang et al, 2006;Kamei, 2009;Sattinger, 1980], we note that processes defined on finite lattices are less well-studied, and have been shown to exhibit unique dynamics due to the influence of boundaries [Chow et al, 1996;Gillis & Golubitsky, 1997;Golubitsky et al, 2004]. Additionally, our model also exhibits nonlocal coupling across the lattice, which is also of interest from the dynamical systems perspective [Gourley et al, 2001;Billingham, 2004;Hamel & Ryzhik, 2014].…”

Synchrony Branching Lemma for Regular Networks

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