Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks

Abstract

Abstract. A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only mechanism that can create such states in a coupled cell system and show that it is not. The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information about the input sets of cells. (The input set of a cell consists of that cell and all cells connected to that cell.) The admissible vector fields for a given graph-the dynamical systems with the corresponding internal dynamics and couplings-are precisely those that are equivariant under the symmetry groupoid. A pattern of synchrony is "robust" if it arises for all admissible vector fields. The first main result shows that robust patterns of synchrony (invariance of "polydiagonal" subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an equivalence relation on cells is "balanced." The second main result shows that admissible vector fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled cell network, the "quotient network." The existence of quotient networks has surprising implications for synchronous dynamics in coupled cell systems.

Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

“…Introduction. Stewart, Golubitsky, and Pivato [10] formalize the definition of a coupled cell system in terms of the symmetry groupoid of an associated coupled cell network and prove three general theorems about such networks. First, a set of cells can be robustly synchronous if and only if the cells are in the same equivalence class of some balanced equivalence relation.…”

“…Third, the restriction of a coupled cell system to a synchrony subspace (or polydiagonal) is a coupled cell system associated to the quotient network. The approach in [10] has two difficulties:…”

“…(Dias and Stewart [3] obtain necessary and sufficient conditions, on a network with a balanced relation, for every quotient system to be a restriction of a cell system corresponding to the original network.) (2) Reasonable networks that are not included in the theory developed in [10] can exhibit patterns of robust synchrony. Examples are linear chains with Neumann boundary conditions considered in Epstein and Golubitsky [5] and square arrays of cells with Neumann boundary conditions considered in Gillis and Golubitsky [6].…”

“…We call this the multiarrow formalism for coupled cell networks. Although the abstract definition of this enlarged class of coupled cell networks is more complicated than the more restrictive definition in [10], the multiarrow formalism has the side benefit that quotient systems are more easily defined in the enlarged class and have more convenient properties.…”

“…A cell is a system of ODEs, and a coupled cell system is a collection of N cells with couplings. As discussed in [10], a class of coupled cell systems is defined by a coupled cell network, which is a (directed, labeled) graph that specifies, among other information, which cells are coupled to which. Two cells of the network are input isomorphic (see [10]) if the dynamics of the cells are specified by the same differential equations, up to a permutation of the variables.…”