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Author(s): Ian Stewart, Martin Golubitsky and Marcus Pivato
Article Title: Symmetry Groupoids and Patterns of Synchrony in Coupled
Cell Networks
Year of publication:2003
Link to published version:http://dx.doi.org/10.1137/S1111111103419896
Publisher statement: None
SIAM J. APPLIED DYNAMICAL SYSTEMS
c
2003 Society for Industrial and Applied Mathematics
Vol. 2, No. 4, pp. 609–646
Symmetry Groupoids and Patterns of Synchrony
in Coupled Cell Networks
∗
Ian Stewart
†
, Martin Golubitsky
‡
, and Marcus Pivato
§
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 in-
formation 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.
Key words. coupled systems, synchrony, groupoids, symmetry
AMS subject classifications. 34C14, 34C15, 20L05
DOI. 10.1137/S1111111103419896
1. Introduction. We use the term cell to indicate a system of ODEs. A coupled cell
system is a set of cells with coupling, that is, a dynamical system whose variables correspond
to cells, such that the output of certain cells affects the time-evolution of other cells. The
salient feature of a coupled cell system is that the output from each cell is considered to be
significant in its own right. A coupled cell system is not merely a system of ODEs but a
system of ODEs equipped with canonical observables—the individual cells (see [8]). From a
mathematical point of view these output signals can be compared, and this observation leads
to a variety of notions of “synchrony.” For surveys, see Boccaletti, Pecora, and Pelaez [2] and
Wang [14].
In this paper we discuss the architecture of a coupled cell system: which cells influence
which, which cells are “identical,” and which couplings are “identical.” We focus on how the
∗
Received by the editors February 19, 2003; accepted for publication (in revised form) by G. Kriegsmann August
29, 2003; published electronically December 22, 2003.
http://www.siam.org/journals/siads/2-4/41989.html
†
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (ins@maths.warwick.ac.uk).
‡
Department of Mathematics, University of Houston, Houston, TX 77204-3476 (mg@uh.edu). The work of this
author was supported in part by NSF grants DMS-0071735 and DMS-0244529 and ARP grant 003652-0032-2001.
§
Department of Mathematics, Trent University, Peterborough, ON, Canada K9L 1Z6 (pivato@xaravve.trentu.ca).
609
610 I. STEWART, M. GOLUBITSKY, AND M. PIVATO
system architecture leads naturally to synchrony. To do this, we must define carefully when
two cells or two couplings are “identical” or “equivalent.” Indeed, the main point of this
paper is to provide a general mathematical foundation for these ideas. This foundation uses
the algebraic structure of groupoids (see Brandt [1] and Higgins [10]) and greatly generalizes
the uses of symmetry in coupled cell systems that we have explored previously [7, 8].
Our conventions do not rule out “two-way” coupling, in which cells A and B both influence
each other. We represent such a state of affairs by having A coupled to B and B coupled to A.
We also do not rule out coupling where cells A and B both influence cell C. Here we consider
both A and B as being coupled to C. We do not assume the effects of A and B to be additive;
in fact, the time-evolution of cell C can in principle be any (smooth) function of the states of
C, A, and B.
In this paper we develop an abstract formalism for coupled cell systems, using simple
examples that have no particular role in applications, but it is worth noting that coupled cell
systems are used to model a variety of physically interesting systems. For examples, see [8]
and references therein. We intend to develop applications of the formalism derived here in
future work.
In this section we illustrate some central issues by discussing several examples.
Two-cell systems. We begin with the simplest system of two identical cells (with coordi-
nates x
1
and x
2
in R
k
). Without making any specific assumption of the form of the “internal
dynamics” of each cell or the form of the “coupling between cells,” the differential equations
for the coupled system have the form
˙x
1
= f(x
1
,x
2
),
˙x
2
= f(x
2
,x
1
);
(1.1)
that is, the same function f governs the dynamics of both cells. There are three issues that
we discuss concerning system (1.1): the graph (diagram, network) associated to a coupled cell
system, symmetry, and synchrony.
Informally, the “network” of a coupled cell system is a finite directed graph whose nodes
represent cells and whose edges represent couplings. Nodes are labeled to indicate “equivalent”
cells, which have the same phase space and the same internal dynamic. Edges are labeled to
indicate “equivalent” couplings. The graph associated to system (1.1) is given in Figure 1.We
think of this graph as representing a pair of systems of differential equations in the following
way. The two cells are indicated by identical symbols—so they have the same state variables.
That is, the coordinates x
1
of cell 1 and x
2
of cell 2 lie in the same phase space R
k
. Since we
can interchange cells 1 and 2 without changing the graph, we assume that the same is true
for the system of differential equations and that they must have the form (1.1). Note that for
this interchange to work, the arrow 1 → 2 must be the same as the arrow 2 → 1.
12
Figure 1. A two-cell network.
SYMMETRY GROUPOIDS AND PATTERNS OF SYNCHRONY 611
The discussion in the previous paragraph can be summarized by the following: the per-
mutation σ(x
1
,x
2
)=(x
2
,x
1
) is a symmetry of the system (1.1). Indeed, more is true: every
system of differential equations on R
k
× R
k
that is equivariant with respect to σ has the form
(1.1). That is, abstractly the study of pairs of identical cells that are identically coupled is
the same as the study of σ-equivariant systems. Two consequences follow from this remark.
First, synchrony in two-cell systems (solutions such that x
1
(t)=x
2
(t) for all time t)isa
robust phenomenon and should not be viewed as surprising. Second, time-periodic solutions
can exhibit a kind of generalized synchrony in which the two cells oscillate a half-period out
of phase.
The first remark can be restated as follows: the diagonal subspace V = {x
1
= x
2
}⊂
R
k
× R
k
is flow-invariant for every system (1.1). This remark can be verified in two ways. By
inspection restrict (1.1)toV, obtaining
˙x
1
= f(x
1
,x
1
),
˙x
1
= f(x
1
,x
1
).
It follows that if the initial conditions for a solution satisfy x
1
(0) = x
2
(0), then x
1
(t)=x
2
(t)
for all time t, and V is flow-invariant. Alternately, we can observe that V is the fixed-point
subspace Fix(σ), and fixed-point subspaces are well known to be flow-invariant.
The second remark is related to general theorems about spatio-temporal symmetries of
time-periodic solutions to symmetric systems of ODEs. There are two types of theorems here:
existence theorems, asserting that certain spatio-temporal symmetries are possible, and bifur-
cation theorems, describing particular scenarios that can generate such solutions. The H/K
theorem [4, 7] is an existence theorem; indeed, it states necessary and sufficient conditions for
periodic solutions with a given spatio-temporal symmetry group to be possible. In particular,
it implies the existence of functions f having time-periodic solutions of period T satisfying
x
2
(t)=x
1
(t + T/2)(1.2)
as long as the phase space of each cell has dimension k ≥ 2. So states with this type of
spatio-temporal pattern can exist. Indeed, they can exist robustly (that is, they can persist
when f is perturbed) and are therefore typical in the appropriate coupled cell systems. In
this case, we can say more: such solutions can arise through Hopf bifurcation. This is a
consequence of the general theory of symmetric Hopf bifurcation, [7, 8, 9]. (Note that when
k = 1, nonconstant periodic solutions satisfying (1.2) must intersect the diagonal V and hence
be in V for all time: this is a contradiction.)
A three-cell network. Consider the three-cell network illustrated in Figure 2. The systems
of differential equations corresponding to this network have the form
˙x
1
= f(x
1
,x
2
),
˙x
2
= g(x
2
,x
1
,x
3
),
˙x
3
= f(x
3
,x
2
),
(1.3)
where g(x
2
,x
1
,x
3
)=g(x
2
,x
3
,x
1
), x
1
,x
3
∈ R
k
, and x
2
∈ R
. Note that all such systems are
equivariant with respect to the permutation τ (x
1
,x
2
,x
3
)=(x
3
,x
2
,x
1
) and that synchronous
612 I. STEWART, M. GOLUBITSKY, AND M. PIVATO
123
Figure 2. A three-cell network with transposition symmetry.
solutions (where x
1
(t)=x
3
(t) for all time t) occur robustly because the “polydiagonal”
subspace W = {x : x
1
= x
3
} is flow-invariant for (1.3).
There are two differences between the three-cell network in Figure 2 and the two-cell
network in Figure 1. First, not all τ -equivariant systems on R
k
× R
× R
k
have the form
(1.3), since in the general τ-equivariant system f can depend nontrivially on both x
1
and x
3
.
So there can be additional structure in coupled cell systems that does not correspond directly
to symmetry. Second, the half-period, out of phase, time-periodic solutions satisfy
x
3
(t)=x
1
(t + T/2) and x
2
(t)=x
2
(t + T/2).(1.4)
In particular, the oscillations in cell 2 are forced by symmetry to occur at twice the frequency
of those in cells 1 and 3. So multirhythms [7] can be forced by the architecture of coupled cell
networks.
Another three-cell network. We now show that robust synchrony is possible in networks
that have no symmetry. Consider the three-cell network in Figure 3. Here we have used two
distinct symbols (square and circle) for cells and three types of arrows for couplings. The
role of these symbols can be seen in the form of the ODE: identical symbols correspond to
identical functions in the appropriate variables.
12
3
Figure 3. A three-cell network without symmetry.
This network has no symmetry, but the network structure forces the “polydiagonal” sub-
space Y = {x : x
1
= x
2
} to be flow-invariant. To verify this point observe that the coupled
cell systems associated with this network have the form
˙x
1
= f(x
1
,x
2
,x
3
),
˙x
2
= f(x
2
,x
1
,x
3
),
˙x
3
= g(x
3
,x
1
),
(1.5)
where x
1
,x
2
∈ R
k
and x
3
∈ R
. Restricting the first two equations to Y yields
˙x
1
= f(x
1
,x
1
,x
3
),
˙x
2
= f(x
1
,x
1
,x
3
),
