Coupled Cell Systems Research Articles

Towards a classification of networks with asymmetric inputs

Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart et al (2003 SIAM J. Appl. Dyn. Syst. 2 609–646), Golubitsky et al (2005 SIAM J. Appl. Dyn. Syst. 4 78–100) and Field (2004 Dyn. Syst. 19 217–243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields—these networks are said to be ordinary differential equation (ODE)-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n − 1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n − 1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

Hidden Symmetries, Coupled Networks and Equivariant Degrees

Hidden symmetries have been previously explored in the context of coupled cell networks and coupled cell systems. These include interior symmetry, quotient symmetry and quotient interior symmetry. We introduce here an equivariant degree theory that incorporates these different forms of hidden symmetry based on lattice structures of synchrony subspaces. The result is a unified theory capable of treating synchrony-breaking Hopf bifurcation problems in coupled cell systems with various forms of hidden symmetries, which leads to full topological classifications of bifurcating branches arising from single bifurcation points under mutual influence of these hidden symmetries.

Synchrony and Antisynchrony in Weighted Networks

We consider weighted coupled cell networks, that is networks where the interactions between any two cells have an associated weight that is a real valued number. Weighted networks are ubiquitous in real-world applications. We consider a dynamical systems perspective by associating with each network a set of continuous dynamical systems, the ones that respect the graph structure of the network. For weighted networks it is natural for the admissible coupled cell systems to have an additive input structure. We present a characterization of the synchrony subspaces and the antisynchrony subspaces for a weighted network depending on the restrictions that are imposed in their admissible input-additive coupled cell systems. These subspaces are flow invariant by those systems and are generalized polydiagonal subspaces, that is, are characterized by conditions on the cell coordinates of the types $x_i = x_j$ and/or $x_k = -x_l$ and/or $x_m=0$. The existence and identification of the synchrony and antisynchrony subspaces for a weighted network are deeply relevant from the applications and dynamics points of view. Our characterization of the synchrony and antisynchrony subspaces of a weighted network follows from our results where we give necessary and sufficient conditions for a generalized polydiagonal to be invariant by the adjacency matrix and/or the Laplacian matrix of the network.

Characterization of fundamental networks

AbstractIn the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

The steady-state lifting bifurcation problem associated with the valency on networks

In this paper we consider homogeneous coupled cell networks with asymmetric inputs — networks where each cell receives exactly one input of each edge type. The coupled cell systems associated with a network are the dynamical systems that respect the network structure. There are subspaces, determined solely by the network structure, that are flow-invariant under any such coupled cell system — the synchrony subspaces. For a homogeneous network with asymmetric inputs, one of the eigenvalues of the Jacobian matrix of any coupled cell system at an equilibrium in the full-synchrony subspace corresponds to the valency of the network. In this work, we study the codimension-one steady-state bifurcations of coupled cell systems with a bifurcation condition associated with the valency. We start by giving an adaptation of the Perron–Frobenius Theorem for the eigenspace associated with the valency showing that the dimension of that eigenspace equals the number of the network source components. A network source component is a strongly connected component of the network whose cells receive inputs only from cells in the component. Each synchrony subspace determines a smaller network called quotient network. The lifting bifurcation problem addresses the issue of understanding when the bifurcation branches of a network can be lifted from one of its quotient networks. We consider the lifting bifurcation problem when the bifurcation condition is associated with the valency. We give sufficient conditions on the number of source components for the answer to the lifting bifurcation problem to be positive and prove that those conditions are necessary and sufficient for a class of networks.

Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems

We prove that steady state bifurcations in finite-dimensional dynamical systems that are symmetric with respect to a monoid representation generically occur along an absolutely indecomposable subrepresentation. This is stated as a conjecture in B. Rink and J. Sanders, "Coupled cell networks and their hidden symmetries", SIAM J. Math. Anal., 46 (2014). It is a generalization of the well-known fact that generic steady state bifurcations in equivariant dynamical systems occur along an absolutely irreducible subrepresentation if the symmetries form a group - finite or compact Lie. Our generalization also includes non-compact symmetry groups. The result has applications in bifurcation theory of homogeneous coupled cell networks as they can be embedded (under mild additional assumptions) into monoid equivariant systems.

Burst synchronization transition of intracellular Ca2+ oscillations in coupled cell systems

Considering four coupled identical and non-identical bursters with different topological types, both of the different synchronization transition paths and influence of the oscillatory patterns on synchronization are studied. Our results reveal that the phenomena of multi-time scale, phase-locking, chaos and anti-phase synchronization transitions can be separately found in the coupled systems by varying the coupling strength. The above analysis of synchronization transition could provide us with a further clue for better understanding the mechanism of intracellular Ca2+ signaling.

Synchrony Branching Lemma for Regular Networks

Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an analogue of the Equivariant Branching Lemma in the context of regular networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sufficient conditions for the existence of steady-state bifurcation branches with some submaximal synchrony. Those conditions only depend on the network structure, but the lattice structure of the balanced colorings is not sufficient to decide which synchrony subspaces support a steady-state bifurcation branch.

Patterns of synchrony for feed-forward and auto-regulation feed-forward neural networks.

We consider feed-forward and auto-regulation feed-forward neural (weighted) coupled cell networks. In feed-forward neural networks, cells are arranged in layers such that the cells of the first layer have empty input set and cells of each other layer receive only inputs from cells of the previous layer. An auto-regulation feed-forward neural coupled cell network is a feed-forward neural network where additionally some cells of the first layer have auto-regulation, that is, they have a self-loop. Given a network structure, a robust pattern of synchrony is a space defined in terms of equalities of cell coordinates that is flow-invariant for any coupled cell system (with additive input structure) associated with the network. In this paper, we describe the robust patterns of synchrony for feed-forward and auto-regulation feed-forward neural networks. Regarding feed-forward neural networks, we show that only cells in the same layer can synchronize. On the other hand, in the presence of auto-regulation, we prove that cells in different layers can synchronize in a robust way and we give a characterization of the possible patterns of synchrony that can occur for auto-regulation feed-forward neural networks.

Topology and bifurcations in Hamiltonian coupled cell systems

ABSTRACTThe coupled cell formalism is a systematic way to represent and study coupled nonlinear differential equations using directed graphs. In this work, we focus on coupled cell systems in which individual cells are also Hamiltonian. We show that some coupled cell systems do not admit Hamiltonian vector fields because the associated directed graphs are incompatible. In broad terms, we prove that only systems with bidirectionally coupled digraphs can be Hamiltonian. Aside from the topological criteria, we also study the linear theory of regular Hamiltonian coupled cell systems, i.e. systems with only one type of node and one type of coupling. We show that the eigenspace at a codimension-one bifurcation from a synchronous equilibrium of a regular Hamiltonian network can be expressed in terms of the eigenspaces of the adjacency matrix of the associated directed graph. We then prove results on steady-state bifurcations and a version of the Hamiltonian Hopf theorem.

Heteroclinic network dynamics on joining coupled cell networks

ABSTRACTWe present a method of combining coupled cell systems to get dynamics supporting robust simple heteroclinic networks given by the product of robust simple heteroclinic networks (cycles). We consider coupled cell networks, with no assumption on symmetry, and combine them via the join operation. Assuming that the dynamics of the component networks supports robust simple heteroclinic cycles or networks, we show that the join dynamics realizes a more complex heteroclinic network given by the product of those cycles or networks. Moreover, the equilibria in the product heteroclinic network correspond to partially synchronous states. Assuming no symmetry for the component coupled cell networks, one of the key points for the existence and robustness of the heteroclinic dynamics is the flow-invariant subspaces forced by the network structure – the synchrony subspaces. The other key point is that the (linear) stability of equilibria in the join dynamics is determined by the (linear) stability of equilibria in the component dynamics. The first point depends only at the network structure of the component networks. The second one depends both at the components network structures and the convenient choice of the join coupling. The proposed method is general and can be applied to the join of symmetric or asymmetric networks. Here, we illustrate it through the join of two asymmetric coupled cell networks where robust simple heteroclinic cycles between fully synchronous equilibria occur. We obtain robust simple heteroclinic networks for the join dynamics between partially synchronized equilibria for the associated join network.

Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics.

We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential equation. Symmetry is one mechanism that creates patterns of synchrony and phase-locking. In general networks, there is another: balanced colorings of the cells. A symmetric network may have anomalous patterns of synchrony and phase-locking that are not consequences of symmetry. We introduce basic notions on coupled cell networks and their associated systems of admissible differential equations. Periodic states also possess spatio-temporal symmetries, leading to phase relations; these are classified by the H/K theorem and its analog for general networks. Systematic general methods for computing the stability of synchronous states exist for symmetric networks, but stability in general networks requires methods adapted to special classes of model equations.

Response Ability to External Signal Enhanced by Biological Spatial Configuration in Coupled Hindmarsh-Rose Neural System

The effect of realistic topology configuration of intercellular connections on the response ability in coupled cell system is numerically investigated by using the Hindmarsh-Rose model. For the proper coupling intensity, we set the control parameter to be near the critical value, and the external stimulus is introduced to the first cell in coupled system. It is found that, on one hand, when the cells are coupled with some proper topological structures, the external stimulus could transmit through the system, and shows better response ability and higher sensitivity. On the other hand, the influence of topological configuration on the synchronous ability and selection effect of neural system are also discussed. Our results display that the topology of coupled system may play an important role in the process of signal propagation, which could help us to understand the coordinated performance of cells in tissue.

A note on fractional feed‐forward networks

We simulate a fractional feed‐forward network. This network consists of three coupled identical ‘cells’ (aka, oscillators). We study the behaviour of the associated coupled cell system for variation of the order of the fractional derivative, 0 < α < 1. We consider the Caputo derivative, approximated by the Grünwald–Letnikov approach, using finite differences of fractional order. There is observed amplification of the small signals by exploiting the nonlinear response of each oscillator near its intrinsic Hopf bifurcation point for each value of α. The value of the Hopf bifurcation point varies with the order of the fractional derivative α. Copyright © 2016 John Wiley & Sons, Ltd.

Synchrony and Elementary Operations on Coupled Cell Networks

Given a finite graph (network), let every node (cell) represent an individual dynamics given by a system of ordinary differential equations, and every arrow (edge) encode the dynamical influence of the tail node on the head node. We have then defined a coupled cell system that is associated with the given network structure. Subspaces that are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure are called synchrony subspaces. They are completely determined by the network structure and form a complete lattice under set inclusions. We analyze the transition of the lattice of synchrony subspaces of a network that is caused by structural changes in the network topology, such as deletion and addition of cells or edges, and rewirings of edges. We give sufficient, and in some cases both sufficient and necessary, conditions under which lattice elements persist or disappear.

Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?

Regular synchrony lattices for product coupled cell networks.

There are several ways for constructing (bigger) networks from smaller networks. We consider here the cartesian and the Kronecker (tensor) product networks. Our main aim is to determine a relation between the lattices of synchrony subspaces for a product network and the component networks of the product. In this sense, we show how to obtain the lattice of regular synchrony subspaces for a product network from the lattices of synchrony subspaces for the component networks. Specifically, we prove that a tensor of subspaces is of synchrony for the product network if and only if the subspaces involved in the tensor are synchrony subspaces for the component networks of the product. We also show that, in general, there are (irregular) synchrony subspaces for the product network that are not described by the synchrony subspaces for the component networks, concluding that, in general, it is not possible to obtain the all synchrony lattice for the product network from the corresponding lattices for the component networks. We also make the following remark concerning the fact that the cartesian and Kronecker products, as graph operations, are quite different, implying that the associated coupled cell systems have distinct structures. Although, the kinds of dynamics expected to occur are difficult to compare, we establish an inclusion relation between the lattices of synchrony subspaces for the cartesian and Kronecker products.

Strange patterns in one ring of Chen oscillators coupled to a ‘buffer’ cell

We study curious dynamical patterns appearing in networks of one ring of cells coupled to a ‘buffer’ cell. Depending on how the cells in the ring are coupled to the ‘buffer’ cell, the full network may have a nontrivial group of symmetries or a nontrivial group of ‘interior’ symmetries. This group is Z3 in the unidirectional case and D3 in the bidirectional case. We simulate the coupled cell systems associated with the networks and obtain steady states, rotating waves, quasiperiodic behavior, and chaos. The different patterns seem to arise through a sequence of Hopf, period-doubling, and period-halving bifurcations. The behavior of the systems with exact symmetry are similar to the ones with ‘interior’ symmetry. The network architecture appears to explain some features, whereas the properties of the Chen oscillator, used to model cells’ internal dynamics, may explain others. We use XPPAUT and MATLAB to numerically compute the relevant states.

On bifurcations in lifts of regular uniform coupled cell networks

A lift of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

The Lattice of Synchrony Subspaces of a Coupled Cell Network: Characterization and Computation Algorithm

Coupled cell systems are networks of dynamical systems (the cells), where the links between the cells are described through the network structure, the coupled cell network. Synchrony subspaces are spaces defined in terms of equalities of certain cell coordinates that are flow-invariant for all coupled cell systems associated with a given network structure. The intersection of synchrony subspaces of a network is also a synchrony subspace of the network. It follows, then, that, given a coupled cell network, its set of synchrony subspaces, taking the inclusion partial order relation, forms a lattice. In this paper we show how to obtain the lattice of synchrony subspaces for a general network and present an algorithm that generates that lattice. We prove that this problem is reduced to obtain the lattice of synchrony subspaces for regular networks. For a regular network we obtain the lattice of synchrony subspaces based on the eigenvalue structure of the network adjacency matrix.