Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. The connection between symmetries and cluster synchronization is experimentally confirmed in the context of real networks with heterogeneities and noise using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony without disturbing the others. Our analysis shows that such behaviour will occur in a wide variety of networks and node dynamics. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behaviour of networks ranging from neural to social.

/pdf/cluster-synchronization-and-isolated-desynchronization-in-2eygmv966p.pdf

Cluster synchronization and isolated desynchronization in complex networks with symmetries

A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that preserve
the network topology, has existed for some time. Global network symmetries
impose strong constraints on the corresponding dynamical systems,
which affect equilibria, periodic states, heteroclinic cycles, and even chaotic
states. In particular, the symmetries of the network can lead to synchrony,
phase relations, resonances, and synchronous or cycling chaos.
Symmetry is a rather restrictive assumption, and a general theory of networks
should be more flexible. A recent generalization of the group-theoretic
notion of symmetry replaces global symmetries by bijections between certain
subsets of the directed edges of the network, the ‘input sets’. Now the symmetry
group becomes a groupoid, which is an algebraic structure that resembles
a group, except that the product of two elements may not be defined. The
groupoid formalism makes it possible to extend group-theoretic methods to
more general networks, and in particular it leads to a complete classification
of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the
network.
Many phenomena that would be nongeneric in an arbitrary dynamical
system can become generic when constrained by a particular network topology.
A network of dynamical systems is not just a dynamical system with
a high-dimensional phase space. It is also equipped with a canonical set of
observables—the states of the individual nodes of the network. Moreover, the
form of the underlying ODE is constrained by the network topology—which
variables occur in which component equations, and how those equations relate
to each other. The result is a rich and new range of phenomena, only a few of
which are yet properly understood.

/pdf/nonlinear-dynamics-of-networks-the-groupoid-formalism-b30bad6eex.pdf

Nonlinear dynamics of networks: the groupoid formalism

http://web.mit.edu/nsl/www/preprints/Polyrhythms05.pdf

Stable concurrent synchronization in dynamic system networks

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

/pdf/mathematical-frameworks-for-oscillatory-network-dynamics-in-4d49dtz09j.pdf

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

http://www.mate.tue.nl/mate/pdfs/2523.pdf

Partial synchronization: from symmetry towards stability

We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.

Stability results for steady, spatially periodic planforms

We continue the study of arrays of coupled identical cells that possess both global and internal symmetries, begun in part I. Here we concentrate on the `direct product' case, for which the symmetry group of the system decomposes as the direct product of the internal group and the global group . Again, the main aim is to find general existence conditions for symmetry-breaking steady-state and Hopf bifurcations by reducing the problem to known results for systems with symmetry or separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and -axial subgroups of the direct product and to relate them to axial and -axial subgroups of the two groups and . We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where and . In particular we show that for Hopf bifurcation the case n = 4 modulo 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n.

https://people.mbi.ohio-state.edu/golubitsky.4/reprintweb-0.5/output/papers/wreath.pdf

Coupled cells with internal symmetry: II. Direct products

Stability Results for Steady, Spatially--Periodic Planforms

When solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to existence near a steady-state bifurcation from a translation-invariant equilibrium.

Planforms in two and three dimensions

In this note we discuss the structure of systems of coupled cells (which we view as systems of ordinary differential equations) where symmetries of the system are obtained through the group L of global permutations of the cells and the group ℒof local internal symmetries of the dynamics in each cell. We show that even when the cells are assumed to be identical with identical coupling, the way that L and ℒ combine to form the total symmetry group of the system Γ depends on properties of the coupling. We illustrate this point by showing how the combination of ℒ with L can lead to a symmetry group Γ that is either a direct product or a wreath product. The symmetry group has strong implications for the dynamics of the system of cells, and the distinction between the two cases is substantial. This has important implications for the modeling of systems by coupled cells.

https://people.mbi.ohio-state.edu/golubitsky.4/reprintweb-0.5/output/papers/nice4.pdf

Benoit Dionne

Papers

Stability results for steady, spatially periodic planforms

Coupled cells with internal symmetry: II. Direct products

Stability Results for Steady, Spatially--Periodic Planforms

Planforms in two and three dimensions

Coupled Cells: Wreath Products and Direct Products