Recent advances in symmetric and network dynamics

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

General optimization framework for accurate and efficient reconstruction of symmetric complex networks from dynamical data

The challenging problem of network reconstruction from dynamical data can in general be formulated as an optimization task of solving multiple linear equations. Existing approaches are of the two types: Point-by-point (PBP) and global methods. The local PBP method is computationally efficient, but the accuracies of its solutions are somehow low, while a global method has the opposite traits: High accuracy and high computational cost. Taking advantage of the network symmetry, we develop a novel framework integrating the advantages of both the PBP and global methods while avoiding their shortcomings: i.e., high reconstruction accuracy is guaranteed, but the computational cost is orders of magnitude lower than that of the global methods in the literature. The mathematical principle underlying our framework is block coordinate descent (BCD) for solving optimization problems, where the various blocks are determined by the network symmetry. The reconstruction framework is validated by numerical examples with a variety of network structures (i.e., sparse and dense networks) and dynamical processes. Our success is a demonstration that the general principle of exploiting symmetry can be extended to tackling the challenging inverse problem or reverse engineering of complex networks. Since solving a large number of linear equations is key to a plethora of problems in science and engineering, our BCD-based network reconstruction framework will find broader applications.

Cluster synchronization induced by manifold deformation

Pinning control of cluster synchronization in a globally connected network of chaotic oscillators is studied. It is found in simulations that when the pinning strength exceeds a critical value, the oscillators are synchronized into two different clusters, one formed by the pinned oscillators and the other one formed by the unpinned oscillators. The numerical results are analyzed by the generalized method of master stability function (MSF), in which it is shown that whereas the method is able to predict the synchronization behaviors of the pinned oscillators, it fails to predict the synchronization behaviors of the unpinned oscillators. By checking the trajectories of the oscillators in the phase space, it is found that the failure is attributed to the deformed synchronization manifold of the unpinned oscillators, which is clearly deviated from that of isolated oscillator under strong pinnings. A similar phenomenon is also observed in the pinning control of cluster synchronization in a complex network of symmetric structures and in the self-organized cluster synchronization of networked neural oscillators. The findings are important complements to the generalized MSF method and provide an alternative approach to the manipulation of synchronization behaviors in complex network systems.

NoLiTiA: An Open-Source Toolbox for Non-linear Time Series Analysis

In many scientific fields including neuroscience, climatology or physics, complex relationships can be described most parsimoniously by non-linear mechanics. Despite their relevance, many neuroscientists still apply linear estimates in order to evaluate complex interactions. This is partially due to the lack of a comprehensive compilation of non-linear methods. Available packages mostly specialize in only one aspect of non-linear time-series analysis and most often require some coding proficiency to use. Here, we introduce NoLiTiA, a free open-source MATLAB toolbox for non-linear time series analysis. In comparison to other currently available non-linear packages, NoLiTiA offers (1) an implementation of a broad range of classic and recently developed methods, (2) an implementation of newly proposed spatially and time-resolved recurrence amplitude analysis and (3) an intuitive environment accessible even to users with little coding experience due to a graphical user interface and batch-editor. The core methodology derives from three distinct fields of complex systems theory, including dynamical systems theory, recurrence quantification analysis and information theory. Besides established methodology including estimation of dynamic invariants like Lyapunov exponents and entropy-based measures, such as active information storage, we include recent developments of quantifying time-resolved aperiodic oscillations. In general, the toolbox will make non-linear methods accessible to the broad neuroscientific community engaged in time series processing.

Projective mechanisms subtending real world phenomena wipe away cause effect relationships

Causal relationships lie at the very core of scientific description of biophysical phenomena. Nevertheless, observable facts involving changes in system shape, dimension and symmetry may elude simple cause and effect inductive explanations. Here we argue that numerous physical and biological phenomena such as chaotic dynamics, symmetry breaking, long-range collisionless neural interactions, zero-valued energy singularities, and particle/wave duality can be accounted for in terms of purely topological mechanisms devoid of causality. We illustrate how simple topological claims, seemingly far away from scientific inquiry (e.g., "given at least some wind on Earth, there must at all times be a cyclone or anticyclone somewhere"; "if one stirs to dissolve a lump of sugar in a cup of coffee, it appears there is always a point without motion"; "at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and barometric pressures") reflect the action of non-causal topological rules. To do so, we introduce some fundamental topological tools and illustrate how phenomena such as double slit experiments, cellular mechanisms and some aspects of brain function can be explained in terms of geometric projections and mappings, rather than local physical effects. We conclude that unavoidable, passive, spontaneous topological modifications may lead to novel functional biophysical features, independent of exerted physical forces, thermodynamic constraints, temporal correlations and probabilistic a priori knowledge of previous cases.

Cluster synchronization in networked nonidentical chaotic oscillators

In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.

Koopman operator and its approximations for systems with symmetries

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, often described by complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces a block diagonal structure in operator approximations, revealing hidden organization. Practically, if the symmetries are known, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.

Symmetries of Chimera States

Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different setwise symmetries in the incoherent oscillators, and in particular, some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related states in spatially extended systems, which seem to inherit the symmetry properties of their counterparts in small networks.

Oscillations in an array of bistable microelectrodes coupled through a globally conserved quantity

The dynamical behavior of an array of microelectrodes is investigated under controlled current conditions during CO electrooxidation, a bistable electrochemical reaction with an S-shaped negative differential resistance (S-NDR) current-potential curve. Under these conditions, the total current constitutes a globally conserved quantity, thus coupling all microelectrodes globally. Upon increasing the total current, the microelectrodes activate one by one, with a single microelectrode being on its intermediate S-NDR current branch and the other ones being either on their passive or their active branches. When a few coupled microelectrodes are activated, the electrochemical system exhibits spontaneous potential oscillations. Mathematical analysis shows that oscillations arise already in a two group approximation of the dynamics, the two groups consisting of 1 electrode and n - 1 electrodes with n ≥ 3, respectively, with each group being described by a single evolution equation. In this minimal representation, oscillations occur when the single electrode is on the intermediate branch and the larger group is on the active branch.

Huygens' clocks revisited

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators

In a recent study about chaos synchronization in complex networks [Nat. Commun. 5, 4079 (2014)NCAOBW2041-172310.1038/ncomms5079], it is shown that a stable synchronous cluster may coexist with vast asynchronous nodes, resembling the phenomenon of a chimera state observed in a regular network of coupled periodic oscillators. Although of practical significance, this new type of state, namely, the isolated-desynchronization state, is hardly observed in practice due to its strict requirements on the network topology. Here, by the strategy of pinning coupling, we propose an effective method for inducing isolated-desynchronization states in symmetric networks of coupled chaotic oscillators. Theoretical analysis based on eigenvalue analysis shows that, by pinning a group of symmetric nodes in the network, there exists a critical pinning strength beyond which the group of pinned nodes can completely be synchronized while the unpinned nodes remain asynchronous. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks with the numerical results in good agreement with the theoretical predictions.

Symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps

We observe a symmetry of Lyapunov exponents in bifurcation structures of one-dimensional maps in which there exists a pair of parameter values in a dynamical system such that two dynamical systems with these paired parameter values have the same Lyapunov exponent. We show that this is a consequence of the presence of an invariant transformation from a dynamical system with one of the two paired parameter values to that with another parameter value, which does not change natures of dynamical systems.

Controlling synchronous patterns in complex networks

Although the set of permutation symmetries of a complex network could be very large, few of them give rise to stable synchronous patterns. Here we present a general framework and develop techniques for controlling synchronization patterns in complex network of coupled chaotic oscillators. Specifically, according to the network permutation symmetry, we design a small-size and weighted network, namely the control network, and use it to control the large-size complex network by means of pinning coupling. We argue mathematically that for any of the network symmetries, there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized. The feasibility of the control method is verified by numerical simulations of both artificial and real-world networks and demonstrated experimentally in systems of coupled chaotic circuits. Our studies show the controllability of synchronous patterns in complex networks of coupled chaotic oscillators.

Variety of regimes of starlike networks of Hénon maps

In this paper we categorize dynamical regimes demonstrated by starlike networks with chaotic nodes. This analysis is done in view of further studying of chaotic scale-free networks, since a starlike structure is the main motif of them. We analyze starlike networks of Hénon maps. They are found to demonstrate a huge diversity of regimes. Varying the coupling strength we reveal chaos, quasiperiodicity, and periodicity. The nodes can be both fully and phase synchronized. The hub node can be either synchronized with the subordinate nodes or oscillate separately from fully synchronized subordinates. There is a range of wild multistability where the zoo of regimes is the most various. One can hardly predict here even a qualitative nature of the expected solution, since each perturbation of the coupling strength or initial conditions results in a new character of dynamics.

Cyclic synchronous patterns in coupled discontinuous maps

Cyclic collective behaviors are commonly observed in biological and neuronal systems, yet the dynamical origins remain unclear. Here, by models of coupled discontinuous map lattices, we investigate the cyclic collective behaviors by means of cluster synchronization. Specifically, we study the synchronization behaviors in lattices of coupled periodic piecewise-linear maps and find that in the nonsynchronous regime the maps can be synchronized into different clusters and, as the system evolves, the synchronous clusters compete with each other and present the recurring process of cluster expanding, shrinking, and switching, i.e., showing the cyclic synchronous patterns. The dynamical mechanisms of cyclic synchronous patterns are explored, and the crucial roles of basin distribution are revealed. Moreover, due to the discontinuity feature of the map, the cyclic patterns are found to be very sensitive to the system initial conditions and parameters, based on which we further propose an efficient method for controlling the cyclic synchronous patterns.