Search

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time auto-regressive processes and continuous-time Ornstein-Uhlenbeck dynamics on networks. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices, and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronisation specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks., Comment: 20 pages, 5 figures

Transfer entropy is an established method for quantifying directed statistical dependencies in neuroimaging and complex systems datasets. The pairwise (or bivariate) transfer entropy from a source to a target node in a network does not depend solely on the local source-target link weight, but on the wider network structure that the link is embedded in. This relationship is studied using a discrete-time linearly-coupled Gaussian model, which allows us to derive the transfer entropy for each link from the network topology. It is shown analytically that the dependence on the directed link weight is only a first approximation, valid for weak coupling. More generally, the transfer entropy increases with the in-degree of the source and decreases with the in-degree of the target, indicating an asymmetry of information transfer between hubs and low-degree nodes. In addition, the transfer entropy is directly proportional to weighted motif counts involving common parents or multiple walks from the source to the target, which are more abundant in networks with a high clustering coefficient than in random networks. Our findings also apply to Granger causality, which is equivalent to transfer entropy for Gaussian variables. Moreover, similar empirical results on random Boolean networks suggest that the dependence of the transfer entropy on the in-degree extends to nonlinear dynamics., Comment: 12 pages, 5 figures

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights., Comment: 19 pages, 8 Figures, 53 References

We analyze the collective dynamics of hierarchically structured networks of densely connected spiking neurons. These networks of sub-networks may represent interactions between cell assemblies or different nuclei in the brain. The dynamical activity pattern that results from these interactions depends on the strength of synaptic coupling between them. Importantly, the overall dynamics of a brain region in the absence of external input, so called ongoing brain activity, has been attributed to the dynamics of such interactions. In our study, two different network scenarios are considered: a system with one inhibitory and two excitatory sub-networks, and a network representation with three inhibitory sub-networks. To study the effect of synaptic strength on the global dynamics of the network, two parameters for relative couplings between these sub-networks are considered. For each case, a co-dimension two bifurcation analysis is performed and the results have been compared to large-scale network simulations. Our analysis shows that Generalized Lotka-Volterra (GLV) equations, well-known in predator-prey studies, yield a meaningful population-level description for the collective behavior of spiking neuronal interaction, which have a hierarchical structure. In particular, we observed a striking equivalence between the bifurcation diagrams of spiking neuronal networks and their corresponding GLV equations. This study gives new insight on the behavior of neuronal assemblies, and can potentially suggest new mechanisms for altering the dynamical patterns of spiking networks based on changing the synaptic strength between some groups of neurons.

We study dynamics of phase-differences (PDs) of coupled oscillators where both the intrinsic frequencies and the couplings vary in time. In case the coupling coefficients are all nonnegative, we prove that the PDs are asymptotically stable if there exists T>0 such that the aggregation of the time-varying graphs across any time interval of length $T$ has a spanning tree. We also consider the situation that the coupling coefficients may be negative and provide sufficient conditions for the asymptotic stability of the PD dynamics. Due to time-variations, the PDs are asymptotic to time-varying patterns rather than constant values. Hence, the PD dynamics can be regarded as a generalisation of the well-known phase-locking phenomena. We explicitly investigate several particular cases of time-varying graph structures, including asymptotically periodic PDs due to periodic coupling coefficients and intrinsic frequencies, small perturbations, and fast-switching near constant coupling and frequencies, which lead to PD dynamics close to a phase-locked one. Numerical examples are provided to illustrate the theoretical results., Comment: 33 pages; 3 figures

We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and behaves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the connection and their respective kernels. Some examples are discussed to illustrate the theory., Comment: 19 pages, 2 figures; added journal ref to title page

As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a huge reduction of dimensionality (number of degrees of freedom) and a change in the level of description. Beyond the mere necessity of developing accurate techniques affording this reduction, there is the question of the correspondence between the initial system and the reduced one. In this paper, we offer a perspective towards a common framework for discussing and understanding multi-level systems exhibiting structures at various spatial and temporal levels. We propose a common foundation and illustrate it with examples from different fields. We also point out the difficulties in constructing such a general setting and its limitations.

We study the stability of networks of multi-agent systems with local pinning strategies and two types of time delays, namely the transmission delay in the network and the pinning delay of the controllers. Sufficient conditions for stability are derived under specific scenarios by computing or estimating the dominant eigenvalue of the characteristic equation. In addition, controlling the network by pinning a single node is studied. Moreover, perturbation methods are employed to derive conditions in the limit of small and large pinning strengths.Numerical algorithms are proposed to verify stability, and simulation examples are presented to confirm the efficiency of analytic results., Comment: 6 pages, 3 figures

We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.

We introduce a family of multi-way Cheeger-type constants $\{h_k^{\sigma}, k=1,2,\ldots, n\}$ on a signed graph $\Gamma=(G,\sigma)$ such that $h_k^{\sigma}=0$ if and only if $\Gamma$ has $k$ balanced connected components. These constants are switching invariant and bring together in a unified viewpoint a number of important graph-theoretical concepts, including the classical Cheeger constant, those measures of bipartiteness introduced by Desai-Rao, Trevisan, Bauer-Jost, respectively, on unsigned graphs,, and the frustration index (originally called the line index of balance by Harary) on signed graphs. We further unify the (higher-order or improved) Cheeger and dual Cheeger inequalities for unsigned graphs as well as the underlying algorithmic proof techniques by establishing their corresponding versions on signed graphs. In particular, we develop a spectral clustering method for finding $k$ almost-balanced subgraphs, each defining a sparse cut. The proper metric for such a clustering is the metric on a real projective space. We also prove estimates of the extremal eigenvalues of signed Laplace matrix in terms of number of signed triangles ($3$-cycles)., Comment: We add more details for the proof of Lemma 6.4, Theorem 6.2, Theorem 6.3. We also explain more details about the control of those various absolute constant appearing in our estimates

A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [Masoller and Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from time-dependent to steady-state synchronization as the width of the delay distribution increases. We also compare the two transitions to synchronization as the coupling strength increases. We use transition probabilities calculated via symbolic analysis and ordinal patterns. We find that, as the coupling strength increases, before the onset of steady-state synchronization the network splits into two clusters which are in anti-phase relation with each other. On the other hand, with increasing delay heterogeneity, no cluster formation is seen at the onset of steady-state synchronization; however, a rather complex unsynchronized state is detected, revealed by a diversity of transition probabilities in the network nodes.

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and traveling wave solutions in the parameter space characterizing the non-locality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of the phase oscillators., Comment: to appear in PRAMANA

We study the existence and stability of synchronous solutions in a continuum field of non-locally coupled identical phase oscillators with distance-dependent propagation delays. We present a comprehensive stability diagram in the parameter space of the system. From the numerical results a heuristic synchronization condition is suggested, and an analytic relation for the marginal stability curve is obtained. We also provide an expression in the form of a scaling relation that closely follows the marginal stability curve over the complete range of the non-locality parameter., Comment: accepted in Phys. Rev. E (2010)

Stochastic point processes with refractoriness appear frequently in the quantitative analysis of physical and biological systems, such as the generation of action potentials by nerve cells, the release and reuptake of vesicles at a synapse, and the counting of particles by detector devices. Here we present an extension of renewal theory to describe ensembles of point processes with time varying input. This is made possible by a representation in terms of occupation numbers of two states: Active and refractory. The dynamics of these occupation numbers follows a distributed delay differential equation. In particular, our theory enables us to uncover the effect of refractoriness on the time-dependent rate of an ensemble of encoding point processes in response to modulation of the input. We present exact solutions that demonstrate generic features, such as stochastic transients and oscillations in the step response as well as resonances, phase jumps and frequency doubling in the transfer of periodic signals. We show that a large class of renewal processes can indeed be regarded as special cases of the model we analyze. Hence our approach represents a widely applicable framework to define and analyze non-stationary renewal processes., Comment: 8 pages, 4 figures

We study synchronization of non-diffusively coupled map networks with arbitrary network topologies, where the connections between different units are, in general, not symmetric and can carry both positive and negative weights. We show that, in contrast to diffusively coupled networks, the synchronous behavior of a non-diffusively coupled network can be dramatically different from the behavior of its constituent units. In particular, we show that chaos can emerge as synchronized behavior although the dynamics of individual units are very simple. Conversely, individually chaotic units can display simple behavior when the network synchronizes. We give a synchronization criterion that depends on the spectrum of the generalized graph Laplacian, as well as the dynamical properties of the individual units and the interaction function. This general result will be applied to coupled systems of tent and logistic maps and to two models of neuronal dynamics. Our approach yields an analytical understanding of how simple model neurons can produce complex collective behavior through the coordination of their actions.

We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d.\ topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links., Comment: 20 pages, 2 figures

We consider complete synchronization of identical maps coupled through a general interaction function and in a general network topology where the edges may be directed and may carry both positive and negative weights. We define mixed transverse exponents and derive sufficient conditions for local complete synchronization. The general non-diffusive coupling scheme can lead to new synchronous behavior, in networks of identical units, that cannot be produced by single units in isolation. In particular, we show that synchronous chaos can emerge in networks of simple units. Conversely, in networks of chaotic units simple synchronous dynamics can emerge; that is, chaos can be suppressed through synchrony.

The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean.

We study the local complete synchronization of discrete-time dynamical networks with time-varying couplings. Our conditions for the temporal variation of the couplings are rather general and include both variations in the network structure and in the reaction dynamics; the reactions could, for example, be driven by a random dynamical system. A basic tool is the concept of Hajnal diameter which we extend to infinite Jacobian matrix sequences. The Hajnal diameter can be used to verify synchronization and we show that it is equivalent to other quantities which have been extended to time-varying cases, such as the projection radius, projection Lyapunov exponents, and transverse Lyapunov exponents. Furthermore, these results are used to investigate the synchronization problem in coupled map networks with time-varying topologies and possibly directed and weighted edges. In this case, the Hajnal diameter of the infinite coupling matrices can be used to measure the synchronizability of the network process. As we show, the network is capable of synchronizing some chaotic map if and only if there exists an integer T>0 such that for any time interval of length T, there exists a vertex which can access other vertices by directed paths in that time interval.

Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.

We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure., Comment: 13 pages, 4 figures

We investigate "chimera" states in a ring of identical phase oscillators coupled in a time-delayed and spatially non-local fashion. We find novel "clustered chimera" states that have spatially distributed phase coherence separated by incoherence with adjacent coherent regions in anti-phase. The existence of such time-delay induced phase clustering is further supported through solutions of a generalized functional self-consistency equation of the mean field. Our results highlight an additional mechanism for cluster formation that may find wider practical applications.

Time-delay systems are, in many ways, a natural set of dynamical systems for natural scientists to study because they form an interface between abstract mathematics and data. However, they are complicated because past states must be sensibly incorporated into the dynamical system. The primary goal of this paper is to begin to isolate and understand the effects of adding time-delay coordinates to a dynamical system. The key results include (i) an analytical understanding regarding extreme points of a time-delay dynamical system framework including an invariance of entropy and the variance of the Kaplan-Yorke formula with simple time re-scalings; (ii) computational results from a time-delay mapping that forms a path between dynamical systems dependent upon the most distant and the most recent past; (iii) the observation that non-trivial mixing of past states can lead to high-dimensional, high-entropy dynamics that are not easily reduced to low-dimensional dynamical systems; (iv) the observed phase transition (bifurcation) between low-dimensional, reducible dynamics and high or infinite-dimensional dynamics; and (v) a convergent scaling of the distribution of Lyapunov exponents, suggesting that the infinite limit of delay coordinates in systems such are the ones we study will result in a continuous or (dense) point spectrum., Comment: 10 pages, 8 figures

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks.

We present a method based on symbolic dynamics for the detection of synchronization in networks of coupled maps and distinguishing between chaotic and random iterations. The symbolic dynamics are defined using special partitions of the phase space which prevent the occurrence of certain symbol sequences related to the characteristics of the dynamics. Synchrony in a large network can be detected using measurements from only a single node by checking for the presence of forbidden symbol transitions. The method utilises a relatively short time series of measurements and hence is computationally very fast. Furthermore, it is robust against parameter uncertainties, is independent of the network size, and does not require knowledge of the connection structure., Comment: 4+ pages, 2 figures (v2: minor grammatical changes)

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building non-synchronizing networks having a prescribed degree distribution., Comment: v2: A new theorem and a numerical example added. To appear in IEEE Trans. Circuits and Systems I: Fundamental Theory and Applications

Networks of weakly nonlinear oscillators are considered with diffusive and time-delayed coupling. Averaging theory is used to determine parameter ranges for which the network experiences amplitude death, whereby oscillations are quenched and the equilibrium solution has a large domain of attraction. The amplitude death is shown to be a common phenomenon, which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions. In addition, when the network consists of dissimilar oscillators, there exist parameter values for which only parts of the network are suppressed. Sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities, coupling coefficients, and connection delays., Comment: 21 pages, 4 figures

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators., Comment: 4 pages, 5 figures

We argue that the coordination of the activities of individual complex agents enables a system to develop and sustain complexity at a higher level. We exemplify relevant mechanisms through computer simulations of a toy system, a coupled map lattice with transmission delays. The coordination here is achieved through the synchronization of the chaotic operations of the individual elements, and on the basis of this, regular behavior at a longer temporal scale emerges that is inaccessible to the uncoupled individual dynamics.

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation., Comment: 5 pages, 5 postscript figures. V2: minor textual amendments