Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Modelling and Synchronisation of Delayed Packet-Coupled Oscillators in Industrial Wireless Sensor Networks

In this paper, a Packet-Coupled Oscillators (PkCOs) synchronisation protocol is proposed for time-sensitive Wireless Sensor Networks (WSNs) based on Pulse-Coupled Oscillators (PCO) in mathematical biology. The effects of delays on synchronisation performance are studied through mathematical modelling and analysis of packet exchange and processing delays. The delay compensation strategy (i.e., feedforward control) is utilised to cancel delays effectively. A simple scheduling function is provided with PkCOs to allocate the packet transmission event to a specified time slot, by configuring reference input of the system to a non-zero value, in order to minimise the possibility of packet collision in synchronised wireless networks. The rigorous theoretical proofs are provided to validate the convergence and stability of the proposed synchronisation scheme. Finally, the simulations and experiments examine the effectiveness of PkCOs with delay compensation and scheduling strategies. The experimental results also show that the proposed PkCOs algorithm can achieve synchronisation with the precision of $26.3\mu s$ ($1$ tick)

Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the influence of stochastic perturbations. We consider transient escapes from a marginally stable "quiescent" equilibrium to a more stable "active" equilibrium. The presence of coupling introduces dependence between the escape processes: for diffusive coupling there is a strongly coupled limit (fast domino regime) where the escapes are strongly synchronised while for intermediate coupling (slow domino regime) without partially escaped stable states, there is still a delayed effect. These regimes can be associated with bifurcations of equilibria in the low-noise limit. In this paper we consider a localized form of non-diffusive (i.e pulse-like) coupling and find similar changes in the distribution of escape times with coupling strength. However we find transition to a slow domino regime that is not associated with any bifurcations of equilibria. We show that this transition can be understood as a codimension-one saddle connection bifurcation for the low-noise limit. At transition, the most likely escape path from one attractor hits the escape saddle from the basin of another partially escaped attractor. After this bifurcation we find increasing coefficient of variation of the subsequent escape times

Diffusive clock synchronization in highly dynamic networks

International audienceThis paper studies the clock synchronization problem in highly dynamic networks. We show that diffusive synchronization algorithms are well adapted to environments in which the network topology may change unpredictably. In a diffusive algorithm, each node repeatedly (i) estimates the clock difference to its neighbors via broadcast of zero-bit messages, and (ii) updates its local clock according to a weighted average of the estimated differences. The system model allows for drifting local clocks, running at possibly different frequencies. We show that having a rooted spanning tree in the network at every time instance suffices to solve clock synchronization. We do not require any stability of the spanning tree, nor do we impose that the links of the spanning tree be known to the nodes. Explicit bounds on the convergence speed are obtained. In particular, our results settle an open question posed by Simeone and Spagnolini to reach clock synchronization in dynamic networks in the presence of nonzero clock drift. We also identify certain reasonable assumptions that allow for a significant higher convergence speed, e.g., bidirectional networks or random graph models

Synchronization-Induced Rhythmicity of Circadian Oscillators in the Suprachiasmatic Nucleus

The suprachiasmatic nuclei (SCN) host a robust, self-sustained circadian pacemaker that coordinates physiological rhythms with the daily changes in the environment. Neuronal clocks within the SCN form a heterogeneous network that must synchronize to maintain timekeeping activity. Coherent circadian output of the SCN tissue is established by intercellular signaling factors, such as vasointestinal polypeptide. It was recently shown that besides coordinating cells, the synchronization factors play a crucial role in the sustenance of intrinsic cellular rhythmicity. Disruption of intercellular signaling abolishes sustained rhythmicity in a majority of neurons and desynchronizes the remaining rhythmic neurons. Based on these observations, the authors propose a model for the synchronization of circadian oscillators that combines intracellular and intercellular dynamics at the single-cell level. The model is a heterogeneous network of circadian neuronal oscillators where individual oscillators are damped rather than self-sustained. The authors simulated different experimental conditions and found that: (1) in normal, constant conditions, coupled circadian oscillators quickly synchronize and produce a coherent output; (2) in large populations, such oscillators either synchronize or gradually lose rhythmicity, but do not run out of phase, demonstrating that rhythmicity and synchrony are codependent; (3) the number of oscillators and connectivity are important for these synchronization properties; (4) slow oscillators have a higher impact on the period in mixed populations; and (5) coupled circadian oscillators can be efficiently entrained by light–dark cycles. Based on these results, it is predicted that: (1) a majority of SCN neurons needs periodic synchronization signal to be rhythmic; (2) a small number of neurons or a low connectivity results in desynchrony; and (3) amplitudes and phases of neurons are negatively correlated. The authors conclude that to understand the orchestration of timekeeping in the SCN, intracellular circadian clocks cannot be isolated from their intercellular communication components