Synchronization Of Pulse-coupled Oscillators Research Articles

Scalable resetting algorithms for synchronization of pulse-coupled oscillators over rooted directed graphs

We study the problem of robust global synchronization of pulse-coupled oscillators (PCOs) over directed graphs. It is known that when the digraphs are strongly connected, global synchronization can be achieved by using a class of deterministic binary set-valued resetting controllers (Poveda and Teel, 2019). However, for large-scale networks, these algorithms are not scalable because some of their tuning parameters have upper bounds of the order O1N, where N is the number of agents. This paper resolves this scalability issue by presenting several new results about global synchronization of PCOs with more general network topologies using the frameworks of deterministic and stochastic hybrid dynamical systems. First, we establish that similar deterministic binary resetting algorithms can achieve robust global and fixed-time synchronization in any rooted acyclic digraph. Moreover, in this case, we show that the synchronization dynamics are now scalable as the tuning parameters of the algorithm are network independent, i.e., of order O(1). However, the algorithms cannot be further extended to all rooted digraphs. We establish this new impossibility result by introducing a counter-example with a particular rooted digraph for which global synchronization cannot be achieved, irrespective of the tuning parameters. Nevertheless, we show that if the binary resetting algorithms are modified by accommodating an Erdös–Renýi type random graph model, then the resulting stochastic resetting dynamics will guarantee global synchronization almost surely for all rooted digraphs and, moreover, the tuning parameters of the dynamics are network independent. Stability and robustness properties of the resetting algorithms are studied using the tools from set-valued hybrid dynamical systems. Numerical simulations are provided at the end of the paper for demonstration of the main results.

Deadlocks in the synchronization of pulse-coupled oscillators on star graphs.

Some convergence proofs for systems of oscillators with inhibitory pulse coupling assume that all initial phases reside in one half of their domain. A violation of this assumption can trigger deadlocks that prevent synchronization. We analyze the conditions for such deadlocks in star graphs, characterizing the domain of initial states leading to deadlocks and deriving its fraction of the state space. The results show that convergence is feasible from a wider range of initial phases. The same type of deadlock occurs in random graphs.

An Attack-Resilient Pulse-Based Synchronization Strategy for General Connected Topologies

Synchronization of pulse-coupled oscillators (PCOs) has gained significant attention recently due to increased applications in sensor networks and wireless communications. Given the distributed and unattended nature of wireless sensor networks, it is imperative to enhance the resilience of pulse-based synchronization against malicious attacks. However, most existing results on resilient PCO synchronization are obtained for all-to-all networks. We propose a new pulse-based synchronization mechanism to improve the resilience of PCO synchronization that is applicable under general connected topologies. Under the proposed synchronization mechanism, we rigorously characterize the condition for stealthy Byzantine attacks and prove that perfect synchronization of legitimate oscillators can be guaranteed in the presence of multiple stealthy Byzantine attackers, irrespective of whether the attackers collude with each other or not. The new mechanism can guarantee resilient synchronization even when the initial phases of legitimate oscillators are widely distributed in a half circle, which is in distinct difference from most existing attack-resilient synchronization algorithms (including the seminal paper from Lamport and Melliar-Smith [1]) that require a priori (almost) synchronization among legitimate oscillators. Numerical simulation results are given to confirm the theoretical results.

Global Synchronization of Pulse-Coupled Oscillator Networks Under Byzantine Attacks

Synchronization of pulse-coupled oscillators (PCOs) has gained significant attention recently due to their increased applications in sensor networks and wireless communications. Given the distributed and unattended nature of wireless sensor networks, it is imperative to enhance the resilience of PCO synchronization against malicious attacks. However, most existing results on attack-resilient pulse-based synchronization are obtained under assumptions of all-to-all coupling topologies or restricted initial phase distributions. In this paper, we propose a new pulse-based synchronization mechanism to improve the attack resilience of PCO synchronization that is applicable to non-all-to-all networks. Under the proposed synchronization mechanism, we prove that perfect synchronization of legitimate oscillators can be guaranteed in the presence of multiple Byzantine attackers who can emit attack pulses arbitrarily without any constraint except that practical bit rate constraint renders the number of pulses from an attacker to be finite. The new mechanism can guarantee synchronization even when the initial phases of all legitimate oscillators are arbitrarily distributed in the entire oscillation period, which is in distinct difference from most existing attack-resilient synchronization approaches (including the seminal paper from Lamport and Melliar-Smith [1]) that require a priori (almost) synchronization among legitimate oscillators. Numerical simulation results are given to confirm the theoretical results.

On the global synchronization of pulse-coupled oscillators interacting on chain and directed tree graphs

Driven by increased applications in biological networks and wireless sensor networks, synchronization of pulse-coupled oscillators (PCOs) has gained increased popularity. However, most existing results address the local synchronization of PCOs with initial phases constrained in a half cycle, and results on global synchronization from any initial condition are very sparse. In this paper, we address global PCO synchronization from an arbitrary phase distribution under chain or directed tree graphs. Our results differ from existing global synchronization studies on decentralized PCO networks in two key aspects: first, our work allows heterogeneous coupling functions, and we analyze the behavior of oscillators with perturbations on their natural frequencies; second, rather than requiring a large enough coupling strength, our results hold under any coupling strength between zero and one, which is crucial because a large coupling strength has been shown to be detrimental to the robustness of PCO synchronization to disturbances.

Attack-Resilient Pulse-Coupled Synchronization

Synchronization of pulse-coupled oscillators is gaining increased popularity in sensor networks and wireless communications due to its simplicity, implicit scalability, and high energy efficiency. However, most existing results are obtained in the absence of malicious attacks. This paper addresses pulse-coupled synchronization resilient to malicious attacks. Not only is the condition for stealthy attacks obtained, but we also prove analytically that pulse-coupled synchronization can be guaranteed despite the presence of a stealthy attacker, even when legitimate oscillators have different initial phases. This is in distinct difference from most of the existing attack-resilient synchronization algorithms (including the seminal paper from Lamport and Melliar-Smith) that require a priori (almost) synchronization among legitimate nodes. Numerical simulations confirm the theoretical results.

Simulation and evaluation of pulse-coupled oscillators in wireless sensor networks

ABSTRACTThe clock in embedded systems usually is driven by a crystal oscillator and implemented via a counter register, such a crystal clock is non-identical and drifting due to the manufacturing tolerance and variation of working conditions. Thus, a common time among distributed wireless sensor nodes, also referred to as Time Synchronization, is required for many time-sensitive wireless applications, such as collaborative condition monitoring, coordinated control and localization. Inspired by fireflies’ behaviour, the Pulse-Coupled Oscillators (PCO) has been proposed for synchronization in complex networks. Since the concurrent transmission of PCO’s Pulses is impossible in Wireless Sensor Networks (WSNs), the desynchronization mechanism is adopted to ensure the implementation of PCO in WSNs. Moreover, due to the uncertainties in radio channels and the complexities of communication protocols and packet-exchange behaviours in wireless networks, it is challenging to have a closed-form solution to the performance of PCO synchronization in WSNs. The realistic software simulation, in particular, the discrete event simulator has been a powerful tool to exam the performance of communication protocols in various scenarios, since an order sequence of well-defined event in time is to represent the behaviour of a complex system. This paper presents the development of a pulse-coupled oscillators time synchronization simulator on the OMNeT++ platform for simulating and studying its behaviour and performance in sensor networks. A clock module with configurable phase and frequency noises, and adjustable and higher resolution is developed to mimic various crystal oscillators in embedded systems, for example, the real-time clock. The developed simulator also supports the full functions devices defined by ZigBee protocol, which allows realistic simulation of multi-hop IEEE 802.15.4 wireless networks. Finally, the intensive simulations of classical PCO with the refractory period in IEEE 802.15.4-based WSNs have been carried out to demonstrate the features and benefits of the developed simulator. It is shown that for the non-identical and time-varying PCO clocks in the WSNs, the achieved synchronization will lose gradually, and the time that maintained synchronization depends on the length of refractory period.

Synchronization of Pulse-Coupled Oscillators and Clocks Under Minimal Connectivity Assumptions

Populations of flashing fireflies, claps of applauding audience, cells of cardiac and circadian pacemakers reach synchrony via event-triggered interactions, referred to as pulse couplings. Synchronization via pulse coupling is widely used in wireless sensor networks, providing clock synchronization with parsimonious packet exchanges. In spite of serious attention paid to networks of pulse coupled oscillators, there is a lack of mathematical results, addressing networks with general communication topologies and general phase-response curves of the oscillators. The most general results of this type (Wang et al., 2012, 2015) establish synchronization of oscillators with a delay-advance phase-response curve over strongly connected networks. In this paper we extend this result by relaxing the connectivity condition to the existence of a root node (or a directed spanning tree) in the graph. This condition is also necessary for synchronization.

Synchronization of Pulse-Coupled Oscillators on (Strongly) Connected Graphs

The synchronization of pulse-coupled oscillators (PCOs) interacting on general (strongly) connected graphs is addressed. It is shown that the network synchronizes if the phases are contained in an arc of length less than $\pi$ , for every (strongly) connected interaction topology. Moreover, global synchronization is feasible in the all-to-all, strongly rooted, and connected bidirectional cases. Examples, counter-examples, and numerical simulations are given to illustrate the analytical findings.

The role of intermediaries in the synchronization of pulse-coupled oscillators

The role of intermediaries in the synchronization of small groups of light controlled oscillators (LCO) is addressed. A single LCO is a two-time-scale phase oscillator. When pulse-coupling two LCOs, the synchronization time decreases monotonously as the coupling strength increases, independent of the initial conditions and frequency detuning. In this work we study numerically the effects that a third LCO induces to the collective behavior of the system. We analyze the new system by dealing with directed heterogeneous couplings among the units. We report a novel and robust phenomenon, absent when coupling two LCOs, which consists of a discontinuous relationship between the synchronization time and coupling strength or initial conditions. The mechanism responsible for the appearance of such discontinuities is discussed.

Synchronization of pulse-coupled oscillators with a refractory period and frequency distribution for a wireless sensor network

The present paper considers the synchronization of globally pulse-coupled oscillators with a refractory period and frequency distribution. The oscillators are capable of achieving time synchronization for a practical wireless sensor network. Furthermore, as a result of the stability analysis of the synchronization, a procedure for designing the oscillators is provided: the determination of the allowable refractory period under a given frequency distribution range. These analytical results are verified by numerical examples.

Unsupervised learning of arbitrarily shaped clusters using ensembles of Gaussian models

We propose a new clustering algorithm, called SyMP, which is based on synchronization of pulse-coupled oscillators. SyMP represents each data point by an Integrate-and-Fire oscillator and uses the relative similarity between the points to model the interaction between the oscillators. SyMP is robust to noise and outliers, determines the number of clusters in an unsupervised manner, and identifies clusters of arbitrary shapes. The robustness of SyMP is an intrinsic property of the synchronization mechanism. To determine the optimum number of clusters, SyMP uses a dynamic and cluster dependent resolution parameter. To identify clusters of various shapes, SyMP models each cluster by an ensemble of Gaussian components. SyMP does not require the specification of the number of components for each cluster. This number is automatically determined using a dynamic intra-cluster resolution parameter. Clusters with simple shapes would be modeled by few components while clusters with more complex shapes would require a larger number of components. The proposed clustering approach is empirically evaluated with several synthetic data sets, and its performance is compared with GK and CURE. To illustrate the performance of SyMP on real and high-dimensional data sets, we use it to categorize two image databases.

A discrete dynamics model for synchronization of pulse-coupled oscillators

Biological information processing systems employ a variety of feature types. It has been postulated that oscillator synchronization is the mechanism for binding these features together to realize coherent perception. A discrete dynamic model of a coupled system of oscillators is presented. The network of oscillators converges to a state where subpopulations of cells become phase synchronized. It has potential applications to describing biological perception as well as for the construction of multifeature pattern recognition systems. It is shown that this model can be used to detect the presence of short line segments in the boundary contour of an object. The Hough transform, which is the standard method for detecting curve segments of a specified shape in an image was found not to be effective for this application. Implementation of the discrete dynamics model of oscillator synchronization is much easier than the differential equation models that have appeared in the literature. A systematic numerical investigation of the convergence properties of the model has been performed and it is shown that the discrete dynamics model can scale up to large number of oscillators.

Threshold effects on synchronization of pulse-coupled oscillators.

The threshold effects of synchronous firing are studied in a class of models. The main result is that for almost all initial conditions, the oscillation system approaches synchrony when the threshold condition is below a critical value.