Consider a distributed network on a finite simple graph @math with diameter @math and maximum degree @math , where each node has a phase oscillator revolving on @math with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on @math synchronizes by time @math if @math is a tree and @math . We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on @math by time @math if @math is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses @math memory per node and converges on any @math with expected worst case running time of @math .
While the methods based on concentration condition could be applied once the system is nearly synchronized or to maintain synchrony against weak fluctuation, a fundamental question must be addressed: This is what Q1 focuses on, which has been answered for some classes of pulse-couplings mainly on complete (all-to-all) graphs or cycles. In their seminal work, Mirollo and Strogatz @cite_15 showed that an excitatory pulse-coupling on complete graphs synchronizes almost all initial configurations. A similar result was derived for an inhibitory pulse-couplings by Klingmayr and Bettstetter @cite_1 . For PCOs on cycle graphs, Wang, and Doyle @cite_11 addressed Q1. More recently, these authors and Teel @cite_40 studied Q1 for PCOs on general topology assuming a global pacemaker. In Theorem and , we give an answer to Q1 in the case of the 4-coupling and adaptive 4-coupling on tree networks.
@cite_40 @cite_15 @cite_1 @cite_11
2218979699 2154953441 2058455332 2030115949
Abstract Pulse-coupled oscillators (PCOs) are limit cycle oscillators coupled by exchanging pulses at discrete time instants. Their importance in biology and engineering has motivated numerous studies aiming to understand the basic synchronization properties of a network of PCOs. In this work, we study synchronization of PCOs subject to a global pacemaker (or global cue) and local interactions between slave oscillators. We characterize solutions and give synchronization conditions using the phase response curve (PRC) as the design element, which is restricted to be of the delay type in the first half of the cycle, interval ( 0 , π ) , and of the advance type in the second half of the cycle, interval ( π , 2 π ) . It is shown that global synchronization is feasible when using an advance-delay PRC if the influence of the global cue is strong enough. Numerical examples are provided to illustrate the analytical findings. A simple model for synchronous firing of biological oscillators based on Peskin's model of the cardiac pacemaker (Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975, pp. 268-278) is studied. The model consists of a population of identical integrate-and-fire oscillators. The coupling between oscillators is pulsatile: when a given oscillator fires, it pulls the others up by a fixed amount, or brings them to the firing threshold, whichever is less. The main result is that for almost all initial conditions, the population evolves to a state in which all the oscillators are firing synchronously. The relationship between the model and real communities of biological oscillators is discussed; examples include populations of synchronously flashing fireflies, crickets that chirp in unison, electrically synchronous pacemaker cells, and groups of women whose menstrual cycles become mutually synchronized. Solutions for time synchronization based on coupled oscillators operate in a self-organizing and adaptive manner and can be applied to various types of dynamic networks. The basic idea was inspired by swarms of fireflies, whose flashing dynamics shows an emergent behavior. This article introduces such a synchronization technique whose main components are “inhibitory coupling” and “self-adjustment.” Based on this new technique, a number of contributions are made. First, we prove that inhibitory coupling can lead to perfect synchrony independent of initial conditions for delay-free environments and homogeneous oscillators. Second, relaxing the assumptions to systems with delays and different phase rates, we prove that such systems synchronize up to a certain precision bound. We derive this bound assuming inhomogeneous delays and show by simulations that it gives a good estimate in strongly-coupled systems. Third, we show that inhibitory coupling with self-adjustment quickly leads to synchrony with a precision comparable to that of excitatory coupling. Fourth, we analyze the robustness against faulty members performing incorrect coupling. While the specific precision-loss encountered by such disturbances depends on system parameters, the system always regains synchrony for the investigated scenarios. The importance of pulse-coupled oscillators (PCOs) in biology and engineering has motivated research to understand basic properties of PCO networks. Despite the large body of work addressing PCOs, a global synchronization result for networks that are more general than all-to-all connected is still unavailable. In this paper we address global synchronization of PCO networks described by cycle graphs. It is shown for the bidirectional cycle case that as the number of oscillators in the cycle grows, the coupling strength must be increased in order to guarantee synchronization for arbitrary initial conditions. For the unidirectional cycle case, the strongest coupling cannot ensure global synchronization yet a refractory period in the phase response curve is sufficient to enable global synchronization. Analytical findings are confirmed by numerical simulations.