Probabilistic convergence guarantees for type-II pulse-coupled oscillators

Abstract

We show that a large class of pulse coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a new classification scheme for Type II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These bounds are then used to develop a simple, fast and rigorous computational analytic technique. These results suggest new methods for the analysis of pulse coupled oscillators, and provide new insights into the operation of biological Type II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.Synchrony in systems of pulse-coupled oscillators (PCOs) is an important feature in physics, biology and engineering. Synchronization can range from being a pathological breakdown, as in epilepsy [1] to one of vital importance, such as in the proper functioning of the heart's sinoatrial node [2,3], to a framework to understand complex systems [4,5]. Additionally, there are attempts to utilize the simplicity of PCO synchronization to synchronize wireless sensor networks [6][7][8][9]. However, many of the idealized models inspired by synchronization are not able to synchronize when the system has a complicated graph structure and time delays -aspects expected in real physical systems. In order to deal with these issues, previous studies have considered oscillators augmented with memory [7,10], infinite spatial density [10] or indegree normalization [5,10]. While these studies have shown linear stability [5], or other forms of local convergence [7,10], global convergence in these settings has either been shown to be impossible [5] or remains unknown.Alternatively, a class of oscillators with Type II phase response curves (PRCs), have been connected to synchronizing behavior theoretically [11,12] and in nature [2,13,14]. The distinguishing feature of oscillators with Type II PRCs is that an oscillator's phase can either be decreased (inhibited) or increased (excited), depending on the internal state of the oscillator. In this paper we focus on PCOs with a particular class of type II phase response curves, introduced in our previous paper [15], which resembles those in nature [2,13] and are well suited for handling complex topologies and time delay. We also show how leveraging the main theorem from [15] allows for a computational analytic routine yielding a fast and rigorous estimate of the convergence probability of a system of PCOs. Furthermore, we provide rigorous lower bounds that guarantee the performance of this computational analytic approach and display how the probability of synchrony converges to 1 in highly connected graphs. This result is of biological relevance to the situations where synchrony is brought about via Type II PRCs, and is a useful guide for the construction of PCOs in sensor nets.Previous work found that a class of Type II PRC, denote...

Convergence of Desynchronization Primitives in Wireless Sensor Networks: A Stochastic Modeling Approach

“…• Impact of measurement noise: Contrary to the proposed model, these previous estimates do not take into account the measurement noise conditions. Independently from this work, and following different analysis and modeling approaches, recent work in synchronization [29], [30] and desynchronization [10] has shown that noise in the desynchronization phase update (e.g., by quantization) and drops (or collisions) of beacon messages can affect the convergence iterations and can lead to convergence iterations that deviate from the estimates obtained based on the ideal (noise-free) model assumed by earlier work. • Generalization to non-uniformly distributed initial firings:…”

“…From (8) and (28) we obtain (29) i.e., the mean values of successive fire message updates remain equidistant after the first firing cycle of a node. The standard deviation of after the update of (28) is (30) Generalizing (28) to the th phase update of the th firing cycle of a node, leads to (31) with i.i.d. random variables, each stemming from Assumption 3.…”

Frequency adjustment and synchrony in networks of delayed pulse-coupled oscillators

“…However, while traditional PCO models provide an excellent tool to study synchronization in idealized settings or with specified network topologies, its application to wireless sensor networks has revealed that when such idealized PCOs are generalized to more realistic settings, they typically have great difficulty synchronizing. In particular, traditional PCO models are especially challenged by the combination of complex network topologies and signal delay [5][6][7][8]; this has naturally led to a number of design questions relevant to both those interested in superior wireless sensor network synchronization protocols and those interested in the theoretical limits of the PCO framework.The design challenge posed by complex network topology and delays has been recently addressed by a variety of specialized PCO models which augment oscillators with: mixtures of inhibition and excitation [5][6][7][8], stochasticity [5], single bits of addition memory [9, 10] or other modifications [11]. These recent PCO models represent a surprisingly large break from traditional PCO studies and from dynamical systems more generally-requiring new analytical techniques, new theoretical goals and new considerations for novelty.However, while these new models have dealt with very difficult settings, they have been unable to address one of the more interesting traditional oscillator questions: can oscillators with heterogeneous frequencies synchronize?…”

“…Finally, to bolster these analytic results, we provide numerical simulation demonstrating the robustness of the system. Similar to [7,8,14,15], consider a PCO model on an undirected graph G = {V, E}, where each oscillator i ∈ V has phase φ i (t) ∈ [0, 1] and speed ω i ∈ [1, 2) such that dφi dt = ω i . When i reaches its terminal phase, φ i (t) = 1, it emits a signal and its phase is reset to 0.…”

Synchronization as Aggregation: Cluster Kinetics of Pulse-Coupled Oscillators

“…Oscillators coupled by sudden pulses have been used to model sensor networks [6][7][8][9][10], earthquakes [11,12], economic booms and busts [13], firing neurons [14,15], and cardiac pacemaker cells [16]. Diverse forms of collective behavior can occur in these pulse-coupled systems, depending on how the oscillators are connected in space.…”