## Convergence Results on Pulse Coupled Oscillator Protocols in Locally Connected Networks

- Lorenzo Ferrari, Anna Scaglione, Reinhard Gentz, Yao-Win Peter Hong
- IEEE/ACM Transactions on Networking
- 2017

- Published 2014

We propose a population model for δ-pulse-coupled oscillators with sparse connectivity. The model is given as an evolution equation for the phase density which take the form of a partial differential equation with a non-local term. We discuss the existence and stability of stationary solutions and exemplify our approach for integrate-and-fire-like oscillators. While for strong couplings, the firing rate of stationary solutions diverges and solutions disappear, small couplings allow for partially synchronous states which emerge at a supercritical Andronov-Hopf bifurcation. The collective dynamics of interacting oscillatory systems has been studied in many different contexts in the natural and life sciences [1–4]. In the thermodynamic limit, evolution equations for the population density proved to be a useful description [5–7], in particular to characterize the stability of synchronous and asynchronous states (see, e.g., [8–17]). Usually, dense or all-to-all-coupled networks are considered for these descriptions. Motivated by natural systems in which constituents interact with few others only, investigations of complex networks have revealed a large influence of the degree and sparseness of connectivity on network dynamics [18–25]. Especially when the knowledge about the connection structure is limited, it suggests itself to assume random connections (as in Erdős-Rényi networks) or random interactions (where excitations are assigned randomly to target oscillators [6, 26–28]). Both approaches often yield comparable dynamics (e.g. [29,30]) whereas random interactions represents a substantial simplification from a mathematical point of view, allowing one to describe the networks in terms of evolution equations for the phase density. These equations are usually posed as starting point for the commonly applied meanor the fluctuation-driven limits. However, rarely are they studied in full although it can be expected that sparseness largely influences the collective dynamics as has been discussed for excitable systems [26]. In this Letter, we propose a population model of δ-pulse coupled oscillators with sparse connectivity, derive the governing equations from a general definition of the density flux, and characterize existence and uniqueness of stationary solutions. For integrate-and-fire-like oscillators, the latter may either disappear with diverging firing rate or lose stability at a supercritical Andronov-Hopf bifurcation (AHB). This is in contrast to the global convergence to complete synchrony for all-to-all coupling that has been shown for finite [8] and for infinite [31] number of oscillators. Consider a population of oscillators n ∈ N with cyclic phases φn(t) ∈ [0, 1) and intrinsic dynamics φ̇n(t) = 1. If for some tf and some oscillator n the phase reaches 1, the oscillator fires and we introduce a phase jump in all oscillators n′ with probability p = m/N [32, 33]. Here, m is the number of recurrent connections per oscillator. The height of the phase jump is defined by the phase response curve ∆(φ) (PRC) (or equivalently by the phase transition curve R(φ)): φn′(t + f ) = φn′(tf ) + ∆ (φn′(tf )) = R(φn′(tf )). (1) The model can be interpreted as an all-to-all coupled network in which connections are not reliable and mediate interactions between oscillators only with a small probability (p). It can also be interpreted as an approximation to an Erdős-Rényi network in which the quenched disorder, imposed by its construction, is replaced by a dynamic p-1 ar X iv :1 40 3. 01 02 v1 [ nl in .C D ] 1 M ar 2 01 4 A. Rothkegel & K. Lehnertz coupling structure which takes the form of an ongoing random influence. For the limit of large sparse networks (N → ∞,m = const.), we represent the network dynamics by a continuity equation for the phase density ρ(φ, t) ∂tρ(φ, t) + ∂φJ(φ, t) = 0 (2) with ρ(φ, t) ≥ 0 and ∫ 1 0 ρ(φ, t)dφ = 1. We assume the probability flux J(φ, t) to be continuous and define both ρ and J at phases φ ∈ [0, 1). Evaluations at φ = 1 are meant as left-sided limits towards φ = 1. J(1, t) is the firing rate. Every oscillator is subject to Poisson excitations ηλ(t) with inhomogeneous rate λ(t) = mJ(1, t) and we can describe its phase variable by the stochastic differential equation ∂tφ(t) = 1 + ηλ(t). To shorten our notation, we will omit in the following the time t as argument of ρ, λ, and J . As we expect R(φ) to be non-invertible and to map intervals to a single phase, we have to take care in which way ρ and J are interpreted at these phases. Given some distribution of oscillators phases, we consider ρ(φ, t)dφ as the fraction of oscillators which are contained in a small interval whose left boundary is fixed to φ. With this definition, ρ(φ, t) is continuous for right-sided limits and the corresponding J(φ, t) is defined by the oscillators which pass an imaginary boundary which is infinitely close to φ and right to φ. The flux can be formalized in the following way: J(φ) = ρ(φ) + λ ∫

@inproceedings{Rothkegel2014SynchronizationIP,
title={Synchronization in populations of sparsely connected pulse-coupled oscillators},
author={Alexander Rothkegel and Klaus Lehnertz},
year={2014}
}