We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also allows us to study the properties of synchrony-optimized networks, and in particular, investigate the role of directed network properties such as nodal in- and out-degrees. For instance, we find that in optimally rewired networks the heterogeneity of the in-degree distribution roughly matches the heterogeneity of the natural frequency distribution, but no such relationship emerges for out-degrees. We also observe that a network's synchronization properties are promoted by a strong correlation between the nodal in-degrees and the natural frequencies of oscillators, whereas the relationship between the nodal out-degrees and the natural frequencies has comparatively little effect. This result is supported by our theory, which indicates that synchronization is promoted by a strong alignment of the natural frequencies with the left singular vectors corresponding to the largest singular values of the Laplacian matrix.
In this paper we investigate the influence of directed motifs on the synchronization properties of Kuramoto oscillators on directed networks. Building different types of sparse directed small world networks similar to the Watts and Strogatz procedure we establish that feed forward loops favour synchronization on directed networks. The paper highlights the importance of local network characteristics for synchronization.
The collective behaviour of neural networks depends on the cellular and synaptic properties of the neurons. The phase-response curve (PRC) is an experimentally obtainable measure of cellular properties that quantifies the shift in the next spike time of a neuron as a function of the phase at which stimulus is delivered to that neuron. The neuronal PRCs can be classified as having either purely positive values (type I) or distinct positive and negative regions (type II). Networks of type 1 PRCs tend not to synchronize via mutual excitatory synaptic connections. We study the synchronization prop…
We present a scalable approach for simplifying the stability analysis of cluster synchronization patterns on directed networks. When a network has directional couplings, decomposition of the coupling matrix into independent blocks (which in turn decouples the variational equation) is no longer adequate to reveal the full relations among perturbation modes. Instead, it is often necessary to introduce directional dependencies among the blocks and establish hierarchies among perturbation modes. For this purpose, we develop an algorithm that finds the simultaneous block upper triangularization of …
In this paper, we discuss distributive synchronization of complex networks in finite time, with a single nonlinear pinning controller. The results apply to heterogeneous dynamic networks, too. Different from many models, which assume the coupling matrix being symmetric (or the connecting graph is undirected), here, the coupling matrix is asymmetric (or the connecting graph is directed
We study the collective dynamics of identical phase oscillators on globally coupled networks whose interactions are asymmetric and mediated by positive and negative couplings. We split the set of oscillators into two interconnected subpopulations. In this setup, oscillators belonging to the same group interact via symmetric couplings while the interaction between subpopulations occurs in an asymmetric fashion. By employing the dimensional reduction scheme of the Ott-Antonsen (OA) theory, we verify the existence of traveling-wave and $π$-states, in addition to the classical fully synchronized a…
We consider optimization of phase response curves for stochastic synchronization of non-interacting limit-cycle oscillators by common Poisson impulsive signals. The optimal functional shape for sufficiently weak signals is sinusoidal, but can differ for stronger signals. By solving the Euler-Lagrange equation associated with the minimization of the Lyapunov exponent characterizing synchronization efficiency, the optimal phase response curve is obtained. We show that the optimal shape mutates from a sinusoid to a sawtooth as the constraint on its squared amplitude is varied.
A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $ω^T L ω$, where $ω$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matr…
In this paper, we propose several models, which can realize synchronization of complex networks in finite time effectively. The results apply to heterogeneous dynamic networks, too. The mechanism of finite time convergence is revealed. Different from many models, which assume the coupling matrix being symmetric (or the connecting graph is undirected), here, the coupling matrix is asymmetric (or the connecting graph is directed).
A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is {\it not} the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an out-flow centrality measure that is equivalent to a reverse PageRank central…
Synchronization of network-coupled dynamical units is important to a variety of natural and engineered processes including circadian rhythms, cardiac function, neural processing, and power grids. Despite this ubiquity, it remains poorly understood how complex network structures and heterogeneous local dynamics combine to either promote or inhibit synchronization. Moreover, for most real-world applications it is impossible to obtain the exact specifications of the system, and there is a lack of theory for how uncertainty affects synchronization. We address this open problem by studying the Sync…
In this paper, we investigate synchronization of coupled second-order linear harmonic oscillators with random noises and time delays. The interaction topology is modeled by a weighted directed graph and the weights are perturbed by white noise. On the basis of stability theory of stochastic differential delay equations, algebraic graph theory and matrix theory, we show that the coupled harmonic oscillators can be synchronized almost surely with perturbation and time delays. Numerical examples are presented to illustrate our theoretical results.
We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Namely, each addition term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncate…
We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusi…
We study experimentally the synchronization patterns in time-delayed directed Boolean networks of excitable systems. We observe a transition in the network dynamics when the refractory time of the individual systems is adjusted. When the refractory time is on the same order-of-magnitude as the mean link time delays or the heterogeneities of the link time delays, cluster synchronization patterns change, or are suppressed entirely, respectively. We also show that these transitions occur when we only change the properties of a small number of nodes identified by their larger in-degree, hence the …
In this technical note, we propose a practicable quantized sampled velocity data coupling protocol for synchronization of a set of harmonic oscillators. The coupling protocol is designed in a quantized way via interconnecting the velocities encoded by a uniform quantizer with a zooming parameter in either a fixed or an adjustable form over a directed communication network. We establish sufficient conditions for the networked harmonic oscillators to converge to a bounded neighborhood of the synchronized orbits with a fixed zooming parameter. We ensure the oscillators to achieve synchronization …
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 wh…
We review some of the recent literature, including Refs. [1-3], on the effects of non-normality on the synchronization of networks of oscillators, and provide numerical evidence that the basin of attraction about the synchronous solution is typically smaller for networks with non-normal Laplacian matrix, compared to networks with a normal Laplacian matrix.
We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients $α_i$ that represent how strongly node $i$ disturbs the synchronization. With this approach, we can quantify the effect of individ…
In this Rapid Communication we investigate spatially constrained networks that realize optimal synchronization properties. After arguing that spatial constraints can be imposed by limiting the amount of `wire' available to connect nodes distributed in space, we use numerical optimization methods to construct networks that realize different trade-offs between optimal synchronization and spatial constraints. Over a large range of parameters such optimal networks are found to have a link length distribution characterized by power law tails $P(l)\propto l^{-α}$, with exponents $α$ increasing as th…
We present an analytical scheme to achieve optimal synchronization in multiplex networks of frustrated and non-frustrated phase oscillators. We derive a multiplex synchrony alignment function (MSAF) for that purpose, the expression of which consists of structural as well as dynamical information of the layers of the multiplex network. Analyzing the MSAF, a set of frequencies (optimal frequencies) is determined to achieve optimal synchronization in the network. Further, using the scheme, we show that perfect synchronization can be achieved in a layer of the multiplex network for given coupling …
From the flashes of fireflies to Josephson junctions and power infrastructure, networks of coupled phase oscillators provide a powerful framework to describe synchronization phenomena in many natural and engineered systems. Most real-world networks are under the influence of noisy, random inputs, potentially inhibiting synchronization. While noise is unavoidable, here we show that there exist optimal noise patterns which minimize desynchronizing effects and even enhance order. Specifically, using analytical arguments we show that in the case of a two-oscillator model, there exists a sharp tran…
We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure.
A recent paper by R. Muolo, T. Carletti, J. P. Gleeson, and M. Asllani [Entropy 23, 36 (2021)] presents a mainly numerical study on the role of non-normality in the synchronization of coupled periodic oscillators, deriving apparent contradictions with the existing literature. Here, we show that their conclusions are artifactual due to a misinterpretation of the master stability function formalism and confirm that the existing literature is correct. We also point to a broader existing literature in which research considered in this paper was correctly addressed numerically, analytically, and ex…
Synchronizing phase frustrated Kuramoto oscillators, a challenge that has found applications from neuronal networks to the power grid, is an eluding problem, as even small phase-lags cause the oscillators to avoid synchronization. Here we show, constructively, how to strategically select the optimal frequency set, capturing the natural frequencies of all oscillators, for a given network and phase-lags, that will ensure perfect synchronization. We find that high levels of synchronization are sustained in the vicinity of the optimal set, allowing for some level of deviation in the frequencies wi…
We develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. Our unified description allows us to identify the resource on which quantum synchronization relies, and to compare quantitatively the synchronization behavior of different limit cycles and signals. We focus on the most elementary quantum system that is able to host a self-sustained oscillation, namely a single spin 1. Despite the spin having no classical analogue, we first show that it can realize the van der Pol limit cycle deep in the quantum regime, which allows us t…
We reply to the recent note "Comment on Synchronization dynamics in non-normal networks: the trade-off for optimality", showing that the authors base their claims mainly on general theoretical arguments that do not necessarily invalidate the adequacy of our previous study. In particular, they do not specifically tackle the correctness of our analysis but instead limit their discussion on the interpretation of our results and conclusions, particularly related to the concept of optimality of network structure related to synchronization dynamics. Nevertheless, their idea of optimal networks is st…
Recent studies have shown that synchronizability of complex networks can be significantly improved by asymmetric couplings, and increase of coupling gradient is always in favor of network synchronization. Here we argue and demonstrate that, for typical complex networks, there usually exists an optimal coupling gradient under which the maximum network synchronizability is achieved. After this optimal value, increase of coupling gradient could deteriorate synchronization. We attribute the suppression of network synchronization at large gradient to the phenomenon of network breaking, and find tha…
We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices. We then show that the solution sets of the two optimization problems coincide and are simultaneously characterized by a simple condition on the Laplacian eigenvalues. Among the optimal networks, we identify a subclass of hierarchical networks, characterized by the absence of feedback loops and the normalization o…
Synchronization, in which individual dynamical units keep in pace with each other in a decentralized fashion, depends both on the dynamical units and on the properties of the interaction network. Yet, the role played by the network has resisted comprehensive characterization within the prevailing paradigm that interactions facilitating pair-wise synchronization also facilitate collective synchronization. Here we challenge this paradigm and show that networks with best complete synchronization, least coupling cost, and maximum dynamical robustness, have arbitrary complexity but quantized total …
The existence and properties of an exact universal excitation waveform for optimal enhancement of directed ratchet transport are deduced from the criticality scenario giving rise to ratchet universality, and confirmed by numerical experiments in the context of a driven overdamped Brownian particle subjected to a vibrating periodic potential. While the universality scenario holds regardless of the waveform of the periodic vibratory excitations involved, it is shown that the enhancement of directed ratchet transport is optimal when the impulse transmitted by those excitations (time integral over…
Coherence resonance (CR), stochastic synchronization (SS), and spike-timing-dependent plasticity (STDP) are ubiquitous dynamical processes in biological neural networks. Whether there exists an optimal network and STDP configuration at which CR and SS are both pronounced is a fundamental question of interest that is still elusive. We expect such a configuration to enable the brain to make synergistic and optimal use of these phenomena to process information efficiently. This paper considers a small-world network of excitable Hodgkin-Huxley neurons driven by channel noise and STDP with an asymm…
In an all-to-all network of integrate-fire oscillators in which there is a disorder in the intrinsic firing rates of the neurons, we show that through spike timing-dependent plasticity the links which have the faster oscillators as presynaptic, tend to be strengthened while the links originated from the slow spiking neurons are weakened. The emergent effective flow of directed connections, introduces the faster neurons as the more influent elements in the network and facilitates synchronization by decreasing the synaptic cost for onset of synchronization.
The preBötzinger Complex, the mammalian inspiratory rhythm generator, encodes inspiratory time as motor pattern. Spike synchronization throughout this sparsely connected network generates inspiratory bursts albeit with variable latencies after preinspiratory activity onset in each breathing cycle. Using preBötC rhythmogenic microcircuit minimal models, we examined the variability in probability and latency to burst, mimicking experiments. Among various physiologically plausible graphs of 1000 point neurons with experimentally determined neuronal and synaptic parameters, directed Erdős-Rényi gr…
Bar-Ilan University（巴伊兰大学）；Indian Institute of Information Technology Raichur（赖布尔印度信息技术学院）；University of Petroleum and Energy Studies（石油与能源研究大学）；Indian Institute of Science Education and Research Thiruvananthapuram（蒂鲁文南特布勒姆印度科学教育与研究学院）
Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we an…
Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchr…
We consider the problem of maximizing the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology. We first extend the well-known master stability formalism to the case of non-diagonalizable networks. We then show that, unless some oscillator is connected to all the others, networks of maximum synchronizability are necessarily non-diagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths. The extension makes the formalism applicable to all possible network structures, w…
We investigate the existence of an optimal interplay between the natural frequencies of a group chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function linking the natural frequencies of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix $L$, the la…
Synchronization of Pulse-Coupled Oscillators and Clocks under Minimal Connectivity Assumptions
Delft University of Technology（代尔夫特理工大学）；Institute for Problems of Mechanical Engineering of the Russian Academy of Sciences（俄罗斯科学院机械问题研究所）；ITMO University（ITMO大学）；University of Groningen（格罗宁根大学）
