Coexistence of Quantized, Time Dependent, Clusters in Globally Coupled Oscillators

Synchronization transitions in phase oscillator populations with partial adaptive coupling

The adaptation underlying many realistic processes plays a pivotal role in shaping the collective dynamics of diverse systems. Here, we untangle the generic conditions for synchronization transitions in a system of coupled phase oscillators incorporating the adaptive scheme encoded by the feedback between the coupling and the order parameter via a power-law function with different weights. We mathematically argue that, in the subcritical and supercritical correlation scenarios, there exists no critical adaptive fraction for synchronization transitions converting from the first (second)-order to the second (first)-order. In contrast to the synchronization transitions previously deemed, the explosive and continuous phase transitions take place in the corresponding regions as long as the adaptive fraction is nonzero, respectively. Nevertheless, we uncover that, at the critical correlation, the routes toward synchronization depend crucially on the relative adaptive weights. In particular, we unveil that the emergence of a range of interrelated scaling behaviors of the order parameter near criticality, manifesting the subcritical and supercritical bifurcations, are responsible for various observed phase transitions. Our work, thus, provides profound insights for understanding the dynamical nature of phase transitions, and for better controlling and manipulating synchronization transitions in networked systems with adaptation.

Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling

The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.

Discontinuous phase transition switching induced by a power-law function between dynamical parameters in coupled oscillators

In biological or physical systems, the intrinsic properties of oscillators are heterogeneous and correlated. These two characteristics have been empirically validated and have garnered attention in theoretical studies. In this paper, we propose a power-law function existed between the dynamical parameters of the coupled oscillators, which can control discontinuous phase transition switching. Unlike the special designs for the coupling terms, this generalized function within the dynamical term reveals another path for generating the first-order phase transitions. The power-law relationship between dynamic characteristics is reasonable, as observed in empirical studies, such as long-term tremor activity during volcanic eruptions and ion channel characteristics of the Xenopus expression system. Our work expands the conditions that used to be strict for the occurrence of the first-order phase transitions and deepens our understanding of the impact of correlation between intrinsic parameters on phase transitions. We explain the reason why the continuous phase transition switches to the discontinuous phase transition when the control parameter is at a critical value.

Deterministic correlations enhance synchronization in oscillator populations with heterogeneous coupling

Synchronization is a critical phenomenon that displays a pivotal role in a wealth of dynamical processes ranging from natural to artificial systems. Here, we untangle the synchronization optimization in a system of globally coupled phase oscillators incorporating heterogeneous interactions encoded by the deterministic-random coupling. We uncover that, within the given restriction, the added deterministic correlations can profoundly enhance the synchronizability in comparison with the uncorrelated scenario. The critical points manifesting the onset of synchronization and desynchronization transitions, as well as the level of phase coherence, are significantly shaped by the increment of deterministic correlations. In particular, we provide an analytical treatment to properly ground the mechanism underlying synchronization enhancement and substantiate that the analytical predictions are in fair agreement with the numerical simulations. This study is a step forward in highlighting the importance of heterogeneous coupling among dynamical agents, which provides insights for control strategies of synchronization in complex systems.

Phase transitions in an adaptive network with the global order parameter adaptation

We consider an adaptive network of Kuramoto oscillators with purely dyadic coupling, where the adaption is proportional to the degree of the global order parameter. We find only the continuous transition to synchronization via the pitchfork bifurcation, an abrupt synchronization (desynchronization) transition via the pitchfork (saddle-node) bifurcation resulting in the bistable region R_{1}. This is a smooth continuous transition to a weakly synchronized state via the pitchfork bifurcation followed by a subsequent abrupt transition to a strongly synchronized state via a second saddle-node bifurcation along with an abrupt desynchronization transition via the first saddle-node bifurcation resulting in the bistable region R_{2} between the weak and strong synchronization. The transition goes from the bistable region R_{1} to the bistable region R_{2}, and transition from the incoherent state to the bistable region R_{2} as a function of the coupling strength for various ranges of the degree of the global order parameter and the adaptive coupling strength. We also find that the phase-lag parameter enlarges the spread of the weakly synchronized state and the bistable states R_{1} and R_{2} to a large region of the parameter space. We also derive the low-dimensional evolution equations for the global order parameters using the Ott-Antonsen ansatz. Further, we also deduce the pitchfork, first and second saddle-node bifurcation conditions, which is in agreement with the simulation results.

Relaxation dynamics of phase oscillators with generic heterogeneous coupling

The coupled phase oscillator model serves as a paradigm that has been successfully used to shed light on the collective dynamics occurring in large ensembles of interacting units. It was widely known that the system experiences a continuous (second-order) phase transition to synchronization by gradually increasing the homogeneous coupling among the oscillators. As the interest in exploring synchronized dynamics continues to grow, the heterogeneous patterns between phase oscillators have received ample attention during the past years. Here, we consider a variant of the Kuramoto model with quenched disorder in their natural frequencies and coupling. Correlating these two types of heterogeneity via a generic weighted function, we systematically investigate the impacts of the heterogeneous strategies, the correlation function, and the natural frequency distribution on the emergent dynamics. Importantly, we develop an analytical treatment for capturing the essential dynamical properties of the equilibrium states. In particular, we uncover that the critical threshold corresponding to the onset of synchronization is unaffected by the location of the inhomogeneity, which, however, does depend crucially on the value of the correlation function at its center. Furthermore, we reveal that the relaxation dynamics of the incoherent state featuring the responses to external perturbations is significantly shaped by all the considered effects, thereby leading to various decaying mechanisms of the order parameters in the subcritical region. Moreover, we untangle that synchronization is facilitated by the out-coupling strategy in the supercritical region. Our study is a step forward in highlighting the potential importance of the inhomogeneous patterns involved in the complex systems, and could thus provide theoretical insights for profoundly understanding the generic statistical mechanical properties of the steady states toward synchronization.

Route to synchronization in coupled phase oscillators with frequency-dependent coupling: Explosive or continuous?

Interconnected dynamical systems often transition between states of incoherence and synchronization due to changes in system parameters. These transitions could be continuous (gradual) or explosive (sudden) and may result in failures, which makes determining their nature important. In this study, we abstract dynamical networks as an ensemble of globally coupled Kuramoto-like phase oscillators with frequency-dependent coupling and investigate the mechanisms for transition between incoherent and synchronized dynamics. The characteristics that dictate a continuous or explosive route to synchronization are the distribution of the natural frequencies of the oscillators, quantified by a probability density function g(ω), and the relation between the coupling strength and natural frequency of an oscillator, defined by a frequency-coupling strength correlation function f(ω). Our main results are conditions on f(ω) and g(ω) that result in continuous or explosive routes to synchronization and explain the underlying physics. The analytical developments are validated through numerical examples.

Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators

Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.

Partial locking in phase-oscillator populations with heterogenous coupling

We consider a variant of the mean-field model of coupled phase oscillators with uniform distribution of natural frequencies. By establishing correlations between the quenched disorder of intrinsic frequencies and coupling strength with both in- and out-coupling heterogeneities, we reveal a generic criterion for the onset of partial locking that takes place in a domain with the coexistence of phase-locked oscillators and drifters. The critical points manifesting the instability of the stationary states are obtained analytically. In particular, the bifurcation mechanism of the equilibrium states is uncovered by the use of frequency-dependent version of the Ott-Antonsen reduction consistently with the analysis based on the self-consistent approach. We demonstrate that both the manner of coupling heterogeneity and correlation exponent have influence on the emergent patterns of partial locking. Our research could find applicability in better understanding the phase transitions and related collective phenomena involving synchronization control in networked systems.

Enlarged Kuramoto model: Secondary instability and transition to collective chaos

The emergence of collective synchrony from an incoherent state is a phenomenon essentially described by the Kuramoto model. This canonical model was derived perturbatively, by applying phase reduction to an ensemble of heterogeneous, globally coupled Stuart-Landau oscillators. This derivation neglects nonlinearities in the coupling constant. We show here that a comprehensive analysis requires extending the Kuramoto model up to quadratic order. This "enlarged Kuramoto model" comprises three-body (nonpairwise) interactions, which induce strikingly complex phenomenology at certain parameter values. As the coupling is increased, a secondary instability renders the synchronized state unstable, and subsequent bifurcations lead to collective chaos. An efficient numerical study of the thermodynamic limit, valid for Gaussian heterogeneity, is carried out by means of a Fourier-Hermite decomposition of the oscillator density.

Contrarians Synchronize beyond the Limit of Pairwise Interactions

We give evidence that a population of pure contrarian globally coupled D-dimensional Kuramoto oscillators reaches a collective synchronous state when the interplay between the units goes beyond the limit of pairwise interactions. Namely, we will show that the presence of higher-order interactions may induce the appearance of a coherent state even when the oscillators are coupled negatively to the mean field. An exact solution for the description of the microscopic dynamics for forward and backward transitions is provided, which entails imperfect symmetry breaking of the population into a frequency-locked state featuring two clusters of different instantaneous phases. Our results contribute to a better understanding of the powerful potential of group interactions entailing multidimensional choices and novel dynamical states in many circumstances, such as in social systems.

Stability of synchronization in simplicial complexes

Various systems in physics, biology, social sciences and engineering have been successfully modeled as networks of coupled dynamical systems, where the links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We show that complete synchronization exists as an invariant solution, and give the necessary condition for it to be observed as a stable state. Moreover, in some relevant instances, such a necessary condition takes the form of a Master Stability Function. This generalizes the existing results valid for pairwise interactions to the case of complex systems with the most general possible architecture.

Discontinuous Transitions and Rhythmic States in the D-Dimensional Kuramoto Model Induced by a Positive Feedback with the Global Order Parameter

From fireflies to cardiac cells, synchronization governs important aspects of nature, and the Kuramoto model is the staple for research in this area. We show that generalizing the model to oscillators of dimensions higher than 2 and introducing a positive feedback mechanism between the coupling and the global order parameter leads to a rich and novel scenario: the synchronization transition is explosive at all even dimensions, whilst it is mediated by a time-dependent, rhythmic, state at all odd dimensions. Such a latter circumstance, in particular, differs from all other time-dependent states observed so far in the model. We provide the analytic description of this novel state, which is fully corroborated by numerical calculations. Our results can, therefore, help untangle secrets of observed time-dependent swarming and flocking dynamics that unfold in three dimensions, and where this novel state could thus provide a fresh perspective for as yet not understood formations.

Universal scaling and phase transitions of coupled phase oscillator populations

The Kuramoto model, which serves as a paradigm for investigating synchronization phenomena of oscillatory systems, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here we generalize a number of classical results by presenting a general framework for analytically capturing the critical scaling properties of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on the coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determine the scaling properties of order parameter above criticality.

Complex networks exhibit intermittent synchronization

The path toward the synchronization of an ensemble of dynamical units goes through a series of transitions determined by the dynamics and the structure of the connections network. In some systems on the verge of complete synchronization, intermittent synchronization, a time-dependent state where full synchronization alternates with non-synchronized periods, has been observed. This phenomenon has been recently considered to have functional relevance in neuronal ensembles and other networked biological systems close to criticality. We characterize the intermittent state as a function of the network topology to show that the different structures can encourage or inhibit the appearance of early signs of intermittency. In particular, we study the local intermittency and show how the nodes incorporate to intermittency in hierarchical order, which can provide information about the node topological role even when the structure is unknown.

The role of timescale separation in oscillatory ensembles with competitive coupling

We study a heterogeneous population consisting of two groups of oscillatory elements, one with attractive and one with repulsive coupling. Moreover, we set different internal timescales for the oscillators of the two groups and concentrate on the role of this timescale separation in the collective behavior. Our results demonstrate that it may significantly modify synchronization properties of the system, and the implications are fundamentally different depending on the ratio between the group timescales. For the slower attractive group, synchronization properties are similar to the case of equal timescales. However, when the attractive group is faster, these properties significantly change and bistability appears. The other collective regimes such as frozen states and solitary states are also shown to be crucially influenced by timescale separation.

Reduction of oscillator dynamics on complex networks to dynamics on complete graphs through virtual frequencies

We consider the synchronization of oscillators in complex networks where there is an interplay between the oscillator dynamics and the network topology. Through a remarkable transformation in parameter space and the introduction of virtual frequencies we show that Kuramoto oscillators on annealed networks, with or without frequency-degree correlation, and Kuramoto oscillators on complete graphs with frequency-weighted coupling can be transformed to Kuramoto oscillators on complete graphs with a rearranged, virtual frequency distribution and uniform coupling. The virtual frequency distribution encodes both the natural frequency distribution (dynamics) and the degree distribution (topology). We apply this transformation to give direct explanations to a variety of phenomena that have been observed in complex networks, such as explosive synchronization and vanishing synchronization onset.

Characterizing nonstationary coherent states in globally coupled conformist and contrarian oscillators

For decades, the description and characterization of nonstationary coherent states in coupled oscillators have not been available. We here consider the Kuramoto model consisting of conformist and contrarian oscillators. In the model, contrarians are chosen from a bimodal Lorentzian frequency distribution and flipped into conformists at random. A complete and systematic analytical treatment of the model is provided based on the Ott-Antonsen ansatz. In particular, we predict and analyze not only the stability of all stationary states (such as the incoherent, the π, and the traveling-wave states), but also that of the two nonstationary states: the Bellerophon and the oscillating-π state. The theoretical predictions are fully supported by extensive numerical simulations.

Brain synchronizability, a false friend

Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network is, given its topological organization, are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. Here, we argue that brain dynamics does not fulfil the formal criteria under which synchronizability is usually quantified and, perhaps more importantly, this measure refers to a global dynamical condition that never holds in the brain (not even in the most pathological conditions), and therefore no neurophysiological conclusions should be drawn based on it. We discuss the meaning of synchronizability and its applicability to neuroscience and propose alternative ways to quantify brain networks synchronization.

Clustering and Bellerophon state in Kuramoto model with second-order coupling

In this paper, clustering in the Kuramoto model with second-order coupling is investigated under the bimodal Lorentzian frequency distribution. By linear stability analysis and the Ott-Antonsen ansatz treatment, the critical coupling strength for the synchronization transition is obtained. The theoretical results are further verified by numerical simulations. It has been revealed that various synchronization paths, including the first- and second-order transitions as well as the multiple bifurcations, exist in this system with different parameters of frequency distribution. In certain parameter regimes, the Bellerophon states are observed and their dynamical features are fully characterized.

Phase synchronization and intermittent behavior in healthy and Alzheimer-affected human-brain-based neural network

We study the dynamical proprieties of phase synchronization and intermittent behavior of neural systems using a network of networks structure based on an experimentally obtained human connectome for healthy and Alzheimer-affected brains. We consider a network composed of 78 neural subareas (subnetworks) coupled with a mean-field potential scheme. Each subnetwork is characterized by a small-world topology, composed of 250 bursting neurons simulated through a Rulkov model. Using the Kuramoto order parameter we demonstrate that healthy and Alzheimer-affected brains display distinct phase synchronization and intermittence properties as a function of internal and external coupling strengths. In general, for the healthy case, each subnetwork develops a substantial level of internal synchronization before a global stable phase-synchronization state has been established. For the unhealthy case, despite the similar internal subnetwork synchronization levels, we identify higher levels of global phase synchronization occurring even for relatively small internal and external coupling. Using recurrence quantification analysis, namely the determinism of the mean-field potential, we identify regions where the healthy and unhealthy networks depict nonstationary behavior, but the results denounce the presence of a larger region or intermittent dynamics for the case of Alzheimer-affected networks. A possible theoretical explanation based on two locally stable but globally unstable states is discussed.

Rhythmic synchronization and hybrid collective states of globally coupled oscillators

Macroscopic rhythms are often signatures of healthy functioning in living organisms, but they are still poorly understood on their microscopic bases. Globally interacting oscillators with heterogeneous couplings are here considered. Thorough theoretical and numerical analyses indicate the presence of multiple phase transitions between different collective states, with regions of bi-stability. Novel coherent phases are unveiled, and evidence is given of the spontaneous emergence of macroscopic rhythms where oscillators’ phases are always found to be self-organized as in Bellerophon states, i.e. in multiple clusters with quantized values of their average frequencies. Due to their rather unconditional appearance, the circumstance is paved that the Bellerophon states grasp the microscopic essentials behind collective rhythms in more general systems of interacting oscillators.

Enhancing network synchronization by phase modulation

Due to time delays in signal transmission and processing, phase lags are inevitable in realistic complex oscillator networks. Conventional wisdom is that phase lags are detrimental to network synchronization. Here we show that judiciously chosen phase lag modulations can result in significantly enhanced network synchronization. We justify our strategy of phase modulation, demonstrate its power in facilitating and enhancing network synchronization with synthetic and empirical network models, and provide an analytic understanding of the underlying mechanism. Our work provides an alternative approach to synchronization optimization in complex networks, with insights into control of complex nonlinear networks.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Asymmetric couplings enhance the transition from chimera state to synchronization

Chimera state has been well studied recently, but little attention has been paid to its transition to synchronization. We study this topic here by considering two groups of adaptively coupled Kuramoto oscillators. By searching the final states of different initial conditions, we find that the system can easily show a chimera state with robustness to initial conditions, in contrast to the sensitive dependence of chimera state on initial conditions in previous studies. Further, we show that, in the case of symmetric couplings, the behaviors of the two groups are always complementary to each other, i.e., robustness of chimera state, except a small basin of synchronization. Interestingly, we reveal that the basin of synchronization will be significantly increased when either the coupling of inner groups or that of intergroups are asymmetric. This transition from the attractor of chimera state to the attractor of synchronization is closely related to both the phase delay and the asymmetric degree of coupling strengths, resulting in a diversity of attractor's patterns. A theory based on the Ott-Antonsen ansatz is given to explain the numerical simulations. This finding may be meaningful for the control of competition between two attractors in biological systems, such as the cardiac rhythm and ventricular fibrillation, etc.

Detection of nonstationary transition to synchronized states of a neural network using recurrence analyses

We study the stability of asymptotic states displayed by a complex neural network. We focus on the loss of stability of a stationary state of networks using recurrence quantifiers as tools to diagnose local and global stabilities as well as the multistability of a coupled neural network. Numerical simulations of a neural network composed of 1024 neurons in a small-world connection scheme are performed using the model of Braun et al. [Int. J. Bifurcation Chaos 08, 881 (1998)IJBEE40218-127410.1142/S0218127498000681], which is a modified model from the Hodgkin-Huxley model [J. Phys. 117, 500 (1952)]. To validate the analyses, the results are compared with those produced by Kuramoto's order parameter [Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin Heidelberg, 1984)]. We show that recurrence tools making use of just integrated signals provided by the networks, such as local field potential (LFP) (LFP signals) or mean field values bring new results on the understanding of neural behavior occurring before the synchronization states. In particular we show the occurrence of different stationary and nonstationarity asymptotic states.

A small change in neuronal network topology can induce explosive synchronization transition and activity propagation in the entire network

We here study explosive synchronization transitions and network activity propagation in networks of coupled neurons to provide a new understanding of the relationship between network topology and explosive dynamical transitions as in epileptic seizures and their propagations in the brain. We model local network motifs and configurations of coupled neurons and analyze the activity propagations between a group of active neurons to their inactive neuron neighbors in a variety of network configurations. We find that neuronal activity propagation is limited to local regions when network is highly clustered with modular structures as in the normal brain networks. When the network cluster structure is slightly changed, the activity propagates to the entire network, which is reminiscent of epileptic seizure propagation in the brain. Finally, we analyze intracranial electroencephalography (IEEG) recordings of a seizure episode from a epilepsy patient and uncover that explosive synchronization-like transition occurs around the clinically defined onset of seizure. These findings may provide a possible mechanism for the recurrence of epileptic seizures, which are known to be the results of aberrant neuronal network structure and/or function in the brain.

Intermittent Bellerophon state in frequency-weighted Kuramoto model

Recently, the Bellerophon state, which is a quantized, time dependent, clustering state, was revealed in globally coupled oscillators [Bi et al., Phys. Rev. Lett. 117, 204101 (2016)]. The most important characteristic is that in such a state, the oscillators split into multiple clusters. Within each cluster, the instantaneous frequencies of the oscillators are not the same, but their average frequencies lock to a constant. In this work, we further characterize an intermittent Bellerophon state in the frequency-weighted Kuramoto model with a biased Lorentzian frequency distribution. It is shown that the evolution of oscillators exhibits periodical intermittency, following a synchronous pattern of bursting in a short period and resting in a long period. This result suggests that the Bellerophon state might be generic in Kuramoto-like models regardless of different arrangements of natural frequencies.

Synchronization and Bellerophon states in conformist and contrarian oscillators

The study of synchronization in generalized Kuramoto models has witnessed an intense boost in the last decade. Several collective states were discovered, such as partially synchronized, chimera, π or traveling wave states. We here consider two populations of globally coupled conformist and contrarian oscillators (with different, randomly distributed frequencies), and explore the effects of a frequency-dependent distribution of the couplings on the collective behaviour of the system. By means of linear stability analysis and mean-field theory, a series of exact solutions is extracted describing the critical points for synchronization, as well as all the emerging stationary coherent states. In particular, a novel non-stationary state, here named as Bellerophon state, is identified which is essentially different from all other coherent states previously reported in the Literature. A robust verification of the rigorous predictions is supported by extensive numerical simulations.