Networks Of Phase Oscillators Research Articles

Intermittent phase dynamics of non-autonomous oscillators through time-varying phase

Oscillatory dynamics pervades the universe, appearing in systems of all scales. Whilst autonomous oscillatory dynamics has been extensively studied and is well understood, the very important problem of non-autonomous oscillatory dynamics is less well understood. Here, we provide a framework for non-autonomous oscillatory dynamics, within which we can define intermittent phenomena such as intermittent phase synchronisation. Moreover, we demonstrate this framework with a coupled pair of non-autonomous phase oscillators as well as a higher-dimensional system comprising of two interacting phase-oscillator networks.

Adaptation rules inducing synchronization of heterogeneous Kuramoto oscillator network with triadic couplings.

A class of adaptation functions is found for which a synchronous mode with different number of phase clusters exists in a network of phase oscillators with triadic couplings. This mode is implemented in a fairly wide range of initial conditions and the maximum number of phase clusters is four. The joint influence of coupling strength and adaptation parameters on synchronization in the network has been studied. The desynchronization transition under variation of the adaptation parameter occurs abruptly and begins with the highest-frequency oscillator, spreading hierarchically to all other elements.

Cyclops states in repulsive theta-neuron networks

Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood. Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored. In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2]. Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.

Синхронизация в сетях фазовых осцилляторов с топологиями связей «кольцо» и «малый мир» при различных видах зависимости частоты осциллятора от его положения в сети

Синхронизация в сетях фазовых осцилляторов с топологиями связей «кольцо» и «малый мир» при различных видах зависимости частоты осциллятора от его положения в сети

Synchronization onset for contrarians with higher-order interactions in multilayer systems.

We investigate the impact of contrarians (via negative coupling) in multilayer networks of phase oscillators having higher-order interactions. We report that the multilayer framework facilitates synchronization onset in the negative pairwise coupling regime. The multilayering strength governs the onset of synchronization and the nature of the phase transition, whereas the higher-order interactions dictate the backward critical coupling. Specifically, the system does not synchronize below a critical value of the multilayering strength. The analytical calculations using the mean-field Ott-Antonsen approach agree with the simulations. The results presented here may be useful for understanding emergent behaviors in real-world complex systems with contrarians and higher-order interactions, such as the brain and social system.

Higher-order interactions in Kuramoto oscillators with inertia.

How do higher-order interactions influence the dynamical landscape of a network of the second-order phase oscillators? We address this question using three coupled Kuramoto phase oscillators with inertia under pairwise and higher-order interactions, in search of various collective states, and new states, if any, that show marginal presence or absence under pairwise interactions. We explore this small network for varying phase lag in the coupling and over a range of negative to positive coupling strength of pairwise as well as higher-order or group interactions. In the extended coupling parameter plane of the network we record several well-known states such as synchronization, frequency chimera states, and rotating waves that appear with distinct boundaries. In the parameter space, we also find states generated by the influence of higher-order interactions: The 2+1 antipodal point and the 2+1 phase-locked states. Our results demonstrate the importantance of the choices of the phase lag and the sign of the higher-order coupling strength for the emergent dynamics of the network. We provide analytical support to our numerical results.

Mean-Field Approximations With Adaptive Coupling for Networks With Spike-Timing-Dependent Plasticity.

Understanding the effect of spike-timing-dependent plasticity (STDP) is key to elucidating how neural networks change over long timescales and to design interventions aimed at modulating such networks in neurological disorders. However, progress is restricted by the significant computational cost associated with simulating neural network models with STDP and by the lack of low-dimensional description that could provide analytical insights. Phase-difference-dependent plasticity (PDDP) rules approximate STDP in phase oscillator networks, which prescribe synaptic changes based on phase differences of neuron pairs rather than differences in spike timing. Here we construct mean-field approximations for phase oscillator networks with STDP to describe part of the phase space for this very high-dimensional system. We first show that single-harmonic PDDP rules can approximate a simple form of symmetric STDP, while multiharmonic rules are required to accurately approximate causal STDP. We then derive exact expressions for the evolution of the average PDDP coupling weight in terms of network synchrony. For adaptive networks of Kuramoto oscillators that form clusters, we formulate a family of low-dimensional descriptions based on the mean-field dynamics of each cluster and average coupling weights between and within clusters. Finally, we show that such a two-cluster mean-field model can be fitted to synthetic data to provide a low-dimensional approximation of a full adaptive network with symmetric STDP. Our framework represents a step toward a low-dimensional description of adaptive networks with STDP, and could for example inform the development of new therapies aimed at maximizing the long-lasting effects of brain stimulation.

Chimeras in phase oscillator networks locally coupled through an auxiliary field: Stability and bifurcations

We study networks in the form of a lattice of nodes with a large number of phase oscillators and an auxiliary variable at each node. The only interactions between nodes are nearest-neighbor. The Ott/Antonsen ansatz is used to derive equations for the order parameters of the phase oscillators at each node, resulting in a set of coupled ordinary differential equations. Chimeras are steady states of these equations, and we follow them as parameters are varied, determining their stability and bifurcations. In two-dimensional domains, we find that spiral wave chimeras and rotating waves have significantly different properties than those in networks with nonlocal coupling.

The synchronizing role of multiplexing noise: Exploring Kuramoto oscillators and breathing chimeras.

The synchronization of spatiotemporal patterns in a two-layer multiplex network of identical Kuramoto phase oscillators is studied, where each layer is a non-locally coupled ring. Particular focus is on the role played by a noisy inter-layer communication. It is shown that modulating the inter-layer coupling strength by uncommon noise has a significant impact on the dynamics of the network, in particular, that modulating the interlayer coupling by noise can counter-intuitively induce synchronization in networks. It is further shown that increasing the noise intensity has many other analogous effects to that of increasing the interlayer coupling strength. For example, the noise intensity can also induce state transitions in a similar way, in some cases causing the layers to completely synchronize within themselves. It is discussed how such disturbances may in many cases be beneficial to multilayer systems. These effects are demonstrated both for white noise and for other kinds of colored noise. A "floating" breathing chimera state is also discovered in this system.

Designing Robust Networks of Coupled Phase Oscillators With Applications to the High Voltage Electric Grid

We propose a mathematical framework for designing robust networks of coupled phase-oscillators by leveraging a vulnerability measure proposed by Tyloo et al. that quantifies the impact of a small perturbation at an individual phase-oscillator's natural frequency to the system's global synchronized frequencies. Given a complex network topology with specific governing dynamics, the proposed framework finds an optimal allocation of edge weights that minimizes such vulnerability measure(s) at the node(s) for which we expect perturbations to occur by solving a tractable semi-definite programming problem. We specify the mathematical model to high voltage electric grids where each node corresponds to a voltage phase angle associated with a bus and edges correspond to transmission lines. Edge weights are determined by the susceptance values along the transmission lines. In this application, frequency synchronization is increasingly challenged by the integration of renewable energy, yet is imperative to the grid's health and functionality. Our framework helps to alleviate this challenge by optimizing the placement of renewable generation and the susceptance values along the transmission lines.

Chaotic chimera attractors in a triangular network of identical oscillators.

A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.

Complex dynamics in adaptive phase oscillator networks.

Networks of coupled dynamical units give rise to collective dynamics such as the synchronization of oscillators or neurons in the brain. The ability of the network to adapt coupling strengths between units in accordance with their activity arises naturally in a variety of contexts, including neural plasticity in the brain, and adds an additional layer of complexity: the dynamics on the nodes influence the dynamics of the network and vice versa. We study a minimal model of Kuramoto phase oscillators including a general adaptive learning rule with three parameters (strength of adaptivity, adaptivity offset, adaptivity shift), mimicking learning paradigms based on spike-time-dependent plasticity. Importantly, the strength of adaptivity allows to tune the system away from the limit of the classical Kuramoto model, corresponding to stationary coupling strengths and no adaptation and, thus, to systematically study the impact of adaptivity on the collective dynamics. We carry out a detailed bifurcation analysis for the minimal model consisting of N=2 oscillators. The non-adaptive Kuramoto model exhibits very simple dynamic behavior, drift, or frequency-locking; but once the strength of adaptivity exceeds a critical threshold non-trivial bifurcation structures unravel: A symmetric adaptation rule results in multi-stability and bifurcation scenarios, and an asymmetric adaptation rule generates even more intriguing and rich dynamics, including a period-doubling cascade to chaos as well as oscillations displaying features of both librations and rotations simultaneously. Generally, adaptation improves the synchronizability of the oscillators. Finally, we also numerically investigate a larger system consisting of N=50 oscillators and compare the resulting dynamics with the case of N=2 oscillators.

Heterogeneous Nucleation in Finite-Size Adaptive Dynamical Networks

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, coevolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite-size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multistep transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

Partial synchronization and community switching in phase-oscillator networks and its analysis based on a bidirectional, weighted chain of three oscillators.

Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities, i.e., a group of nodes displaying a similar dynamical process. We have investigated both analytically and numerically the development of a dynamical community structure, where the community is referred to as a group of nodes synchronized in frequency, in networks of phase oscillators. We first demonstrate that using a few example networks, the community structure changes when network connectivity or interaction strength is varied. In particular, we found that community switching, i.e., a portion of oscillators change the group to which they synchronize, occurs for a range of parameters. We then propose a three-oscillator model: a bidirectional, weighted chain of three Kuramoto phase oscillators, as a theoretical framework for understanding the community formation and its variation. Our analysis demonstrates that the model shows a variety of partially synchronized patterns: oscillators with similar natural frequencies tend to synchronize for weak coupling, while tightly connected oscillators tend to synchronize for strong coupling. We obtain approximate expressions for the critical coupling strengths by employing a perturbative approach in a weak coupling regime and a geometric approach in strong coupling regimes. Moreover, we elucidate the bifurcation types of transitions between different patterns. Our theory might be useful for understanding the development of partially synchronized patterns in a wider class of complex networks than community structured networks.

Periodic orbits in the Ott–Antonsen manifold

In their seminal paper (2008 Chaos 18 037113), Ott and Antonsen showed that large groups of phase oscillators driven by a certain type of common force display low dimensional long-term dynamics, which is described by a small number of ordinary differential equations. This fact was later used as a simplifying reduction technique in many studies of synchronisation phenomena occurring in networks of coupled oscillators and in neural networks. Most of these studies focused mainly on partially synchronised states corresponding to the equilibrium-type dynamics in the so called Ott–Antonsen manifold. Going beyond this paradigm, here we propose a universal approach for the efficient analysis of partially synchronised states with non-equilibrium periodic collective dynamics. Our method is based on the observation that the Poincaré map of the complex Riccati equation, which describes the dynamics in the Ott–Antonsen manifold, coincides with the well-known Möbius transformation. To illustrate the possibilities of our method, we use it to calculate a complete bifurcation diagram of travelling chimera states in a ring network of phase oscillators with asymmetric nonlocal coupling.

Heterogeneous phase oscillators encoded by simplexes under coupling strength symmetry breaking: Optimal synchronization and abrupt desynchronization

Recent advances demonstrate that the collective dynamics of heterogeneous phase oscillators ensemble encoded by simplexes play an important role in the storage of information in the human brain. In this letter, we discuss the collective dynamics of heterogeneous phase oscillator networks encoded by simplexes under coupling strength symmetry breaking. Specifically, the strength of the asymmetric coupling between the oscillators is redefined by the degree of the oscillators under 1- and 2-simplex encoding and the heterogeneity parameter. Here, we discuss the effect of different degrees of coupling strength symmetry breaking on the synchronization capability of the system. In particular, we find that proper coupling strength symmetry breaking is beneficial to enhance the synchronization capability of the system. We capture an interesting phenomenon in heterogeneous phase oscillator networks that distinguish symmetric coupling under 1- and 2-simplex encoding, i.e., the system exhibits significant desynchronization behavior under strong symmetry breaking of the coupling strength. Finally, for a fixed heterogeneity parameter and coupling strength assignment probability, we give the optimal intrinsic frequency assignment scheme based on the spectral decomposition of the composite Laplace matrix of the underlying network. Theoretically, these findings may help us to better understand the collective dynamics in heterogeneous phase oscillator networks encoded by simplexes under asymmetric coupling that are prevalent in the real world. In particular, they provide inspiration and guidance for optimizing the synchronization of heterogeneous phase oscillator networks encoded by simplexes under coupling strength symmetry breaking.

Flexible patterns of information transfer in frustrated networks of phase oscillators

Brain networks are characterized by flexible patterns of pairwise correlations and information exchange between different brain regions. Such dynamic patterns are crucial for an efficient response of the brain to environmental and cognitive demands. We here propose that the collective oscillations in the brain can provide a mechanism to control dynamical interactions and the exchange of information across brain networks. In particular, we show that the phase difference between oscillatory activities in different brain regions determines the transmission of neural signals. To further corroborate this, we study a network of coupled oscillators with repulsive couplings and show that the amount of information transfer between the nodes is determined by the phase differences. The emergence of multiple (locally) stable states due to the frustration makes it possible to change the patterns of information transfer between the nodes by means of the switching between different stable states. Our results indicate that frustration can be the mechanism through which large-scale brain networks control the effective connectivity and the routes for the information transfer between different brain regions.

Forecasting macroscopic dynamics in adaptive Kuramoto network using reservoir computing.

Forecasting a system's behavior is an essential task encountering the complex systems theory. Machine learning offers supervised algorithms, e.g., recurrent neural networks and reservoir computers that predict the behavior of model systems whose states consist of multidimensional time series. In real life, we often have limited information about the behavior of complex systems. The brightest example is the brain neural network described by the electroencephalogram. Forecasting the behavior of these systems is a more challenging task but provides a potential for real-life application. Here, we trained reservoir computer to predict the macroscopic signal produced by the network of phase oscillators. The Lyapunov analysis revealed the chaotic nature of the signal and reservoir computer failed to forecast it. Augmenting the feature space using Takkens' theorem improved the quality of forecasting. RC achieved the best prediction score when the number of signals coincided with the embedding dimension estimated via the nearest false neighbors method. We found that short-time prediction required a large number of features, while long-time prediction utilizes a limited number of features. These results refer to the bias-variance trade-off, an important concept in machine learning.

Dynamics of phase oscillator networks with synaptic weight and structural plasticity

We study the dynamics of Kuramoto oscillator networks with two distinct adaptation processes, one varying the coupling strengths and the other altering the network structure. Such systems model certain networks of oscillatory neurons where the neuronal dynamics, synaptic weights, and network structure interact with and shape each other. We model synaptic weight adaptation with spike-timing-dependent plasticity (STDP) that runs on a longer time scale than neuronal spiking. Structural changes that include addition and elimination of contacts occur at yet a longer time scale than the weight adaptations. First, we study the steady-state dynamics of Kuramoto networks that are bistable and can settle in synchronized or desynchronized states. To compare the impact of adding structural plasticity, we contrast the network with only STDP to one with a combination of STDP and structural plasticity. We show that the inclusion of structural plasticity optimizes the synchronized state of a network by allowing for synchronization with fewer links than a network with STDP alone. With non-identical units in the network, the addition of structural plasticity leads to the emergence of correlations between the oscillators’ natural frequencies and node degrees. In the desynchronized regime, the structural plasticity decreases the number of contacts, leading to a sparse network. In this way, adding structural plasticity strengthens both synchronized and desynchronized states of a network. Second, we use desynchronizing coordinated reset stimulation and synchronizing periodic stimulation to induce desynchronized and synchronized states, respectively. Our findings indicate that a network with a combination of STDP and structural plasticity may require stronger and longer stimulation to switch between the states than a network with STDP only.

Heterogeneous phase oscillators under symmetry breaking of coupling strength: Optimal synchronization and desynchronization under strong gain

Revealing the collective dynamics of heterogeneous oscillator networks under asymmetric coupling is a great challenge. In this study, we discuss optimal synchronization and desynchronization under strong gain in heterogeneous phase oscillator networks with symmetry breaking of coupling strength. Firstly, for a fixed coupling strength, we discuss how the heterogeneity parameter affects the synchronization ability of the system. Secondly, for different values of the heterogeneity parameter, we propose the concepts of gain factor and weakening factor. We find that the synchronization ability of the system is greatly weakened due to the presence of the weakening factor. On the contrary, the presence of the gain factor allows the system to quickly transition to a completely synchronized state even at a small coupling strength. Thirdly, we find an interesting phenomenon that distinguishes the symmetrically coupled heterogeneous phase oscillator networks: the system under symmetry breaking of coupling strength exhibits a significant desynchronization behavior at strong gain. Finally, for a fixed heterogeneity parameter, we give the optimal intrinsic frequency assignment scheme based on the spectral decomposition of the Laplace matrix of the underlying network. Theoretically, these findings may help us to better understand the collective dynamics in heterogeneous phase oscillator networks under asymmetric coupling that are prevalent in the real world. In particular, they provide inspiration and guidance for optimizing the synchronization of heterogeneous phase oscillator networks under symmetry breaking of coupling strength.