System Of Phase Oscillators Research Articles

Competition for synchronization in a phase oscillator system

A system of phase oscillators with a Central Oscillator (CO) and a set of n Peripheral Oscillators (POs) is considered. Feed-forward and feedback connections between the CO and POs are determined by two interaction functions which are assumed to be smooth, odd, and periodic. To describe the competition of POs for synchronization with the CO, we study the asymptotic stability of fixed points corresponding to in-phase synchronization of a group of k POs, while other POs are in anti-phase with the CO. It is shown that stability conditions can be formulated in terms of four parameters that describe the slopes of the interaction functions at zero and half-period points. Analytical description of stability in terms of the regions in 4-dimensional parameter space is given. Combining stability analysis with the detailed study of geometry of invariant manifolds, the bifurcations of fixed points are investigated. We show that various dynamical regimes such as multistability, heteroclinic orbits, and chaos are possible. Analytical stability conditions for global synchronization of POs with the CO are formulated for the systems with local connections between POs. It is shown that synchronization in a large system with local connections becomes unstable even under weak desynchronizing influence from the CO. The application of the results to modeling in neuroscience and, in particular, for modeling visual attention is discussed.

The effects of time delays on synchronization properties in a network of neural mass models

Patterns of (de)synchronization are key to neural functioning. We asked what general mechanisms are responsible for the emergence and disappearance of these (de)synchronization patterns. To answer this question we analytically revealed potential effects of physiologically motivated time delays in neural populations. We investigated a network of Neural Mass Models (NMM), here Freeman's seminal model that has successfully been used to model alpha-oscillations [1] and EEG activity in the olfactory system [2] and visual cortex [3]. In contrast to these local applications we were interested in distributed activities over larger networks and hence coupled arbitrarily many NMMs involving the aforementioned time delays. That is, we represented activity in distinct brain areas by means of oscillatory excitatory/inhibitory pairs and consider network behavior as representative for global brain activity. We characterized synchronization as the appearance of clustered stationary phase probability density functions (PDFs). In general, this requires a phase dynamics description of the network, which can be established by applying a slowly varying amplitude approximation (see also [4]), together with a continuity equation covering the PDF's evolution. In fact we succeeded to show the existence of synchronized phase PDFs even for small inter area coupling strengths in the case of both zero and constant finite delays, whereas synchronization disappeared when delays were distributed throughout the network. Homogeneity of the effective coupling matrices (D) appearing in the phase dynamics equation is crucial in this respect as it is required for the existence of clustered phase PDFs. Loss of homogeneity in D can be caused by distributed delays as well as heterogeneity in the structural coupling matrix, leading to the conclusion that the mechanism modulating desynchronization is equal for both cases. The effect of distributed delays is consistent with analytic results in phase oscillator systems [5,6]. This also holds for the drop in oscillation frequency for increased time delay values [5,7] that we observed in simulation results. These two findings validate the description of network behavior in terms of its phase dynamics. In biologically more realistic networks time delays have been shown to determine network synchronization properties in a similar way [8]. Furthermore delays have turned out to be crucial in the behavior of resting state networks [9]. Our results suggest that structural heterogeneity may be compensated for by an appropriate distribution of time delays, thereby explaining the necessity of time delays in RSNs to establish biological plausible synchronization patterns.

Complex macroscopic behavior in systems of phase oscillators with adaptive coupling

Using recent dimensionality reduction techniques in large systems of coupled phase oscillators exhibiting bistability, we analyze the complex macroscopic behavior arising when the coupling between oscillators is allowed to evolve slowly as a function of either macroscopic or local system properties. For example, we observe macroscopic excitability and intermittent synchrony in a system of time-delayed Kuramoto oscillators with Hebbian and anti-Hebbian learning. We demonstrate the robustness of our findings by considering systems with increasing complexity, including time-delayed oscillators with adaptive network structure and community interaction, as well as a system with bimodally distributed frequencies.

Nonuniversal Transitions to Synchrony in the Sakaguchi-Kuramoto Model

We investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (Sakaguchi-Kuramoto model). We show that for certain unimodal frequency distributions there appear unusual types of synchronization transitions, where synchrony can decay with increasing coupling, incoherence can regain stability for increasing coupling, or multistability between partially synchronized states and/or the incoherent state can appear. Our method is a bifurcation analysis based on a frequency dependent version of the Ott-Antonsen method and allows for a universal description of possible synchronization transition scenarios for any given distribution of natural frequencies.

Stepwise transition to higher degrees of coherence in a random network of phase oscillators

We consider a model system of phase oscillators which are connected in a random network. The network favors the connection of oscillators with close values of phases. We extend the order parameter used in the study of synchronization of phase oscillators and define generalized order parameters for the model system. We investigate the equilibrium properties of the model and reveal a phenomenon of stepwise transitions to higher degrees of coherence as the system goes through a series of second-order phase transitions for the order parameters. We also discuss a possible realization of the model in real physical systems.

Persistent fluctuations in synchronization rate in globally coupled oscillators with periodic external forcing

A system of phase oscillators with repulsive global coupling and periodic external forcing undergoing asynchronous rotation is considered. The synchronization rate of the system can exhibit persistent fluctuations depending on parameters and initial phase distributions, and the amplitude of the fluctuations scales with the system size for uniformly random initial phase distributions. Using the Watanabe-Strogatz transformation that reduces the original system to low-dimensional macroscopic equations, we show that the fluctuations are collective dynamics of the system corresponding to low-dimensional trajectories of the reduced equations. It is argued that the amplitude of the fluctuations is determined by the inhomogeneity of the initial phase distribution, resulting in system-size scaling for the random case.

Bifurcations in phase oscillator networks with a central element

A system of phase oscillators with identical natural frequencies and the star-like architecture of connections is considered. Interaction functions are described by two terms of Fourier expansion. Bifurcation analysis of small systems containing 3 or 4 oscillators has been performed. The results are summarized in bifurcation diagrams that provide a full description of the boundaries between regions with different dynamics and the types of bifurcations that lead to the changes in the topology of phase space. The bifurcations include changes of fixed point stability and formation (destruction) of limit and heteroclinic cycles. For the system with 4 oscillators chaotic behaviour has been investigated. The results can be useful to control system dynamics through an appropriate choice and variation of parameter values. The generalization of the results to the systems with an arbitrary number of oscillators and application of the results in computational neuroscience are discussed.

Multiscale dynamics in communities of phase oscillators

We investigate the dynamics of systems of many coupled phase oscillators with heterogeneous frequencies. We suppose that the oscillators occur in M groups. Each oscillator is connected to other oscillators in its group with "attractive" coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is "repulsive," i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen, Chaos 18, 037113 (2008). We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the M groups. We find a manifold L of neutrally stable equilibria, and we show that all other equilibria are unstable. For M ≥ 3, L has dimension M - 2, and for M = 2, it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling parameters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the M groups on the manifold L. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.

Analog CMOS circuit implementation of a pulse-coupled phase oscillator system and observation of synchronization phenomena

Analog CMOS circuit implementation of a system of pulse-coupled phase oscillators is proposed. A CMOS circuit that achieves the dynamics of pulse-coupled oscillators has been designed and fabricated using a 0.25-µm CMOS technology. The proposed oscillator circuits with continuous-time operation interact with each other via a pulse at each firing time. Update of the oscillator state is achieved by integrating the phase sensitivity function with the pulse width time span. The phase sensitivity function is generated by the combination of binary functions, while the function consists of three-values {-1,0,1}. Introducing a zero-value span in the function leads to fast synchronization and robustness to parameter fluctuation due to LSI device mismatches, which facilitates VLSI implementation. Using the fabricated CMOS circuit, we have observed not only in- and anti-phase but also out-of-phase synchronization.

Synchronization in lattice-embedded scale-free networks

In this work, we study the effects of embedding a system of non-linear phase oscillators in a two-dimensional scale-free lattice. In order to analyze the effects of the embedding, we consider two different topologies. On the one hand, we consider a scale-free complex network where no constraint on the length of the links is taken into account. On the other hand, we use a method recently introduced for embedding scale-free networks in regular Euclidean lattices. In this case, the embedding is driven by a natural constraint of minimization of the total length of the links in the system. We analyze and compare the synchronization properties of a system of non-linear Kuramoto phase oscillators, when interactions between the oscillators take place in these networks. First, we analyze the behavior of the Kuramoto order parameter and show that the onset of synchronization is lower for non-constrained lattices. Then, we consider the behavior of the mean frequency of the oscillators as a function of the natural frequency for the two different networks and also for different values of the scale-free exponent. We show that, in contrast to non-embedded lattices that present a mean-field-like behavior characterized by the presence of a single cluster of synchronized oscillators, in embedded lattices the presence of a diversity of synchronized clusters at different mean frequencies can be observed. Finally, by considering the behavior of the mean frequency as a function of the degree, we study the role of hubs in the synchronization properties of the system.

Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)

In a recent paper by Ott and Antonsen [Chaos 19, 023117 (2009)], it was shown for the case of Lorentzian distributions of oscillator frequencies that the dynamics of a very general class of large systems of coupled phase oscillators time-asymptotes to a particular simplified form given by Ott and Antonsen [Chaos 18, 037113 (2008)]. This comment extends this previous result to a broad class of oscillator distribution functions.

Multistable Attractors in a Network of Phase Oscillators with Three-Body Interactions

Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with a three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in an asymptotic state can vary continuously within some range depending on the initial phase pattern.

Spectral properties of chimera states

Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a spatiotemporal pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that, in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.

Designing the Dynamics of Globally Coupled Oscillators

A method for designing cluster states with prescribed stability is presented for coupled phase oscillator systems with all-to-all coupling. We determine criteria for the coupling function that ensure the existence and stability of a large variety of clustered configurations. We show that such criteria can be satisfied by choosing Fourier coefficients of the coupling function. We demonstrate that using simple trigonometric and localized coupling functions one can realize arbitrary patterns of stable clusters and that the designed systems are capable of performing finite state computation. The design principles may be relevant when engineering complex dynamical behavior of coupled systems, e.g. the emergent dynamics of artificial neural networks, coupled chemical oscillators and robotic swarms. Subject Index: 034, 044, 054, 055

Learning of Spatio–Temporal Codes in a Coupled Oscillator System

In this paper, we consider a learning strategy that allows one to transmit information between two coupled phase oscillator systems (called teaching and learning systems) via frequency adaptation. The dynamics of these systems can be modeled with reference to a number of partially synchronized cluster states and transitions between them. Forcing the teaching system by steady but spatially nonhomogeneous inputs produces cyclic sequences of transitions between the cluster states, that is, information about inputs is encoded via a "winnerless competition" process into spatio-temporal codes. The large variety of codes can be learned by the learning system that adapts its frequencies to those of the teaching system. We visualize the dynamics using "weighted order parameters (WOPs)" that are analogous to "local field potentials" in neural systems. Since spatio-temporal coding is a mechanism that appears in olfactory systems, the developed learning rules may help to extract information from these neural ensembles.

Long time evolution of phase oscillator systems

It is shown, under weak conditions, that the dynamical evolution of large systems of globally coupled phase oscillators with Lorentzian distributed oscillation frequencies is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of the system states. This manifold was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behaviors of the order parameters of these systems can be obtained by restriction to the reduced manifold.

PHASE DYNAMICS OF COMPLEX-VALUED NEURAL NETWORKS AND ITS APPLICATION TO TRAFFIC SIGNAL CONTROL

Complex-valued Hopfield networks which possess the energy function are analyzed. The dynamics of the network with certain forms of an activation function is de-composable into the dynamics of the amplitude and phase of each neuron. Then the phase dynamics is described as a coupled system of phase oscillators with a pair-wise sinusoidal interaction. Therefore its phase synchronization mechanism is useful for the area-wide offset control of the traffic signals. The computer simulations show the effectiveness under the various traffic conditions.

Strongly asymmetric clustering in systems of phase oscillators

In this paper, we look at clustering in systems of globally coupled identical phase oscillators. In particular, we extend and apply techniques developed earlier to study stable clustering behavior involving clusters of greatly differing size. We discuss the bifurcations in which these asymmetric cluster states are created, and how these relate to bifurcations of the synchronized state. Because of the simplicity of systems of phase oscillators, it is possible to say a significant amount about asymmetric clustering analytically. We apply some of the theory developed to one particular system, and illustrate how the techniques can be used to find behavior which might otherwise be missed.

Synchronization in Oscillator Systems with a Central Element and Phase Shifts

A system of many phase oscillators is studied in the case of a star-like interaction through a central element (the analogue of a common field) with phase shifts in the connections. Equations for the frequency of full andpartial synchronization are derived. A numerical solution to the equations allows the evaluation of the role of the phase shifts in the dynamical behavior of the system.

System of Phase Oscillators with Diagonalizable Interaction

We consider a system of N phase oscillators having randomly distributed natural frequencies and diagonalizable interactions among the oscillators. We show that, in the limit of $N \to \infty$, all solutions of such a system are incoherent with probability one for any strength of coupling, which implies that there is no sharp transition from incoherence to coherence as the coupling strength is increased, in striking contrast to Kuramoto's (special) oscillator system.