Collective phase synchronization in locally coupled limit-cycle oscillators

How to grow an oscillators' network with enhanced synchronization

We study a way to set the natural frequency of a newly added oscillator in a growing network to enhance synchronization. Population growth is one of the typical features of many oscillator systems for which synchronization is required to perform their functions properly. Despite this significance, little has been known about synchronization in growing systems. We suggest effective growing schemes to enhance synchronization as the network grows under a predetermined rule. Specifically, we find that a method based on a link-wise order parameter outperforms that based on the conventional global order parameter. With simple solvable examples, we verify that the results coincide with intuitive expectations. The numerical results demonstrate that the approximate optimal values from the suggested method show a larger synchronization enhancement in comparison with other naïve strategies. The results also show that our proposed approach outperforms others over a wide range of coupling strengths.

Synchronization in multiplex models of neuron-glial systems: Small-world topology and inhibitory coupling

In this work, we investigate the impact of mixed coupling on synchronization in a multiplex oscillatory network. The network mimics the neural-glial systems by incorporating interacting slow ("glial") and fast ("neural") oscillatory layers. Connections between the "glial" elements form a regular periodic structure, in which each element is connected to the eight other neighbor elements, whereas connections among "neural" elements are represented by Watts-Strogatz networks (from regular and small-world to random Erdös-Rényi graph) with a matching mean node degree. We find that the random rewiring toward small-world topology readily yields the dynamics close to that exhibited for a completely random graph, in particular, leading to coarse-graining of dynamics, suppressing multi-stability of synchronization regimes, and the onset of Kuramoto-type synchrony in both layers. The duration of transient dynamics in the system measured by relaxation times is minimized with the increase of random connections in the neural layer, remaining substantial only close to synchronization-desynchronization transitions. "Inhibitory" interactions in the "neural" subnetwork layer undermine synchronization; however, the strong coupling with the "glial" layer overcomes this effect.

Two order parameters for the Kuramoto model on complex networks

We investigate the behavior of two different order parameters for the Kuramoto model in the desynchronized phase. Since the primary role of the order parameter is to distinguish different phases, we focus on the ability to discern the desynchronized phase from the synchronized one on complex networks with the size N. From the exact derivation of the difference between two order parameters, Δ, on a star network, we find that these order parameters disagree in the desynchronized phase. We also show that the hub plays an important role and provide an analytic conjecture on the condition that the two order parameters agree with each other as N→∞. The conjecture is numerically confirmed.

Multiplexing topologies and time scales: The gains and losses of synchrony

Inspired by the recent interest in collective dynamics of biological neural networks immersed in the glial cell medium, we investigate the frequency and phase order, i.e., Kuramoto type of synchronization in a multiplex two-layer network of phase oscillators of different time scales and topologies. One of them has a long-range connectivity, exemplified by the Erdős-Rényi random network, and supports both kinds of synchrony. The other is a locally coupled two-dimensional lattice that can reach frequency synchronization but lacks phase order. Drastically different layer frequencies disentangle intra- and interlayer synchronization. We find that an indirect but sufficiently strong coupling through the regular layer can induce both phase order in the originally nonsynchronized random layer and global order, even when an isolated regular layer does not manifest it in principle. At the same time, the route to global synchronization is complex: an initial onset of (partial) synchrony in the regular layer, when its intra- and interlayer coupling is increased, provokes the loss of synchrony even in the originally synchronized random layer. Ultimately, a developed asynchronous dynamics in both layers is abruptly taken over by the global synchrony of both kinds.

Spatial uniformity in the power-grid system

Robust synchronization is indispensable for stable operation of a power grid. Recently, it has been reported that a large number of decentralized generators, rather than a small number of large power plants, provide enhanced synchronization together with greater robustness against structural failures. In this paper, we systematically control the spatial uniformity of the geographical distribution of generators and conclude that the more uniformly generators are distributed, the more enhanced synchronization occurs. In the presence of temporal failures of power sources, we observe that spatial uniformity helps the power grid to recover stationarity in a shorter time. We also discuss practical implications of our results in designing the structure of a power-grid network.

The role of dimensionality in neuronal network dynamics

Recent results from network theory show that complexity affects several dynamical properties of networks that favor synchronization. Here we show that synchronization in 2D and 3D neuronal networks is significantly different. Using dissociated hippocampal neurons we compared properties of cultures grown on a flat 2D substrates with those formed on 3D graphene foam scaffolds. Both 2D and 3D cultures had comparable glia to neuron ratio and the percentage of GABAergic inhibitory neurons. 3D cultures because of their dimension have many connections among distant neurons leading to small-world networks and their characteristic dynamics. After one week, calcium imaging revealed moderately synchronous activity in 2D networks, but the degree of synchrony of 3D networks was higher and had two regimes: a highly synchronized (HS) and a moderately synchronized (MS) regime. The HS regime was never observed in 2D networks. During the MS regime, neuronal assemblies in synchrony changed with time as observed in mammalian brains. After two weeks, the degree of synchrony in 3D networks decreased, as observed in vivo. These results show that dimensionality determines properties of neuronal networks and that several features of brain dynamics are a consequence of its 3D topology.

Parallel discrete-event simulation schemes with heterogeneous processing elements

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

Extended finite-size scaling of synchronized coupled oscillators

We present a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with nonidentical natural frequencies in terms of the Kuramoto model. This provides a comprehensive view of phase synchronization. In particular, we extend finite-size scaling (FSS) in the steady state to dynamics, determine critical exponents, and find the critical coupling strength. The dynamic scaling approach enables us to measure not only the FSS exponent associated with the correlation volume in finite systems but also thermodynamic critical exponents. Based on the extended FSS theory, we also discuss how the sampling of natural frequencies and thermal noise affect dynamic scaling, which is numerically confirmed.

Synchronization in interdependent networks

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength J(I)) is ferromagnetically intercoupled (the strength J) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength J(II)). In the absence of the internetwork coupling (J=0), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network (J(I)≪1) the increase of J suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as J(I) is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of J(I). The nonmonotonic change of the critical value of J(II) is also compared with the result from the numerical RG calculation.

Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions

Recent experiments in one- and two-dimensional microfluidic arrays of droplets containing Belousov-Zhabotinsky reactants show a rich variety of spatial patterns [M. Toiya et al., J. Phys. Chem. Lett. 1, 1241 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest-neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2π/3. We report the existence of domains with either clockwise or anticlockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.

Paths to globally generalized synchronization in scale-free networks

We apply the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators, including Hénon maps, logistic maps, and Lorentz oscillators. As the coupling strength epsilon between nodes of the network is increased, transitions from partially to globally generalized synchronization and intermittent behaviors near the synchronization thresholds, are found. The generalized synchronization starts from the hubs of the network and then spreads throughout the whole network with the increase of epsilon . Our result is useful for understanding the synchronization process in complex networks.

Entrainment transition in populations of random frequency oscillators

The entrainment transition of coupled random frequency oscillators is revisited. The Kuramoto model (global coupling) is shown to exhibit unusual sample-dependent finite-size effects leading to a correlation size exponent nu=5/2. Simulations of locally coupled oscillators in d dimensions reveal two types of frequency entrainment: mean-field behavior at d>4 and aggregation of compact synchronized domains in three and four dimensions. In the latter case, scaling arguments yield a correlation length exponent nu=2/(d-2), in good agreement with numerical results.

Continuous and discontinuous phase transitions and partial synchronization in stochastic three-state oscillators

We investigate both continuous (second-order) and discontinuous (first-order) transitions to macroscopic synchronization within a single class of discrete, stochastic (globally) phase-coupled oscillators. We provide analytical and numerical evidence that the continuity of the transition depends on the coupling coefficients and, in some nonuniform populations, on the degree of quenched disorder. Hence, in a relatively simple setting this class of models exhibits the qualitative behaviors characteristic of a variety of considerably more complicated models. In addition, we study the microscopic basis of synchronization above threshold and detail the counterintuitive subtleties relating measurements of time-averaged frequencies and mean-field oscillations. Most notably, we observe a state of suprathreshold partial synchronization in which time-averaged frequency measurements from individual oscillators do not correspond to the frequency of macroscopic oscillations observed in the population.

Effects of disorder on synchronization of discrete phase-coupled oscillators

We study synchronization in populations of phase-coupled stochastic three-state oscillators characterized by a distribution of transition rates. We present results on an exactly solvable dimer as well as a systematic characterization of globally connected arrays of N types of oscillators (N=2,3,4) by exploring the linear stability of the nonsynchronous fixed point. We also provide results for globally coupled arrays where the transition rate of each unit is drawn from a uniform distribution of finite width. Even in the presence of transition rate disorder, numerical and analytical results point to a single phase transition to macroscopic synchrony at a critical value of the coupling strength. Numerical simulations make possible further characterization of the synchronized arrays.

Critical behavior and synchronization of discrete stochastic phase-coupled oscillators

Synchronization of stochastic phase-coupled oscillators is known to occur but difficult to characterize because sufficiently complete analytic work is not yet within our reach, and thorough numerical description usually defies all resources. We present a discrete model that is sufficiently simple to be characterized in meaningful detail. In the mean-field limit, the model exhibits a supercritical Hopf bifurcation and global oscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, the model undergoes a continuous phase transition that we characterize numerically using finite-size scaling analysis. In particular, we explicitly rule out multistability and show that the onset of global synchrony is marked by signatures of the XY universality class. Our numerical results cover dimensions d=2, 3, 4, and 5 and lead to the appropriate XY classical exponents beta and nu, a lower critical dimension dlc=2, and an upper critical dimension duc=4.

Universality of synchrony: critical behavior in a discrete model of stochastic phase-coupled oscillators

We present the simplest discrete model to date that leads to synchronization of stochastic phase-coupled oscillators. In the mean field limit, the model exhibits a Hopf bifurcation and global oscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, the model undergoes a continuous phase transition which we characterize numerically using finite-size scaling analysis. In particular, the onset of global synchrony is marked by signatures of the XY universality class, including the appropriate classical exponents beta and nu, a lower critical dimension d(lc) = 2, and an upper critical dimension d(uc) = 4.

Collective synchronization in spatially extended systems of coupled oscillators with random frequencies

We study collective behavior of locally coupled limit-cycle oscillators with random intrinsic frequencies, spatially extended over d -dimensional hypercubic lattices. Phase synchronization as well as frequency entrainment are explored analytically in the linear (strong-coupling) regime and numerically in the nonlinear (weak-coupling) regime. Our analysis shows that the oscillator phases are always desynchronized up to d=4 , which implies the lower critical dimension dP(l) =4 for phase synchronization. On the other hand, the oscillators behave collectively in frequency (phase velocity) even in three dimensions (d=3) , indicating that the lower critical dimension for frequency entrainment is dF(l)=2 . Nonlinear effects due to the periodic nature of limit-cycle oscillators are found to become significant in the weak-coupling regime: So-called runaway oscillators destroy the synchronized (ordered) phase and there emerges a fully random (disordered) phase. Critical behavior near the synchronization transition into the fully random phase is unveiled via numerical investigation. Collective behavior of globally coupled oscillators is also examined and compared with that of locally coupled oscillators.