Networks Of Phase Oscillators Research Articles

Analysis of remote synchronization in complex networks

A novel regime of synchronization, called remote synchronization, where the peripheral nodes form a phase synchronized cluster not including the hub, was recently observed in star motifs [Bergner et al., Phys. Rev. E 85, 026208 (2012)]. We show the existence of a more general dynamical state of remote synchronization in arbitrary networks of coupled oscillators. This state is characterized by the synchronization of pairs of nodes that are not directly connected via a physical link or any sequence of synchronized nodes. This phenomenon is almost negligible in networks of phase oscillators as its underlying mechanism is the modulation of the amplitude of those intermediary nodes between the remotely synchronized units. Our findings thus show the ubiquity and robustness of these states and bridge the gap from their recent observation in simple toy graphs to complex networks.

Analysis of a solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction

We study a phase oscillator network on a circle with an infinite-range interaction. First, we treat the Mexican-hat interaction with the zeroth and first Fourier components. We give detailed derivations of the auxiliary equations for the phases and self-consistent equations for the amplitudes. We solve these equations and characterize the nontrivial solutions in terms of order parameters and the rotation number. Furthermore, we derive the boundaries of the bistable regions and study the bifurcation structures in detail. Expressions for location-dependent resultant frequencies and entrained phases are also derived. Secondly, we treat a different interaction that is composed of mth and nth Fourier components, where m<n, and we study its nontrivial solutions. In both cases, the results of numerical simulations agree quite well with the theoretical results.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Explosive synchronization of complex networks with different chaotic oscillators

Recent studies have shown that explosive synchronization transitions can be observed in networks of phase oscillators [Gómez-Gardeñes J, Gómez S, Arenas A and Moreno Y 2011 Phys. Rev. Lett. 106 128701] and chaotic oscillators [Leyva I, Sevilla-Escoboza R, Buldú J M, Sendiña-Nadal I, Gómez-Gardeñes J, Arenas A, Moreno Y, Gómez S, Jaimes-Reátegui R and Boccaletti S 2012 Phys. Rev. Lett. 108 168702]. Here, we study the effect of different chaotic dynamics on the synchronization transitions in small world networks and scale free networks. The continuous transition is discovered for Rössler systems in both of the above complex networks. However, explosive transitions take place for the coupled Lorenz systems, and the main reason is the abrupt change of dynamics before achieving complete synchronization. Our results show that the explosive synchronization transitions are accompanied by the change of system dynamics.

Computing with complex-valued networks of phase oscillators

A computational paradigm based on phase-locked states of oscillators embedded in a complex-valued weighted network is introduced, and its efficiency and scalability in performing single- and multiple-input Boolean and non-Boolean operations are demonstrated. The paradigm introduces phase rotation mechanisms in each link of the network, without the need for external control loops, or adaptive mechanisms. We show that the proposed computation allows outperforming standard implementations of Boolean logic, by reducing the computational cost of operations, and permitting the construction of memory elements with single dynamical units.

Explosive transitions to synchronization in networks of phase oscillators

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. The occurrence of a first-order phase transition to synchronization of an ensemble of networked phase oscillators was reported, so far, for very particular network architectures. Here, we show how a sharp, discontinuous transition can occur, instead, as a generic feature of networks of phase oscillators. Precisely, we set conditions for the transition from unsynchronized to synchronized states to be first-order, and demonstrate how these conditions can be attained in a very wide spectrum of situations. We then show how the occurrence of such transitions is always accompanied by the spontaneous setting of frequency-degree correlation features. Third, we show that the conditions for abrupt transitions can be even softened in several cases. Finally, we discuss, as a possible application, the use of this phenomenon to express magnetic-like states of synchronization.

Self-arrangement of cellular circadian rhythms through phase-resetting in plant roots

We discovered a striped pattern of gene expression with circadian rhythms in growing plant roots using bioluminescent imaging of gene expression. Our experimental analysis revealed that the stripe wave in the bioluminescent image originated at the root tip and was caused by a continuous phase resetting of circadian oscillations. Some complex stripe waves containing arrhythmic regions were also observed. We succeeded in describing the formation mechanisms of these patterns using a growing phase oscillator network with a phase-resetting boundary condition.

Disorder-induced dynamics in a pair of coupled heterogeneous phase oscillator networks

We consider a pair of coupled heterogeneous phase oscillator networks and investigate their dynamics in the continuum limit as the intrinsic frequencies of the oscillators are made more and more disparate. The Ott/Antonsen Ansatz is used to reduce the system to three ordinary differential equations. We find that most of the interesting dynamics, such as chaotic behaviour, can be understood by analysing a gluing bifurcation of periodic orbits; these orbits can be thought of as "breathing chimeras" in the limit of identical oscillators. We also add Gaussian white noise to the oscillators' dynamics and derive a pair of coupled Fokker-Planck equations describing the dynamics in this case. Comparison with simulations of finite networks of oscillators is used to confirm many of the results.

Noise-induced synchronization in small world networks of phase oscillators

A small-world (SW) network of similar phase oscillators, interacting according to the Kuramoto model, is studied numerically. It is shown that deterministic Kuramoto dynamics on SW networks has various stable stationary states. This can be attributed to the so-called defect patterns in an SW network, which it inherits from deformation of helical patterns in its regular parent. Turning on an uncorrelated random force causes vanishing of the defect patterns, hence increasing the synchronization among oscillators for moderate noise intensities. This phenomenon, called stochastic synchronization, is generally observed in some natural networks such as the brain neural network.

Complex networks: when random walk dynamics equals synchronization

Synchrony prevalently emerges from the interactions of coupled dynamical units. For simple systems such as networks of phase oscillators, the asymptotic synchronization process is assumed to be equivalent to a Markov process that models standard diffusion or random walks on the same network topology. In this paper, we analytically derive the conditions for such equivalence for networks of pulse-coupled oscillators, which serve as models for neurons and pacemaker cells interacting by exchanging electric pulses or fireflies interacting via light flashes. We find that the pulse synchronization process is less simple, but there are classes of, e.g., network topologies that ensure equivalence. In particular, local dynamical operators are required to be doubly stochastic. These results provide a natural link between stochastic processes and deterministic synchronization on networks. Tools for analyzing diffusion (or, more generally, Markov processes) may now be transferred to pin down features of synchronization in networks of pulse-coupled units such as neural circuits.

Solvable Model of a Phase Oscillator Network on a Circle with Infinite-Range Mexican-Hat-Type Interaction

We describe a solvable model of a phase oscillator network on a circle with infinite-range Mexican-hat-type interaction. We derive self-consistent equations of the order parameters and obtain three non-trivial solutions characterized by the rotation number. We also derive relevant characteristics such as the location-dependent distributions of the resultant frequencies of desynchronized oscillators. Simulation results closely agree with the theoretical ones.

Phase oscillator neural network as artificial central pattern generator for robots

We design an artificial central pattern generator and we exemplify its integration using different prototypic virtual robot models. The artificial central pattern generator is modeled through a network of phase oscillators. Virtual robots are controlled by this central pattern generator using an interface that interpolates stored angular states of their target poses. Finally, we consider control signals coming from the robot's environment to the central pattern generator and to the interface between the central pattern generator and the robot. We show that the control signals can change the dynamics of the system in order to modify the robot's movement patterns. We study three cases: changing the frequency of the pattern sequences, switching between gaits of locomotion and the reorientation of a robotic entity.

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.

A dynamical study of pulse-coupled oscillators in the brain

In 1967, Winfree [1] proposed a novel mathematical approach to describe phenomena of collective synchrony in nature (i.e. flashing of fireflies, clapping in a theatre, alpha rhythms, etc.) using a large coupled network of phase oscillators with a diversity of natural frequencies. By analyzing this large heterogeneous network from a “mean field” approach, the spontaneous synchrony can be understood as a critical phase transition similar to most statistical mechanical systems. In this work, we employ this approach to model the phase transitions and bifurcation structures of a large network of pulse-coupled theta neurons [2] by appropriate choice of Winfree's response and functions, the latter of which is parameterized by a parameter n [1]. As this parameter increases, the influence function approximates the behavior of a pulse-coupled synapse. Assuming a Lorentzian distribution of natural frequencies of width Δ and mean value ω0, taking the thermodynamic limit, and employing the Ott-Antonsen reduction method [3], the collective dynamics of the pulse-coupled network can be analytically reduced to a single low-dimensional dynamical equation for the mean field parameter z(t). We analyze the bifurcation diagrams for different values of the sharpness parameter n. We find that more complex behavior is apparent with increasing sharpness of the influence function, and that equilibria outside the physically relevant region (within the unit circle) affect the transient dynamics of z(t) inside the circle. Further, we find that some level of coherence always exists in the network for non-zero coupling, in contrast to other mean field coupled phase oscillator networks [1,4]. Most interestingly, heterogeneity is observed to suppress complexity in the collective/macroscopic behavior. As the network becomes more homogeneous (∆→0), more complex dynamic including aperiodic and multistability emerge from the macroscopic mean field. Figure 1 A sample bifurcation diagram showing the complex structure of fixed points at various locations in parameter space, for a sharpness parameter of (n = 7). Several representative phase portraits from several distinct region of parameter space are included. ...

Onset of synchronization in complex networks of noisy oscillators

We study networks of noisy phase oscillators whose nodes are characterized by random degrees counting the number of their connections. Both these degrees and the natural frequencies of the oscillators are distributed according to a given probability density. Replacing the randomly connected network by an all-to-all coupled network with weighted edges allows us to formulate the dynamics of a single oscillator coupled to the mean field and to derive the corresponding Fokker-Planck equation. From the latter we calculate the critical coupling strength for the onset of synchronization as a function of the noise intensity, the frequency distribution, and the first two moments of the degree distribution. Our approach is applied to a dense small-world network model, for which we calculate the degree distribution. Numerical simulations prove the validity of the replacement. We also test the applicability to more sparsely connected networks and formulate homogeneity and absence of correlations in the degree distribution as limiting factors of our approach.

Design of oscillator networks with enhanced synchronization tolerance against noise

Can synchronization properties of a network of identical oscillators in the presence of noise be improved through appropriate rewiring of its connections? What are the optimal network architectures for a given total number of connections? We address these questions by running the optimization process, using the stochastic Markov Chain Monte Carlo method with replica exchange, to design networks of phase oscillators with increased tolerance against noise. As we find, the synchronization of a network, characterized by the Kuramoto order parameter, can be increased up to 40%, as compared to that of the randomly generated networks, when the optimization is applied. Large ensembles of optimized networks are obtained, and their statistical properties are investigated.

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.

Chimeras in random non-complete networks of phase oscillators

We consider the simplest network of coupled non-identical phase oscillators capable of displaying a "chimera" state (namely, two subnetworks with strong coupling within the subnetworks and weaker coupling between them) and systematically investigate the effects of gradually removing connections within the network, in a random but systematically specified way. We average over ensembles of networks with the same random connectivity but different intrinsic oscillator frequencies and derive ordinary differential equations (ODEs), whose fixed points describe a typical chimera state in a representative network of phase oscillators. Following these fixed points as parameters are varied we find that chimera states are quite sensitive to such random removals of connections, and that oscillations of chimera states can be either created or suppressed in apparent bifurcation points, depending on exactly how the connections are gradually removed.

Multistability of twisted states in non-locally coupled Kuramoto-type models

A ring of N identical phase oscillators with interactions between L-nearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q-twisted states, where the phase difference between neighboring oscillators on the ring is 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [Chaos 16, 015103 (2006)] In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2πq/N in one sector of the ring, -2πq/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞.

Splay States in Finite Pulse-Coupled Networks of Excitable Neurons

The emergence and stability of splay states is studied in fully coupled finite networks of $N$ excitable quadratic integrate-and-fire neurons, connected via synapses modeled as pulses of finite amplitude and duration. For such synapses, by introducing two distinct types of synaptic events (pulse emission and termination), we were able to write down an exact event-driven map for the system and to evaluate the splay state solutions. For $M$ overlapping postsynaptic potentials, the linear stability analysis of the splay state should also take in account, besides the actual values of the membrane potentials, the firing times associated with the $M$ previous pulse emissions. As a matter of fact, it was possible, by introducing $M$ complementary variables, to rephrase the evolution of the network as an event-driven map and to derive an analytic expression for the Floquet spectrum. We find that, independently of $M$, the splay state is marginally stable with $N-2$ neutral directions. Furthermore, we have identified a family of periodic solutions surrounding the splay state and sharing the same neutral stability directions. In the limit of $\delta$-pulses, it is still possible to derive an event-driven formulation for the dynamics; however, the number of neutrally stable directions associated with the splay state becomes $N$. Finally, we prove a link between the results for our system and a previous theory [S. Watanabe and S. H. Strogatz, Phys. D, 74 (1994), pp. 197--253] developed for networks of phase oscillators with sinusoidal coupling.