Phase Synchronization Of Oscillations Research Articles

Synchronization of phase oscillators with chemical coupling: Removal of oscillators and external feedback control

In many applications of biological interest spatially localized phase oscillators present a non-local coupling characterized by the diffusion of a mediating chemical. The latter is both produced and absorbed by the phase oscillators according to their dynamical behavior. We investigate numerically phase and frequency synchronization in a model for chemical coupling of uniform phase oscillators with natural frequencies randomly distributed according to some statistical distribution. We study the effect of removal of oscillators, caused by lesions, on the synchronization properties of the network. The partial or complete suppression of synchronization is also investigated through the influence of a time-delayed feedback external signal.

Dynamics and Synchronization of Complex-Valued Ring Networks

Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0–1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.

Synchronization of phase oscillators under asymmetric and bimodal distributions of natural frequencies

Abstract We report on the setting of synchronization in ensembles of phase oscillators under an asymmetric and bimodal distribution of initial frequencies. In particular, by both the Ott–Antonsen reduction and direct numerical simulations we give evidence that the system exhibits four different transition routes towards synchronization, including continuous, discontinuous, and hybrid transitions, depending on the choice of the parameters of the frequency distribution. As the coupling strength increases, several coherent states emerge, such as two types of travelling wave states and the Bellerophon state, which are carefully characterized.

Communication between Brain Areas Based on Nested Oscillations.

Unraveling how brain regions communicate is crucial for understanding how the brain processes external and internal information. Neuronal oscillations within and across brain regions have been proposed to play a crucial role in this process. Two main hypotheses have been suggested for routing of information based on oscillations, namely communication through coherence and gating by inhibition. Here, we propose a framework unifying these two hypotheses that is based on recent empirical findings. We discuss a theory in which communication between two regions is established by phase synchronization of oscillations at lower frequencies (<25 Hz), which serve as temporal reference frame for information carried by high-frequency activity (>40 Hz). Our framework, consistent with numerous recent empirical findings, posits that cross-frequency interactions are essential for understanding how large-scale cognitive and perceptual networks operate.

Synchronization of phase oscillators with coupling mediated by a diffusing substance

We investigate the transition to phase and frequency synchronization in a one-dimensional chain of phase oscillator “cells” where the coupling is mediated by the local concentration of a chemical which can diffuse in the inter-oscillator medium and it is both secreted and absorbed by the oscillator “cells”, influencing their dynamical behavior. This coupling has the advantage of having a tunable parameter which makes it possible to pass continuously from a global (all-to-all) to a local (nearest-neighbor) coupling form. We have verified that synchronous behavior depends on the coupling strength and coupling length.

Synchronization of phase oscillators with frequency-weighted coupling.

Recently, the first-order synchronization transition has been studied in systems of coupled phase oscillators. In this paper, we propose a framework to investigate the synchronization in the frequency-weighted Kuramoto model with all-to-all couplings. A rigorous mean-field analysis is implemented to predict the possible steady states. Furthermore, a detailed linear stability analysis proves that the incoherent state is only neutrally stable below the synchronization threshold. Nevertheless, interestingly, the amplitude of the order parameter decays exponentially (at least for short time) in this regime, resembling the Landau damping effect in plasma physics. Moreover, the explicit expression for the critical coupling strength is determined by both the mean-field method and linear operator theory. The mechanism of bifurcation for the incoherent state near the critical point is further revealed by the amplitude expansion theory, which shows that the oscillating standing wave state could also occur in this model for certain frequency distributions. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

Synchronization of Phase Oscillators in Networks with Certain Frequency Sequence

Synchronization of Kuramoto phase oscillators arranged in real complex neural networks is investigated. It is shown that the synchronization greatly depends on the sets of natural frequencies of the involved oscillators. The influence of network connectivity heterogeneity on synchronization depends particularly on the correlation between natural frequencies and node degrees. This finding implies a potential application that inhibiting the effects caused by the changes of network structure can be balanced out nicely by choosing the correlation parameter appropriately.

Is auditory discrimination mature by middle childhood? A study using time‐frequency analysis of mismatch responses from 7 years to adulthood

Behavioural and electrophysiological studies give differing impressions of when auditory discrimination is mature. Ability to discriminate frequency and speech contrasts reaches adult levels only around 12 years of age, yet an electrophysiological index of auditory discrimination, the mismatch negativity (MMN), is reported to be as large in children as in adults. Auditory ERPs were measured in 30 children (7 to 12 years), 23 teenagers (13 to 16 years) and 32 adults (35 to 56 years) in an oddball paradigm with tone or syllable stimuli. For each stimulus type, a standard stimulus (1000 Hz tone or syllable [ba]) occurred on 70% of trials, and one of two deviants (1030 or 1200 Hz tone, or syllables [da] or [bi]) equiprobably on the remaining trials. For the traditional MMN interval of 100–250 ms post-onset, size of mismatch responses increased with age, whereas the opposite trend was seen for an interval from 300 to 550 ms post-onset, corresponding to the late discriminative negativity (LDN). Time-frequency analysis of single trials revealed that the MMN resulted from phase-synchronization of oscillations in the theta (4–7 Hz) range, with greater synchronization in adults than children. Furthermore, the amount of synchronization was significantly correlated with frequency discrimination threshold. These results show that neurophysiological processes underlying auditory discrimination continue to develop through childhood and adolescence. Previous reports of adult-like MMN amplitudes in children may be artefactual results of using peak measurements when comparing groups that differ in variance.

Synchronization of phase oscillators with heterogeneous coupling: A solvable case

We consider an extension of Kuramoto’s model of coupled phase oscillators where oscillator pairs interact with different strengths. When the coupling coefficient of each pair can be separated into two different factors, each one associated to an oscillator, Kuramoto’s theory for the transition to synchronization can be explicitly generalized, and the effects of coupling heterogeneity on synchronized states can be analytically studied. The two factors are respectively interpreted as the weight of the contribution of each oscillator to the mean field, and the coupling of each oscillator to that field. We explicitly analyze the effects of correlations between those weights and couplings, and show that synchronization can be completely inhibited when they are strongly anti-correlated. Numerical results validate the theory, but suggest that finite-size effect are relevant to the collective dynamics close to the synchronization transition, where oscillators become entrained in synchronized frequency clusters.

EXACT SOLUTIONS AND DYNAMICS OF GLOBALLY COUPLED OSCILLATORS

We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto–Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the porous medium, Burgers and extended Burgers systems depending on the degree of singularity of the coupling. We show that porous medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daido's order function are given in terms of elliptic functions.

Clustering and the synchronization of oscillator networks

By manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks with Poisson and scale-free degree distributions. For both types of networks, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, they show an additional, advanced transition instead of a single synchronization threshold. This cluster-enhanced synchronization of hubs may be relevant to the brain that is scale-free and highly clustered.

Bifurcation Study of the Kuramoto Transition in Random Oscillator Networks

We study the frequency synchronization of phase oscillators in random networks. We investigate the kinetic equations of the oscillator distribution function using the continuum approximation. From a linear analysis of this model, we find that the unstable eigenfunction is proportional to k. We also derive an amplitude equation of the unstable modes employing center manifold reduction. We find that the coefficients of the Taylor expansion of the amplitude equation always diverge for random scale-free networks. The results of numerical studies are consistent with these analyses.