Noise-induced synchronization in small world networks of phase oscillators

Stochastic Kuramoto oscillators with inertia and higher-order interactions

The impact of noise in coupled oscillators with pairwise interactions has been extensively explored. Here, we study stochastic second-order coupled Kuramoto oscillators with higher-order interactions and show that as noise strength increases, the critical points associated with synchronization transitions shift toward higher coupling values. By employing the perturbation analysis, we obtain an expression for the forward critical point as a function of inertia and noise strength. Further, for overdamped systems, we show that as noise strength increases, the first-order transition switches to second-order even for higher-order couplings. We include a discussion on the nature of critical points obtained through Ott-Antonsen ansatz.

Fate of vortex-synchronized state in oscillator networks with node defects

We investigate synchronization behaviors of a Kuramoto oscillator network with a two-dimensional square-lattice configuration. We show that the oscillator network can reach a phase-locking vortex synchronized state in the long time limit starting from random initial oscillator phases sampled according to the von Mises distribution characterized by a zero mean and a finite concentration parameter. We further reveal that the stability of the vortex synchronized state is sensitive to the presence of local node defects, in contrast to the usual knowledge that oscillator networks should exhibit robustness against local perturbations. Moreover, we explore the behaviors of the vortex synchronized state in networks with an additional temporal white noise on the oscillator phases or a spatial noise due to randomly distributed oscillator frequencies. Interestingly, we find that the vortex synchronized state can become immune to local node defects when the variance of spatial noise is above a certain threshold, suggesting a beneficial role of usually unwanted spatial noise in protecting vortex-synchronized networks.

Enhancing Synchronization by Optimal Correlated Noise

From the flashes of fireflies to Josephson junctions and power infrastructure, networks of coupled phase oscillators provide a powerful framework to describe synchronization phenomena in many natural and engineered systems. Most real-world networks are under the influence of noisy, random inputs, potentially inhibiting synchronization. While noise is unavoidable, here we show that there exist optimal noise patterns which minimize desynchronizing effects and even enhance order. Specifically, using analytical arguments we show that in the case of a two-oscillator model, there exists a sharp transition from a regime where the optimal synchrony-enhancing noise is perfectly anticorrelated, to one where the optimal noise is correlated. More generally, we then use numerical optimization methods to demonstrate that there exist anticorrelated noise patterns that optimally enhance synchronization in large complex oscillator networks. Our results may have implications in networks such as power grids and neuronal networks, which are subject to significant amounts of correlated input noise.

Contrariety and inhibition enhance synchronization in a small-world network of phase oscillators

We numerically study Kuramoto model synchronization consisting of the two groups of conformist-contrarian and excitatory-inhibitory phase oscillators with equal intrinsic frequency. We consider random and small-world (SW) topologies for the connectivity network of the oscillators. In random networks, regardless of the contrarian to conformist connection strength ratio, we found a crossover from the π-state to the blurred π-state and then a continuous transition to the incoherent state by increasing the fraction of contrarians. However, for the excitatory-inhibitory model in a random network, we found that for all the values of the fraction of inhibitors, the two groups remain in phase and the transition point of fully synchronized to an incoherent state reduced by strengthening the ratio of inhibitory to excitatory links. In the SW networks we found that the order parameters for both models do not show monotonic behavior in terms of the fraction of contrarians and inhibitors. Up to the optimal fraction of contrarians and inhibitors, the synchronization rises by introducing the number of contrarians and inhibitors and then falls. We discuss that the nonmonotonic behavior in synchronization is due to the weakening of the defects already formed in the pure conformist and excitatory agent model in SW networks. We found that in SW networks, the optimal fraction of contrarians and inhibitors remain unchanged for the rewiring probabilities up to ∼0.15, above which synchronization falls monotonically, like the random network. We also showed that in the conformist-contrarian model, the optimal fraction of contrarians is independent of the strength of contrarian links. However, in the excitatory-inhibitory model, the optimal fraction of inhibitors is approximately proportional to the inverse of inhibition strength.

Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.

Stochastic synchronization of dynamics on the human connectome

Synchronization is a collective mechanism by which oscillatory networks achieve their functions. Factors driving synchronization include the network's topological and dynamical properties. However, how these factors drive the emergence of synchronization in the presence of potentially disruptive external inputs like stochastic perturbations is not well understood, particularly for real-world systems such as the human brain. Here, we aim to systematically address this problem using a large-scale model of the human brain network (i.e., the human connectome). The results show that the model can produce complex synchronization patterns transitioning between incoherent and coherent states. When nodes in the network are coupled at some critical strength, a counterintuitive phenomenon emerges where the addition of noise increases the synchronization of global and local dynamics, with structural hub nodes benefiting the most. This stochastic synchronization effect is found to be driven by the intrinsic hierarchy of neural timescales of the brain and the heterogeneous complex topology of the connectome. Moreover, the effect coincides with clustering of node phases and node frequencies and strengthening of the functional connectivity of some of the connectome's subnetworks. Overall, the work provides broad theoretical insights into the emergence and mechanisms of stochastic synchronization, highlighting its putative contribution in achieving network integration underpinning brain function.

Coherent Dynamics Enhanced by Uncorrelated Noise

Synchronization is a widespread phenomenon observed in physical, biological, and social networks, which persists even under the influence of strong noise. Previous research on oscillators subject to common noise has shown that noise can actually facilitate synchronization, as correlations in the dynamics can be inherited from the noise itself. However, in many spatially distributed networks, such as the mammalian circadian system, the noise that different oscillators experience can be effectively uncorrelated. Here, we show that uncorrelated noise can in fact enhance synchronization when the oscillators are coupled. Strikingly, our analysis also shows that uncorrelated noise can be more effective than common noise in enhancing synchronization. We first establish these results theoretically for phase and phase-amplitude oscillators subject to either or both additive and multiplicative noise. We then confirm the predictions through experiments on coupled electrochemical oscillators. Our findings suggest that uncorrelated noise can promote rather than inhibit coherence in natural systems and that the same effect can be harnessed in engineered systems.

Random pulse induced synchronization and resonance in uncoupled non-identical neuron models

Random pulses contribute to stochastic resonance in neuron models, whereas common random pulses cause stochastic-synchronized excitation in uncoupled neuron models. We studied concurrent phenomena contributing to phase synchronization and stochastic resonance following induction by a weak common random pulse in uncoupled non-identical Hodgkin-Huxley type neuron models. The common random pulse was selected from a gamma distribution and the degree of synchronization depended on the corresponding shape parameter. Specifically, a low shape parameter of the weak random pulse induced well-synchronized spiking in uncoupled neuron models, whereas a high shape parameter of the weak random pulse or a weak periodic pulse caused low degrees of synchronization. These were improved by concurrent inputs of periodic and random pulses with high shape parameters. Finally, the output pulse was synchronized with the periodic pulse, and the common random pulse revealed periodic responses in the present neuron models.

Two-network Kuramoto-Sakaguchi model under tempered stable Lévy noise

We examine a model of two interacting populations of phase oscillators labeled "blue" and "red." To this we apply tempered stable Lévy noise, a generalization of Gaussian noise where the heaviness of the tails parametrized by a power law exponent α can be controlled by a tempering parameter λ. This system models competitive dynamics, where each population seeks both internal phase synchronization and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase lag values and the shape of the noise distribution can lead to steady or noisy behavior. Comparing the analytic and numerical studies shows that the bulk behavior of the system can be effectively described by dynamics in the presence of tilted ratchet potentials. Generally, changes in α away from the Gaussian noise limit 1<α<2 disrupt the locking between blue and red, while increasing λ acts to restore it. However, we observe that with further decreases of α to small values α≪1, with λ≠0, locking between blue and red may be restored. This is seen analytically in a restoration of metastability through the ratchet mechanism, and numerically in transitions between periodic and noisy regions in a fitness landscape using a measure of noise. This nonmonotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent and provides a mechanism for guiding the collective behavior of such a complex competitive dynamical system.

Fisher information and criticality in the Kuramoto model of nonidentical oscillators

We use the Fisher information to provide a lens on the transition to synchronization of the Kuramoto model of nonidentical frequencies on a variety of undirected graphs. We numerically solve the equations of motion for a N=400 complete graph and N=1000 small-world, scale-free, uniform random, and random regular graphs. For large but finite graphs of small average diameter the Fisher information F as a function of coupling shows a peak closely coinciding with the critical point as determined by Kuramoto's order parameter or synchronization measure r. However, for graphs of larger average diameter the position of the peak in F differs from the critical point determined by estimates of r. On the one hand, this is a finite-size effect even at N=1000; however, we show across a range of topologies that the Fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase-synchronized clusters, often directly from metastable to stable frequency synchronization. Solving explicitly for a two-cluster ansatz subject to Gaussian noise shows that the Fisher infomation peaks at such a transition. We discuss the implications for Fisher information as an indicator for edge-of-chaos phenomena in finite-coupled oscillator systems.