Kuramoto Order Parameter Research Articles

Two Proper Metrics for Quantifying Remote Synchronization of Identical Oscillator Systems

Remote synchronization (RS) has attracted considerable attention in nonlinear oscillator systems during the past decade. Two metrics, τdirect and τindirect, are proposed in this study to measure RS and identify its associated type in systems of identical oscillators, addressing limitations of traditional indicators used for nonidentical oscillators. We confirm these two metrics' correctness by comparing the identified stable regions with the actual regions of the obtained system through numerical integration, consistent with each other perfectly. Furthermore, these metrics remain valid for systems of nonidentical oscillators and yield conclusions consistent with the traditional Kuramoto order parameters, which are commonly employed to quantify RS. It is expected that the two metrics will enhance the exploration of RS properties in future studies.

Criticality and partial synchronization analysis in Wilson-Cowan and Jansen-Rit neural mass models.

Synchronization is a phenomenon observed in neuronal networks involved in diverse brain activities. Neural mass models such as Wilson-Cowan (WC) and Jansen-Rit (JR) manifest synchronized states. Despite extensive research on these models over the past several decades, their potential of manifesting second-order phase transitions (SOPT) and criticality has not been sufficiently acknowledged. In this study, two networks of coupled WC and JR nodes with small-world topologies were constructed and Kuramoto order parameter (KOP) was used to quantify the amount of synchronization. In addition, we investigated the presence of SOPT using the synchronization coefficient of variation. Both networks reached high synchrony by changing the coupling weight between their nodes. Moreover, they exhibited abrupt changes in the synchronization at certain values of the control parameter not necessarily related to a phase transition. While SOPT was observed only in JR model, neither WC nor JR model showed power-law behavior. Our study further investigated the global synchronization phenomenon that is known to exist in pathological brain states, such as seizure. JR model showed global synchronization, while WC model seemed to be more suitable in producing partially synchronized patterns.

On the higher-order smallest ring-star network of Chialvo neurons under diffusive couplings.

Network dynamical systems with higher-order interactions are a current trending topic, pervasive in many applied fields. However, our focus in this work is neurodynamics. We numerically study the dynamics of the smallest higher-order network of neurons arranged in a ring-star topology. The dynamics of each node in this network is governed by the Chialvo neuron map, and they interact via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through multi-body interactions. We deploy the higher-order coupling strength as the primary bifurcation parameter. We start by analyzing our model using standard tools from dynamical systems theory: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of disparate chaotic attractors. We also observe an interesting route to chaos from a fixed point via period-doubling and the appearance of cyclic quasiperiodic closed invariant curves. Furthermore, we numerically observe the existence of codimension-1 bifurcation points: saddle-node, period-doubling, and Neimark-Sacker. We also qualitatively study the typical phase portraits of the system, and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure, respectively. Finally, we study the synchronization behavior among the neurons using the cross correlation coefficient and the Kuramoto order parameter. We conjecture that unfolding these patterns and behaviors of the network model will help us identify different states of the nervous system, further aiding us in dealing with various neural diseases and nervous disorders.

Negative emotions disrupt intentional synchronization during group sensorimotor interaction.

Emotions play a fundamental role in human interactions and trigger responses in physiological, psychological, and behavioral modalities. Interpersonal coordination often entails attunement between individuals in various modalities. Previous research has elucidated the mechanisms of interpersonal synchronization and the emotions aroused by joint action: cardiac activity aligns in disputing marital couples, spectators share enjoyment in observing live dance performances, and joint finger-tapping evokes positive emotions. However, little is known about the impact of emotions on intentional interpersonal synchronization. To address this problem, we conducted an experiment in 2022 asking 60 participants to engage in a three-way finger-tapping synchronization task. We systematically induced emotional states (positive, neutral, and negative) with social comparison feedback using success-failure manipulations. An analysis of behavior synchronization using the Kuramoto order parameter revealed that negative emotion induction significantly diminished time spent in synchrony compared to positive induction. Moreover, the results exposed incremental struggles in attaining higher levels of synchronization (Q2-Q3) after the induction of negative emotions. These outcomes further substantiate the necessity of integrating the indices of agents' emotions into interpersonal synchronization and coordination models. We discuss the implications of this work for research on interpersonal emotion in joint action and applied outcomes in emotion-aware technologies and interventions. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Symmetry-breaking higher-order interactions in coupled phase oscillators

This paper investigates the route to synchronization in an extension of the Kuramoto model, which incorporates higher order interactions and do not preserve rotational symmetry, resulting in interesting dynamical scenarios. Using the standard Kuramoto order parameter and the Ott–Antonsen ansatz, the transition points corresponding to different dynamical characteristics are identified and the extent of synchrony is quantified. These results indicate that the system exclusively undergoes explosive transitions and dynamical states are dependent on the mean natural frequency. Interestingly, for a certain range of parameter values, an unsteady yet partially synchronized state characterized by an oscillatory order parameter is observed. The analytical results are shown to agree well with the numerical simulations.

Electroencephalogram-Based Metastability in Mild Cognitive Impairment Alzheimer's Disease.

Background: In this study, we analyze metastability, a feature of brain dynamics in subjects experiencing mild cognitive impairment Alzheimer's disease (MCI-AD) under eyes open (EO) and eyes closed (EC) conditions. Alzheimer's disease (AD) is a critically prolonged brain disorder that interrupts neural synchronization and desynchronization. Thus, studying metastability under EO and EC conditions would help in understanding the cortical dynamics and its impact in early-stage AD. Methods: Metastability is investigated using three methods namely frequency variance analysis, Kuramoto order parameter, and through meta-state activation patterns. Frequency variance estimated from 21 electroencephalogram (EEG) channels was clustered into three regions namely anterior, central, and posterior to study the regional metastability analysis. Global metastability was assessed from Kuramoto order parameter and meta-state activation patterns by collating the 21 EEG channels. Results: Reduction in metastability was observed in central regions of MCI-AD subjects through the study of frequency variance analysis. There was a marked reduction in global metastability in the patient group under the resting EO condition. Reduction in meta-state activation properties such as temporal activation sequence complexity, modularity, and leap size in MCI-AD condition under the EO condition indicates an overall reduction in brain flexibility. Conclusion: Taken together, the study infers an underlying structural change in neuronal dynamics influencing the reduction of metastability under the MCI-AD condition. The study further revealed that this reduction in metastability is more pronounced in the EO condition.

Dynamics of oscillator populations with disorder in the coupling phase shifts

We study populations of oscillators, all-to-all coupled by means of quenched disordered phase shifts. While there is no traditional synchronization transition with a nonvanishing Kuramoto order parameter, the system demonstrates a specific order as the coupling strength increases. This order is characterized by partial phase locking, which is put into evidence by the introduced novel correlation order parameter, which is shown to grow monotonically with the coupling strength, and via frequency entrainment by following concentration of the oscillators frequencies. Simulations with phase oscillators, Stuart–Landau oscillators, and chaotic Roessler oscillators demonstrate similar scaling of the correlation order parameter with the coupling and the system size and also similar behavior of the frequencies with maximal entrainment (at which the standard deviation of the frequencies is reduced by a factor close to four) at some finite coupling.

Exploration of field-like torque and field-angle tunability in coupled spin-torque nano oscillators for synchronization.

We investigate the influence of field-like torque and the direction of the external magnetic field on a one-dimensional array of serially connected spin-torque nano oscillators (STNOs), having free layers with perpendicular anisotropy, to achieve complete synchronization between them by analyzing the associated Landau-Lifshitz-Gilbert-Slonczewski equation. The obtained results for synchronization are discussed for the cases of 2, 10, and 100 oscillators separately. The roles of the field-like torque and the direction of the external field on the synchronization of the STNOs are explored through the Kuramoto order parameter. While the field-like torque alone is sufficient to bring out global synchronization in the system made up of a small number of STNOs, the direction of the external field is also needed to be slightly tuned to synchronize the one-dimensional array of a large number of STNOs. The formation of complete synchronization through the construction of clusters within the system is identified for the 100 oscillators. The large amplitude synchronized oscillations are obtained for small to large numbers of oscillators. Moreover, the tunability in frequency for a wide range of currents is shown for the synchronized oscillations up to 100 spin-torque oscillators. In addition to achieving synchronization, the field-like torque increases the frequency of the synchronized oscillations. The transverse Lyapunov exponents are deduced to confirm the stable synchronization in coupled STNOs due to the field-like torque and to validate the results obtained in the numerical simulations. The output power of the array is estimated to be enhanced substantially due to complete synchronization by the combined effect of field-like torque and tunability of the field-angle.

Synchronization and decoherence in a self-excited inertia-wheel multiple rigid-body dynamical system.

We investigate the synchronization and decoherence of a self-excited inertia wheel multiple rigid-body dynamical system. We employ an Euler-Lagrange formulation to derive a nondimensional state space that governs the dynamics of a coupled pendula array where each element incorporates an inertia wheel. The dynamical system exhibits multiple equilibria, periodic limit-cycle oscillations, quasiperiodic, and chaotic oscillations and rotations. We make use of a combined approach including a singular perturbation multiple time scale and numerical bifurcation methodologies to determine the existence of synchronized and decoherent solutions in both weakly and strongly nonlinear regimes, respectively. The analysis reveals that synchronous oscillations are in-phase, whereas quasiperiodic oscillations are anti-phase. Furthermore, the non-stationary rotations are found to exhibit combinations of oscillations and rotations of the individual elements that are asynchronous. A Kuramoto order parameter analysis of representative solutions in various bifurcation regimes reveals the existence of chimera-like solutions where two elements are synchronized, whereas the third is desynchronized. Moreover, synchronous solutions were found to coexist with stable chimera solutions with a constant phase difference between the oscillators.

High expectations on phase locking: Better quantifying the concentration of circular data.

The degree to which unimodal circular data are concentrated around the mean direction can be quantified using the mean resultant length, a measure known under many alternative names, such as the phase locking value or the Kuramoto order parameter. For maximal concentration, achieved when all of the data take the same value, the mean resultant length attains its upper bound of one. However, for a random sample drawn from the circular uniform distribution, the expected value of the mean resultant length achieves its lower bound of zero only as the sample size tends to infinity. Moreover, as the expected value of the mean resultant length depends on the sample size, bias is induced when comparing the mean resultant lengths of samples of different sizes. In order to ameliorate this problem, here, we introduce a re-normalized version of the mean resultant length. Regardless of the sample size, the re-normalized measure has an expected value that is essentially zero for a random sample from the circular uniform distribution, takes intermediate values for partially concentrated unimodal data, and attains its upper bound of one for maximal concentration. The re-normalized measure retains the simplicity of the original mean resultant length and is, therefore, easy to implement and compute. We illustrate the relevance and effectiveness of the proposed re-normalized measure for mathematical models and electroencephalographic recordings of an epileptic seizure.

Chaotic chimera attractors in a triangular network of identical oscillators.

A prominent type of collective dynamics in networks of coupled oscillators is the coexistence of coherently and incoherently oscillating domains known as chimera states. Chimera states exhibit various macroscopic dynamics with different motions of the Kuramoto order parameter. Stationary, periodic and quasiperiodic chimeras are known to occur in two-population networks of identical phase oscillators. In a three-population network of identical Kuramoto-Sakaguchi phase oscillators, stationary and periodic symmetric chimeras were previously studied on a reduced manifold in which two populations behaved identically [Phys. Rev. E 82, 016216 (2010)1539-375510.1103/PhysRevE.82.016216]. In this paper, we study the full phase space dynamics of such three-population networks. We demonstrate the existence of macroscopic chaotic chimera attractors that exhibit aperiodic antiphase dynamics of the order parameters. We observe these chaotic chimera states in both finite-sized systems and the thermodynamic limit outside the Ott-Antonsen manifold. The chaotic chimera states coexist with a stable chimera solution on the Ott-Antonsen manifold that displays periodic antiphase oscillation of the two incoherent populations and with a symmetric stationary chimera solution, resulting in tristability of chimera states. Of these three coexisting chimera states, only the symmetric stationary chimera solution exists in the symmetry-reduced manifold.

The role of the fitness model in the suppression of neuronal synchronous behavior with three-stage switching control in clustered networks

In this work, we investigate the synchronization of neuronal activity through a model of a clustered network formed by scale-free subnetworks. These simulate the areas of the cerebral cortex and capture the spatial distribution of the vertices. The growth of the scale-free subnetworks takes place according to the fitness model, and the architecture of the clustered network presents internal and external links to simulate connections inside and between cortical areas. The corticocortical connections are established according to a human connectivity matrix obtained through experimental data. The model also considers electrical and chemical synapses. A two-dimensional map, in the bursting regime, simulates the dynamic behavior of the neuron. The high synchronization of neuronal activity is revealed by the Kuramoto order parameter. To corroborate the analyses using the order parameter, we calculate a suppression measure. In order to suppress this synchronization, we propose a three-stage switching control as a function of the delayed mean-field in each subnetwork. This suppressor agent is effective in two scenarios, whether applied according to the spatial distribution of neurons or in the emitting hubs of the subnetworks. Our results show that the fitness model has a relevant role in the study of neuronal activity suppression, allied to the application of a three-stage switching control by means of a time-delayed feedback method being an efficient way to suppress synchronization in clustered networks.

Effects of feedback control in small-world neuronal networks interconnected according to a human connectivity map

In this work, we consider a phenomenological two-dimensional discrete model coupled in a structure of a clustered network to investigate the suppression of neuronal synchronization in a complex network. We constructed a network according to a weighted human connectivity matrix and an adjacency matrix that carries small-world properties. The coupling between neurons is inserted through a chemical synapse term and a neuronal activation function. The neuronal synchronization is measured by the Kuramoto order parameter. We intend to achieve the suppression of burst phase synchronization, and for that, we use a mathematical tool, based on the technique of deep brain stimulation, which consists of applying an external signal to the network after a certain time, causing the bursts to desynchronize. Our results are efficient when applying the feedback method both in the global network and in the cortical regions. We have seen that for cortical regions, synchronization is more difficult to suppress, however, by slightly increasing the perturbation, we are able to achieve the desired effect. This shows that it is possible to use the control efficiently in isolated cortical areas, therefore, we present a new alternative to conventional methods, avoiding to apply the control to the entire network to obtain the same results.

Basins of attraction of chimera states on networks.

Networks of identical coupled oscillators display a remarkable spatiotemporal pattern, the chimera state, where coherent oscillations coexist with incoherent ones. In this paper we show quantitatively in terms of basin stability that stable and breathing chimera states in the original two coupled networks typically have very small basins of attraction. In fact, the original system is dominated by periodic and quasi-periodic chimera states, in strong contrast to the model after reduction, which can not be uncovered by the Ott-Antonsen ansatz. Moreover, we demonstrate that the curve of the basin stability behaves bimodally after the system being subjected to even large perturbations. Finally, we investigate the emergence of chimera states in brain network, through inducing perturbations by stimulating brain regions. The emerged chimera states are quantified by Kuramoto order parameter and chimera index, and results show a weak and negative correlation between these two metrics.

Mirroring of synchronization in a bi-layer master-slave configuration of Kuramoto oscillators.

The phenomenon of mirroring of synchronization is investigated in dynamically dissimilar, unidirectionally coupled, bi-layer master-slave configuration of globally coupled Kuramoto oscillators. The dynamics of the master layer depends solely on the distribution of the natural frequencies of its oscillators. On the other hand, the slave layer dynamics depends not only on the distribution of the natural frequencies of its oscillators but also on the unidirectional coupling with the master layer. The standard Kuramoto order parameter is used to study synchronization in the individual layers and of the bi-layer network. A transition to a completely mirroring state is observed in the dynamics of the slave layer, as the mirroring coefficient in the unidirectional coupling is increased. We derive analytically and verify numerically the conditions for the slave layer to fully mimic the synchronization properties of the master layer. It is further shown that while the master and slave layers are individually synchronized, the bi-layer network exhibits a state of frustrated synchronization.

MultiSyncPy: A Python package for assessing multivariate coordination dynamics

In order to support the burgeoning field of research into intra- and interpersonal synchrony, we present an open-source software package: multiSyncPy. Multivariate synchrony goes beyond the bivariate case and can be useful for quantifying how groups, teams, and families coordinate their behaviors, or estimating the degree to which multiple modalities from an individual become synchronized. Our package includes state-of-the-art multivariate methods including symbolic entropy, multidimensional recurrence quantification analysis, coherence (with an additional sum-normalized modification), the cluster-phase ‘Rho’ metric, and a statistical test based on the Kuramoto order parameter. We also include functions for two surrogation techniques to compare the observed coordination dynamics with chance levels and a windowing function to examine time-varying coordination for most of the measures. Taken together, our collation and presentation of these methods make the study of interpersonal synchronization and coordination dynamics applicable to larger, more complex and often more ecologically valid study designs. In this work, we summarize the relevant theoretical background and present illustrative practical examples, lessons learned, as well as guidance for the usage of our package – using synthetic as well as empirical data. Furthermore, we provide a discussion of our work and software and outline interesting further directions and perspectives. multiSyncPy is freely available under the LGPL license at: https://github.com/cslab-hub/multiSyncPy, and also available at the Python package index.

Synchronization in cilia carpets: multiple metachronal waves are stable, but one wave dominates

Carpets of actively bending cilia represent arrays of biological oscillators that can exhibit self-organized metachronal synchronization in the form of traveling waves of cilia phase. This metachronal coordination supposedly enhances fluid transport by cilia carpets. Using a multi-scale model calibrated by an experimental cilia beat pattern, we predict multi-stability of wave modes. Yet, a single mode, corresponding to a dexioplectic wave, has predominant basin-of-attraction. Similar to a ‘dynamic’ Mermin–Wagner theorem, relaxation times diverge with system size, which rules out global order in infinite systems. In finite systems, we characterize a synchronization transition as function of quenched frequency disorder, using generalized Kuramoto order parameters. Our framework termed Lagrangian mechanics of active systems allows to predict the direction and stability of metachronal synchronization for given beat patterns.

Controlling phase synchrony in the mean field coupled Hindmarsh–Rose neurons

The phase synchronization in a network of mean field coupled Hindmarsh–Rose neurons and the control of phase synchrony by an external input has been analyzed in this work. The analysis of interspike interval, with varying coupling strength, reveals the dynamical change induced in each neuron in the network. The bursting phase lines depict that mean field coupling induces phase synchrony in excitatory mode and desynchrony in inhibitory mode. The coefficient of variability, in spatial and temporal domain, signifies the deviations in firing times of neurons, in a collective manner. The Kuramoto order parameter quantifies the intermittent and complete phase synchrony, induced by excitatory mean field coupling. The capability of external input, in the form of spikes, to control the intermittent and complete phase synchrony has been analyzed. The coefficient of variability and Kuramoto order parameter has been studied by varying the amplitude, pulse width and frequency of the input. The studies have shown that high-frequency spike input, with optimum amplitude and pulse width, has high desynchronizing ability, which is substantiated by the parameter space analysis. The control of synchrony in the network of neurons may find application in rectifying neural disorders.

Attracting Poisson chimeras in two-population networks.

Chimera states, i.e., dynamical states composed of coexisting synchronous and asynchronous oscillations, have been reported to exist in diverse topologies of oscillators in simulations and experiments. Two-population networks with distinct intra- and inter-population coupling have served as simple model systems for chimera states since they possess an invariant synchronized manifold in contrast to networks on a spatial structure. Here, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. First, we elucidate how the Kuramoto order parameter of the finite-sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. These findings suggest that it is suitable to classify the chimera states according to their order parameter dynamics, and therefore, we define Poisson and non-Poisson chimera states. We then perform a Lyapunov analysis of these two types of chimera states, which yields insight into the full stability properties of the chimera trajectories as well as of collective modes. In particular, our analysis also confirms that Poisson chimeras are neutrally stable. We then introduce two types of "perturbation" that act as small heterogeneities and render Poisson chimeras attracting: A topological variation via the simplest nonlocal intra-population coupling that keeps the network symmetries and the allowance of amplitude variations in the globally coupled two-population network; i.e., we replace the phase oscillators by Stuart-Landau oscillators. The Lyapunov spectral properties of chimera states in the two modified networks are investigated, exploiting an approach based on network symmetry-induced cluster pattern dynamics of the finite-size network.

Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression

Obtaining coarse-grained models that accurately incorporate finite-size effects is an important open challenge in the study of complex, multi-scale systems. We apply Langevin regression, a recently developed method for finding stochastic differential equation (SDE) descriptions of realistically-sampled time series data, to understand finite-size effects in the Kuramoto model of coupled oscillators. We find that across the entire bifurcation diagram, the dynamics of the Kuramoto order parameter are statistically consistent with an SDE whose drift term has the form predicted by the Ott–Antonsen ansatz in the N→∞ limit. We find that the diffusion term is nearly independent of the bifurcation parameter, and has a magnitude decaying as N−1/2, consistent with the central limit theorem. This shows that the diverging fluctuations of the order parameter near the critical point are driven by a bifurcation in the underlying drift term, rather than increased stochastic forcing.