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The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity. [ABSTRACT FROM AUTHOR]

Abstract We employ Kuramoto model to assess the presence of synchronization in individuals who fulfill a cooperation task. Our input data is a couple of signals obtained from functional Near-Infrared Spectroscopy Data Acquisition and Pre-processing technology that is used to capture the brain activity of an individual by measuring the oxyhemoglobin (HbO) level. We consider 1 min signal for individuals in three distinct states: (i) rest; (ii) before a disturb happens; (iii) after the disturbance. We estimate global and local order parameters synchronization with the purpose to compare the conditions of reaching a synchronous state in the networks corresponding to different states for distinct individuals and hemispheres of the prefrontal cortices of same individual. Experimental results confirmed once more that coherent state is reached not for same conditions in both individuals and hemispheres of the prefrontal cortices. Furthermore, condition changes even for different events. The computation of the effective frequencies for each degree class indicates clearly the network difference in rest, before and after disturb. Finally, we investigate the dynamic connectivity matrix and consider the similarity between distinct prefrontal cortices over time.

We discuss an analog simulation of the Kuramoto model of coupled phase oscillators that can be constructed in the laboratory by coupling a set of light-emitting diodes (LEDs) through a microcontroller. The components of the circuit, namely a WS2812B LED strip, Arduino UNO R3, and a standard power supply, are inexpensive and can be sourced locally. By increasing the strength of the coupling, the ensemble of LEDs can be made to flash in synchrony, and the extent of synchronization can be quantified and measured. [ABSTRACT FROM AUTHOR]

We study actions of Möbius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe–Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotation after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators’ phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the “hidden” variable that is not visible on macroscopic level. More precisely, we demonstrate that the chimera model is an intriguing example of the classical system that exhibits the holonomy in fiber bundles of the group of Möbius transformations. [ABSTRACT FROM AUTHOR]

The dynamics of synaptic interactions within spiking neuron networks play a fundamental role in shaping emergent collective behavior. This paper studies a finite-size network of quadratic integrate-and-fire neurons interconnected via a general synaptic function that accounts for synaptic dynamics and time delays. Through asymptotic analysis, we transform this integrate-and-fire network into the Kuramoto-Sakaguchi model, whose parameters are explicitly expressed via synaptic function characteristics. This reduction yields analytical conditions on synaptic activation rates and time delays determining whether the synaptic coupling is attractive or repulsive. Our analysis reveals alternating stability regions for synchronous and partially synchronous firing, dependent on slow synaptic activation and time delay. We also demonstrate that the reduced microscopic model predicts the emergence of synchronization, weakly stable cyclops states, and non-stationary regimes remarkably well in the original integrate-and-fire network and its theta neuron counterpart. Our reduction approach promises to open the door to rigorous analysis of rhythmogenesis in networks with synaptic adaptation and plasticity.

Orthogonal group synchronization is the problem of estimating n elements Z1,..., Zn from the r x r orthogonal group given some relative measurements Rij = ZiZ 1 j. The least-squares formulation is nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the dimension of the optimization problem from O(n) to O(n2). Alternatively, Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at dimension O(n²). For smaller relaxations, the problem structure matters. It has been observed in the robotics literature that, for simultaneous localization and mapping problems, it seems sufficient to increase the dimension by a small constant multiple over the original. We partially explain this. This also has implications for Kuramoto oscillators. Specifically, we minimize the least-squares cost function in terms of estimators Y1,...,Yn. For p ≥ r, each Yi is relaxed to the Stiefel manifold St(r, p) of r x p matrices with orthonormal rows. The available measurements implicitly define a (connected) graph G on n vertices. In the noiseless case, we show that, for all connected graphs G, second-order critical points are globally optimal as soon as p ≥ r+2. (This implies that Kuramoto oscillators on St(r, p) synchronize for all p ≥ r + 2.) This result is the best possible for general graphs; the previous best known result requires 2p ≥ 3(r +1). For p >r +2, our result is robust to modest amounts of noise (depending on p and G). Our proof uses a novel randomized choice of tangent direction to prove (near-)optimality of second-order critical points. Finally, we partially extend our noiseless landscape results to the complex case (unitary group); we show that there are no spurious local minima when 2p ≥ 3r. [ABSTRACT FROM AUTHOR]

Several studies have shown that coordination among neural ensembles is a key to understand human cognition. A well charted path is to identify coordination states associated with cognitive functions from spectral changes in the oscillations of EEG or MEG. A growing number of studies suggest that the tendency to switch between coordination states, sculpts the dynamic repertoire of the brain and can be indexed by a measure known as metastability. In this article, we characterize perturbations in the metastability of global brain network dynamics following Transcranial Magnetic Stimulation that could quantify the duration for which information processing is altered. Thus allowing researchers to understand the network effects of brain stimulation, standardize stimulation protocols and design experimental tasks. We demonstrate the effect empirically using publicly available datasets and use a digital twin (a whole brain connectome model) to understand the dynamic principles that generate such observations. We observed a significant reduction in metastability, concurrent with an increase in coherence following single-pulse TMS reflecting the existence of a window where neural coordination is altered. The reduction in complexity was validated by an additional measure based on the Lempel-Ziv complexity of microstate labeled EEG data. Interestingly, higher frequencies in the EEG signal showed faster recovery in metastability than lower frequencies. The digital twin shed light on how the phase resetting introduced by the single-pulse TMS in local cortical networks can propagate globally, giving rise to changes in metastability and coherence. [ABSTRACT FROM AUTHOR]

Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of "rational" interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium. [ABSTRACT FROM AUTHOR]

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The purpose of clustering is to identify distributions and patterns within unlabelled datasets. Since the proposal of the original synchronization clustering (SynC) algorithm in 2010, synchronization clustering has become a significant research direction. This paper proposes a shrinking synchronization clustering (SSynC) algorithm utilizing a linear weighted Vicsek model. SSynC algorithm is developed from SynC algorithm and a more effective synchronization clustering (ESynC) algorithm. Through analysis and comparison, we find that SSynC algorithm demonstrates superior synchronization effect compared to SynC algorithm, which is based on an extensive Kuramoto model. Additionally, it exhibits similar effect to ESynC algorithm, based on a linear version of Vicsek model. In the simulations, a comparison is conducted between several synchronization clustering algorithms and classical clustering algorithms. Through experiments using some artificial datasets, eight real datasets and three picture datasets, we observe that compared to SynC algorithm, SSynC algorithm not only achieves a better local synchronization effect but also requires fewer iterations and incurs lower time costs. Furthermore, when compared to ESynC algorithm, SSynC algorithm obtains reduced time costs while achieving nearly the same local synchronization effect and the same number of iterations. Extensive comparison experiments with some class clustering algorithms demonstrate the effectiveness of SSynC algorithm. [ABSTRACT FROM AUTHOR]

We propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or column)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender’s index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. [ABSTRACT FROM AUTHOR]

Several studies have shown that coordination among neural ensembles is a key to understand human cognition. A well charted path is to identify coordination states associated with cognitive functions from spectral changes in the oscillations of EEG or MEG. A growing number of studies suggest that the tendency to switch between coordination states, sculpts the dynamic repertoire of the brain and can be indexed by a measure known as metastability. In this article, we characterize perturbations in the metastability of global brain network dynamics following Transcranial Magnetic Stimulation that could quantify the duration for which information processing is altered. Thus allowing researchers to understand the network effects of brain stimulation, standardize stimulation protocols and design experimental tasks. We demonstrate the effect empirically using publicly available datasets and use a digital twin (a whole brain connectome model) to understand the dynamic principles that generate such observations. We observed a significant reduction in metastability, concurrent with an increase in coherence following single-pulse TMS reflecting the existence of a window where neural coordination is altered. The reduction in complexity was validated by an additional measure based on the Lempel-Ziv complexity of microstate labeled EEG data. Interestingly, higher frequencies in the EEG signal showed faster recovery in metastability than lower frequencies. The digital twin shed light on how the phase resetting introduced by the single-pulse TMS in local cortical networks can propagate globally, giving rise to changes in metastability and coherence.

The synchronization of multiple oscillators serves as the central mechanism for maintaining stable circadian rhythms in physiology and behavior. Aging and disease can disrupt synchronization, leading to changes in the periodicity of circadian activities. While our understanding of the circadian clock under synchronization has advanced significantly, less is known about its behavior outside synchronization, which can also fall within a predictable domain. These states not only impact the stability of the rhythms but also modulate the period length. In C57BL/6 mice, aging, diseases, and removal of peripheral circadian oscillators often result in lengthened behavioral circadian periods. Here, we show that these changes can be explained by a surprisingly simple mathematical relationship: the frequency is the reciprocal of the period, and its distribution becomes skewed when the period distribution is symmetric. The synchronized frequency of a population in the skewed distribution and the macroscopic frequency of combined oscillators differ, accounting for some of the atypical circadian period outputs observed in networks without synchronization. Building on this finding, we investigate the dynamics of circadian outputs in the context of aging and disease, where synchronization is weakened. [ABSTRACT FROM AUTHOR]

We prove the complete and partial phase synchronization for the Kuramoto oscillators with distributed time-delays and inertia effect. Our results assert that the Kuramoto models incorporated with a small variation of distributed time-delays and inertia effect still exhibit synchronization. This shows the robustness of the synchronization phenomena of the original Kuramoto model in the perturbation of time-delay and inertia effects. We also present several numerical experiments supporting our main results and exhibiting interesting patterns. [ABSTRACT FROM AUTHOR]

The mammalian circadian clock is located in the suprachiasmatic nucleus (SCN) and consists of a network of coupled neurons, which are entrained to the environmental light-dark cycle. The phase coherence of the neurons is plastic and driven by the duration of daylight. With aging, the capacity to behaviorally adapt to seasonal changes in photoperiod reduces. The mechanisms underlying photoperiodic adaptation are largely unknown, but are important to unravel for the development of novel interventions to improve the quality of life of the elderly. We analyzed the phase coherence of single-cell PERIOD2::LUCIFERASE (PER2::LUC) expression rhythms in the SCN of young and old mice entrained to either long or short photoperiod. The phase coherence was used as input to a 2-community noisy Kuramoto model to estimate the coupling strength between and within neuronal subpopulations. The model revealed a correlation between coupling strength and photoperiod-induced changes in the phase relationship among neurons, suggesting a functional link. We found that the SCN of young mice adapts in coupling strength over a large range, with weak coupling in long photoperiod (LP) and strong coupling in short photoperiod (SP). In aged mice, we also found weak coupling in LP, but a reduced capacity to reach strong coupling in SP. The inability to respond with an increase in coupling strength suggests that manipulation of photoperiod is not a suitable strategy to enhance clock function with aging. We conclude that the inability of aged mice to reach strong coupling contributes to deficits in behavioral adaptation to seasonal changes in photoperiod. [ABSTRACT FROM AUTHOR]

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov–Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity. [ABSTRACT FROM AUTHOR]

We studied the collective behaviors of the time-delayed Kuramoto model with frustration under general network topology. For the generalized Kuramoto model with the graph diameter no greater than two and under a sufficient regime in terms of small time delay and frustration and large coupling strength, we showed that the complete frequency synchronization occurs exponentially fast when the initial configuration is distributed in a half circle. We also studied a complete network, which is a small perturbation of all-to-all coupling, as well as presented sufficient frameworks leading to the exponential emergence of frequency synchronization for the initial data confined in a half circle. [ABSTRACT FROM AUTHOR]

The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes. [ABSTRACT FROM AUTHOR]

We study asymptotic behaviors of the second-order Lohe matrix model on the unitary group. For the emergent behaviors, we show that a solution to the Lohe matrix model with identical Hamiltonians always converges to equilibrium using a gradient flow approach. Moreover, we establish the uniform-in-time stability with respect to initial data using the complete aggregation estimate, which can be applied to obtain the corresponding mean-field kinetic equation for the one-oscillator distribution function on the phase space. In fact, we rigorously show that Wasserstein-2 distance between two solutions in different regimes converges to zero in any finite time interval, when the number of oscillators tends to infinity. For the asymptotic emergent dynamics of the kinetic equation, we use an energy estimate and temporal evolution for the diameter to provide sufficient frameworks leading to the complete and practical aggregations for identical and non-identical Hamiltonians, respectively. [ABSTRACT FROM AUTHOR]

Alzheimer's disease (AD) is a degenerative brain disease, and the condition is difficult to assess. In the past, numerous brain dynamics models have made remarkable contributions to neuroscience and the brain from the microcosmic to the macroscopic scale. Recently, large-scale brain dynamics models have been developed based on dual-driven multimodal neuroimaging data and neurodynamics theory. These models bridge the gap between anatomical structure and functional dynamics and have played an important role in assisting the understanding of the brain mechanism. Large-scale brain dynamics have been widely used to explain how macroscale neuroimaging biomarkers emerge from potential neuronal population level disturbances associated with AD. In this review, we describe this emerging approach to studying AD that utilizes a biophysically large-scale brain dynamics model. In particular, we focus on the application of the model to AD and discuss important directions for the future development and analysis of AD models. This will facilitate the development of virtual brain models in the field of AD diagnosis and treatment and add new opportunities for advancing clinical neuroscience. [ABSTRACT FROM AUTHOR]

At present, inverted pendulum model (IPM) has been recognized by increasing researchers due to its capability to capture the characteristics of pedestrian lateral excitation. However, the IPM still has a downside that has been continuously questioned: the velocity proportional pedestrian load coefficient c p is inconsistent with the on-site observations in the case of pedestrian walking with normal frequency. This problem is most likely resulted from that the IPM ignores the synchronization effect. The IPM is derived from indoor single-person experiment and cannot consider the synchronization in the crowd. Synchronization effects are classified into two types： pedestrian–bridge and pedestrian–pedestrian synchronizations. Pedestrian–bridge synchronization is stimulated by bridge vibration, whereas pedestrian–pedestrian synchronization is stimulated by surrounding crowd. In this study, a novel model for pedestrian-induced lateral vibration of footbridge is proposed. It uses the IPM as the basis and adopts a binary coupled Kuramoto model that originally derived from the collective synchronization phenomena to overcome the limitation in IPM. This model can consider the pedestrian–bridge and pedestrian–pedestrian synchronizations, which provides a new way to reveal the mechanism of pedestrian-induced lateral vibration of footbridge. The proposed method is verified through a case study of the London Millennium Bridge, and some useful conclusions are obtained. [ABSTRACT FROM AUTHOR]

Many systems of physical and biological interest are characterized by assemblies of phase oscillators whose interaction is mediated by a diffusing chemical. The coupling effect results from the fact that the local concentration of the mediating chemical affects both its production and absorption by each oscillator. Since the chemical diffuses through the medium in which the oscillators are embedded, the coupling among oscillators is non-local; it considers all the oscillators depending on their relative spatial distances. We considered a mathematical model for this coupling, when the diffusion time is arbitrary with respect to the characteristic oscillator periods, yielding a system of coupled nonlinear integro-differential equations which can be solved using Green functions for appropriate boundary conditions. In this paper, we show numerical solutions of these equations for three finite domains: a linear one-dimensional interval, a rectangular, and a circular region, with absorbing boundary conditions. From the numerical solutions, we investigate phase and frequency synchronization of the oscillators, with respect to changes in the coupling parameters for the three considered geometries. [ABSTRACT FROM AUTHOR]

The Kuramoto model was developed to describe the coupling of oscillators, motivated by the natural synchronization phenomena. We are interested in modeling an epileptic seizure considering it as the synchronization of action potentials using and modifying this model. In this article, we propose to modify this model, changing the constant coupling force for a function with logistic growth to simulate the onset and epileptic seizure level in an adult male rat caused by the administration of lithium–pilocarpine. Later, we select some frequencies and their respective amplitude values using an algorithm based on the fast Fourier transform (FFT) from an electroencephalography signal when the rat is in basal conditions. Then, we take these values as the natural frequencies of the oscillators in the modified Kuramoto model, considering every oscillator as a single neuron to simulate the emergence of an epileptic seizure numerically by increasing the synchronization value in the coupling function. Finally, using Dynamic Time Warping algorithm, we compare the simulated signal by the Kuramoto model with an FFT approximation of the epileptic seizure. [ABSTRACT FROM AUTHOR]

We present a model of a sub-THz frequency spectrum analyzer based on the antiferromagnet/non–magnetic metal heterostructure. The considered structure operates in the active self–oscillation regime with the high Q–factor (closed to $10^{6}$) and is phase–locked to the analyzed signal, while the frequency of the oscillator is linearly tuned by the spin–polarized current flowing through the non–magnetic metal. We find an analytical expression for the duration of the phase–locking interval by analyzing the generalized Kuramoto model with inertia. We show that the speed of the described spectrum analyzer is comparable to and exceeds the speed of modern commercial spectrum analyzers. The presented results can be helpful for practically realizing the room–temperature, tunable, sensitive, and high–speed sub-THz frequency spectrum analyzers. [ABSTRACT FROM AUTHOR]

In this paper, we present a survey of recent progress on the emergent behaviors of stochastic particle models which arise from the modeling of collective dynamics. Collective dynamics of interacting autonomous agents is ubiquitous in nature, and it can be understood as a formation of concentration in a state space. The jargons such as aggregation, herding, flocking and synchronization describe such concentration phenomena. Recently it became one of the emerging topics in the applied mathematics community due to possible engineering applications and close relation with nonlocal partial differential equations. When an autonomous agent system interacts with unknown environment as an open system, the effects of hidden and unidentified interactions between the environment and the autonomous system are often realized by stochastic noises in agent dynamics, and the temporal evolution of the autonomous system results in stochastic collective models. From the viewpoint of dynamical systems theory, it is very interesting how collective dynamics emerges from initial state. As concrete examples, we consider four specific stochastic collective models (stochastic Winfree and Kuramoto models for synchronization, the stochastic Cucker–Smale model for flocking, and a first-order stochastic nonlinear consensus model), and we also briefly review the state-of-the-art results for these models on the emergence of collective dynamics and discuss their applications in finance and optimization. [ABSTRACT FROM AUTHOR]

Lots of studies have been carried out on characteristic of epileptic Electroencephalograph (EEG). However, traditional EEG characteristic research methods lack exploration of spatial information. To study the characteristics of epileptic EEG signals from the perspective of the whole brain, this paper proposed combination methods of multi-channel characteristics from timefrequency and spatial domains. This paper was from two aspects: Firstly, signals were converted into 2D Hilbert Spectrum (HS) images which reflected the timefrequency characteristics by Hilbert-Huang Transform (HHT). These images were identified by Convolutional Neural Network (CNN) model whose sensitivity was 99.8%, accuracy was 98.7%, specificity was 97.4%, F1-score was 98.7%, and AUCROC was 99.9%. Secondly, the multi-channel signals were converted into brain networks which reflected the spatial characteristics by Symbolic Transfer Entropy (STE) among different channels EEG. And the results show that there are different network properties between ictal and interictal phase and the signals during the ictal enter the synchronization state more quickly, which was verified by Kuramoto model. To summarize, our results show that there was different characteristics among channels for the ictal and interictal phase, which can provide effective physical non-invasive indicators for the identification and prediction of epileptic seizures. [ABSTRACT FROM AUTHOR]

We describe the transient regime of synchronizing flows defined over a network, introducing a codification of the paths towards synchronization. In all cases, the collection of paths toward synchronization defines a combinatorial structure: the transition diagram. In this paper, besides introducing this general framework, we study in full detail the case of completely connected networks where a monotonic convergence towards synchronization takes place. For these systems we are able to supply a detailed analytical description of the transition diagram describing both, the Laplacian flow and a special case of the Kuramoto model on the same network. This constitutes a first instance of a full symbolic description of the path towards synchronization and illustrates the hidden complexity of the transient regime. [ABSTRACT FROM AUTHOR]

Coupled phase oscillators are adopted as powerful platforms in studying synchrony behaviors emerged in various systems with rhythmic dynamics. Much attention has been focused on coupled time-continuous oscillators described by differential equations. In this paper, we study the synchronization dynamics of networks of coupled circle maps as the discrete version of the Kuramoto model. Despite of its simplicity in mathematical form, it is found that discreteness may induce many interesting synchronization behaviors. Multiple synchronization and desynchronization transitions of both phases and frequencies are found with varying the coupling among circle-map oscillators. The mechanisms of these transitions are interpreted in terms of the mean-field approach, where collective bifurcation cascades are revealed for coupled circle-map oscillators. [ABSTRACT FROM AUTHOR]

Summary: The standard van Neumann computer excels at many things. However, it can be very inefficient in solving optimization problems with a large solution space. For that reason, a novel analog approach, the oscillator‐based Ising machine, has been proposed as a better alternative for dealing with such problems. In this work, we review the concept of oscillator‐based Ising machine and address how optimization problems can be mapped onto such machines when the quadratic unconstrained binary optimization (QUBO) formulation is given. Furthermore, we provide an ideal circuit that can be used in combination with the wave digital concept for real‐time simulated annealing. The functionality of this circuit is explained on the basis of a Lyapunov stability analysis. The latter also provides an answer for the decision‐making question: When has the Ising machine solved a mapped problem? At the end, we provide emulation results demonstrating the correlation between functionality and stability. Our results are based on an eigenvalue analysis of the linearized system and demonstrate that all eigenvalues converge to the left complex half plane, once an optimization problem has been optimally solved. This implies that the system enters an attractor region and is hence forced to eventually converge to the optimal solution. [ABSTRACT FROM AUTHOR]

This paper explores how adaptive creativity is continuously generated and sustained in living systems. The philosophical frame and motivations for this investigation will be introduced by juxtaposing an actual creative process with current cybernetic efforts to automate creativity. Past and present process philosophers that have critiqued the implicit commitments of these contemporary techniques will set the stage for further investigations. The litmus test of progress in this investigation will be measured against the extension of two concepts: virtuality, as introduced by Gilbert Simondon (On the Mode of the Existence of Technical Objects), and relevant noise, as introduced by Bacigalupi ("Semiogenesis: A Dynamic System Approach to Agency and Structure"). To refine the concepts of virtuality and relevant noise for our purposes, a rigorous theoretical model will be proposed whose intent is to explain veritable unbounded creativity. This model will then serve as a heuristic lens to explore additional extant models, such as the Kuramoto model (Strogatz), that are used to explain empirical observations of adaptive and creative behaviors in a diversity of biological phenomena. Based on these models of biological creativity and the critique of the contemporary cybernetic project, this paper will conclude by outlining the ethical implications of our culture's current commitments to a cybernetic world view and how we might evolve beyond it. [ABSTRACT FROM AUTHOR]

We study the emergent dynamics and global well‐posedness of the matrix‐valued integro‐differential equation which can be derived from the continuum limit of the lattice Lohe group model. The lattice Lohe group model corresponds to the generalized high‐dimensional Kuramoto model. The solution to the lattice Lohe group model can be cast as a simple function‐valued solution to the continuum Lohe group model. We first construct a local classical solution to the continuum Lohe group model, and then we find an invariant set and derive a global well‐posedness in some sufficient frameworks formulated in terms of initial data, system functions, and system parameters. We also show that phase‐locked states can emerge from the admissible class of initial data in a large coupling regime. Moreover, we show that sequence of simple functions obtained from the solutions of the lattice Lohe group model converges to a local classical solution to the continuum Lohe group model in supremum norm. [ABSTRACT FROM AUTHOR]

The Kuramoto model is a classical model used for the describing of synchronization in populations of oscillatory units. In the present paper we study the Kuramoto model with delay with a focus on the distribution of the oscillators' frequencies. We consider a series of rational distributions which allow us to reduce the population dynamics to a set of several delay differential equations. We use the bifurcation analysis of these equations to study the transition from the asynchronous to synchronous state. We demonstrate that the form of the frequency distribution may play a substantial role in synchronization. In particular, for Lorentzian distribution the delay prevents synchronization, while for other distributions the delay can facilitate synchronization. [ABSTRACT FROM AUTHOR]

Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. In this paper, we present a procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each bounded connected component of a (real) atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the n = 4 case. We also apply our method to design an algorithm to compute the real dimension of algebraic sets, the original motivation for this research. We compare the efficiency of our method to existing methods to compute the real dimension on a benchmark family. [ABSTRACT FROM AUTHOR]

We study fast and slow clustering dynamics of Cucker–Smale ensemble with internal phase dynamics via the Cucker–Smale–Kuramoto (in short, CSK) model. The CSK model describes the emergent dynamics of flocking particles with phase dynamics. It consists of the Cucker–Smale flocking model and the Kuramoto model, and their interplay is registered in the communication weight function between particles. We present a sufficient framework for mono-cluster flocking and complete synchronization in terms of system parameters and initial data. In particular, the mono-cluster flocking will emerge exponentially fast (fast dynamics). On the other hand, when initial spatial-velocity configuration is close to the bi-cluster flocking, we also provide a sufficient framework leading to convergence of bi-cluster flocking algebraically slow depending on the decay rate of the communication weight (slow flocking dynamics). We also provide several numerical examples and compare them with analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we study flocking of a Kuramoto type model with local interaction functions. This model can be used to describe dynamics of an ensemble of autonomous agents moving in the plane with unit speed. Suppose that each agent only interacts with its neighbors. Under some conditions on initial data and model parameters, we obtain a sufficient framework for flocking, and further show that the overall system achieves a consensus state at an exponential rate. [ABSTRACT FROM AUTHOR]

Purpose of this work is studying of the dynamics of a power grid model that results from the expansion of a highly centralized grid, i.e. a hub-cluster, by adding a small subgrid. The main attention is paid to the study of possible power grid operation regimes and their characteristics. Methods. Numerical simulation of power grid operation, the dynamics of which is described by the Kuramoto equations with inertia, is used. Results. Various power grid operation regimes and the boundaries of their existence in the parameter space are given. The main characteristics of these regimes, such as the probability of realization and the magnitude of oscillations of regime variables, are considered. The conditions for safe connection to hub-cluster power grids are obtained. Conclusion. The dynamics of power grid consisting of two subgrids and its operation regimes are considered. Based on the characteristics of these regimes, their safety for subgrids is determined. The results obtained made it possible to formulate conditions for a safe connection to hub-cluster power grids.

In the paper Ameli and Samani (Nonlinear Dyn https://doi.org/10.1007/s11071-022-07703-0, 2022), the authors formulate low-dimensional evolutions of the macroscopic order parameters in the generalized Kuramoto model, in which the quenched disorder of the heterogeneous natural frequencies and coupling strength are correlated via a weighted absolute value function. The authors state that the collective dynamics, as well as the global bifurcation of various attractors, can be delineated in the framework of the low-dimensional manifold. We argue that such low-dimensional descriptions for the frequency-weighted coupling are not correct in general. This contradiction is explained from several aspects, including the Ott-Antonsen reduction and the forward (backward) critical points corresponding to the onset (vanishing) of synchrony. Remarkably, we uncover that the singularity of the frequency-weighted coupling forbids the analytical continuation, but can vastly simplify the coherent behaviors of the system. Importantly, we justify that our analysis can be extended to a wide class of systems involving the frequency-weighted coupling scheme. [ABSTRACT FROM AUTHOR]

The interplay between system dynamics and topological structure plays a crucial role in nonlinear system and modern network science. In this paper, we intend to investigate the effects of both heterogeneity and asymmetric coupling on the collective behaviors of Kuramoto oscillators, and reveal the processes how the synchronization transits to multi-cluster or disorder states. We find that the complete synchronization and anti-phase synchronization states in a star motif change into the remote synchronization state, owing to the disparity in the values of oscillators' natural frequencies, and further become into multi-cluster due to the existence of multiple hubs in complete bipartite network, and finally achieve disorder state in complex networks. These findings are helpful to understand the forming processes of remote synchronization and general multi-cluster in coupled systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete synchronization of the Kuramoto model with general network containing a spanning tree, when the initial phases are distributed in an open half circle. As lack of uniform coercivity in general digraph, in order to capture the dissipation structure on a general network, we apply the strongly connected graph decomposition criteria in Ha et al. [Math. Models Methods Appl. Sci., 30 (2020)] to yield a hierarchical structure, which will induce estimates similar to the hypo-coercivity estimates. This drives the phase diameter into an invariant region confined in quarter circle in finite time under a large coupling regime. Then the uniformly small bound- edness of the diameter eventually leads to the emergence of exponentially fast synchronization. [ABSTRACT FROM AUTHOR]

We consider the temporal asymptotic behavior of the all-to-all coupled Kuramoto model with inertia and time-periodic natural frequencies. Due to the inertial effect, there are three cases of the dynamical ensemble with respect to the coupling strength; large coupling, near boundary, and small coupling. For each case, we present the asymptotic behavior of the solution to the inertial Kuramoto model with periodic natural frequencies: the solutions commonly consist of a macroscopic phase, a mean-centered-periodic solution, and an exponential decay term. The macroscopic phase is a drift-type term determined by initial data and natural frequencies, and the mean-centered-periodic solution is a standing wave independent of initial data. We provide sufficient conditions for the existence of a mean-centered-periodic solution with a time-periodic phase difference between nodes for each case and its exponential stability. We also provide several simulations to confirm our mathematical results. [ABSTRACT FROM AUTHOR]

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.

We consider a Kuramoto model of four-coupled oscillators of nonidentical initial frequencies. Under the influence of coupling, the oscillators fall into a full phase locked state of a common frequency when the coupling strength surpasses a threshold value. We find numerically the parameters that control this distinguishable coupling constant at the moment the oscillators transit into an entire frequency synchronization when a complete phase lock state takes place. We are able to set apart a recognizable phase condition at the fully locked state. This phase condition helps to derive an analytic formula to calculate the coupling factor as soon as the oscillators depart to a full phase locking state. The explicit expression of the edge coupling factor is given in terms of the initial frequencies of the four oscillators. The formula valid for calculating the distinct coupling allows to find mathematical expressions to calculate the phase differences when the four-coupled phase oscillators are strictly at the full phase locked state and have a common frequency synchronization. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to discuss whether NEV enterprises can achieve synchronous effects with the whole Chinese automobile industry in terms of growth mode. In this paper, we study the development of new energy vehicles from the perspective of ecosystem. Growth mechanisms and synchronization effects also exist in new energy enterprise populations, just like biological populations in natural ecosystems. Here, we propose a moving logistic model to analyze the growth mechanism of new energy vehicle enterprises and obtain serial data of intrinsic growth rate, internal inhibition coefficient, and theoretical maximum sales volume. The intrinsic growth rate and theoretical maximum sales volume show an initial trend of decline followed by recovery. The evaluation results of coupling degree and synchronization of the new energy vehicle population were obtained through the measurement of the Kuramoto model and its derivative model. The coupling degree of the new energy vehicle population is not high, and the synchronization effect fluctuates and oscillates. The change trend of synchronization effect is similar to that of intrinsic growth rate and theoretical maximum sales volume. This phenomenon shows that the new energy vehicle population has been significantly affected by changes in the external market environment. The analysis method of enterprise growth mechanism based on the moving logistic model and the measurement method of coupling degree and synchronization effect based on the Kuramoto model and its derivative models proposed in this paper effectively achieve the research objectives of this paper. [ABSTRACT FROM AUTHOR]

The use of EEG to simultaneously record multiple brains (i.e., hyperscanning) during social interactions has led to the discovery of inter-brain coupling (IBC). IBC is defined as the neural synchronization between people and is considered to be a marker of social interaction. IBC has previously been observed across different frequency bands, including theta [4-7 Hz]. Given the proximity of this frequency range with behavioral rhythms, models have been able to combine IBC in theta with sensorimotor coordination patterns. Interestingly, empirical EEG-hyperscanning results also report the emergence of IBC in the gamma range [>30 Hz]. Gamma oscillations' fast and transient nature makes a direct link between gamma-IBC and other (much slower) interpersonal dynamics difficult, leaving gamma-IBC without a plausible model. However, at the intrabrain level, gamma activity is coupled with the dynamics of lower frequencies through cross-frequency coupling (CFC). This paper provides a biophysical explanation, through the simulation of neural data, for the emergence of gamma inter-brain coupling using a Kuramoto model of four oscillators divided into two separate (brain) units. By modulating both the degree of inter-brain coupling in the theta band (i.e., between-units coupling) and CFC (i.e., intraunit theta-gamma coupling), we provide a theoretical explanation of the observed gamma-IBC phenomenon in the EEG-hyperscanning literature. [ABSTRACT FROM AUTHOR]