Networks Of Coupled Oscillators Research Articles

Fixed-Time Synchronization of Coupled Oscillator Networks with a Pacemaker.

This paper investigates the fixed-time synchronization problem of a Kuramoto-oscillator network in the presence of a pacemaker. Based on the framework of the cyber-physical system (CPS), fixed-time synchronization criteria of such network are presented respectively for identical and non-identical oscillators. In virtue of Lyapunov stability analyses, sufficient conditions are deduced for achieving phase agreement and frequency synchronization for arbitrary initial phases and/or frequencies under distributed control strategies. Theoretical analysis shows that synchronization can be achieved in a fixed time, which is unrelated to initial phases/frequencies. Furthermore, the upper bounds of synchronization time are also obtained. Finally, the numerical simulations also verify the effectiveness of the derived results.

Synchronization and stability in automotive transportation networks

AbstractThe aim of this study was to analyze and evaluate synchronization and stability issues for the planning, operation, and control of processes in automotive transportation networks. We modeled transportation networks by using a coupled oscillator network based on a modified Kuramoto model. Each transport process was mapped as a phase oscillator indicating the transport time, period, and delay. The method could be applied to complex networks where it has not been possible to find analytical solutions. The novel generic oscillator approach was then applied to two transport topologies, namely, milk‐run and just‐in‐sequence transport, based on real‐world problems from a German car manufacturer. We conducted a detailed study of the parameter regions where different synchronization regimes occurred and investigated how the topology influenced the stability and dynamic behavior of transport networks. In particular, we focused on how transport period offsets and transport delays affected synchronous states. We showed that by the introduction of a transport synchronization matrix, the synchronization states in a transport network could be represented in a compact and comprehensive manner. Moreover, thresholds for round‐trip stability could be calculated by analyzing the phase decoupling of a milk‐run. These results were used for the vehicle route planning of milk‐runs with synchronization constraints. Furthermore, the influence of the time delay of a track and trace system on the transport synchronization was analyzed. Finally, for the subsequent investigation of a just‐in‐sequence transport network, we showed how an adaptive control mechanism could re‐synchronize an out‐of‐tune delivery process.

Graphop mean-field limits and synchronization for the stochastic Kuramoto model

Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator's phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov-Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence-coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.

Stability of modulated signals in the damped mechanical network of discontinuous coupled system oscillators with irrational nonlinearities

Stability of modulated signals in the damped mechanical network of discontinuous coupled system oscillators with irrational nonlinearities

Grass-roots optimization of coupled oscillator networks.

Despite the prevalence of biological and physical systems for which synchronization is critical, existing theory for optimizing synchrony depends on global information and does not sufficiently explore local mechanisms that enhance synchronization. Thus, there is a lack of understanding for the self-organized, collective processes that aim to optimize or repair synchronous systems, e.g., the dynamics of paracrine signaling within cardiac cells. Here we present "grass-roots" optimization of synchronization, which is a multiscale mechanism in which local optimizations of smaller subsystems cooperate to collectively optimize an entire system. Considering models of cardiac tissue and a power grid, we show that grass-roots-optimized systems are comparable to globally optimized systems, but they also have the added benefit of being robust to targeted attacks or subsystem islanding. Our findings motivate and support further investigation into the physics of local mechanisms that can support self-optimization for complex systems.

Linear Frequency Modulation of NbO2-Based Nanoscale Oscillator With Li-Based Electrochemical Random Access Memory for Compact Coupled Oscillatory Neural Network.

Oscillatory neural network (ONN)-based classification of clustered data relies on frequency synchronization to injected signals representing input data, showing a more efficient structure than a conventional deep neural network. A frequency tunable oscillator is a core component of the network, requiring energy-efficient, and area-scalable characteristics for large-scale hardware implementation. From a hardware viewpoint, insulator-metal transition (IMT) device-based oscillators are attractive owing to their simple structure and low power consumption. Furthermore, by introducing non-volatile analog memory, non-volatile frequency programmability can be obtained. However, the required device characteristics of the oscillator for high performance of coupled oscillator have not been identified. In this article, we investigated the effect of device parameters of IMT oscillator with non-volatile analog memory on coupled oscillators network for classification of clustered data. We confirmed that linear conductance response with identical pulses is crucial to accurate training. In addition, considering dispersed clustered inputs, a wide synchronization window achieved by controlling the hold voltage of the IMT shows resilient classification. As an oscillator that satisfies the requirements, we evaluated the NbO2-based IMT oscillator with non-volatile Li-based electrochemical random access memory (Li-ECRAM). Finally, we demonstrated a coupled oscillator network for classifying spoken vowels, achieving an accuracy of 85%, higher than that of a ring oscillator-based system. Our results show that an NbO2-based oscillator with Li-ECRAM has the potential for an area-scalable and energy-efficient network with high performance.

Oscillator-Network-Based Ising Machine.

With the slowdown of Moore’s law, many emerging electronic devices and computing architectures have been proposed to sustain the performance advancement of computing. Among them, the Ising machine is a non-von-Neumann solver that has received wide attention in recent years. It is capable of solving intractable combinatorial optimization (CO) problems, which are difficult to be solve using conventional digital computers. In fact, many CO problems can be mapped to finding the corresponding ground states of Ising model. At present, Ising machine prototypes based on different physical principles, such as emerging memristive oscillators, have been demonstrated, among which the Ising Hamiltonian solver based on the coupled oscillator network simultaneously holds the advantages of room-temperature operation, compact footprint, low power consumption, and fast speed to solution. This paper comprehensively surveys the recent developments in this important field, including the types of oscillators, the implementation principle of the Ising model, and the solver’s performance. Finally, methods to further improve the performance have also been suggested.

Stochastic Stability Analysis for Stochastic Coupled Oscillator Networks with Bidirectional Cross-Dispersal.

It is well known that stochastic coupled oscillator network (SCON) has been widely applied; however, there are few studies on SCON with bidirectional cross-dispersal (SCONBC). This paper intends to study stochastic stability for SCONBC. A new and suitable Lyapunov function for SCONBC is constructed on the basis of Kirchhoff's matrix tree theorem in graph theory. Combining stochastic analysis skills and Lyapunov method, a sufficient criterion guaranteeing stochastic stability for the trivial solution of SCONBC is provided, which is associated with topological structure and coupling strength of SCONBC. Furthermore, some numerical simulation examples are given in order to illustrate the validity and practicability of our results.

Mean-field analysis of Stuart-Landau oscillator networks with symmetric coupling and dynamical noise.

This paper presents analyses of networks composed of homogeneous Stuart-Landau oscillators with symmetric linear coupling and dynamical Gaussian noise. With a simple mean-field approximation, the original system is transformed into a surrogate system that describes uncorrelated oscillation/fluctuation modes of the original system. The steady-state probability distribution for these modes is described using an exponential family, and the dynamics of the system are mainly determined by the eigenvalue spectrum of the coupling matrix and the noise level. The variances of the modes can be expressed as functions of the eigenvalues and noise level, yielding the relation between the covariance matrix and the coupling matrix of the oscillators. With decreasing noise, the leading mode changes from fluctuation to oscillation, generating apparent synchrony of the coupled oscillators, and the condition for such a transition is derived. Finally, the approximate analyses are examined via numerical simulation of the oscillator networks with weak coupling to verify the utility of the approximation in outlining the basic properties of the considered coupled oscillator networks. These results are potentially useful for the modeling and analysis of indirectly measured data of neurodynamics, e.g., via functional magnetic resonance imaging and electroencephalography, as a counterpart of the frequently used Ising model.

Mesoscopic model reduction for the collective dynamics of sparse coupled oscillator networks.

The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques that capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to partial synchronization. Our model reduction builds upon the method of collective coordinates. We first show that standard collective coordinate reduction has difficulties capturing this bifurcation. We identify a particular topological structure at bifurcation consisting of a main synchronized cluster, the oscillator that desynchronizes at bifurcation, and an intermediary node connecting them. Utilizing this structure and ensemble averages, we derive an analytic expression for the mismatch between the true bifurcation from global to partial synchronization and its estimate calculated via the collective coordinate approach. This allows to calibrate the standard collective coordinate approach without prior knowledge of which node will desynchronize. We introduce a second mesoscopic reduction, utilizing the same particular topological structure, which allows for a quantitative dynamical description of the phases near bifurcation. The mesoscopic reductions significantly reduce the computational complexity of the collective coordinate approach, reducing from O(N2) to O(1). We perform numerical simulations for Erdős-Rényi networks and for modified Barabási-Albert networks demonstrating remarkable quantitative agreement at and close to bifurcation.

Effect of diluted connectivities on cluster synchronization of adaptively coupled oscillator networks

Synchronization in networks of oscillatory units is an emergent phenomenon that has been observed in various systems, from power grids to ensembles of nerve cells. Many real-world networks have adaptive properties, meaning that their connectivities change with time, depending on the dynamical state of the system. Networks of adaptively coupled oscillators show various synchronization phenomena, such as hierarchical multifrequency clusters, traveling waves, or chimera states. While these self-organized patterns have been previously studied on all-to-all coupled networks, this work extends the investigations towards more complex networks, analyzing the influence of random network topologies for various degrees of dilution of the connectivities. Using numerical and analytical approaches, we investigate the robustness of multicluster states on networks of adaptively coupled Kuramoto-Sakaguchi oscillators against the random dilution of the underlying network topology. We utilize the master stability approach for adaptive networks in order to highlight the interplay between adaptivity and topology. With this, we show the robustness of multifrequency cluster states to diluted connectivities.

Hubs, diversity, and synchronization in FitzHugh-Nagumo oscillator networks: Resonance effects and biophysical implications

Using the FitzHugh-Nagumo equations to represent the oscillatory electrical behavior of β-cells, we develop a coupled oscillator network model with cubic lattice topology, showing that the emergence of pacemakers or hubs in the system can be viewed as a natural consequence of oscillator population diversity. The optimal hub to nonhub ratio is determined by the position of the diversity-induced resonance maximum for a given set of FitzHugh-Nagumo equation parameters and is predicted by the model to be in a range that is fully consistent with experimental observations. The model also suggests that hubs in a β-cell network should have the ability to "switch on" and "off" their pacemaker function. As a consequence, their relative amount in the population can vary in order to ensure an optimal oscillatory performance of the network in response to environmental changes, such as variations of an external stimulus.

Collective Synchronous Spiking in a Brain Network of Coupled Nonlinear Oscillators.

A network of propagating nonlinear oscillatory modes (waves) in the human brain is shown to generate collectively synchronized spiking activity (hypersynchronous spiking) when both amplitude and phase coupling between modes are taken into account. The nonlinear behavior of the modes participating in the network are the result of the nonresonant dynamics of weakly evanescent cortical waves that, as shown recently, adhere to an inverse frequency-wave number dispersion relation when propagating through an inhomogeneous anisotropic media characteristic of the brain cortex. This description provides a missing link between simplistic models of synchronization in networks of small amplitude phase coupled oscillators and in networks built with various empirically fitted models of pulse or amplitude coupled spiking neurons. Overall the phase-amplitude coupling mechanism presented in the Letter shows significantly more efficient synchronization compared to current standard approaches and demonstrates an emergence of collective synchronized spiking from subthreshold oscillations that neither phase nor amplitude coupling alone are capable of explaining.

A new approach to bifurcations in the Kuramoto model

The Kuramoto model is a standard model for the dynamics of coupled oscillator networks. In particular, it is used to study long time behavior such as phase-locking where all oscillators rotate at a common frequency with fixed angle differences. It has been observed, and been proved rigorously, that phase-locking occurs if the coupling between oscillators is sufficiently strong i.e. is larger than a certain threshold value called the critical coupling. However, if the coupling is below the critical coupling, then many interesting dynamics can occur. In this paper, we study bifurcations in the Kuramoto model on a ring network using a novel vector flow. We derive criteria for certain bifurcations, including bifurcations involving stable phase-locked solutions, and construct concrete examples of arbitrary size. We also note that our examples can be used to demonstrate bistability in the Kuramoto model in some cases.

Cardiac Muscle Cell‐Based Coupled Oscillator Network for Collective Computing

Biocomputing Biocomputing is an emerging field aimed at creating computers from biological materials. In article number 2000253, Pinar Zorlutuna and co-workers present a biocomputing system constructed using heart muscle cells to natively solve a prototypical constrained optimization problem. A biological computing fabric is created by micropatterning a coupled network of oscillating cardiac cells, which is used to solve a 3-node graph coloring problem.

Phase Drift in Networks of Coupled Colpitts Oscillators

In modern times, satellite-based global positioning and navigation systems, such as the GPS, include precise time-keeping devices, e.g. atomic clocks, which are crucial for navigation and for a wide range of economic and industrial applications. However, precise timing might not be available when the environment renders satellite equipment inoperable. In response to this critical need, we have been carrying out, over the past three years, theory and preliminary experiments [Buono et al., 2018a; Buono et al., 2018b; Palacios et al., 2020], towards developing a novel and inexpensive precision timing device that can function independently of GPS availability. The fundamental idea is to exploit collective behavior generated by networks of coupled nonlinear oscillators. Common sense may suggest that synchronized oscillations may lead to higher accuracy. Previous works show, however, that it is not synchronization but rather, traveling wave patterns, in which consecutive oscillators are out of phase by a constant amount, that can better reduce the negative effects of noise and material imperfections which cause phase drift. In this work we advance the state-of-art in the network-based concept by studying, mainly computationally, collective behavior in networks of Colpitts oscillators. These type of oscillators are chosen because they offer a wide range of advantages (such as the ability to tune up the oscillations over a broad frequency range). The results highlight the regions of parameter space, including coupling strength, where traveling wave patterns have the largest basins of attraction and the ability to reduce phase drift by a [Formula: see text] scaling law, where [Formula: see text] is the number of oscillators in the network. The results should also provide guidelines for follow-up design and fabrication tasks of a network-based technology for precision timing.

A Probabilistic Compute Fabric Based on Coupled Ring Oscillators for Solving Combinatorial Optimization Problems

Nondeterministic polynomial time hard (NP-hard) combinatorial optimization problems (COPs) are intractable to solve using a traditional computer as the time to find a solution increases very rapidly with the number of variables. An efficient alternative computing method uses coupled spin networks to solve COP. This work presents a first-of-its-kind coupled ring oscillator (ROSC)-based scalable probabilistic Ising computer to solve NP-hard COPs. An integrated coupled oscillator network was designed with 560 ROSCs that mimic a coupled spin network. Each ROSC can be coupled to any of its neighbors using programmable back-to-back (B2B) inverter-based coupling mechanism. The ROSC-based spins and B2B inverter-based coupling were optimized to work under a wide range of system noise as well as voltage and temperature variations. Randomly generated 1000 max-cut problems were mapped and solved in the hardware. The integrated Ising computer produced satisfactory solutions of max-cut problems when compared with commercial software running on a CPU. Experiments show that the integrated CMOS-based Ising computer can find the solution to NP-hard problems with an accuracy of 82%-100%. In addition, the repeated measurements of the same problem showed that the Ising computer can traverse through several local minima to find high-quality solutions under various voltage and temperature variation conditions. The experimental results show that ROSCs are a potential candidate for a dedicated hardware accelerator aiming to solve a wide range of COPs.

Cardiac Muscle Cell‐Based Coupled Oscillator Network for Collective Computing

Current rate of data generation and the need for real‐time data analytics can benefit from new computational approaches where computation proceeds in a massively parallel way while being scalable and energy efficient. Biological systems arising from interaction of living cells can provide such pathways for sustainable computing. Current designs for biocomputing leveraging the information processing units of the cells, such as DNA, gene, or protein circuitries, are inherently slow (hours to days speed) and, therefore, are primarily being considered for archival storage of information. On the contrary, electrically active cells that can synchronize in milliseconds and can be connected as networks to perform massively parallel tasks can transform biocomputing and lead to novel ways of high throughput information processing. Herein, coupled oscillator networks made of living cardiac muscle cells, or bio‐oscillators, is explored as collective computing components for solving computationally hard problems. An empirically validated circuit compatible macromodel is developed for the bio‐oscillators and the fibroblast cells acting as coupling elements, to faithfully reproduce the synchronization dynamics of the network and it is shown that such bio‐oscillator network can be scaled up to hundreds of nodes and be used to solve computationally hard problems faster than traditional heuristics‐based Boolean algorithms.

Experimental Investigation of the Dynamics of Coupled Oscillators as Ising Machines

Coupled electronic oscillator networks, under second harmonic injection, have recently been shown to behave as Ising machines capable of solving computationally hard combinatorial optimization problems. In this work, we experimentally investigate the dynamical properties of a reconfigurable network of up to 30 oscillators ( $\equiv $ spins) configured as an Ising machine. Specifically, we analyze the characteristics of the solutions to the Ising model produced by the oscillators and show that as the system evolves towards the ground state through the high-dimensional phase space, it gets trapped in local minima resulting in sub-optimal solutions. Moreover, the exact local minima where the system gets trapped also changes implying that the trajectory of evolution of the system also changes with each trial. Finally, we illustrate experimentally how an appropriately designed annealing scheme can help the coupled oscillators escape a local minimum and attain a lower energy state.

Small-world network structure mimics spontaneous synchronization in epileptic seizures

Model using complex coupled oscillator networks show the impact of neural network structure in seizure-related synchronization.