Networks Of Coupled Oscillators Research Articles

Strong coupling yields abrupt synchronization transitions in coupled oscillators

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions—such as synchrony and rotating wave solutions—following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation—or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. In particular, we show that such conditions can be understood by considering the relative contributions of higher harmonics to the phase dynamics. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system. Published by the American Physical Society 2024

Aging transitions of multimodal oscillators in multilayer networks

When individual oscillators age and become inactive, the collective dynamics of coupled oscillators is often affected as well. Depending on the fraction of inactive oscillators or cascading failures that percolate from crucial information exchange points, the critical shift toward macroscopic inactivity in coupled oscillator networks is known as the aging transition. Here, we study this phenomenon in two overlayed square lattices that together constitute a multilayer network, whereby one layer is populated with slow Poincaré oscillators and the other with fast Rulkov neurons. Moreover, in this multimodal setup, the excitability of fast oscillators is influenced by the phase of slow oscillators that are gradually inactivated toward the aging transition in the fast layer. Through extensive numerical simulations, we find that the progressive inactivation of oscillators in the slow layer nontrivially affects the collective oscillatory activity and the aging transitions in the fast layer. Most counterintuitively, we show that it is possible for the intensity of oscillatory activity in the fast layer to progressively increase to up to 100%, even when up to 60% of units in the slow oscillatory layer are inactivated. We explain our results with a numerical analysis of collective behavior in individual layers, and we discuss their implications for biological systems. Published by the American Physical Society 2024

An integrated coupled oscillator network to solve optimization problems

Solving combinatorial optimization problems is essential in scientific, technological, and engineering applications, but can be very time and energy-consuming using classical algorithms executed on digital processors. Oscillator-based Ising machines offer a promising alternative by exploiting the analog coupling between electrical oscillators to solve such optimization problems more efficiently. Here we present the design and the capabilities of our scalable approach to solve Ising and quadratic unconstrained binary optimization problems. This approach includes routable oscillator connections to simplify the time-consuming embedding of the problem into the oscillator network. Our manufactured silicon chip, featuring 1440 oscillators implemented in a 28 nm technology, demonstrates the ability to solve optimization problems in 950 ns while consuming typically 319 μW per node. A frequency, phase, and delay calibration ensures robustness against manufacturing variations. The system is evaluated with multiple sets of benchmark problems to analyze the sensitivity for parameters such as the coupling strength or frequency.

Higher-Order Network Interactions Through Phase Reduction for Oscillators with Phase-Dependent Amplitude

Coupled oscillator networks provide mathematical models for interacting periodic processes. If the coupling is weak, phase reduction—the reduction of the dynamics onto an invariant torus—captures the emergence of collective dynamical phenomena, such as synchronization. While a first-order approximation of the dynamics on the torus may be appropriate in some situations, higher-order phase reductions become necessary, for example, when the coupling strength increases. However, these are generally hard to compute and thus they have only been derived in special cases: This includes globally coupled Stuart–Landau oscillators, where the limit cycle of the uncoupled nonlinear oscillator is circular as the amplitude is independent of the phase. We go beyond this restriction and derive second-order phase reductions for coupled oscillators for arbitrary networks of coupled nonlinear oscillators with phase-dependent amplitude, a scenario more reminiscent of real-world oscillations. We analyze how the deformation of the limit cycle affects the stability of important dynamical states, such as full synchrony and splay states. By identifying higher-order phase interaction terms with hyperedges of a hypergraph, we obtain natural classes of coupled phase oscillator dynamics on hypergraphs that adequately capture the dynamics of coupled limit cycle oscillators.

Multi-type synchronization for coupled van der Pol oscillator systems with multiple coupling modes.

In this paper, we investigate synchronous solutions of coupled van der Pol oscillator systems with multiple coupling modes using the theory of rotating periodic solutions. Multiple coupling modes refer to two or three types of coupling modes in van der Pol oscillator networks, namely, position, velocity, and acceleration. Rotating periodic solutions can represent various types of synchronous solutions corresponding to different phase differences of coupled oscillators. When matrices representing the topology of different coupling modes have symmetry, the overall symmetry of the oscillator system depends on the intersection of the symmetries of the different topologies, determining the type of synchronous solutions for the coupled oscillator network. When matrices representing the topology of different coupling modes lack symmetry, if the adjacency matrices representing different coupling modes can be simplified into structurally identical quotient graphs (where weights can be proportional) through the same external equitable partition, the symmetry of the quotient graph determines the synchronization type of the original system. All these results are consistent with multi-layer networks where connections between different layers are one-to-one.

Synchronization through frequency shuffling.

A wide variety of engineered and natural systems are modeled as networks of coupled nonlinear oscillators. In nature, the intrinsic frequencies of these oscillators are not constant in time. Here, we probe the effect of such a temporal heterogeneity on coupled oscillator networks through the lens of the Kuramoto model. To do this, we shuffle repeatedly the intrinsic frequencies among the oscillators at either random or regular time intervals. What emerges is the remarkable effect that frequent shuffling induces earlier onset (i.e., at a lower coupling) of synchrony among the oscillator phases. Our study provides a novel strategy to induce and control synchrony under resource constraints. We demonstrate our results analytically and in experiments with a network of Wien Bridge oscillators with internal frequencies being shuffled in time.

Distributed Network of Coupled Oscillators With Multiple Limit Cycles

Distributed Network of Coupled Oscillators With Multiple Limit Cycles

Ultralow Power Electronic Analog of a Biological Fitzhugh-Nagumo Neuron.

Here, we introduce an electronic circuit that mimics the functionality of a biological spiking neuron following the Fitzhugh-Nagumo (FN) model. The circuit consists of a tunnel diode that exhibits negative differential resistance (NDR) and an active inductive element implemented by a single MOSFET. The FN neuron converts a DC voltage excitation into voltage spikes analogous to biological action potentials. We predict an energy cost of 2 aJ/cycle through detailed simulation and modeling for these FN neurons. Such an FN neuron is CMOS compatible and enables ultralow power oscillatory and spiking neural network hardware. We demonstrate that FN neurons can be used for oscillator-based computing in a coupled oscillator network to form an oscillator Ising machine (OIM) that can solve computationally hard NP-complete max-cut problems while showing robustness toward process variations.

Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions.

We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.

Coherent olfactory bulb gamma oscillations arise from coupling independent columnar oscillators.

Spike timing-based representations of sensory information depend on embedded dynamical frameworks within neuronal networks that establish the rules of local computation and interareal communication. Here, we investigated the dynamical properties of olfactory bulb circuitry in mice of both sexes using microelectrode array recordings from slice and in vivo preparations. Neurochemical activation or optogenetic stimulation of sensory afferents evoked persistent gamma oscillations in the local field potential. These oscillations arose from slower, GABA(A) receptor-independent intracolumnar oscillators coupled by GABA(A)-ergic synapses into a faster, broadly coherent network oscillation. Consistent with the theoretical properties of coupled-oscillator networks, the spatial extent of zero-phase coherence was bounded in slices by the reduced density of lateral interactions. The intact in vivo network, however, exhibited long-range lateral interactions that suffice in simulation to enable zero-phase gamma coherence across the olfactory bulb. The timing of action potentials in a subset of principal neurons was phase-constrained with respect to evoked gamma oscillations. Coupled-oscillator dynamics in olfactory bulb thereby enable a common clock, robust to biological heterogeneities, that is capable of supporting gamma-band spike synchronization and phase coding across the ensemble of activated principal neurons.NEW & NOTEWORTHY Odor stimulation evokes rhythmic gamma oscillations in the field potential of the olfactory bulb, but the dynamical mechanisms governing these oscillations have remained unclear. Establishing these mechanisms is important as they determine the biophysical capacities of the bulbar circuit to, for example, maintain zero-phase coherence across a spatially extended network, or coordinate the timing of action potentials in principal neurons. These properties in turn constrain and suggest hypotheses of sensory coding.

Event-Triggered Pinning Impulsive Control for Cluster Synchronization of Coupled Genetic Oscillator Networks With Proportional Delay

Event-Triggered Pinning Impulsive Control for Cluster Synchronization of Coupled Genetic Oscillator Networks With Proportional Delay

Associative memory by virtual oscillator network based on single spin-torque oscillator

A coupled oscillator network may be able to perform an energy-efficient associative memory operation. However, its realization has been difficult because inhomogeneities unavoidably arise among the oscillators during fabrication and lead to an unreliable operation. This issue could be resolved if the oscillator network were able to be formed from a single oscillator. Here, we performed numerical simulations and theoretical analyses on an associative memory operation that uses a virtual oscillator network based on a spin-torque oscillator. The virtual network combines the concept of coupled oscillators with that of feedforward neural networks. Numerical experiments demonstrate successful associations of 60-pixel patterns with various memorized patterns. Moreover, the origin of the associative memory is shown to be forced synchronization driven by feedforward input, where phase differences among oscillators are fixed and correspond to the colors of the pixels in the pattern.

Numerical Method for Finding Synchronous Solutions of the Coupled Oscillator Networks

Numerical Method for Finding Synchronous Solutions of the Coupled Oscillator Networks

Superconducting-Oscillatory Neural Network With Pixel Error Detection for Image Recognition

Brain-inspired oscillatory neural networks (ONNs) utilize coupled oscillators to emulate biological neuronal dynamics. ONNs are naturally suitable for image/pattern recognition. Many hardware implementations of ONNs have been explored recently by using analog or digital CMOS systems, which can be limited by Moore's law. In this work, we used one of the advanced superconducting technologies for coupled oscillator networks to further improve the energy efficiency and processing speed. Inductively coupled ring oscillators using rapid single flux quantum (RSFQ) technology were designed and simulated. These Josephson junction (JJ) based oscillators can operate as fast as tens of GHz but only consume a energy of several aJ per operation. Excluding cryo-cooling factors, the estimated power consumption of our RSFQ oscillator is hundreds of nW at a frequency of 12.5 GHz, which is multiple times smaller than a typical CMOS ring oscillator. Furthermore, a system for pattern recognition with error detection was designed based on these oscillators. Synchronization dynamics in a pair of coupled oscillators were used to identify whether a test pixel matches a reference pixel. A network comprised of 16 pairs of oscillators were wired together, along with synchronization detectors and a fluxon integrator to demonstrate pattern recognition function. This work was performed using standard JJ models and demonstrated through WRspice and JoSIM software simulations. The circuits and systems proposed in this work aim to provide insights into the use of superconducting technology for implementing ONNs and may serve as basic building blocks for the design of more complex oscillatory computational networks in image processing and beyond. Single oscillator modeling and systematic network function exploration will be performed in future work.

Review of recent progress on solid‐state millimeter‐wave and terahertz signal sources

SummaryThe research and development on solid‐state source generators have been making progress on chip, aiming to generate low noise, high output power, and high DC‐to‐RF energy efficiency in the millimeter‐wave (mmW) and terahertz (THz) domains. In particular, their full integration in silicon technologies, such as advanced complementary‐metal‐oxide‐semiconductor processes, has motivated the design of low cost, high performance, and miniature integrated circuits (ICs). This manuscript focuses on providing a general overview of recent progress on solid‐state signal source operating beyond 60 GHz till the THz region, from the perspective of operating principle, design metrics, state‐of‐the‐art performance, testing results, and their applications in emerging mmW/THz systems. Finally, with the use of fully integrated source generators, several design arts are highlighted to showcase the recent advances of ICs toward millimeter‐wave and THz radiation, imaging, coupled oscillator network (CON), and communication. Potential future research directions enabled by mmW/THz signal sources will be envisioned.

Stasis in heterogeneous networks of coupled oscillators: discontinuous transition with hysteresis

We consider a heterogeneous ensemble of dynamical systems in that individually are either attracted to fixed points (and are termed inactive) or to limit cycles (in which case they are termed active). These distinct states are separated by bifurcations that are controlled by a single parameter. Upon coupling them globally, we find a discontinuous transition to global inactivity (or stasis) when the proportion of inactive components in the ensemble exceeds a threshold: there is a first–order phase transition from a globally oscillatory state to global oscillation death. There is hysteresis associated with these phase transitions. Numerical results for a representative system are supported by analysis using a system-reduction technique and different dynamical regimes can be rationalised through the corresponding bifurcation diagrams of the reduced set of equations.

Continuum Limits of Coupled Oscillator Networks Depending on Multiple Sparse Graphs

The continuum limit provides a useful tool for analyzing coupled oscillator networks. Recently, Medvedev (Commun Math Sci 17(4):883–898, 2019) gave a mathematical foundation for such an approach when the networks are defined on single graphs which may be dense or sparse, directed or undirected, and deterministic or random. In this paper, we consider coupled oscillator networks depending on multiple graphs, and extend his results to show that the continuum limit is also valid in this situation. Specifically, we prove that the initial value problem (IVP) of the corresponding continuum limit has a unique solution under general conditions and that the solution becomes the limit of those to the IVP of the networks in some adequate meaning. Moreover, we show that if solutions to the networks are stable or asymptotically stable when the node number is sufficiently large, then so are the corresponding solutions to the continuum limit, and that if solutions to the continuum limit are asymptotically stable, then so are the corresponding solutions to the networks in some weak meaning as the node number tends to infinity. These results can also be applied to coupled oscillator networks with multiple frequencies by regarding the frequencies as a weight matrix of another graph. We illustrate the theory for three variants of the Kuramoto model along with numerical simulations.

Designing Robust Networks of Coupled Phase Oscillators With Applications to the High Voltage Electric Grid

We propose a mathematical framework for designing robust networks of coupled phase-oscillators by leveraging a vulnerability measure proposed by Tyloo et al. that quantifies the impact of a small perturbation at an individual phase-oscillator's natural frequency to the system's global synchronized frequencies. Given a complex network topology with specific governing dynamics, the proposed framework finds an optimal allocation of edge weights that minimizes such vulnerability measure(s) at the node(s) for which we expect perturbations to occur by solving a tractable semi-definite programming problem. We specify the mathematical model to high voltage electric grids where each node corresponds to a voltage phase angle associated with a bus and edges correspond to transmission lines. Edge weights are determined by the susceptance values along the transmission lines. In this application, frequency synchronization is increasingly challenged by the integration of renewable energy, yet is imperative to the grid's health and functionality. Our framework helps to alleviate this challenge by optimizing the placement of renewable generation and the susceptance values along the transmission lines.

Large‐Scale Cardiac Muscle Cell‐Based Coupled Oscillator Network for Vertex Coloring Problem

Modern computers require an exponential increase in resources when solving computationally hard problems, motivating the need for an alternative computing platform to solve such problems in an energy‐efficient manner. Vertex coloring, a nondeterministic polynomial time (NP‐hard) combinatorial optimization problem, is one such problem. Herein, an experimental demonstration of using cardiac cell‐based bio‐oscillator network coupling dynamics to solve a vertex coloring problem in various scales of graphs using a simple cell patterning method to construct scalable and controlled cardiac cell networks is presented. Although there are limitations to using these cardiac cells as oscillators, such as their low frequency compared to complementary metal–oxide–semiconductor (CMOS) oscillators, that result in longer processing times, the accuracy in large graph instances, the significantly less amount of energy consumption, and the ease of fabrication and potential to extend this system to massively parallel 3D structures make the bio‐oscillators a promising new platform for collective computing applications.

Dual Control of Coupled Oscillator Networks

Robust coordination and organization in large ensembles of nonlinear oscillatory units play a vital role in a wide range of natural and engineered system. The control of self-organizing network-coupled systems has recently seen significant attention, but largely in the context of modifying or augmenting existing structures. This leaves a gap in our understanding of reactive control, where and how to design direct interventions, and what we may learn about structure and dynamics from such control strategies. Here we study reactive control of coupled oscillator networks and demonstrate dual control strategies, i.e., two different mechanisms for control, that may each be implemented on their own and interchangeably to achieve synchronization. These diverse strategies exploit different network properties, with the first directly targeting oscillators that are challenging to entrain, and the second focusing on oscillators with a strong influence on others. Thus, in addition to presenting alternative strategies for network control, the distinct control sets illuminate the oscillators' dynamical and structural roles within the system. The applicability of dual control is demonstrated using both synthetic and real networks.