Binary system modes of matrix-coupled multidimensional oscillators

This paper presents the results of a study of synchronization processes in oscillatory neural networks of various structures based on the Kuramoto model. The estimates of synchronization processes occurring in the oscillatory networks are examined. The results of studying the practical application of oscillatory networks for solving cluster analysis problems are presented.

Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood. Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored. In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2]. Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.

The growing number of edge devices in everyday life generates a considerable amount of data that current AI algorithms, like artificial neural networks, cannot handle inside edge devices with limited bandwidth, memory, and energy available. Neuromorphic computing, with low-power oscillatory neural networks (ONNs), is an alternative and attractive solution to solve complex problems at the edge. However, ONN is currently limited with its fully-connected recurrent architecture to solve auto-associative memory problems. In this work, we use an alternative two-layer bidirectional ONN architecture. We introduce a two-layer feedforward ONN architecture to perform image edge detection, using the ONN to replace convolutional filters to scan the image. Using an HNN Matlab emulator and digital ONN design simulations, we report efficient image edge detection from both architectures using various size filters (3 × 3, 5 × 5, and 7 × 7) on black and white images. In contrast, the feedforward architectures can also perform image edge detection on gray scale images. With the digital ONN design, we also assess latency performances and obtain that the bidirectional architecture with a 3 × 3 filter size can perform image edge detection in real-time (camera flow from 25 to 30 images per second) on images with up to 128 × 128 pixels while the feedforward architecture with same 3 × 3 filter size can deal with 170 × 170 pixels, due to its faster computation.

The Kuramoto model in complex networks

Development of neuromorphic systems based on new nanoelectronics materials and devices is of immediate interest for solving the problems of cognitive technology and cybernetics. Computational modeling of two- and three-oscillator schemes with thermally coupled VO2-switches is used to demonstrate a novel method of pattern storage and recognition in an impulse oscillator neural network (ONN), based on the high-order synchronization effect. The method allows storage of many patterns, and their number depends on the number of synchronous states Ns. The modeling demonstrates attainment of Ns of several orders both for a three-oscillator scheme Ns ~ 650 and for a two-oscillator scheme Ns ~ 260. A number of regularities are obtained, in particular, an optimal strength of oscillator coupling is revealed when Ns has a maximum. Algorithms of vector storage, network training, and test vector recognition are suggested, where the parameter of synchronization effectiveness is used as a degree of match. It is shown that, to reduce the ambiguity of recognition, the number coordinated in each vector should be at least one unit less than the number of oscillators. The demonstrated results are of a general character, and they may be applied in ONNs with various mechanisms and oscillator coupling topology.

Contents: Preface. Part I: Oscillations in Single Neurons and Local Networks. G. Gross, J.M. Kowalski, B.K. Rhoades, Spontaneous and Evoked Oscillations in Cultured Mammalian Neuronal Networks. D.C. Tam, Detection of Oscillations and Synchronous Firing in Neurons. D.A. Baxter, H.A. Lechner, C.C. Canavier, R.J. Butera, Jr., A.A. DeFraneschi, J.W. Clark, Jr., J.H. Byrne, Coexisting Stable Oscillatory States in Single Cells and Multicellular Neuronal Oscillators. M. Stemmler, M. Usher, C. Koch, Oscillatory Local Field Potentials. G.J. Mpitsos, J.P. Edstrom, Computations Neurons Perform in Networks: Inside Versus Outside and Lessons Learned From a Sixteenth-Century Shoemaker. Part II: Oscillations in Cortical and Cortical/Subcortical Systems. I. Soltesz, The Interplay of Intrinsic and Synaptic Membrane Currents in Delta, Theta and 40-Hz Oscillations. E. Thomas, Dynamics of Low-Frequency Oscillations in a Model Thalamocortical Network. M.E. Jackson, L.J. Cauller, Towards the Function of Reciprocal Corticocortical Connections: Computational Modeling and Electrophysiological Studies. D. Young, An Oscillatory Model of Cortical Neural Processing. G.L. Yuen, Response Synchrony, APG Theory, and Motor Control. Part III: Oscillatory Models in Perception, Memory, and Cognition. D. Horn, I. Opher, Temporal Segmentation and Binding in Oscillatory Neural Systems. A. Grunewald, S. Grossberg, Perceptual Framing and Cortical Synchronization. D. DeMaris, Attention, Depth Gestalts, and Spatially Extended Chaos in the Perception of Ambiguous Figures. G. Borisyuk, R. Borisyuk, Y. Kazanovich, G. Strong, Oscillatory Neural Networks: Modeling Binding and Attention by Synchronization of Neural Activity. B. Baird, T. Troyer, F. Eeckman, Attentional Network Streams of Synchronized 40-Hz Activity in a Cortical Architecture of Coupled Oscillatory Associative Memories. Part IV: Applications of Synchronized and Chaotic Oscillations. G.E. Mobus, P.S. Fisher, Foraging Search at the Edge of Chaos. A.G. Brown, S. Collins, An Oscillatory Associative Memory Analogue Architecture. N.S. Park, D. Robertson, K. Stenning, Symbolic Knowledge Encoding Using a Dynamic Binding Mechanism and an Embedded Inference Mechanism. A. Jagota, X. Wang, Oscillations in Discrete and Continuous Hopfield Networks. S. Wolpert, Modeling Neural Oscillation Using VLSI-Based Neuromimes.

Alternative paradigms to the von Neumann computing scheme are currently arousing huge interest. Oscillatory neural networks (ONNs) using emerging phase-change materials like VO2 constitute an energy-efficient, massively parallel, brain-inspired, in-memory computing approach. The encoding of information in the phase pattern of frequency-locked, weakly coupled oscillators makes it possible to exploit their rich non-linear dynamics and their synchronization phenomena for computing. A single fully connected ONN layer can implement an auto-associative memory comparable to that of a Hopfield network, hence Hebbian learning rule is the most widely adopted method for configuring ONNs for such applications, despite its well-known limitations. An extensive amount of literature is available about learning in Hopfield networks, with information regarding many different learning algorithms that perform better than the Hebbian rule. However, not all of these algorithms are useful for ONN training due to the constraints imposed by their physical implementation. This paper evaluates different learning methods with respect to their suitability for ONNs. It proposes a new approach, which is compared against previous works. The proposed method has been shown to produce competitive results in terms of pattern recognition accuracy with reduced precision in synaptic weights, and to be suitable for online learning.

A scenario frequently encountered in real-world complex systems is the temporary failure of a few components. For spatially extended chaotic systems whose functionality hinges on the collective dynamics of the interacting components, a viable approach to dealing with the failures is replacing the malfunctioning components with their backups, so that the collective dynamics of the systems can be precisely maintained for a duration long enough to resolve the problem. Here, taking the chaotic dynamics of the paradigmatic Kuramoto model as an example and considering the scenario of oscillator failures, we propose substituting the failed oscillators with digital twins trained by the data measured from the system evolution. Specifically, leveraging the technique of an adaptable reservoir computer (RC) in machine learning, we demonstrate that a single, small-size RC can substitute any oscillator in the Kuramoto model such that the time evolution of the synchronization order parameter of the repaired system is identical to that of the original system for a certain time period. The performance of adaptable RC is evaluated in various contexts, and it is found that the sustaining period is influenced by multiple factors, including the size of the training dataset, the overall coupling strength of the system, and the number of substituted oscillators. Additionally, though the synchronization order parameter diverges from the ground truth in the long-term running, the functional networks of the oscillators are still faithfully sustained by the machine substitutions.

We devised a new method of data mining for a large-scale database. In the method, a network of locally coupled phase oscillators subject to Kuramoto's model substitutes for given multivariate data to generate major features through phase locking of the oscillators, i.e., phase transition of the data set. We applied the method to the national database of care needs certification for the Japanese public long-term care insurance program, and found three major patterns in the aging process of the frail elderly. This work revealed the latent utility of Kuramoto's model for data processing.

Introduction. Recent studies into the properties of spintronic oscillators have led to broadening their scope of practical application as devices for generating and processing signals. The practical implementation of spintronic oscillators is, however, significantly limited by their low power capacity, thus requiring synchronization between devices.Aim. Determination of conditions for the implementation of the synchronous regime of two antiferromagnetic spintronic oscillators coupled by a common current.Materials and methods. To simplify the numerical simulation of a system of coupled resistively antiferromagnetic oscillators, the method of multiple-time-scale analysis was used. This allowed a system of Kuramoto equations to be considered instead of the original system. To determine the locking band of the Kuramoto model, the homoclinic trajectory approximation method was applied.Results. A system of Kuramoto equation for the phases of partial oscillators under the influence of the inertial term and phase shift was obtained. Expressions describing the locking and synchronization band as functions of the system parameters (bias currents and sizes) were derived. The numerically simulated Kuramoto model was used to determine the bands of the synchronous and asynchronous regimes.Conclusion. The results of numerical simulations of the system of Kuramoto equations and the Adler equation for two coupled spintronic oscillators agree well with the theoretically calculated values of locking and synchronization ranges. The scheme for reducing the model of antiferromagnetic oscillators to a Kuramoto model can be further extended to the case of a larger number of coupled oscillators, which will simplify computational experiments and significantly reduce the time required for numerical simulations.

Effective desynchronization can be exploited as a tool for probing the functional significance of synchronized neural activity underlying perceptual and cognitive processes or as a mild treatment for neurological disorders like Parkinson's disease. In this article we show that pulse-based desynchronization techniques, originally developed for networks of globally coupled oscillators (Kuramoto model), can be adapted to networks of coupled neurons with dendritic dynamics. Compared to the Kuramoto model, the dendritic dynamics significantly alters the response of the neuron to the stimulation. Under medium stimulation amplitude a bistability of the response of a single neuron is observed. When stimulated at some initial phases, the neuron displays only modulations of its firing, whereas at other initial phases it stops oscillating entirely. Significant alterations in the duration of stimulation-induced transients are also observed. These transients endure after the end of the stimulation and cause maximal desynchronization to occur not during the stimulation, but with some delay after the stimulation has been turned off. To account for this delayed desynchronization effect, we have designed a new calibration procedure for finding the stimulation parameters that result in optimal desynchronization. We have also developed a new desynchronization technique by low frequency entrainment. The stimulation techniques originally developed for the Kuramoto model, when using the new calibration procedure, can also be applied to networks with dendritic dynamics. However, the mechanism by which desynchronization is achieved is substantially different than for the network of Kuramoto oscillators. In particular, the addition of dendritic dynamics significantly changes the timing of the stimulation required to obtain desynchronization. We propose desynchronization stimulation for experimental analysis of synchronized neural processes and for the therapy of movement disorders.

Symbol decoding in multiple-input multiple-output (MIMO) wireless communication systems requires the deployment of fast, energy-efficient computing hardware deployable at the edge. The brute-force and exact maximum likelihood (ML) decoder, solved on conventional classical digital hardware to decode MIMO symbols, has exponential time complexity. Approximate classical solvers implemented on the same hardware have polynomial time complexity at the best. In this article, we design an alternative ring-oscillator-based coupled oscillator array (also known as oscillatory neural network (ONN)) to act as an oscillator Ising machine (OIM) and heuristically solve the ML-based MIMO detection problem. Complementary metal oxide semiconductor (CMOS) technology is used to design the ring oscillators, and ferroelectric field effect transistor (FeFET) technology is chosen as the non-volatile memory (NVM) coupling element (X) between the oscillators in this CMOS + X OIM design. For this purpose, we experimentally report high linear range of conductance variation (1 µS to 60 µS) with programming voltage pulses in a HfO 2 -based FeFET device fabricated at 28 nm high-K/ metal gate (HKMG) CMOS technology node. We incorporate the conductance modulation characteristic in SPICE simulation of the ring oscillators connected in an all-to-all fashion through a crossbar array of these FeFET devices. We show that the above range of conductance variation of FeFET is suitable to obtain best OIM performance, thereby making FeFET a suitable NVM device for this application. Our SPICE simulations show that there is no significant performance drop for symbol detection up to MIMO array sizes of 90 transmitting and 90 receiving antennas. Our simulations, combined with analytical treatment using Kuramoto model of oscillators, predict that this designed classical analog OIM, if implemented experimentally, will offer logarithmic scaling of computation time with MIMO size, thereby offering huge improvement (in terms of computation speed) over exact and approximate classical solvers run on conventional digital hardware.

This paper demonstrates a coupled Schmitt trigger oscillator based oscillator neural network (SMT-ONN) for pattern recognition applications. Unlike previous ONN models, the SMT-ONN can be easily realized in both hardware and software levels. A mathematical model of the Schmitt Trigger Oscillator as well as the corresponding CMOS circuit are presented to validate the mathematical model. The SMT-ONN can realize the pattern recognition task by considering the convergence time and frequency as the recognition indicators. A Kuramoto model based frequency synchronization approach is utilized, and simulation results indicate less than 160 ms convergence time and close frequency match for a simplified pattern recognition application.

In the classic Kuramoto system of coupled two-dimensional rotators, chimera states characterized by the coexistence of synchronous and asynchronous groups of oscillators are long-lived because the average lifetime of these states increases exponentially with the system size. Recently, it was discovered that, when the rotators in the Kuramoto model are three-dimensional, the chimera states become short-lived in the sense that their lifetime scales with only the logarithm of the dimension-augmenting perturbation. We introduce transverse-stability analysis to understand the short-lived chimera states. In particular, on the unit sphere representing three-dimensional (3D) rotations, the long-lived chimera states in the classic Kuramoto system occur on the equator, to which latitudinal perturbations that make the rotations 3D are transverse. We demonstrate that the largest transverse Lyapunov exponent calculated with respect to these long-lived chimera states is typically positive, making them short-lived. The transverse-stability analysis turns the previous numerical scaling law of the transient lifetime into an exact formula: the "free" proportional constant in the original scaling law can now be precisely determined in terms of the largest transverse Lyapunov exponent. Our analysis reinforces the speculation that in physical systems, chimera states can be short-lived as they are vulnerable to any perturbations that have a component transverse to the invariant subspace in which they live.

Oscillatory Neural Networks (ONNs) are an emerging computing paradigm that encodes information in the phase difference of coupled oscillators. Traditionally, ONNs have been investigated using homogeneous frequency oscillators. However, physical hardware implementations are inherently subject to frequency mismatches, device variability, and nonuniformities, which are typically perceived as undesirable for circuit performance. This work challenges this assumption by inquiring whether frequency discrepancies and multifrequency oscillators can be harnessed for performing computation. First, the frequency variation threshold that oscillators can withstand to compute is derived. Next, for frequency variations beyond such a threshold, a multifrequency ONN computing approach is introduced and validated in both hardware implementation and software simulations by performing pattern retrieval. The results demonstrate computation with nonuniform frequency oscillators and highlight the feasibility of their physical hardware implementation.