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In this work, we experimentally demonstrate an integrated circuit (IC) of 30 relaxation oscillators with reconfigurable capacitive coupling to solve the NP-Hard maximum cut (Max-Cut) problem. We show that under the influence of an external second-harmonic injection signal, the oscillator phases exhibit a bipartition that can be used to calculate a high-quality approximate Max-Cut solution. Leveraging the all-to-all reconfigurable coupling architecture, we experimentally evaluate the computational properties of the oscillators using randomly generated graph instances of varying size and edge density ( $\eta $ ). Furthermore, comparing the Max-Cut solutions with the optimal values, we show that the oscillators (after simple postprocessing) produce a Max-Cut that is within 99% of the optimal value in 28 of the 36 measured graphs; importantly, the oscillators are particularly effective in dense graphs with the Max-Cut being optimal in seven out of nine measured graphs with $\eta =0.8$ . Our work marks a step toward creating an efficient, room-temperature-compatible non-Boolean hardware-based solver for hard combinatorial optimization problems.

A pair of adjacent beating flagella can be synchronized via hydrodynamic interactions at low Reynolds numbers. Here, we investigate the factors that determine the relaxation time for elastohydrodynamic synchronization by using the phase reduction theory. The results show that the relaxation time is determined primarily by the natural frequency of a flagellum as well as the coupling intensity between the two flagella. After rescaling with both the natural frequency and the coupling intensity, the relaxation curves are found to be close to each other. This indicates that the number of beatings required for elastohydrodynamic synchronization remains almost unchanged regardless of the waveforms under a common coupling intensity.

We study the dynamical invariant for dissipative three coupled oscillators mainly from the quantum mechanical point of view. It is known that there are many advantages of the invariant quantity in elucidating mechanical properties of the system. We use such a property of the invariant operator in quantizing the system in this work. To this end, we first transform the invariant operator to a simple one by using a unitary operator in order that we can easily manage it. The invariant operator is further simplified through its diagonalization via three-dimensional rotations parameterized by three Euler angles. The coupling terms in the quantum invariant are eventually eliminated thanks to such a diagonalization. As a consequence, transformed quantum invariant is represented in terms of three independent simple harmonic oscillators which have unit masses. Starting from the wave functions in the transformed system, we have derived the full wave functions in the original system with the help of the unitary operators.

Winfree oscillators are phase oscillator models of neurons, characterized by their phase response curve and pulsatile interaction function. We use the Ott/Antonsen ansatz to study large heterogeneous networks of Winfree oscillators, deriving low-dimensional differential equations which describe the evolution of the expected state of networks of oscillators. We consider the effects of correlations between an oscillator's in-degree and out-degree, and between the in- and out-degrees of an “upstream” and a “downstream” oscillator (degree assortativity). We also consider correlated heterogeneity, where some property of an oscillator is correlated with a structural property such as degree. We finally consider networks with parameter assortativity, coupling oscillators according to their intrinsic frequencies. The results show how different types of network structure influence its overall dynamics.

In this work we present an in-memory computing platform based on coupled VO2 oscillators fabricated in a crossbar configuration on silicon. Compared to existing platforms, the crossbar configuration promises significant improvements in terms of area density and oscillation frequency. Further, the crossbar devices exhibit low variability and extended reliability, hence, enabling experiments on 4-coupled oscillator. We demonstrate the neuromorphic computing capabilities using the phase relation of the oscillators. As an application, we propose to replace digital filtering operation in a convolutional neural network with oscillating circuits. The concept is tested with a VGG13 architecture on the MNIST dataset, achieving performances of 95% in the recognition task.

The synchronization of coupled oscillators is a phenomenon found throughout nature. Mechanical oscillators are paradigmatic examples, but synchronizing their nanoscaled versions is challenging. We report synchronization of the mechanical dynamics of a pair of optomechanical crystal cavities that, in contrast to previous works performed in similar objects, are intercoupled with a mechanical link and support independent optical modes. In this regime they oscillate in antiphase, which is in agreement with the predictions of our numerical model that considers reactive coupling. We also show how to temporarily disable synchronization of the coupled system by actuating one of the cavities with a heating laser, so that both cavities oscillate independently. Our results can be upscaled to more than two cavities and pave the way towards realizing integrated networks of synchronized mechanical oscillators., This work was supported by the European Commission project PHENOMEN (H2020-EU-713450), the Spanish Severo Ochoa Excellence program, and the MINECO project PHENTOM (FIS2015-70862-P). D. N. U., G. A., and M. F. C. gratefully acknowledge the support of a Ramón y Cajal postdoctoral fellowship (RYC-2014-15392), a BIST studentship, and a Severo Ochoa studentship, respectively.

Fano resonances and Rabi splittings are routinely reported in the scientific literature. Asymmetric resonance lineshapes are usually associated with Fano resonances, and two split peaks in the spectrum are often attributed to a Rabi splitting. True Fano resonances and Rabi splittings are unequivocal signatures of coherent coupling between subsystems. However, can the same spectral lineshapes characterizing Fano resonances and Rabi splittings arise from a purely incoherent sum of intensities? Here we answer this question through experiments with a tunable Fabry-P\'erot cavity containing a CsPbBr\textsubscript{3} perovskite crystal. By measuring the transmission and photoluminescence of this system using microscope objectives with different numerical aperture ($NA$), we find that even a modest $NA=0.4$ can artificially generate Fano resonances and Rabi splittings. We furthermore show that this modest $NA$ can obscure the anti-crossing of a bona fide strongly coupled light-matter system. Through transfer matrix calculations we confirm that these spectral artefacts are due to the incoherent sum of transmitted intensities at different angles captured by the $NA$. Our results are relevant to the wide nanophotonics community, characterizing dispersive optical systems with high numerical aperture microscope objectives. We conclude with general guidelines to avoid pitfalls in the characterization of such optical systems., Comment: 6 pages, 5 figures

Cross-frequency coupling (CFC) refers to the nonlinear interaction between oscillations in different frequency bands, and it is a rather ubiquitous phenomenon that has been observed in a variety of physical and biophysical systems. In particular, the coupling between the phase of slow oscillations and the amplitude of fast oscillations, referred as phase-Amplitude coupling (PAC), has been intensively explored in the brain activity recorded from animals and humans. However, the interpretation of these CFC patterns remains challenging since harmonic spectral correlations characterizing nonsinusoidal oscillatory dynamics can act as a confounding factor. Specialized signal processing techniques are proposed to address the complex interplay between spectral harmonicity and different types of CFC, not restricted only to PAC. For this, we provide an in-depth characterization of the time locked index (TLI) as a tool aimed to efficiently quantify the harmonic content of noisy time series. It is shown that the proposed TLI measure is more robust and outperforms traditional phase coherence metrics (e.g., phase locking value, pairwise phase consistency) in several aspects. We found that a nonlinear oscillator under the effect of additive noise can produce spurious CFC with low spectral harmonic content. On the other hand, two coupled oscillatory dynamics with independent fundamental frequencies can produce true CFC with high spectral harmonic content via a rectification mechanism or other post-interaction nonlinear processing mechanisms. These results reveal a complex interplay between CFC and harmonicity emerging in the dynamics of biologically plausible neural network models and more generic nonlinear and parametric oscillators. We show that, contrary to what is usually assumed in the literature, the high harmonic content observed in nonsinusoidal oscillatory dynamics is neither a sufficient nor necessary condition to interpret the associated CFC patterns as epiphenomenal. There is mounting evidence suggesting that the combination of multimodal recordings, specialized signal processing techniques, and theoretical modeling is becoming a required step to completely understand CFC patterns observed in oscillatory rich dynamics of physical and biophysical systems. Fil: Dellavale Clara, Hector Damian. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Universidad Nacional de Cuyo; Argentina Fil: Velarde, Osvaldo Matias. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Universidad Nacional de Cuyo; Argentina Fil: Mato, German. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Universidad Nacional de Cuyo; Argentina Fil: Urdapilleta, Eugenio. Comisión Nacional de Energía Atómica. Centro Atómico Bariloche; Argentina. Comisión Nacional de Energía Atómica. Gerencia del Área de Energía Nuclear. Instituto Balseiro; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Patagonia Norte; Argentina. Universidad Nacional de Cuyo; Argentina

Whereas ergodic theories relate to limiting cases of infinite thermal reservoirs and infinitely long times, some ergodicity tendencies may appear also for finite reservoirs and time durations. These tendencies are here explored and found to exist, but only for extremely long times and very soft ergodic criteria. &ldquo, Weak ergodicity breaking&rdquo, is obviated by a judicious time-weighting, as found in a previous work [Found. Phys. (2015) 45: 673&ndash, 690]. The treatment is based on an N-oscillator (classical) and an N-spin (quantal) model. The showing of ergodicity is facilitated by pictorial presentations.

Precise spectroscopy of oscillating fields plays a significant role in many fields. Here, we propose an experimentally feasible scheme to measure the frequency of a fast-oscillating field using a single-qubit sensor. By invoking a stable classical clock, the signal phase correlations between successive measurements enable us to extract the target frequency with extremely high precision. In addition, we integrate dynamical decoupling technique into the framework to suppress the influence of slow environmental noise. Our framework is feasible with a variety of atomic and single solid-state spin systems within the state-of-the-art experimental capabilities as a versatile tool for quantum spectroscopy., acceptedVersion

We establish the similarity in two model-based reconstructions of the coupling between the polar magnetic fields of the Sun represented by the solar faculae time series. The reconstructions are inferred from the pair of the coupled oscillators modelled with the Van der Pol and Kuramoto equations. They are associated with the substantial simplification of solar dynamo models and, respectively, a simple ad hoc model reproducing the phenomenon of synchronization. While the polar fields are synchronized, both of the reconstruction procedures restore couplings, which attain moderate values and follow each other rather accurately as the functions of time. We also estimate the evolution of the phase difference between the polar fields and claim that they tend to move apart more quickly than approach each other.

A pulse traveling on a uniform nondissipative chain of $N$ masses connected by springs is soon destructured by dispersion. Here it is shown that a proper modulation of the masses and the elastic constants makes it possible to obtain a periodic dynamics and a perfect transmission of any kind of pulse between the chain ends, since the initial configuration evolves to its mirror image in the half period. This makes the chain to behave as a Newton's cradle. By a known algorithm based on orthogonal polynomials one can numerically solve the general inverse problem leading from the spectrum to the dynamical matrix and then to the corresponding mass-spring sequence, so yielding all possible ``perfect cradles''. As quantum linear systems obey the same dynamics of their classical counterparts, these results also apply to the quantum case: for instance, a wavefunction localized at one end would evolve to its mirror image at the opposite chain end., 15 pages, 7 figures, 4 tables

We present the mechanical model of an array of elastic filaments and simulate the response with different mechanical couplings. This class of systems is inspired by robust and elegant solutions for locomotion mechanics that have emerged in several small-scale biological entities in the form of beating protrusions, such as cellular cilia and eukaryotic flagella. The collective dynamics of cilia arrays reveals important features such as array alignments, two-phase asymmetric beating of individual filaments, and the emergence of metachronal coordination, which make them suitable for bio-inspired terrestrial and aquatic locomotion. The model presented here is the basis for further developments towards the design of terrestrial and aquatic locomotion systems for general purpose robotic devices.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 123130 (2020) and may be found at https://doi.org/10.1063/5.0021420, We study patterns of partial synchronization in a network of FitzHugh���Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability. Among these networks, a small-world network with intermediate rewiring probability best mimics the findings achieved with the simulations using the empirical structural connectivity. For the other network topologies, either no spontaneously occurring epileptic-seizure-related synchronization phenomena can be observed in the simulated dynamics, or the overall degree of synchronization remains high throughout the simulation. This indicates that a topology with some balance between regularity and randomness favors the self-initiation and self-termination of episodes of seizure-like strong synchronization.

This paper deals with an optimal control problem associated with the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation, This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement Nº 694126-DyCon). The work of both authors was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the Air Force Office of Scientific Research (AFOSR) under Award Nº FA9550-18-1-0242. The work of EZ was partially funded by the Alexander von Humboldt-Professorship program, the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex, the Grant ICON-ANR-16-ACHN-0014 of the French ANR and the Transregio 154 Project Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks of the German DFG

International audience; A laser diode subjected to optoelectronic feedback in which some light is converted to photocurrent that is fed back into the laser injection terminals can display periodic oscillations in its optical intensity. We demonstrate experimental and numerical evidence that, as the feedback level is varied, a step-wise change in the relaxation oscillation frequency manifests itself in the output optical intensity. These transitions in the dynamics can either correspond to an abrupt jump between two limit cycles, associated with a subcritical torus bifurcation of a limit cycle, or to a progressive switching mediated by an intermediate quasiperiodic state. Finally, when ramping the feedback level down, hysteresis is observed in the sequence of switching events between the limit cycles. Such devices are of interest for photonic microwave sources for wavelength-division multiplexed radio-over-fiber systems.

Collective behaviors of coupled oscillators are an important issue in the field of nonlinear dynamics and complex networks, which have attracted much attention in recent years. For some systems satisfying certain symmetries, the dynamics of high-dimensional space can be reduced to that of low-dimensional subspace in terms of the Watanabe-Strogatz theory. In this paper, the dynamics of the transition to synchronization on the coupled star networks are studied, and we identify a two-step phase transition in the system both from the macroscopic and from the microscopic viewpoints. Theoretically, the low-dimensional order-parameter dynamical equation is developed to get analytical insights, and the roles of the coupled hubs in the synchronization process are uncovered. Physically, the two-phase transition steps correspond to different bifurcations in phase space. Theoretical analysis is examined by performing the numerical simulations, and the results and analysis proposed in this paper are helpful to understand the phase transition in general heterogenous networks.

International audience; We consider a system of two masses connected by linear springs and in contact with a belt moving at a constant velocity. One of the masses can collide with a fixed rigid obstacle. The contact forces between the masses and the belt are given by Coulomb’s laws. Moreover, when the colliding mass is in contact with the obstacle, we assume that a perfect elastic impact occurs. Several periodic orbits including contact against the fixed obstacle followed by slip and stick phases are obtained in analytical form.

Oscillatory neural networks with nano-oscillators and synapse devices are a promising alternative to implement neuromorphic systems owing to its fast recognition speed and low power consumption. In this paper, we demonstrate a compact frequency storable oscillator using nanoscale two-terminal NbO2 insulator-metal-transition devices along with TaOx-based resistive switching memory (RRAM) devices. By controlling RRAM resistance, we realized a wide range of analog oscillation frequencies. The synchronization window of two coupled oscillators, which is a key parameter for determining pattern recognition, increases with the increasing coupling capacitance and decreasing RRAM resistance of the reference oscillator. The simple device structure (metal-NbO2-metal-TaOx-metal), small device area (4F2), and frequency storability of NbO2-based coupled oscillator device show a strong potential for future integrated neuromorphic device application.

The Complex Ginzburg–Landau equation is studied assuming a directed network of coupled oscillators. The asymmetry makes the spectrum of the Laplacian operator complex, and it is ultimately responsible for the onset of a generalized class of topological instability, reminiscent of the Benjamin–Feir type. The analysis is initially carried out for a specific class of networks, characterized by a circulant adjacency matrix. This allows us to delineate analytically the domain in the parameter space for which the generalized instability occurs. We then move forward to considering the family of non linear oscillators coupled via a generic direct, though balanced, graph. The characteristics of the emerging patterns are discussed within a self-consistent theoretical framework.

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 30, 051104 (2020) and may be found at https://doi.org/10.1063/5.0008341., Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh–Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer networks. We show that including randomness in the connectivity structure either of the remote network layers or of the relay layer increases the range of interlayer coupling strength where relay synchronization can be observed.

A control method for desynchronizing an array of mean-field coupled FitzHugh–Nagumo-type oscillators is described. The technique is based on applying an adjustable DC voltage source to the coupling node. Both, numerical solution of corresponding nonlinear differential equations and hardware experiments with a nonlinear electrical circuit have been performed.

Dynamical systems are widespread, with examples in physics, chemistry, biology, population dynamics, communications, climatology and social science. They are rarely isolated but generally interact with each other. These interactions can be characterized by coupling functions—which contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how each interaction occurs. Coupling functions can be used, not only to understand, but also to control and predict the outcome of the interactions. This theme issue assembles ground-breaking work on coupling functions by leading scientists. After overviewing the field and describing recent advances in the theory, it discusses novel methods for the detection and reconstruction of coupling functions from measured data. It then presents applications in chemistry, neuroscience, cardio-respiratory physiology, climate, electrical engineering and social science. Taken together, the collection summarizes earlier work on coupling functions, reviews recent developments, presents the state of the art, and looks forward to guide the future evolution of the field. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.

Interacting dynamical systems abound in nature, with examples ranging from biology and population dynamics, through physics and chemistry, to communications and climate. Often their states, parameters and functions are time-varying, because such systems interact with other systems and the environment, exchanging information and matter. A common problem when analysing time-series data from dynamical systems is how to determine the length of the time window for the analysis. When one needs to follow the time-variability of the dynamics, or the dynamical parameters and functions, the time window needs to be resolved first. We tackled this problem by introducing a method for adaptive determination of the time window for interacting oscillators, as modeled and scaled for the cardiorespiratory interaction. By investigating a system of coupled phase oscillators and utilizing the Dynamical Bayesian Inference method, we propose a procedure to determine the time window and the propagation parameter of the covariance matrix. The optimal values are determined so that the inferred parameters follow the dynamics of the actual ones and at the same time the error of the inference represented by the covariance matrix is minimal. The effectiveness of the methodology is presented on a system of coupled limit-cycle oscillators and on the cardiorespiratory interaction. Three cases of cardiorespiratory interaction were considered-measurement with spontaneous free breathing, one with periodic sine breathing and one with a-periodic time-varying breathing. The results showed that the cardiorespiratory coupling strength and similarity of form of coupling functions have greater values for slower breathing, and this variability follows continuously the change of the breathing frequency. The method can be applied effectively to other time-varying oscillatory interactions and carries important implications for analysis of general dynamical systems.

In this manuscript we introduce cubic bubbles that are pinned to 3D printed millimetric frames immersed in water. Cubic bubbles are more stable over time and space than standard spherical bubbles, while still allowing large oscillations of their faces. We found that each face can be described as a harmonic oscillator coupled to the other ones. These resonators are coupled by the gas inside the cube but also by acoustic interactions in the liquid. We provide an analytical model and 3D numerical simulations predicting the resonance with a very good agreement. Acoustically, cubic bubbles prove to be good monopole sub-wavelength emitters, with non-emissive secondary surfaces modes., Comment: 5 pages with 4 figures, supplementary material with two figures. To appear in Physical Review Letters, with Editors' suggestion

A natural problem associated with coupling of non-autonomous oscillators and synchronization is described. It leads to a question about the boundedness of a domain associated with a derived quadratic form. Properties of real symmetric matrices with bounded domains are developed, and unboundedness is translated into satisfiability of a certain Lyapunov-like matrix form. Eventually the real symmetric matrices associated with unbounded domains are explicitly characterized in terms of inertia explicit matrices. A consequence of the characterization is that large, irreducible matrices are likely to have bounded domains.

Individual biological oscillators can synchronize to generate a collective rhythm. During vertebrate development, mobile cells exchange signals to synchronize a rhythmic pattern generator that makes the embryonic segments. Previous theoretical works have shown that cell mobility can enhance synchronization of coupled oscillators when signal exchange is instantaneous. However, in vertebrate segmentation, the exchange of signals is thought to comprise delays from signal sending and processing, which could alter the effect of mobility on synchronization. Here, we study synchronization dynamics of mobile phase oscillators in the presence of coupling delays. We find that mobility can speed up synchronization when coupling delays are present. We derive an analytical expression for the characteristic time of synchronization dynamics, which is in very good agreement with numerical simulations. This analytical expression suggests a subdivision of the mobility range into different dynamical regimes and reveals that, with delayed coupling, synchronization is enhanced at a lower mobility rate than with instantaneous coupling. We argue that these results may be relevant to the synchronization of mobile oscillators in vertebrate segmentation. Fil: Petrungaro, Gabriela. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina Fil: Uriu, Koichiro. Kanazawa University; Japón Fil: Morelli, Luis Guillermo. Institut Max Planck fur Molekulare Physiologie; Alemania. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigación en Biomedicina de Buenos Aires - Instituto Partner de la Sociedad Max Planck; Argentina

This paper presents a new method for evaluating the synchronization of quasi-periodic oscillations of two oscillators, termed &ldquo, chimeric synchronization&rdquo, The family of metrics is proposed to create a neural network information converter based on a network of pulsed oscillators. In addition to transforming input information from digital to analogue, the converter can perform information processing after training the network by selecting control parameters. In the proposed neural network scheme, the data arrives at the input layer in the form of current levels of the oscillators and is converted into a set of non-repeating states of the chimeric synchronization of the output oscillator. By modelling a thermally coupled VO2-oscillator circuit, the network setup is demonstrated through the selection of coupling strength, power supply levels, and the synchronization efficiency parameter. The distribution of solutions depending on the operating mode of the oscillators, sub-threshold mode, or generation mode are revealed. Technological approaches for the implementation of a neural network information converter are proposed, and examples of its application for image filtering are demonstrated. The proposed method helps to significantly expand the capabilities of neuromorphic and logical devices based on synchronization effects.

The study of large arrays of coupled excitable systems has largely benefited from a technique proposed by Ott and Antonsen, which results in a low dimensional system of equations for the system&rsquo, s order parameter. In this work, we show how to explicitly introduce a variable describing the global synaptic activation of the network into these family of models. This global variable is built by adding realistic synaptic time traces. We propose that this variable can, under certain conditions, be a good proxy for the local field potential of the network. We report experimental, in vivo, electrophysiology data supporting this claim.

We present cointegration analysis as a method to infer the network structure of a linearly phase coupled oscillating system. By defining a class of oscillating systems with interacting phases, we derive a data generating process where we can specify the coupling structure of a network that resembles biological processes. In particular we study a network of Winfree oscillators, for which we present a statistical analysis of various simulated networks, where we conclude on the coupling structure: the direction of feedback in the phase processes and proportional coupling strength between individual components of the system. We show that we can correctly classify the network structure for such a system by cointegration analysis, for various types of coupling, including uni-/bi-directional and all-to-all coupling. Finally, we analyze a set of EEG recordings and discuss the current applicability of cointegration analysis in the field of neuroscience.

Many biological and neural systems can be seen as networks of interacting periodic processes. Importantly, their functionality, i.e., whether these networks can perform their function or not, depends on the emerging collective dynamics of the network. Synchrony of oscillations is one of the most prominent examples of such collective behavior and has been associated both with function and dysfunction. Understanding how network structure and interactions, as well as the microscopic properties of individual units, shape the emerging collective dynamics is critical to find factors that lead to malfunction. However, many biological systems such as the brain consist of a large number of dynamical units. Hence, their analysis has either relied on simplified heuristic models on a coarse scale, or the analysis comes at a huge computational cost. Here we review recently introduced approaches, known as the Ott–Antonsen and Watanabe–Strogatz reductions, allowing one to simplify the analysis by bridging small and large scales. Thus, reduced model equations are obtained that exactly describe the collective dynamics for each subpopulation in the oscillator network via few collective variables only. The resulting equations are next-generation models: Rather than being heuristic, they exactly link microscopic and macroscopic descriptions and therefore accurately capture microscopic properties of the underlying system. At the same time, they are sufficiently simple to analyze without great computational effort. In the last decade, these reduction methods have become instrumental in understanding how network structure and interactions shape the collective dynamics and the emergence of synchrony. We review this progress based on concrete examples and outline possible limitations. Finally, we discuss how linking the reduced models with experimental data can guide the way towards the development of new treatment approaches, for example, for neurological disease.

Neurons can anticipate incoming signals by exploiting a physiological mechanism that is not well understood. This article offers a novel explanation on how a receiver neuron can predict the sender's dynamics in a unidirectionally-coupled configuration, in which both sender and receiver follow the evolution of a multi-scale excitable system. We present a novel theoretical viewpoint based on a mathematical object, called canard, to explain anticipation in excitable systems. We provide a numerical approach, which allows to determine the transient effects of canards. To demonstrate the general validity of canard-mediated anticipation in the context of excitable systems, we illustrate our framework in two examples, a multi-scale radio-wave circuit (the van der Pol model) that inspired a caricature neuronal model (the FitzHugh-Nagumo model) and a biophysical neuronal model (a 2-dimensional reduction of the Hodgkin-Huxley model), where canards act as messengers to the senders' prediction. We also propose an experimental paradigm that would enable experimental neuroscientists to validate our predictions. We conclude with an outlook to possible fascinating research avenues to further unfold the mechanisms underpinning anticipation. We envisage that our approach can be employed by a wider class of excitable systems with appropriate theoretical extensions., E.K.E. was supported by the ERC Advanced Grant NerVi No. 227747. S.R. was supported by Ikerbasque (The Basque Foundation for Science) and from the Basque Excellence Research Centers (BERC).

Interacting dynamical systems are widespread in nature. The influence that one such system exerts on another is described by a coupling function; and the coupling functions extracted from the time-series of interacting dynamical systems are often found to be time-varying. Although much effort has been devoted to the analysis of coupling functions, the influence of time-variability on the associated dynamics remains largely unexplored. Motivated especially by coupling functions in biology, including the cardiorespiratory and neural delta-alpha coupling functions, this paper offers a contribution to the understanding of effects due to time-varying interactions. Through both numerics and mathematically rigorous theoretical consideration, we show that for time-variable coupling functions with time-independent net coupling strength, transitions into and out of phase- synchronization can occur, even though the frozen coupling functions determine phase-synchronization solely by virtue of their net coupling strength. Thus the information about interactions provided by the shape of coupling functions plays a greater role in determining behaviour when these coupling functions are time-variable. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.

We investigate the dynamical transitions in a network of nonlocally coupled Stuart-Landau oscillators with a combination of attractive and repulsive couplings. The competing interaction between the couplings plays a crucial role in many realistic situations, particularly in neuronal systems. We report that the employed attractive and repulsive couplings induce imperfect amplitude mediated chimera state which emerges as an intermediate between the oscillatory dynamics and the oscillation death state. Each oscillator in the synchronized and desynchronized groups constituting the imperfect amplitude mediated chimera drifts between both the homogeneous and inhomogeneous oscillations as a function of time. To distinguish the homogeneous and inhomogeneous oscillations, we use the finite-time average of each oscillator. The observed distinct dynamical states are further classified by finding the strength of the inhomogeneous oscillators in the corresponding dynamical states. We also find that the number of clusters in the cluster oscillation death states exponentially decays as a function of the coupling range and obeys a power law relation. Finally, we confirm the robustness of the observed amplitude mediated chimera state by introducing a Gaussian white noise in the system.

Recurrence analysis and its quantifiers are strongly dependent on the evaluation of the vicinity threshold parameter, i.e., the threshold to regard two points close enough in phase space to be considered as just one. We develop a new way to optimize the evaluation of the vicinity threshold in order to assure a higher level of sensitivity to recurrence quantifiers to allow the detection of even small changes in the dynamics. It is used to promote recurrence analysis as a tool to detect nonstationary behavior of time signals or space profiles. We show that the ability to detect small changes provides information about the present status of the physical process responsible to generate the signal and offers mechanisms to predict future states. Here, a higher sensitive recurrence analysis is proposed as a precursor, a tool to predict near future states of a particular system, based on just (experimentally) obtained signals of some available variables of the system. Comparisons with traditional methods of recurrence analysis show that the optimization method developed here is more sensitive to small variations occurring in a signal. The method is applied to numerically generated time series as well as experimental data from physiology.

Almost-dispersionless pulse transfer between the extremal masses of a uniform harmonic spring-mass chain of arbitrary length can be induced by suitably modifying two masses and their spring's elastic constant at both extrema of the chain. It is shown that a deviation (or a pulse) imposed to the first mass gives rise to a wave packet that, after a time of the order of the chain length, almost perfectly reproduces the same deviation (pulse) at the opposite end, with an amplitude loss that is as small as 1.3 % in the infinite-length limit; such a dynamics can continue back and forth again for several times before dispersion cleared the effect. The underlying coherence mechanism is that the initial condition excites a bunch of normal modes with almost equal frequency spacing. This constitutes a possible mechanism for efficient energy transfer, e.g., in nanofabricated structures., 16 pages, 10 figures, 2 tables

A dynamic system showing stable rhythmic activity can be represented by the dynamics of phase oscillators. This would provide a useful mathematical framework through which one can understand the system's dynamic properties. A recent study proposed a Bayesian approach capable of extracting the underlying phase dynamics directly from time-series data of a system showing rhythmic activity. Here we extended this method to spike data that otherwise provide only limited phase information. To determine how this method performs with spike data, we applied it to simulated spike data generated by a realistic neuronal network model. We then compared the estimated dynamics obtained based on the spike data with the dynamics theoretically derived from the model. The method successfully extracted the modeled phase dynamics, particularly the interaction function, when the amount of available data was sufficiently large. Furthermore, the method was able to infer synaptic connections based on the estimated interaction function. Thus, the method was found to be applicable to spike data and practical for understanding the dynamic properties of rhythmic neural systems.

In this article we prove the stability of mean field systems as the Winfree model in the synchronized state. The model is governed by the coupling strength parameter $\kappa$ and the natural frequency of each oscillator. The stability is proved independently of the number of os-cillators and the distribution of the natural frequencies. In order to prove the main result, we introduce the positive invariant cone and we start by studying the linearized system. The method can be applied to others mean field models as the Kuramoto model.

The synchronous dynamics of an array of excitable oscillators, coupled via a generic graph, is studied. Non homogeneous perturbations can grow and destroy synchrony, via a self-consistent instability which is solely instigated by the intrinsic network dynamics. By acting on the characteristic time-scale of the network modulation, one can make the examined system to behave as its (partially) averaged analog. This result if formally obtained by proving an extended version of the averaging theorem, which allows for partial averages to be carried out. As a byproduct of the analysis, oscillation death are reported to follow the onset of the network driven instability., Comment: To appear in Europhys. Lett

We suggest the main principals and functional units of the parallel chemical computer, namely, (i) a generator (which is a network of coupled oscillators) of oscillatory dynamic modes, (ii) a unit which is able to recognize these modes (a ‘reader’) and (iii) a decision-making unit, which analyses the current mode, compares it with the external signal and sends a command to the mode generator to switch it to the other dynamical regime. Three main methods of the functioning of the reader unit are suggested and tested computationally: (a) the polychronization method, which explores the differences between the phases of the generator oscillators; (b) the amplitude method which detects clusters of the generator and (c) the resonance method which is based on the resonances between the frequencies of the generator modes and the internal frequencies of the damped oscillations of the reader cells. Pro and contra of these methods have been analysed.

The self-consistent method, first introduced by Kuramoto, is a powerful tool for the analysis of the steady states of coupled oscillator networks. For second-order oscillator networks complications to the application of the self-consistent method arise because of the bistable behavior due to the co-existence of a stable fixed point and a stable limit cycle, and the resulting complicated boundary between the corresponding basins of attraction. In this paper, we report on a self-consistent analysis of second-order oscillators which is simpler compared to previous approaches while giving more accurate results in the small inertia regime and close to incoherence. We apply the method to analyze the steady states of coupled second-order oscillators and we introduce the concepts of margin region and scaled inertia. The improved accuracy of the self-consistent method close to incoherence leads to an accurate estimate of the critical coupling corresponding to transitions from incoherence.

The intersection between the two concepts of structural control and defectiveness is discussed. Two simple oscillators differently connected by serial spring-dashpot arrangement are used to simply simulate technically relevant cases: dissipatively coupled adjacent free-standing structures, structures equipped by TMD and base-isolated structures. Eigensolution loci of the two classes of systems are tracked against one or more significant parameters to determine the potential benefits realized by different combinations of stiffness and viscosity. In both studied cases, codimension-two manifolds in the four-parameter space corresponding to coalescing eigenvalues are determined by analytical expressions. Conditions to discern semi-simple eigenvalues from defective ones confirm that the latter is the generic case laying in a two-parameter space while the former span a one-parameter subspace. The knowledge of the location of the defective systems in the parameter space permits to determine regions with specific dynamical properties useful for control design purpose.

We study the dynamics of mobile, locally coupled identical oscillators in the presence of coupling delays. We find different kinds of chimera states, in which coherent in-phase and anti-phase domains coexist with incoherent domains. These chimera states are dynamic and can persist for long times for intermediate mobility values. We discuss the mechanisms leading to the formation of these chimera states in different mobility regimes. This finding could be relevant for natural and technological systems composed of mobile communicating agents., 10 pages, 8 figures, draft version

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

We study coupled quantum systems as the working media of thermodynamic machines. Under a suitable phase-space transformation, the coupled systems can be expressed as a composition of independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. We also show that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncoupled system, thereby showing that the quantum correlations do not help in optimal work extraction. Further, we study two explicit examples, coupled spin-$1/2$ systems and coupled quantum oscillators with analogous interactions. Interestingly, for particular kind of interactions, the efficiency of the coupled oscillators outperforms that of the coupled spin-$1/2$ systems when they work as heat engines. However, for the same interaction, the coefficient of performance behaves in a reverse manner, while the systems work as the refrigerator. Thus the same coupling can cause opposite effects in the figures of merit of heat engine and refrigerator., Comment: 19 pages, 8 figures

Damping of spikes in an array of coupled oscillators by injection of sinusoidal current is studied both electronically and numerically. The effect is investigated using an array consisting of thirty mean-field coupled FitzHugh–Nagumo-type oscillators. The results are considered as a possible mechanism of the deep brain stimulation used to avoid the symptoms of the Parkinson's disease.

International audience; We investigate the dynamics of a two degrees-of-freedom oscillator excited by dry friction. The system consists of two masses connected by linear springs and in contact with a belt moving at a constant velocity. The contact forces between the masses and the belt are given by Coulomb's laws. Several periodic orbits including slip and stick phases are obtained. In particular, the existence of periodic orbits involving a part where one of the masses moves at a higher speed than the belt is proved.

We study destabilization mechanisms of spiral coherence-incoherence patterns known as spiral chimera states that form on a two-dimensional lattice of nonlocally coupled phase oscillators. For this purpose we employ the linearization of the Ott--Antonsen equation that is valid in the continuum limit and perform a detailed two-parameter stability analysis of a $D_4$-symmetric chimera state, i.e., a four-core spiral state. We identify fold, Hopf, and parity-breaking bifurcations as the main mechanisms whereby spiral chimeras can lose stability. Beyond these bifurcations we find new spatio-temporal patterns, in particular quasiperiodic chimeras and $D_2$-symmetric spiral chimeras, as well as drifting states.

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed.

The detection of directional couplings between dynamics based onmeasured spike trains is a crucial problem in the understanding of many different systems. In particular, in neuroscience it is important to assess the connectivity between neurons.One of the approaches that can estimate directional coupling from the analysis of point processes is the nonlinear interdependence measure L. Although its efficacy has already been demonstrated, it still needs to be tested under more challenging and realistic conditions prior to an application to real data. Thus, in this paper we use the Hindmarsh-Rose model system to test the method in the presence of noise and for different spiking regimes.We also examine the influence of different parameters and spike train distances. Our results show that the measure L is versatile and robust to various types of noise, and thus suitable for application to experimental data. We acknowledge funding from the European Union Horizon 2020 research and innovation programme under theMarie Skłodowska-Curie Grant Agreement No. 642563 “Complex Oscillatory Systems: Modeling and Analysis” (COSMOS), (I.M., T.K., R.G.A.), and from the Volkswagen Foundation, the Spanish Ministry of Economy and Competitiveness Grant No. FIS2014-54177-R and the CERCA Programme of the Generalitat de Catalunya (R.G.A).