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The synchronization of oscillator ensembles is pervasive throughout nonlinear science, from classical or quantum mechanics to biology, to human assemblies. Traditionally, the main focus has been the identification of threshold parameter values for the transition to synchronization as well as the nature of such transition. Here, we show that considering an oscillator lattice as a discrete growing interface provides unique insights into the dynamical process whereby the lattice reaches synchronization for long times. Working on a generalization of the Kuramoto model that allows for odd or non-odd couplings, we elucidate synchronization of oscillator lattices as an instance of generic scale invariance, whereby the system displays space-time criticality, largely irrespective of parameter values. The critical properties of the system (like scaling exponent values and the dynamic scaling Ansatz which is satisfied) fall into universality classes of kinetically rough interfaces with columnar disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang (KPZ) equations, for Kuramoto (odd) coupling and generic (non-odd) couplings, respectively. From the point of view of kinetic roughening, the critical properties we find turn out to be quite innovative, especially concerning the statistics of the fluctuations as characterized by their probability distribution function (PDF) and covariance. While the latter happens to be that of the Larkin model irrespective of the symmetry of the coupling, in the generic non-odd coupling case the PDF turns out to be the Tracy-Widom distribution associated with the KPZ nonlinearity. This brings the synchronization of oscillator lattices into a remarkably large class of strongly-correlated, low-dimensional (classical and quantum) systems with strong universal fluctuations., 26 pages, 18 figures

Rhythmic activities that alternate between coherent and incoherent phases are ubiquitous in chemical, ecological, climate, or neural systems. Despite their importance, general mechanisms for their emergence are little understood. In order to fill this gap, we present a framework for describing the emergence of recurrent synchronization in complex networks with adaptive interactions. This phenomenon is manifested at the macroscopic level by temporal episodes of coherent and incoherent dynamics that alternate recurrently. At the same time, the dynamics of the individual nodes do not change qualitatively. We identify asymmetric adaptation rules and temporal separation between the adaptation and the dynamics of individual nodes as key features for the emergence of recurrent synchronization. Our results suggest that asymmetric adaptation might be a fundamental ingredient for recurrent synchronization phenomena as seen in pattern generators, e.g., in neuronal systems.

For the study of symmetry-breaking phenomena in neuronal networks, simplified versions of the FitzHugh-Nagumo model are widely used. In this paper, these phenomena are investigated in a network of FitzHugh-Nagumo oscillators taken in the form of the original model and it is found that it exhibits diverse partial synchronization patterns that are unobserved in the networks with simplified models. Apart from the classical chimera, we report a new type of chimera pattern whose incoherent clusters are characterized by spatial random swings among a few fixed periodic attractors. Another peculiar hybrid state is found that combines the features of this chimera state and a solitary state such that the main coherent cluster is interspersed with some nodes with identical solitary dynamics. In addition, oscillation death including chimera death emerges in this network. A reduced model of the network is derived to study oscillation death, which helps explaining the transition from spatial chaos to oscillation death via the chimera state with a solitary state. This study deepens our understanding of chimera patterns in neuronal networks.

Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable substrate potential is examined by analyzing response function, the largest Lyapunov exponent, and Poincaré sections for two neighboring particles. The obtained results show that the largest Lyapunov exponent, besides being used for investigating integral quantities, can be used for detecting microchanges in chain configuration of both damped Frenkel-Kontorova model with inertial term and its strictly overdamped limit. Slight changes in relative positions of the particles are registered through jumps of the largest Lyapunov exponent in the pinning regime. The occurrence of such jumps is highly dependent on type of commensurate structure and deformation of substrate potential. The obtained results also show that the minimal force required to initiate collective motion of the chain is not dependent on the number of Lyapunov exponent jumps in the pinning regime. These jumps are also registered in the sliding regime, where they are a consequence of a more complex structure of largest Lyapunov exponent on the step. © 2023 American Physical Society. Preprint: [https://doi.org/10.48550/arXiv.2209.08929]

Presplošen naslov diplomske naloge, definiran na začetku poti, bi bilo bolje preoblikovati v "Avtomatsko odkrivanje modelov obsežne možganske aktivnosti", ki natančneje opiše področje obravnave. Naši možgani organizirajo vrsto funkcij, ki jih potrebujemo za učinkovito zadovoljevanje (osnovnih) življenjskih potreb. To je mogoče le z natančno koordinacijo med posameznimi možganskimi regijami. Eden izmed predlaganih mehanizmov koordinacije je sklopljenost oz. časovna sinhronizacija možganskih ritmov med lokalnimi populacijami nevronov, ki jo najpreprosteje modeliramo z omrežjem med seboj sklopljenih oscilatorjev. V diplomski nalogi nas je zanimalo, kako natančno lahko z metodama avtomatske identifikacije sistemov (ProGED in SINDy) odkrijemo tako strukturo kot parametre izbranih modelov oscilatorjev. Izbrali smo pet sistemov oscilatorjev in pet parov sklopljenih oscilatorjev. S simulacijo sistemov smo pridobili sintetične vhodne podatke, nadalje modificirane z različnimi nivoji šuma in opazljivosti. Metodi sta bili uspešni pri rekonstrukciji posameznega oscilatorja, dokler je bilo v podatkih manj kot 1 % šuma. ProGED kaže dobro rekonstrukcijo modelov tudi pri 10 % šumu, medtem ko SINDy spodleti zaradi uporabe na šum občutljivejšega numeričnega odvajanja. Iz istega razloga SINDy ne more odkrivati modelov le z delno opazljivimi podatki, medtem ko ProGED lahko, a v praksi z večjo napako rekonstrukcije. Obe metodi sta imeli tudi pri najčistejših in celovitih podatkih velike težave pri identifikaciji para sklopljenih oscilatorjev, na kar je najbolj vplivalo večje število spremenljivk stanja in neznanih parametrov. V splošnem ugotavljamo, da so trenutne implementacije klasičnih metod avtomatske identifikacije dinamičnih sistemov premalo skalabilne, da bi bile uporabne za modeliranje kompleksnejših M/EEG podatkov, kjer bi bilo vključenih več deset ali sto med seboj sklopljenih oscilatorjev z relativno veliko stopnjo šuma. The overly general title of the thesis, defined at the beginning of the journey, would be better reformulated to "Automatic system identification of large-scale brain activity models", which more precisely describes the scope of this work. Our brain orchestrates the various functions that we need to effectively meet (basic) life needs. This is possible only through precise coordination between individual brain regions. One of the proposed coordination mechanisms is coupling or a temporal synchronization of brain rhythms between local populations of neurons, which can be most simply modeled by a network of coupled oscillators. In the diploma thesis, we were interested in how accurately can we discover both the structure and the parameters of the selected oscillator models with the methods of automatic system identification (ProGED and SINDy). We selected ten dynamical systems, five single oscillators and five pairs of coupled oscillators. By simulating the systems, we obtained synthetic input data, further modified with different levels of noise and observability. Both methods were successful in reconstructing a single oscillator as long as there was less than 1 % noise in the data. ProGED shows good model reconstruction even at 10 % noise, while SINDy fails due to the use of more noise-sensitive numerical derivation. For the same reason, SINDy cannot detect models with only partially observable data, while ProGED can, but in practice with a larger reconstruction error. Even with the cleanest and most complete data, both methods had great difficulty in identifying a pair of coupled oscillators, which was mostly influenced by a large number of state variables and unknown parameters. Overall, we conclude that current implementations of classical automated identification methods for dynamic systems are insufficiently scalable to be useful for modeling more complex M/EEG data involving dozens or hundreds of coupled oscillators with a relatively high level of noise.

We consider a model for an N × N lattice network of weakly coupled neural oscilla- tors with periodic boundary conditions (2D square torus), where the coupling between neurons is assumed to be within a von Neumann neighborhood of size r, denoted as von Neumann r-neighborhood. Using the phase model reduction technique, we study the existence of cluster solutions with constant phase differences (Ψh, Ψv) between adjacent oscillators along the horizontal and vertical directions in our network, where Ψh and Ψv are not necessarily to be identical. Applying the Kronecker production representation and the circulant matrix theory, we develop a novel approach to analyze the stability of cluster solutions with constant phase difference (i.e., Ψh,Ψv are equal). We begin our analysis by deriving the precise conditions for stability of such cluster solutions with von Neumann 1-neighborhood and 2 neighborhood couplings, and then we generalize our result to von Neumann r-neighborhood coupling for arbitrary neighborhood size r ≥ 1. This developed approach for the stability analysis indeed can be extended to an arbitrary coupling in our network. Finally, numerical simulations are used to validate the above analytical results for various values of N and r by considering an inhibitory network of Morris-Lecar neurons.

While the Von-Neumann architecture-based digital computing framework has been the foundation for modern information processing, there are certain classes of computational problems that are still considered intractable to solve using digital computers. Such problems entail exponentially increasing computing resources with increasing problem size. This has motivated the exploration of alternate computing paradigms. Ising machines have emerged as a promising candidate for solving such computationally challenging problems e.g., the NP-Hard problem, Maximum Cut (Max-Cut) of a graph. However, the effectiveness of this paradigm and its ability to outperform state-of-the-art digital computers will ultimately be decided by the area, scalability, energy efficiency, and performance of the underlying hardware implementation. Current implementations using CMOS, optical phenomena as well as qubits have limitations in terms of these performance parameters. In this dissertation, the electronic oscillator-based Ising machine (OIM) is investigated due to its promise of CMOS compatibility, compactness, and room-temperature operation. Prior works on coupled oscillator Ising machines have focused on small-scale prototype implementations (4-8 oscillators), while larger designs have been limited to solving only planar and sparse graphs. Therefore, to establish the coupled oscillator platform as a promising candidate for realizing physical Ising machines, its scalability needs to be evaluated- which remains missing and is the primary focus of this dissertation. To accomplish this, a 672 oscillator-based Ising machine IC (65nm GP CMOS technology) with 30,896 programmable coupling elements is developed. Using this platform, a fundamental trade-off between the solution quality and the time-to-compute is experimentally revealed. Further, we evaluate methods to overcome this tradeoff and propose a hybrid (OIM + heuristic local search) approach. Our projections show that the hybrid approach offers 3-100× improvement in experimentally measured time-to-compute at iso-solution quality, compared to a digital algorithm. Finally, we expand on the dynamical system model to enable solving a broad range of problems requiring multi-valued ‘spin’ states such as the traveling salesman problem, the graph coloring problem, the Hamiltonian path/cycle problem, etc.

These are experimental time series (three repeats) for a pair of Minati-Frasca single-transistor chaotic oscillators, coupled through a band-pass filter having tunable settings. They are provided to support the replication of the results reported in the associated publication, as well as any further public-domain academic research in the field of chaotic oscillators, distributed sensing, and related aspects, in compliance with the specified license terms and all applicable legal clauses. The following reference must be cited when using these data: Minati L, Li B, Bartels J, Li Z, Frasca M, Ito H, Incomplete synchronization of chaos under frequency-limited coupling: observations in single-transistor microwave oscillators, Chaos, Solitons and Fractals 165 (2022) 112854, DOI 10.1016/j.chaos.2022.112854. This work was partially supported by JSPS KAKENHI Grant Number 19H02191 and JST-SPRING Grant Number JPMJSP2106. L.M. conceived the study in collaboration with M.F. when based in Trento, Italy, later performing the experiments and preparing the manuscript while in Tokyo, Japan.

The functionalization of internal resonance (IR) is theoretically and experimentally demonstrated on a nonlinear hybrid vibration energy harvester (HVEH) based on piezoelectric (PE) and electromagnetic (EM) transductions. This nonlinear phenomenon is tuned by adjusting the gaps between the moving magnets of the structure, enabling 1:1 and 2:1 IR. The experimental results prove that the activation of 2:1 IR with a realistic excitation amplitude allows the improvement of both the frequency bandwidth (BW) and the harvested power (HP) by 300% and 100%, respectively compared to the case away from IR. These remarkable results open the way towards a very large scale integration of coupled resonators with simultaneous internal resonances.

Topological signals, i.e., dynamical variables defined on nodes, links, triangles, etc. of higher-order networks, are attracting increasing attention. However, the investigation of their collective phenomena is only at its infancy. Here we combine topology and nonlinear dynamics to determine the conditions for global synchronization of topological signals defined on simplicial or cell complexes. On simplicial complexes we show that topological obstruction impedes odd dimensional signals to globally synchronize. On the other hand, we show that cell complexes can overcome topological obstruction and in some structures signals of any dimension can achieve global synchronization.

Brain oscillations emerge during sensory and cognitive processes and have been classified into different frequency bands. Yet, even within the same frequency band and between nearby brain locations, the exact frequencies of brain oscillations can differ. These frequency differences (detuning) have been largely ignored and play little role in current functional theories of brain oscillations. This contrasts with the crucial role that detuning plays in synchronization theory, as originally derived in physical systems. Here, we propose that detuning is equally important to understand synchronization in biological systems. Detuning is a critical control parameter in synchronization, which is not only important in shaping phase-locking, but also in establishing preferred phase relations between oscillators. We review recent evidence that frequency differences between brain locations are ubiquitous and essential in shaping temporal neural coordination. With the rise of powerful experimental techniques to probe brain oscillations, the contributions of exact frequency and detuning across neural circuits will become increasingly clear and will play a key part in developing a new understanding of the role of oscillations in brain function.

Brain oscillations emerge during sensory and cognitive processes and have been classified into different frequency bands. Yet, even within the same frequency band and between nearby brain locations, the exact frequencies of brain oscillations can differ. These frequency differences (detuning) have been largely ignored and play little role in current functional theories of brain oscillations. This contrasts with the crucial role that detuning plays in synchronization theory, as originally derived in physical systems. Here, we propose that detuning is equally important to understand synchronization in biological systems. Detuning is a critical control parameter in synchronization, which is not only important in shaping phase-locking, but also in establishing preferred phase relations between oscillators. We review recent evidence that frequency differences between brain locations are ubiquitous and essential in shaping temporal neural coordination. With the rise of powerful experimental techniques to probe brain oscillations, the contributions of exact frequency and detuning across neural circuits will become increasingly clear and will play a key part in developing a new understanding of the role of oscillations in brain function.

Biological processes are governed by the expression of proteins, and for some proteins, their level of expression can fluctuate periodically over time (i.e., they oscillate). Many oscillatory proteins (e.g., cell cycle proteins and those from the HES family of transcription factors) are connected in complex ways, often within large networks. This complexity can be elucidated by developing intuitive mathematical models that describe the underlying critical aspects of the relationships between these processes. Here, we provide a mathematical explanation of a recently discovered biological phenomenon: the phasic position of the gene Hes1’s oscillatory expression at the beginning of the cell cycle of an individual human breast cancer stem cell can have a predictive value on how long that cell will take to complete a cell cycle. We use a two-component model of coupled oscillators to represent Hes1 and the cell cycle in the same cell with minimal assumptions. Inputting only the initial phase angles, we show that this model is capable of predicting the dynamic mitosis to mitosis behaviour of Hes1 and predicting cell cycle length patterns as found in real-world experimental data. Moreover, we discover that bidirectional coupling between Hes1 and the cell cycle is critical within the system for the data to be reproduced and that nonfixed asymmetry in the interactions between the oscillators is required. The phase dynamics we present here capture the complex interplay between Hes1 and the cell cycle, helping to explain nongenetic cell cycle variability, which has critical implications in cancer treatment contexts.

There is ample evidence that situation characteristics and people’s momentary states are meaningfully related (e.g., Horstmann et al., 2021; Sherman et al., 2015). Moreover, there are sizable individual differences in these contingencies (Kuper et al., 2022). However, this work leaves open several questions on the transactions (Rauthmann, 2021) between situation characteristics and personality states over time. Most studies focus on concurrent within-person associations rather than explicitly examining changes over time (but see Rauthmann et al., 2016; Hecht et al., 2022). Moreover, previous work used classical linear models rather than dynamic system models that may better reflect the nature of the processes at hand. In the present project, we aim to investigate non-linear dynamics between changes in personality states and situation characteristics using both person-specific and multilevel change-as-outcome models (Butner et al., 2015; Danvers et al., 2020). Specifically, we will predict subsequent changes in both personality states and situation characteristics by each other and their interaction. In addition to examining average effects across participants, we will focus on individual differences in these relations as well as the associations of these individual differences among each other.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022, Tutor: Albert Díaz Guilera, The Kuramoto model is one of the most representative models of coupled phase oscillators, commonly used in the description of synchronization phenomenon in complex systems. The aim of this report is to study, from a mathematical and computational point of view, three linear conversions of the Kuramoto model in the case of identical oscillators with global coupling. A comparison with the original Kuramoto dynamics is also conducted

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.

In this work, we propose a dynamical systems perspective on the modeling of sepsis and its organ-damaging consequences. We develop a functional two-layer network model for sepsis based upon the interaction of parenchymal cells and immune cells via cytokines, and the coevolutionary dynamics of parenchymal, immune cells, and cytokines. By means of the simple paradigmatic model of phase oscillators in a two-layer system, we analyze the emergence of organ threatening interactions between the dysregulated immune system and the parenchyma. We demonstrate that the complex cellular cooperation between parenchyma and stroma (immune layer) either in the physiological or in the pathological case can be related to dynamical patterns of the network. In this way we explain sepsis by the dysregulation of the healthy homeostatic state (frequency synchronized) leading to a pathological state (desynchronized or multifrequency cluster) in the parenchyma. We provide insight into the complex stabilizing and destabilizing interplay of parenchyma and stroma by determining critical interaction parameters. The coupled dynamics of parenchymal cells (metabolism) and nonspecific immune cells (response of the innate immune system) is represented by nodes of a duplex layer. Cytokine interaction is modeled by adaptive coupling weights between nodes representing immune cells (with fast adaptation timescale) and parenchymal cells (slow adaptation timescale), and between pairs of parenchymal and immune cells in the duplex network (fixed bidirectional coupling). The proposed model allows for a functional description of organ dysfunction in sepsis and the recurrence risk in a plausible pathophysiological context.

Biology is suffused with rhythmic behaviour, and interacting biological oscillators often synchronize their rhythms with one another. Colonies of some ant species are able to synchronize their activity to fall into coherent bursts, but models of this phenomenon have neglected the potential effects of intrinsic noise and interspecific differences in individual-level behaviour. We investigated the individual and collective activity patterns of two Leptothorax ant species. We show that in one species ( Leptothorax sp. W), ants converge onto rhythmic cycles of synchronized collective activity with a period of about 20 min. A second species ( Leptothorax crassipilis ) exhibits more complex collective dynamics, where dominant collective cycle periods range from 16 min to 2.8 h. Recordings that last 35 h reveal that, in both species, the same colony can exhibit multiple oscillation frequencies. We observe that workers of both species can be stimulated by nest-mates to become active after a refractory resting period, but the durations of refractory periods differ between the species and can be highly variable. We model the emergence of synchronized rhythms using an agent-based model informed by our empirical data. This simple model successfully generates synchronized group oscillations despite the addition of noise to ants' refractory periods. We also find that adding noise reduces the likelihood that the model will spontaneously switch between distinct collective cycle frequencies.

This research explores the effect of synaptic pruning on a ring-shaped neural network of non-locally coupled FitzHugh–Nagumo (FHN) oscillators. The neurons in the pruned region synchronize with each other, and they repel the coherent domain of the chimera states. Furthermore, the width of the pruned region decides the precision and efficiency of the control effect on the position of coherent domains. This phenomenon gives a systematic comprehension of the relation between pruning and synchronization in neural networks from a new aspect that has never been addressed. An explanation of this mechanism is also given.

We analyze the influence of music in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report an increase of coherence between the global dynamics in our network and the input signal induced by a specific music song. We show that the level of coherence depends crucially on the frequency band. We compare our results with experimental data, which also describe global neural synchronization between different brain regions in the gamma-band range and its increase just before transitions between different parts of the musical form (musical high-level events). The results also suggest a separation in musical form-related brain synchronization between high brain frequencies, associated with neocortical activity, and low frequencies in the range of dance movements, associated with interactivity between cortical and subcortical regions., Comment: 26 pages, 10 figures. arXiv admin note: text overlap with arXiv:2103.13804

Studying the stability of synchronization of coupled oscillators is one of the prominent topics in network science. However, in most cases, the computational cost of complex network analysis is challenging because they consist of a large number of nodes. This study includes overcoming this obstacle by presenting a method for reducing the dimension of a large-scale network, while keeping the complete region of stable synchronization unchanged. To this aim, the first and last non-zero eigenvalues of the Laplacian matrix of a large network are preserved using the eigen-decomposition method and Gram-Schmidt orthogonalization. The method is only applicable to undirected networks and the result is a weighted undirected network with smaller size. The reduction method is studied in a large-scale a small-world network of Sprott-B oscillators. The results show that the trend of the synchronization error is well maintained after node reduction for different coupling schemes.

The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott-Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work, we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a Fourier-Hermite decomposition of the oscillator density. The Fourier-Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the "enlarged Kuramoto model" (endowed with nonpairwise interactions). In both systems, we obtain new results inaccessible through direct numerical integration of populations. I.L. acknowledges support from Universidad de Cantabria and the Government of Cantabria under the Concepción Arenal programme.

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, co-evolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite size adaptive network of heterogeneous phase oscillators. Depending on the nature of defects in the internal frequency distribution, we observe either an abrupt single-step transition to full synchronization or a more gradual multi-step transition. This observation has a striking resemblance to heterogeneous nucleation. We develop a mean-field approach to study the interplay between adaptivity and nodal heterogeneity and describe the dynamics of multicluster states and their role in determining the character of the phase transition. Our work provides a theoretical framework for studying the interplay between adaptivity and nodal heterogeneity.

The appearance of chaos along the “charged” resonance circuit branch of the current-voltage characteristics is examined in a model of intrinsic Josephson junctions shunted by resistive, inductive and capacitive circuit elements. Detailed numerical simulations of the electric charge density, current-voltage characteristics and various chaos indicators, such as Lyapunov exponents and Poincaré sections are performed over a wide range of relevant system parameters. In mapping the parameter space, several distinctly different regions are found: some completely regular, while others are dominated by chaos, where the rc-frequency determines the appearance and properties of those regions. At higher values of the rc-frequency chaos appears at first over relatively narrow regions at the lower-current end of the rc-branch. As the rc-frequency is lowered, the chaotic region at the lowered-current end of the rc-branch may becomes wider, depending sensitively on the combination of system parameters. Examination of the metric entropy and maximal Kaplan–Yorke dimension shows that the dimensions of the chaotic attractors associated with the chaos do not plateau, as in the overdamped case, but surprisingly increase indefinitely with the number of junctions, indicating that the system is capable of truly high-dimensional chaos. The onset of the chaos along the rc-branch is found to occur via a two-frequency quasi-periodic route. Our results provided a guide for specific parameter combinations that could minimize the chaos, thereby making certain applications potentially more viable.

Honey bees live in colonies containing tens of thousands of workers that coordinate their activities to produce efficient colony-level behavior. In free-foraging colonies, nest bees are entrained to the forager daily phase of activity even when experiencing conflicting light-dark illumination regime, but little is known on the cues mediating this potent social synchronization. We monitored locomotor activity in an array of individually caged bees in which we manipulated the contact with neighbour bees. We used circular statistics and coupling function analyses to estimate the degree of social synchronization. We found that young bees in cages connected to cages housing foragers showed stronger rhythms, better synchronization with each other, higher coupling strength, and a phase more similar to that of the foragers compared to similar bees in unconnected cages. These findings suggest that close distance contacts are sufficient for social synchronization or that cage connection facilitated the propagation of time-giving social cues. Coupling strength was higher for bees placed on the same tray compared with bees at a similar distance but on a different tray, consistent with the hypothesis that substrate borne vibrations mediate phase synchronization. Additional manipulation of the contact between cages showed that social synchronization is better among bees in cages connected with tube with a single mesh partition compared to sealed tubes consistent with the notion that volatile cues act additively to substrate borne vibrations. These findings are consistent with self-organization models for social synchronization of activity rhythms and suggest that the circadian system of honey bees evolved remarkable sensitivity to non-photic, non-thermal, time giving entraining cues enabling them to tightly coordinate their behavior in the dark and constant physical environment of their nests.

14 pags., 2 figs., 2 apps., We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint., Bogar Díaz acknowledges support from the CONEX-Plus programme funded by Universidad Carlos III de Madrid and the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 801538. This work was partially supported by DGAPAPAPIIT Grant No. IN105422.

This paper discusses the application of coupled reactance-less memristor-based oscillators (MBO) with binary output signals in oscillatory networks. A class of binary-coupled memristor oscillators provides simple integration with standard CMOS logic elements. Combining MBOs with binary logic elements ensures the operation of complex information processing algorithms. The analysis of the simplest networks based on MBOs is performed. The typical reactance-less MBO with current and potential inputs is considered. The output responses for input control signals are analyzed. It is shown that the current input signal impacts primarily the rate of memristor resistance variation, while the potential input signal changes the thresholds. The exploit of the potential input for the synchronization of coupled MBOs and current control input in order to provide the necessary encoding of information is suggested. The example of the application of coupled MBOs in oscillatory networks is given, and results of simulation are presented.

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations.

Electromagnetic field oscillations produced by the brain are increasingly being viewed as causal drivers of consciousness. Recent research has highlighted the importance of the body’s various endogenous rhythms in organizing these brain-generated fields through various types of entrainment. We expand this approach by examining evidence of extracerebral shared oscillations between the brain and other parts of the body, in both humans and animals. We then examine the degree to which these data support one of General Resonance Theory’s (GRT) principles: the Slowest Shared Resonance (SSR) principle, which states that the combination of micro- to macro-consciousness in coupled field systems is a function of the slowest common denominator frequency or resonance. This principle may be utilized to develop a spatiotemporal hierarchy of brain-body shared resonance systems. It is predicted that a system’s SSR decreases with distance between the brain and various resonating structures in the body. The various resonance relationships examined, including between the brain and gastric neurons, brain and sensory organs, and brain and spinal cord, generally match the predicted SSR relationships, empirically supporting this principle of GRT.

International audience; The collective behavior of many coupled oscillator systems is currently being explored for the implementation of different non-conventional computing paradigms. In particular, VO 2 based nano-oscillators have been proposed to implement oscillatory neural networks that can serve as associative memories, useful in pattern recognition applications. Although the dynamics of a pair of coupled oscillators have already been extensively analyzed, in this brief, the topic is addressed more practically. Firstly, for the application mentioned above, each oscillator needs to be initialized in a given phase to represent the input pattern. We demonstrate the impact of this initialization mechanism on the final phase relationship of the oscillators. Secondly, such oscillatory networks are based on frequency synchronization, in which the impact of variability is critical. We carried out a comprehensive mathematical analysis of a pair of coupled oscillators taking into account both issues, which is a first step towards the design of the oscillatory neural networks for associative memory applications.

An approach to solve the inverse problem of the reconstruction of the network of time-delay oscillators from their time series is proposed and studied in the case of the nonstationary connectivity matrix. Adaptive couplings have not been considered yet for this particular reconstruction problem. The problem of coupling identification is reduced to linear optimization of a specially constructed target function. This function is introduced taking into account the continuity of the nonlinear functions of oscillators and does not exploit the mean squared difference between the model and observed time series. The proposed approach allows us to minimize the number of estimated parameters and gives asymptotically unbiased estimates for a large class of nonlinear functions. The approach efficiency is demonstrated for the network composed of time-delayed feedback oscillators with a random architecture of constant and adaptive couplings in the absence of a priori knowledge about the connectivity structure and its evolution. The proposed technique extends the application area of the considered class of methods.

The computation of rank ordering plays a fundamental role in cognitive tasks and offers a basic building block for computing arbitrary digital functions. Spiking neural networks have been demonstrated to be capable of identifying the largest k out of N analog input signals through their collective nonlinear dynamics. By finding partial rank orderings, they perform k-winners-take-all computations. Yet, for any given study so far, the value of k is fixed, often to k equal one. Here we present a concept for spiking neural networks that are capable of (re)configurable computation by choosing k via one global system parameter. The spiking network acts via pulse-suppression induced by inhibitory pulse-couplings. Couplings are proportional to each units' state variable (neuron voltage), constituting an uncommon but straightforward type of leaky integrate-and-fire neural network. The result of a computation is encoded as a stable periodic orbit with k units spiking at some frequency and others at lower frequency or not at all. Orbit stability makes the resulting analog-to-digital computation robust to sufficiently small variations of both, parameters and signals. Moreover, the computation is completed quickly within a few spike emissions per neuron. These results indicate how reconfigurable k-winners-take-all computations may be implemented and effectively exploited in simple hardware relying only on basic dynamical units and spike interactions resembling simple current leakages to a common ground.

Oscillator neural networks (ONNs) are a promising hardware option for artificial intelligence. With an abundance of theoretical treatments of ONNs, few experimental implementations exist to date. In contrast to prior publications of only building block functionality, we report a practical experimental demonstration of neural computing using an ONN. The arrays contain 26 CMOS ring oscillators in the GHz range of frequencies are tuned by image data and convolution kernels. Synchronization of oscillators results in an analog output voltage approximating convolution neural network operation.

Coupled-mode theory is applied to obtain an analytic form of scattering parameters for a class of transmission line metamaterials with antisymmetric split-rings. The same structure is modeled with equivalent circuit, which includes electric and magnetic coupling with the line and inter-resonator coupling. Modified even/odd analysis is used to obtain scattering parameters from the equivalent circuit. These two methods are shown to be equivalent in a narrow band, and their constants related. The obtained results are compared with full-wave simulations and measurements, and it is shown that both methods give accurate approximation in one octave frequency band. The derived analytic expressions are suitable for study of resonant phenomena, with potential practical applications for filters, phase shifters, delay lines, and sensors.

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior In this work we extend the investigation of coupled discrete phase oscillators to circular networks with non-global coupling, using analytic approximations and simulation. For the later, the chosen method is an event-driven simulation. Its general implementation allows for the investigation of arbitrary graphs, and is used to investigate small-world networks. At the same time, a mean-field (MF) approximation for small-world networks is introduced, which predicts the stability of travelling waves at positive coupling, where usually the globally synchronized solution would be observed. In the zero disorder limit the MF recovers the mean-field approximation proposed in previous works, but here travelling waves are found to be stable even when the underlying graph has some disorder. The wave solutions compete with global oscillations as well as with each other when there is more than one stable wave number, leading to spontaneous transitions between wave numbers. Finite systems will always be subject to these fluctuations, but larger systems are more robust since noise becomes smaller relative to wave period and amplitude. Preliminary simulations and scaling analyses indicated that wave solutions did not lose stability if interaction range and system size are increased in the same proportion, a property which is captured by the MF approximation. To further probe its validity, we tested other predictions such as wave stability in the presence of disorder, the later introduced through rewiring the base graph. Another finding is that the speed of propagation of such waves should increase with increasing natural frequencies, which is verified in simulation. Neste trabalho estendemos o estudo de sistemas de osciladores acoplados para redes circulares de acoplamento não global utilizando modelagem matemática e simulações. O método simulacional é do tipo ``event driven'', e a sua implementação genérica permitiu a investigação de grafos de conectividade arbitrários de modo que redes de mundo pequeno também puderam ser investigadas. Também desenvolvemos uma teoria de campo médio para descrever o comportamento coletivo dos osciladores acoplados nessas redes de mundo pequeno, que prevê a estabilidade de ondas viajantes em regimes de acoplamento positivo. Em geral nessas condições se observa uma sincronia global entre osciladores, mas o alcance não global das interações permite a sobrevivência de tais estados. No entanto, a descrição também prevê a estabilidade de ondas mesmo quando desordem de alcance global é introduzida no sistema através de um algorítimo de reconexão do grafo subjacente. Nesse regime a solução de ondas viajantes está constantemente competindo com outros estados estáveis, como oscilações globais ou ondas viajantes com número de onda diferente (quando há mais de um número de onda estável). Isso faz com que o sistema nunca atinga propriedades macroscópicas estáveis, mesmo depois de longos tempos. Sistemas finitos sempre estarão sujeitos a esse tipo de regime, com a criação e aniquilação espontânea de números de onda, mas se tornam mais estáveis quando o tamanho do sistema cresce, tornando a intensidade das flutuações nas frentes de onda pequenas em relação a amplitude das ondas. Inicialmente, as simulações indicaram que soluções de ondas viajantes permanecem estáveis se o tamanho do sistema cresce proporcionalmente ao alcance das interações, uma propriedade que foi capturada pela aproximação de campo médio. Para fortalecer a validade de tal aproximação, algumas de suas previsões foram testadas, como a estabilidade de ondas viajantes na presença de desordem introduzida pelo processo de reconexão. Outro resultado é a modulação da velocidade de propagação das ondas através de vieses macroscópicos introduzidos nas frequências naturais de oscilação das unidades microscópicas do sistema, que também foi verificado em simulação.

Many animals benefit from synchronizing their daily activities with conspecifics. In this hybrid paper, we first review recent literature supporting and extending earlier evidence for a lack of clear relationship between the level of sociality and social entrainment of circadian rhythms. Social entrainment is specifically potent in social animals that live in constant environments in which some or all individuals do not experience the ambient day-night cycles. We next focus on highly social honeybees in which there is good evidence that social cues entrain the circadian clocks of nest bees and can override the influence of conflicting light-dark cycles. The current understanding of social synchronization in honeybees is consistent with self-organization models in which surrogates of forager activity, such as substrate-borne vibrations and colony volatiles, entrain the circadian clocks of bees dwelling in the dark cavity of the nest. Finally, we present original findings showing that social synchronization is effective even in an array of individually caged callow bees placed on the same substrate and is improved for bees in connected cages. These findings reveal remarkable sensitivity to social time-giving cues and show that bees with attenuated rhythms (weak oscillators) can nevertheless be socially synchronized to a common phase of activity. This article is part of the theme issue ‘Synchrony and rhythm interaction: from the brain to behavioural ecology’.

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.

We analyze the influence of an external sound source in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report synchronization patterns, induced by the frequency of the sound source. We show that the level of synchrony can be enhanced by choosing the frequency of the sound source and its amplitude as control parameters for synchronization patterns. We discuss a minimum model elucidating the modalities of the influence of music on the human brain., 6 Figures

We analyze the influence of an external sound source in a network of FitzHugh–Nagumo oscillators with empirical structural connectivity measured in healthy human subjects. We report synchronization patterns, induced by the frequency of the sound source. We show that the level of synchrony can be enhanced by choosing the frequency of the sound source and its amplitude as control parameters for synchronization patterns. We discuss a minimum model elucidating the modalities of the influence of music on the human brain.

Asymmetric gait is associated with pain, injury, and reduced stability in patient populations. Data from side by side walking suggest that unintentional synchronization with an external cue may reduce gait asymmetry. Two types of asymmetric gait were examined here: (1) mass imbalance between limbs to simulate single limb amputation and (2) restriction of plantarflexion during toe-off to simulate reduced propulsion from neurological impairment. Twenty-five healthy participants walked normally and with simulated gait asymmetry on a custom-designed treadmill that oscillated in the vertical direction via pneumatic actuation (amplitude: 2 cm, frequency: participant’s preferred step frequency). Swing Time Asymmetry (STA) and Phase Coordination Index (PCI) both increased significantly with the application of unilateral mass and plantarflexion restriction (p &lt, 0.001). However, walking with simulated asymmetry did not alter unintentional synchronization with the treadmill motion. Further, oscillation of the treadmill did not improve STA or PCI while walking with simulated asymmetry. Analysis of synchronized step clusters using the Weibull survival function revealed that synchronization with the platform persisted for longer durations when compared with data from side by side walking. These results suggest that walking on a vertically oscillating surface may not be an effective approach for improving gait asymmetry.

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.

The generation of flows over long distances is crucial for the functioning of many living organisms. A widespread mechanism for generating such flows is via the contraction of the actomyosin cortex pumping the surrounding fluid. Even in the absence of a "stimulator" such as the heart, it is found that the dynamics of the actomyosin cortex can self-organize to give rise to coordinated flows over long distances. Like E. coli or C.elegans, Physarum polycephalum is one of the model organisms intensely studied by the scientific community. In its vegetative phase, called plasmodium, this organism consists of thousands of undifferentiated cells that have fused to form a multinucleated giant cell that can reach macroscopic sizes (several tens of cm²). This organism then develops a tubular network in which oscillating flows are generated by the active contraction of the membranous layer surrounding the tubular elements. The role of this transport network is twofold : on the one hand, it supplements diffusive flows that are too weak to transport nutrients and various biochemical signals on such scales. On the other hand, it allows the transport of the body mass of the organism by playing on the soil-gel transition of the cytoplasm. P. Polycephalum thus appears to be one of the simplest organisms with a transport network. Despite this apparent simplicity, the formation of this tubular network shares many similarities with the development of the vascular network of more evolved organisms, or that leading to the irrigation of tumors. In particular, we can clearly identify two phases in the development of the network : first, a phase of growth of a densely reticulated primary network, then a phase of reorganization during which the primary network becomes hierarchical, with some veins growing at the expense of others that disappear. From a physical point of view, plasmodium behaves like an active gel, which at short times generates waves of contraction inducing peristaltic movements of the fluid, and at long times controls its sol-gel transition to grow, move, and modify the architecture of its network. This thesis work focuses on the characterization of the contractile activity in P. Polycephalum, and its coupling with the cytoplasmic flows crossing the network (at short times) and its evolution (at long times). The non-neural adaptation of contractile activity and flows to the external environment (via an unknown feedback loop) is suspected to be at the origin of the organism’s ability to perform complex tasks such as preferential migration or the calculation of the shortest paths between food sources. Characterizing the nature of this unknown self-organizing mechanism is therefore a step towards understanding the emergence of animal cognition. Using a transmission imaging device, I have characterized the contractile patterns observed in phases of stagnation and growth of the plasmodium, and shown that the latter requires a coordination of contractile activity over large distances from the organism, which lacks a heart and a nervous system. Then, in a work mixing experiments and modelling, I have highlighted the effect of spatial confinement on the contractile activity, on the flow patterns this activity generates within the cytoplasm and on the reorganization of the tubular network. Finally, from the observations of reorganization and numerical analysis of the tubular networks formed, I propose a model of network hierarchization whose principle is based on the minimization of flows’ viscous dissipation.; La génération d’écoulements sur de grandes distances est cruciale pour le fonctionnement de nombreux organismes vivants. Un mécanisme très répandu pour générer de tels écoulements est via la contraction du cortex d’actomyosine pompant le fluide environnant. Même en l’absence d’un “stimulateur” tel que le cœur, on constate que la dynamique du cortex d’actomyosine peut s’auto-organiser pour donner lieu à des écoulements coordonnés sur de grandes distances. À l’instar de E. coli ou C. elegans, Physarum polycephalum fait partie des organismes modèles intensément étudiés par la communauté scientifique. Dans sa phase végétative, appelé plasmode, cet organisme est constitué de milliers de cellules indifférenciées qui ont fusionné pour former une cellule géante multinucléée pouvant atteindre des tailles macroscopiques (plusieurs dizaines de cm²). Cet organisme développe alors un réseau tubulaire dans lequel des écoulements oscillants sont générés par la contraction active de la couche membraneuse entourant les éléments tubulaires. Le rôle de ce réseau de transport est double : d’une part il suppléé aux écoulements diffusifs qui sont trop faibles pour transporter les nutriments et les divers signaux biochimiques sur de telles échelles. D’autre part il permet de transporter la masse corporelle de l’organisme en jouant sur la transition sol-gel du cytoplasme. P. Polycephalum apparaît donc comme l’un des organismes les plus simples pourvus d’un réseau de transport. En dépit de cette apparente simplicité, la formation de ce réseau tubulaire partage de nombreux points communs avec le développement du réseau vasculaire d’organismes plus évolués, ou celui conduisant à l’irrigation de tumeurs. En particulier, on peut identifier clairement deux phases dans le développement du réseau : tout d’abord une phase de croissance d’un réseau primaire densément réticulé, puis une phase de réorganisation au cours de laquelle le réseau primaire se hiérarchise, certaines veines grossissant aux dépens d’autres qui disparaissent. D’un point de vue physique, le plasmodium se comporte comme un gel actif, qui aux temps courts génère des ondes de contraction induisant les mouvements péristaltiques du fluide, et aux temps longs contrôle sa transition sol-gel pour croître, se mouvoir et modifier l’architecture de son réseau. Ce travail de thèse porte sur la caractérisation de l’activité contractile chez P. Polycephalum, et de son couplage avec les écoulements cytoplasmiques traversant le réseau (aux temps courts) et de l’évolution de celui-ci (aux temps longs). L’adaptation non-neurale de l’activité contractile et des écoulements à l’environnement extérieur (via une boucle de rétroaction inconnue) est soupçonnée d’être à l’origine de la capacité de l’organisme à réaliser des tâches complexes telles que la migration préférentielle ou le calcul des plus courts chemins entre des sources de nourritures. La caractérisation de la nature de ce mécanisme auto-organisateur inconnu constitue donc un pas vers la compréhension de l’émergence de la cognition animale. À l’aide d’un dispositif d’imagerie en transmission, j’ai caractérisé les motifs contractiles observés dans desphases de stagnation et de croissance du plasmode et montré que cette dernière nécessite une coordination de l’activité contractile sur de grandes distances de la part de l’organisme, pourtant dépourvu de cœur et de système nerveux. Puis, dans un travail mêlant expériences et modélisations, j’ai mis en évidence l’effet du confinement spatial sur l’activité contractile, sur les écoulements que celle ci génère au sein du cytoplasme et sur la réorganisation du réseau tubulaire. Enfin, à partir des observations de réorganisation et de l’analyse numérique des réseaux tubulaires formés je propose un modèle de hiérarchisation de réseaux dont le principe repose sur la minimisation de la dissipation visqueuse due aux écoulements.

Nitrergic nerves have been proposed to play a critical role in the orchestration of peristaltic activities throughout the gastrointestinal tract. In the present study, we investigated the role of nitric oxide, using spatiotemporal mapping, in peristaltic activity of the wholeex vivomouse intestine. We identified a propulsive motor pattern in the form of propagating myogenic contractions, that are clustered by the enteric nervous system into a minute rhythm that is dependent on nitric oxide. The cluster formation was abolished by TTX, lidocaine and nitric oxide synthesis inhibition, whereas the myogenic contractions, occurring at the ICC-MP initiated slow wave frequency, remained undisturbed. Cluster formation, inhibited by block of nitric oxide synthesis, was fully restored in a highly regular rhythmic fashion by a constant level of nitric oxide generated by sodium nitroprusside; but the action of sodium nitroprusside was inhibited by lidocaine indicating that it was relying on neural activity, but not rhythmic nitrergic nerve activity. Hence, distention-induced activity of cholinergic nerves and/or a co-factor within nitrergic nerves such as ATP is also a requirement for the minute rhythm. Cluster formation was dependent on distention but was not evoked by a distention reflex. Block of gap junction conductance by carbenoxolone, dose dependently inhibited, and eventually abolished clusters and contraction waves, likely associated, not with inhibition of nitrergic innervation, but by abolishing ICC network synchronization. An intriguing feature of the clusters was the presence of bands of rhythmic inhibitions at 4–8 cycles/min; these inhibitory patches occurred in the presence of tetrodotoxin or lidocaine and hence were not dependent on nitrergic nerves. We propose that the minute rhythm is generated by nitric oxide-induced rhythmic depolarization of the musculature via ICC-DMP.

Oscillatory Neural Networks (ONNs) are currently arousing interest in the research community for their potential to implement very fast, ultra-low-power computing tasks by exploiting specific emerging technologies. From the architectural point of view, ONNs are based on the synchronization of oscillatory neurons in cognitive processing, as occurs in the human brain. As emerging technologies, VO2 and memristive devices show promising potential for the efficient implementation of ONNs. Abundant literature is now becoming available pertaining to the study and building of ONNs based on VO2 devices and resistive coupling, such as memristors. One drawback of direct resistive coupling is that physical resistances cannot be negative, but from the architectural and computational perspective this would be a powerful advantage when interconnecting weights in ONNs. Here we solve the problem by proposing a hardware implementation technique based on differential oscillatory neurons for ONNs (DONNs) with VO2-based oscillators and memristor-bridge circuits. Each differential oscillatory neuron is made of a pair of VO2 oscillators operating in anti-phase. This way, the neurons provide a pair of differential output signals in opposite phase. The memristor-bridge circuit is used as an adjustable coupling function that is compatible with differential structures and capable of providing both positive and negative weights. By combining differential oscillatory neurons and memristor-bridge circuits, we propose the hardware implementation of a fully connected differential ONN (DONN) and use it as an associative memory. The standard Hebbian rule is used for training, and the weights are then mapped to the memristor-bridge circuit through a proposed mapping rule. The paper also introduces some functional and hardware specifications to evaluate the design. Evaluation is performed by circuit-level electrical simulations and shows that the retrieval accuracy of the proposed design is comparable to that of classic Hopfield Neural Networks. Unión Europea H2020 grant 871501 “NeurONN,” Spanish Ministry of Economy and Competitivity grant PID2019-105556GB-C31 (NANOMIND) (with support from the European Regional Development Fund).