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We consider the effect of an external periodic force on chimera states in the phase oscillator model proposed in [Phys. Rev. Lett, v. 101, 00319007 (2008)]. Using the Ott – Antonsen reduction, the dynamical equations for the global order parameter characterizing the degree of synchronization are constructed. The frequency locking by an external periodic force region is constructed. The possibility of stable chimeras synchronization and unstable chimeras stabilization is established. The instability development of the chimera states leads to the appearance of breather chimeras or complete synchronization. [ABSTRACT FROM AUTHOR]

Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase shifts as a function of the phase at which the input is received. When the pulse is weak, it is customary to linearize around zero pulse strength. The result is called the infinitesimal phase response function. These ideas have been used extensively in theoretical biology, and also in some areas of engineering. I give examples showing that the infinitesimal phase response function may predict that two oscillators, as they exchange pulses back and fourth, will converge to synchrony, yet this is false when the exact phase response function is used, for all positive interaction strengths. For short, the analogue of the Hartman–Grobman theorem that one might expect to hold at first sight is invalid. I give a condition under which the prediction derived using the infinitesimal phase response function does hold for the exact phase response function when interactions are sufficiently weak but of positive strength. However, I argue that this condition may often fail to hold.

It is known that the gait changes in accordance with the travel velocity by exploiting the body dynamics in the study of quadruped robots. For instance, some studies attained passive dynamic quadruped gaits such as a pace with a rigid trunk and a trot with a roll joint. The quadruped gait by the decentralized phase oscillator was reported, which exhibited a gait transition due to the intrinsic oscillator frequency. The robots had relatively short legs, which provides a stable upright posture, and the locomotion would not be influenced strongly by body dynamics. In this study, the quadruped RW06-Duo based on the passive dynamic walking mechanism, in which CoM is located at a high position, is developed and attains quadruped locomotion by the phase oscillator. The limb joints except for the knee (elbow) can rotate freely and passively; therefore, intralimb coordination is attained passively, and the phase oscillator and spine stiffness provide interlimb coordination. RW06-Duo exhibits different gaits according to travel velocity and demonstrates the gait transition from pace to trot or DS walk for the same oscillator frequency and stiffness. [ABSTRACT FROM AUTHOR]

This work introduces a methodology for studying synchronization in adaptive networks with heterogeneous plasticity (adaptation) rules. As a paradigmatic model, we consider a network of adaptively coupled phase oscillators with distance-dependent adaptations. For this system, we extend the master stability function approach to adaptive networks with heterogeneous adaptation. Our method allows for separating the contributions of network structure, local node dynamics, and heterogeneous adaptation in determining synchronization. Utilizing our proposed methodology, we explain mechanisms leading to synchronization or desynchronization by enhanced long-range connections in nonlocally coupled ring networks and networks with Gaussian distance-dependent coupling weights equipped with a biologically motivated plasticity rule.

This study aims to generate 3-D dynamic bipedal walking using two oscillators of a CPG approach based on its body dynamics. First, a 3-D biped, which inherited the aspect of the previous passive dynamic walker, was developed that contained flat feet with pressure sensors, ankle joints composed of ball joints surrounded by coil springs, telescopic knees, and actuated hip joints using a direct drive (DD) motor. The torso was controlled by the DD motors using a PD control scheme. Next, the Owaki–Ishiguro method was used with ground contact information in the phase oscillator to the knee joint. Finally, experimental tests were conducted to elucidate the effectiveness of the biped. A 3-D dynamic bipedal gait of RW06 was developed using knee oscillation based on the phase oscillator. Bipedal walking was excited only by the knee oscillation; therefore, the gait was strongly affected by the body dynamics. [ABSTRACT FROM AUTHOR]

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks.

A system consisting of interconnected networks, or a network of networks (NoN), appears diversely in many real-world systems, including the brain. In this study, we consider NoNs consisting of heterogeneous phase oscillators and investigate how the topology of subnetworks affects the global synchrony of the network. The degree of synchrony and the effect of subnetwork topology are evaluated based on the Kuramoto order parameter and the minimum coupling strength necessary for the order parameter to exceed a threshold value, respectively. In contrast to an isolated network in which random connectivity is favorable for achieving synchrony, NoNs synchronize with weaker interconnections when the degree distribution of subnetworks is heterogeneous, suggesting the major role of the high-degree nodes. We also investigate a case in which subnetworks with different average natural frequencies are coupled to show that direct coupling of subnetworks with the largest variation is effective for synchronizing the whole system. In real-world NoNs like the brain, the balance of synchrony and asynchrony is critical for its function at various spatial resolutions. Our work provides novel insights into the topological basis of coordinated dynamics in such networks. [ABSTRACT FROM AUTHOR]

Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry S N are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive—and we characterize generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ϵ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ϵ > 0 . Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with S N symmetry. For fixed N and taking the limit 0 < ϵ ≪ λ ≪ 1 , we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto–Sakaguchi system of coupled oscillators. The next order of approximation generically includes terms with up to four interacting phases, regardless of N . Using a normalization that maintains nontrivial interactions in the limit N → ∞ , we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size. [ABSTRACT FROM AUTHOR]

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear--for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience. [ABSTRACT FROM AUTHOR]

Bacterial cells maintain their characteristic cell size over many generations. Several rod-shaped bacteria, such as Escherichia coli and the cyanobacteria Synechococcus elongatus , divide after adding a constant length to their length at birth. Through this division control known as the adder mechanism, perturbation in cell length due to physiological fluctuation decays over generations at a rate of 2 − 1 per cell division. However, previous experiments have shown that the circadian clock in cyanobacteria reduces cell division frequency at a specific time of day under constant light. This circadian gating should modulate the division control by the adder mechanism, but its significance remains unknown. Here we address how the circadian gating affects cell length, doubling time, and cell length stability in cyanobacteria by using mathematical models. We show that a cell subject to circadian gating grows for a long time, and gives birth to elongated daughter cells. These elongated daughter cells grow faster than the previous generation, as elongation speed is proportional to cell length and divide in a short time before the next gating. Hence, the distributions of doubling time and cell length become bimodal, as observed in experimental data. Interestingly, the average doubling time over the population of cells is independent of gating because the extension of doubling time by gating is compensated by its reduction in the subsequent generation. On the other hand, average cell length is increased by gating, suggesting that the circadian clock controls cell length. We then show that the decay rate of perturbation in cell length depends on the ratio of delay in division by the gating τ G to the average doubling time τ 0 as 2 τ G / τ 0 − 1. We estimated τ G ≈ 2. 5 , τ 0 ≈ 13. 6 hours, and τ G / τ 0 ≈ 0. 18 from experimental data, indicating that a long doubling time in cyanobacteria maintains the decay rate similar to that of the adder mechanism. Thus, our analysis suggests that the acquisition of the circadian clock during evolution did not impose a constraint on cell size homeostasis in cyanobacteria. [Display omitted] • Cyanobacterial cell division regulated by a circadian clock is studied. • The circadian clock causes two cell populations with different doubling times. • Average cell length is increased by the clock, but average doubling time is not. • Cell size homeostasis depends on the delay in cell division by the circadian clock. • The circadian clock was not an evolutionally constraint on cell size homeostasis. [ABSTRACT FROM AUTHOR]

CPG (Central pattern generator) is a dynamical system of coupled nonlinear oscillators or neural networks inspired by a control mechanism in animal bodies. Without any rhythmic inputs, the CPG has the ability to produce oscillatory patterns. This paper presents a novel structure of a CPG network which can produce rhythmic motion that imitates movement of animals such as snake and lamprey. The focus is on the locomotion control of a snake-like robot, where phase oscillator has been adopted as the dynamical model to control the harmonic motion of the CPG network. There are two main points addressed in this paper: (1) simple network structure of unidirectional coupling oscillators, and (2) a single parameter to control the body shape and to control the forward and backward movement of the snake-like robot. The proposed CPG network is designed to have a simple structure with less complexity, less mathematical computation, fast convergence speed and exhibit limit cycle behavior. In addition, a new parameter, τ is introduced to control the smoothness of the CPG output as well as the speed of the snake-like robot. Simulation and experimental results show that the proposed CPG network can be used to control the serpentine locomotion of a snake-like robot. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a model to study the effects of coupling-frequency correlations on synchronization in complete graphs. When the linear correlation is adopted, we find a symmetric network where frequencies of the oscillators are distributed in a bipolar way, having values either −1 or +1. In the network, the oscillators either all drift or all phase-locked. The behavior can separate qualitatively two other types of correlations, where slow and fast oscillators can remain unsynchronized respectively. It is obvious that the weighting exponent plays an important role. Besides, the numerical simulation results indicate that the linear correlation has the best performance in synchronization ability among three types of correlations in view of the average node cost. [Copyright &y& Elsevier]

Abstract: The Kuramoto model is modified by introducing a negative coupling strength, which is a generalization of the original one. Among the abundant dynamics, the clustering phenomenon of the modified Kuramoto model is analyzed in detail. After clustering appears in a network of coupled oscillators, the nodes are split into several clusters by their phases, in which the phases difference within each cluster is less than a threshold and larger than a threshold between different clusters. We show that this interesting phenomenon can be applied to identify the complete sub-graphs and further applied to graph coloring problems. Simulations on test beds of graph coloring problems have illustrated and verified the scheme. [Copyright &y& Elsevier]

Abstract: Almost all organisms show some kind of time periodicity in their behavior. In mammals, the neurons of the suprachiasmatic nucleus form a biological clock regulating the activity–inactivity cycle of the animal. The main question is how this clock is able to entrain to the natural 24 h light–dark cycle by which it is stimulated. Such a system is usually modeled as a collection of mutually coupled two-state (active–inactive) phase oscillators with an external stimulus (Zeitgeber). In this article however, we investigate the entrainment of a single pacer cell to the ensemble of other pacer cells. Moreover the stimulus of the ensemble is taken to be periodic. The pacer cell interacts with its environment by phase delay at the end of its activity interval and phase advance at the end of its inactivity interval. We develop a mathematical model for this system, naturally leading to a circle map depending on parameters like the intrinsic period and phase delay and advance. The existence of resonance tongues in a circle map shows that an individual pacer cell is able to synchronize with the ensemble. We furthermore show how the parameters in the model can be related to biological observable quantities. Finally we give several directions of further research. [Copyright &y& Elsevier]

Abstract: In this Letter, self-organization of directed networks is surveyed. Inspired from the results in neural networks research, we propose an asymmetric coupling scheme with simple edge deleting rules. Results show that all-to-all networks can be organized into scale-free networks with feed-forward structures. Corresponding analysis is also given. [ABSTRACT FROM AUTHOR]

Synchronization of coupled oscillators is by now a very well studied subject. A large number of analytical and computational tools are available for the treatment of experimental results. This article focuses on a method recently proposed to construct a phase model from experimental data. The advantage of this method is that it extracts a phase model in a noninvasive manner without any prior knowledge of the experimental system. The aim of the present research is to apply this methodology to a network of electronic genetic oscillators. These electronic circuits mimic the dynamics of a synthetic genetic oscillator, called "repressilator", which is capable of synthesizing autonomous biological rhythms. The obtained phase model is shown to be capable of recovering the route leading to synchronization of genetic oscillators. The predicted onset point of synchronization agrees quite well with the experiment. This encourages further application of the present method to synthetic biological systems. [ABSTRACT FROM AUTHOR]

The role of relative spike timing on sensory coding and stochastic dynamics of small pulse-coupled oscillator networks is investigated physiologically and mathematically, based on the small biological eye network of the marine invertebrate Hermissenda. Without network interactions, the five inhibitory photoreceptors of the eye network exhibit quasi-regular rhythmic spiking; in contrast, within the active network, they display more irregular spiking but collective network rhythmicity. We investigate the source of this emergent network behavior first analyzing the role of relative input to spike–timing relationships in individual cells. We use a stochastic phase oscillator equation to model photoreceptor spike sequences in response to sequences of inhibitory current pulses. Although spike sequences can be complex and irregular in response to inputs, we show that spike timing is better predicted if relative timing of spikes to inputs is accounted for in the model. Further, we establish that greater noise levels in the model serve to destroy network phase-locked states that induce non-monotonic stimulus rate-coding, as predicted in Butson and Clark (J Neurophysiol 99:146–154, 2008a; J Neurophysiol 99:155–165, 2008b). Hence, rate-coding can function better in noisy spiking cells relative to non-noisy cells. We then study how relative input to spike–timing dynamics of single oscillators contribute to network-level dynamics. Relative timing interactions in the network sharpen the stimulus window that can trigger a spike, affecting stimulus encoding. Also, we derive analytical inter-spike interval distributions of cells in the model network, revealing that irregular Poisson-like spike emission and collective network rhythmicity are emergent properties of network dynamics, consistent with experimental observations. Our theoretical results generate experimental predictions about the nature of spike patterns in the Hermissenda eye. [ABSTRACT FROM AUTHOR]

A three-layer network model of oscillatory associative memory is proposed. The network is capable of storing binary images, which can be retrieved upon presenting an appropriate stimulus. Binary images are encoded in the form of the spatial distribution of oscillatory phase clusters in-phase and anti-phase relative to a reference periodic signal. The information is loaded into the network using a set of interlayer connection weights. A condition for error-free pattern retrieval is formulated, delimiting the maximal number of patterns to be stored in the memory (storage capacity). It is shown that the capacity can be significantly increased by generating an optimal alphabet (basis pattern set). The number of stored patterns can reach values of the network size (the number of oscillators in each layer), which is significantly higher than the capacity of conventional oscillatory memory models. The dynamical and information characteristics of the retrieval process based on the optimal alphabet, including the size of “attraction basins“ and the input pattern distortion admissible for error-free retrieval, are investigated. [ABSTRACT FROM AUTHOR]

Abstract: The synchronization process of coupled phase oscillators is further investigated in this Letter. Coupled with a new adaptive scheme, the network can be stable in a middle state which oscillates between the synchronization state and the asynchronous state, namely there exists a third steady state. We call this novel phenomenon ‘order-oscillation’. Related results and detailed analysis are given at last. [Copyright &y& Elsevier]

Abstract: Spatially heterogeneous intensities of environmental signals are common in nature, being caused, e.g., by rugged or curved surfaces leading to varying angles of incidence and intensities. In this work, we perform numerical studies of one-dimensional arrays of coupled phase oscillators driven by a periodic signal with spatially heterogeneous amplitude, considering both random and gradual amplitude distributions of the driving. We compare the effects of global and next-neighbor interactions, respectively, on the mutual and forced synchronization in the array. Weak global coupling leads to full mutual synchronization for all studied driving configurations. The degree of external synchronization follows a majority rule, depending on the number of externally entrained oscillators in the uncoupled case. The effects of next-neighbor coupling depend on the spatial distribution of the driving amplitude. For random distributions, local interactions show the same qualitative effects as global coupling. In contrast, for gradual distributions and large driving heterogeneities, next-neighbor coupling is detrimental to both mutual and external synchronization. We discuss these observations with respect to fundamental aspects of heterogeneity and variability of dynamical systems, as well as the intercellular synchronization of circadian oscillators. [Copyright &y& Elsevier]

Neural reliability and stochastic synchronization are remarkable features of real neurons with important consequences for neural computation. Both phenomena are general properties of any device with a resetting threshold, such as neurons. However, certain characteristics of the single neuron dynamics can notably improve neural reliability and stochastic synchronization. Here we show that neural resonators synchronize more reliably and more robustly than neural integrators. This suggests that neurons conveying sensory information in a spike-timing code are likely to be resonators, as supported by our recent studies on reliability and stochastic synchronization in the olfactory bulb. [Copyright &y& Elsevier]