Search

We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrodinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj–Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj–Pikovsky model. We examine the stability of trajectories belonging to invariant manifolds by means of numerical evaluation of Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics on invariant manifolds and conservation of phase volume of the Hamiltonian system. We demonstrate the complexity of dynamics with results of numerical simulations.

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 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.

The Kuramoto model, which serves as a paradigm for investigating synchronization phenomena of oscillatory systems, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here we generalize a number of classical results by presenting a general framework for analytically capturing the critical scaling properties of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on the coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determine the scaling properties of order parameter above criticality.

Power grid networks, as well as neuronal networks with synaptic plasticity, describe real-world systems of tremendous importance for our daily life. The investigation of these seemingly unrelated types of dynamical networks has attracted increasing attention over the past decade. In this paper, we provide insight into the fundamental relation between these two types of networks. For this, we consider well-established models based on phase oscillators and show their intimate relation. In particular, we prove that phase oscillator models with inertia can be viewed as a particular class of adaptive networks. This relation holds even for more general classes of power grid models that include voltage dynamics. As an immediate consequence of this relation, we discover a plethora of multicluster states for phase oscillators with inertia. Moreover, the phenomenon of cascading line failure in power grids is translated into an adaptive neuronal network.

Phase oscillator systems with global sine-coupling are known to exhibit low-dimensional dynamics. In this paper, such characteristics are extended to phase oscillator systems driven by Cauchy noise. The low-dimensional dynamics solution agreed well with the numerical simulations of noise-driven phase oscillators in the present study. The low-dimensional dynamics of identical oscillators with Cauchy noise coincided with those of heterogeneous oscillators with Cauchy-distributed natural frequencies. This allows for the study of noise-driven identical oscillator systems through heterogeneous oscillators without noise and vice versa., Comment: 12 pages, 2 figures

We investigate the transition to phase and frequency synchronization in a one-dimensional chain of phase oscillator “cells” where the coupling is mediated by the local concentration of a chemical which can diffuse in the inter-oscillator medium and it is both secreted and absorbed by the oscillator “cells”, influencing their dynamical behavior. This coupling has the advantage of having a tunable parameter which makes it possible to pass continuously from a global (all-to-all) to a local (nearest-neighbor) coupling form. We have verified that synchronous behavior depends on the coupling strength and coupling length.

A coupled phase-oscillator model consists of phase-oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the non-synchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality class is reduced to one and the critical exponent is shared in the considered models having coupling functions up to the second harmonics with unimodal and symmetric natural frequency distributions., Comment: 9 pages, 8 figures

Interaction mechanism in the acoustic communication of actual animals is investigated by combining mathematical modeling and empirical data. Here we use a deterministic mathematical model (a phase oscillator model) to describe the interaction mechanism underlying the choruses of male Japanese tree frogs (Hyla japonica) in which the male frogs attempt to avoid call overlaps with each other due to acoustic communication. The mathematical model with a general interaction term is identified by a Bayesian approach from multiple audio recordings on the choruses of three male frogs. The identified model qualitatively reproduces the stationary and dynamical features of the empirical data, supporting the validity of the model identification. In addition, we quantify the magnitude of attention paid among the male frogs from the identified model, and then analyze the relationship between the attention and behavioral parameters by using a statistical model. The analysis demonstrates the biologically valid relationship about the negative correlation between the attention and inter-frog distance, and also indicates the existence of a behavioral strategy that the male frogs selectively pay attention towards a less attractive male frog so as to utilize the advantage of their attractiveness for effective mate attraction.

This paper presents the design of a phase-based robust oscillator for wearable robots that assists the human performing periodic or repetitive tasks. The robustness of the phase oscillator controller is evaluated by finding bounds for perturbations that guaranteed the stability of the output. Then, the Lyapunov redesign method is applied to construct a robust controller using a bounding function which can handle the uncertainties such as noise and perturbations in the overall human-robot system. The robust controller produces a bounded control signal to modify the amplitude and frequency of the resulting second-order oscillator to modulate the stiffness and damping properties. In this paper, the focus is put on the wearable robot that assists human hip joint while performing periodic activities such as walking. The proposed approach is verified through a simple pendulum experiment. The results show that a better limit cycle can be obtained with Lyapunov redesigned phase oscillator which controls the radial spread of the steady state. Finally, the potential of the proposed approach for hip assistance in a healthy subject wearing HeSa (Hip Exoskeleton for Superior Assistance) during periodic activities are discussed.

We review chimera patterns, which consist of coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics in networks of identical oscillators. We focus on chimera states involving amplitude as well as phase dynamics, complex topologies like small-world or hierarchical (fractal), noise, and delay. We show that a plethora of novel chimera patterns arise if one goes beyond the Kuramoto phase oscillator model. For the FitzHugh-Nagumo system, the Van der Pol oscillator, and the Stuart-Landau oscillator with symmetry-breaking coupling various multi-chimera patterns including amplitude chimeras and chimera death occur. To test the robustness of chimera patterns with respect to changes in the structure of the network, regular rings with coupling range R, small-world, and fractal topologies are studied. We also address the robustness of amplitude chimera states in the presence of noise. If delay is added, the lifetime of transient chimeras can be drastically increased.

Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov exponents constructed so far, we show that weak chimeras that are chaotic can exist in the limit of vanishing coupling between coupled populations of phase oscillators. We present numerical evidence that positive Lyapunov exponents can persist for a positive measure set of this inter-population coupling strength.

In this paper, we present a quadrature-phase oscillator that can provide four output voltages while operating from a single 0.5 V supply. The oscillator is based on two cross-coupled modified differential pair cells and provides signals with a phase difference of ± 180 ° or ± 90 ° depending on the chosen output nodes. A test chip with an active area of 0.175 mm2 was designed and fabricated in a 65-nm CMOS process and its measurement results show a phase noise of −119.5 dBc/Hz at 1 MHz offset from a carrier frequency of 1.967 GHz while consuming 6.15mW. Finally, experimental results show a FoM of 185.5 dB for the core oscillator and a quadrature-phase error of less than ± 0.5 ° .

(c) 2019 American Physical Society, We study correspondence between a phase oscillator network with distributed natural frequencies and a classical XY model at finite temperatures with the same random and frustrated interactions used in the Sherrington-Kirkpatrick model. We perform numerical calculations of the spin glass order parameter q and the distributions of the local fields P(R), where R is the amplitude of the local field. As a result, we find that the parameter dependences of P(R) in both models agree fairly well in all ranges of parameters in the spin glass phase and those of q agree at least for lower values of parameters in the spin glass phase, if parameters are normalized by using the previously obtained correspondence relation between two models with the same other types of interactions. Furthermore, we numerically calculate the time evolution of quantities such as the instantaneous local field in the phase oscillator network in order to study the roles of synchronous and asynchronous oscillators. We also study the self-consistent equation of the local fields in the oscillator network and XY model derived by the mean-field approximation.

Synchronisation and stability under periodic oscillatory driving are well-understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counter-intuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronisation where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilisation phenomenon is numerically observed. Our findings help support the case that in general, deterministic non-autonomous perturbation is a very good candidate for stabilising complex dynamics., to be published in Phys. Rev. E

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.

Purpose– This paper aims to develop a robofish with oscillating pectoral fins, and control it to mimic the bionic prototype by central pattern generators (CPGs).Design/methodology/approach– First, the oscillation characteristics of the cownose ray were analyzed quantitatively. Second, a robofish with multi-joint pectoral fins was developed according to the bionic morphology and kinematics. Third, the improved phase oscillator was established, which contains a spatial asymmetric coefficient and a temporal asymmetric coefficient. Moreover, the CPG network is created to mimic the cownose ray and accomplish three-dimensional (3D) motions. Finally, the experiments were done to test the authors ' works.Findings– The results demonstrate that the CPGs is effective to control the robofish to imitate the cownose ray realistically. In addition, the robofish is able to accomplish 3D motions of high maneuverability, and change among different swimming modes quickly and smoothly.Originality/value– The research provides the method to develop a robofish from both 3D morphology and kinematics. The motion analysis and CPG control make sure that the robofish has the features of high maneuverability and camouflage. It is useful for military underwater applications and underwater detections in narrow environments. Second, this work lays the foundation for the autonomous 3D control. Moreover, the robotic fish can be taken as a scientific tool for the fluid bionics research.

In our previous work, we studied an interconnected bursting neuron model for insect locomotion, and its corresponding phase oscillator model, which at high speed can generate stable tripod gaits with three legs off the ground simultaneously in swing, and at low speed can generate stable tetrapod gaits with two legs off the ground simultaneously in swing. However, at low speed several other stable locomotion patterns, that are not typically observed as insect gaits, may coexist. In the present paper, by adding heterogeneous external input to each oscillator, we modify the bursting neuron model so that its corresponding phase oscillator model produces only one stable gait at each speed, specifically: a unique stable tetrapod gait at low speed, a unique stable tripod gait at high speed, and a unique branch of stable transition gaits connecting them. This suggests that control signals originating in the brain and central nervous system can modify gait patterns.

Functional oscillator networks, such as neuronal networks in the brain, exhibit switching between metastable states involving many oscillators. We give exact results how such global dynamics can arise in paradigmatic phase oscillator networks: higher-order network interaction gives rise to metastable chimeras - localized frequency synchrony patterns - which are joined by heteroclinic connections. Moreover, we illuminate the mechanisms that underly the switching dynamics in these experimentally accessible networks.

For a system of coupled identical phase oscillators with full permutation symmetry, any broken symmetries in dynamical behaviour must come from spontaneous symmetry breaking, i.e. from the nonlinear dynamics of the system. The dynamics of phase differences for such a system depends only on the coupling (phase interaction) function $g(\varphi)$ and the number of oscillators $N$. This paper briefly reviews some results for such systems in the case of general coupling $g$ before exploring two cases in detail: (a) general two harmonic form: $g(\varphi)=q\sin(\varphi-\alpha)+r\sin(2\varphi-\beta)$ and $N$ small (b) the coupling $g$ is odd or even. We extend previously published bifurcation analyses to the general two harmonic case, and show for even $g$ that the dynamics of phase differences has a number of time-reversal symmetries. For the case of even $g$ with one harmonic it is known the system has $N-2$ constants of the motion. This is true for $N=4$ and any $g$, while for $N=4$ and more than two harmonics in $g$, we show the system must have fewer independent constants of the motion., Comment: 30 pages, 9 figures

We design an artificial central pattern generator and we exemplify its integration using different prototypic virtual robot models. The artificial central pattern generator is modeled through a network of phase oscillators. Virtual robots are controlled by this central pattern generator using an interface that interpolates stored angular states of their target poses. Finally, we consider control signals coming from the robot's environment to the central pattern generator and to the interface between the central pattern generator and the robot. We show that the control signals can change the dynamics of the system in order to modify the robot's movement patterns. We study three cases: changing the frequency of the pattern sequences, switching between gaits of locomotion and the reorientation of a robotic entity.

Analog CMOS circuit implementation of a system of pulse-coupled phase oscillators is proposed. A CMOS circuit that achieves the dynamics of pulse-coupled oscillators has been designed and fabricated using a 0.25-µm CMOS technology. The proposed oscillator circuits with continuous-time operation interact with each other via a pulse at each firing time. Update of the oscillator state is achieved by integrating the phase sensitivity function with the pulse width time span. The phase sensitivity function is generated by the combination of binary functions, while the function consists of three-values {-1,0,1}. Introducing a zero-value span in the function leads to fast synchronization and robustness to parameter fluctuation due to LSI device mismatches, which facilitates VLSI implementation. Using the fabricated CMOS circuit, we have observed not only in- and anti-phase but also out-of-phase synchronization.

Almost all organisms show some kind of time periodicity in their behavior. Especially in mammals the neurons of the suprachiasmatic nucleus form a biological clock regulating the activity-inactivity cycle of the animal. This clock is stimulated by the natural 24-hour light-dark cycle. In our model of this system we consider each neuron as a so called phase oscillator, coupled to other neurons for which the light-dark cycle is a Zeitgeber. To simplify the model we first take an externally stimulated single phase oscillator. The first part of the phase interval is called the active state and the remaining part is the inactive state. Without external stimulus the oscillator oscillates with its intrinsic period. An external stimulus, be it from activity of neighboring cells or the periodic daylight cycle, acts twofold, it may delay the change form active to inactive and it may advance the return to the active state. The amount of delay and advance depends on the strength of the stimulus. We use a circle map as a mathematical model for this system. This map depends on several parameters, among which the intrinsic period and phase delay and advance. In parameter space we find Arnol'd tongues where the system is in resonance with the Zeitgeber. Thus already in this simplified system we find entrainment and synchronization. Also some other phenomena from biological experiments and observations can be related to the dynamical behavior of the circle map.

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.

The behavior of human circadian rhythms could be interpreted within the two-oscillator regime: one for the circadian pacemaker driving temperature/plasma melatonin rhythm and the other for the sleep–wake rhythm, tentatively called the SCN (suprachiasmatic nucleus) oscillator and non-SCN oscillator, respectively. Recently, the existence of the non-SCN oscillator was demonstrated through showing the possibility of non-photic entrainments by the shifted sleep schedule, and its interactions with the SCN oscillator were disclosed through analyzing the re-entrainment processes. Based on these experimental results at the behavioral level, we developed a phase oscillator model consisting of mutually coupled SCN and non-SCN oscillators, and an extra-oscillator representing an overt sleep–wake rhythm. Our model successfully reproduced the experimental results of the non-photic entrainments. In addition, our phase oscillator model and the widely accepted twoprocess model were shown to share a possible mechanism underlying interactions found between the SCN and non-SCN oscillators during the re-entrainment processes. Therefore, it was suggested that the interactions indicated essential dynamics of the SCN and non-SCN oscillators. Then, a possible mechanism underlying the interactions at the behavioral level was reduced into the conditions of coupling strength and intrinsic angular velocity in the coupled realistic oscillators. These conditions led us to the hypothesis at the organ level that a part of the SCN oscillator is non-photically entrained through being coupled with the non-SCN oscillator. Of course, its reality should be assessed further. This top-down approach is one of practical strategies for modeling the hierarchical biological system.

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.

We study the reliability of phase oscillator networks in response to fluctuating inputs. Reliability means that an input elicits essentially identical responses upon repeated presentations, regardless of the network’s initial condition. Single oscillators are well known to be reliable. We show in this paper that unreliable behavior can occur in a network as small as a coupled oscillator pair in which the signal is received by the first oscillator and relayed to the second with feedback. A geometric explanation based on shear-induced chaos at the onset of phase-locking is proposed. We treat larger networks as decomposed into modules connected by acyclic graphs, and give a mathematical analysis of the acyclic parts. Moreover, for networks in this class, we show how the source of unreliability can be localized, and address questions concerning downstream propagation of unreliability once it is produced.

We introduce an adaptation algorithm by which an ensemble of coupled oscillators with attractive and repulsive interactions is induced to adopt a prescribed synchronized state. While the performance of adaptation is controlled by measuring a macroscopic quantity, which characterizes the achieved degree of synchronization, adaptive changes are introduced at the microscopic level of the interaction network, by modifying the configuration of repulsive interactions. This scheme emulates the distinct levels of selection and mutation in biological evolution and learning.

Oscillator networks have been developed in order to perform specific tasks related to image processing. Here we analytically investigate the existence of synchronism in a pair of phase oscillators that are short-range dynamically coupled. Then, we use these analytical results to design a network able of detecting border of black-and-white figures. Each unit composing this network is a pair of such phase oscillators and is assigned to a pixel in the image. The couplings among the units forming the network are also dynamical. Border detection emerges from the network activity.

Behavioral diversity is one of the universal features for all animals but missing in artificial machines. Behavioral diversity enables animals to explore alternative behaviors and avoid to getting stuck in a dead-end situation due to severe environmental changes. Plasmodium of true slime mold (Physarum polycephalum) is one of the splendid models to investigate behavioral diversity of animals. We introduce a constructive approach to understand the versatile and adaptive behaviors of the plasmodium using a simulation model. The results obtained shed a new light on how to design artificial system in ways that allow behavioral diversity and purposeful behavior, e.g., chemotaxis or phototaxis.

Consider a distributed network on a finite simple graph $G=(V,E)$ with diameter $d$ and maximum degree $\Delta$, where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on $G$ synchronizes by time $51d$ if $G$ is a tree and $\Delta \le 3$. We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on $G$ by time $83d$ if $G$ is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses $O(\log \Delta)$ memory per node and converges on any $G$ with expected worst case running time of $O(|V|+(d^{5}+\Delta^{2})\log |V|)$., Comment: 41 pages, 21 figures, preprint. Definition of the adaptive 4-coupling is presented as a pesudocode. Simulation of the adaptive 4-coupling modulo $M=64$ is added in Figure 1, SIAM Journal on Applied Dynamical Systems, 2018

A quantitative comparison of various classes of oscillators (integrate-and-fire, Winfree, and Kuramoto-Daido type) is performed in the weak-coupling limit for a fully connected network of identical units. An almost perfect agreement is found, with only tiny differences among the models. We also show that the regime of self-consistent partial synchronization is rather general and can be observed for arbitrarily small coupling strength in any model class. As a byproduct of our study, we are able to show that an integrate-and-fire model with a generic pulse shape can be always transformed into a similar model with $\ensuremath{\delta}$ pulses and a suitable phase response curve.

Collective effects in one-dimensional ciliar arrays are studied analytically and numerically. A new phase oscillator description for ciliar motion is introduced which depends only on a single parameter. It allows the systematic study of hydrodynamic interactions between cilia exhibiting arbitrary beating patterns. It is shown that if the hydrodynamic interactions do not alter the beating pattern of the cilia no synchronization of ciliar motion occurs. This is in particular the case in arrays with low ciliar densities. But hydrodynamic interactions can lead to formation of a collective (metachronal) wave which is stable for periodic boundary conditions. In finite arrays free boundaries destroy this collective motion. The dispersion relation for metachronal waves is found to be non-universal, i.e., to depend crucially on the microscopic details of the ciliar beating pattern.

We model spiking neurons in locus coeruleus (LC), a brain nucleus involved in modulating cognitive performance, and compare with recent experimental data. Extracellular recordings from LC of monkeys performing target detection and selective attention tasks show varying responses dependent on stimuli and performance accuracy. From membrane voltage and ion channel equations, we derive a phase oscillator model for LC neurons. Average spiking probabilities of a pool of cells over many trials are then computed via a probability density formulation. These show that: (1) Post-stimulus response is elevated in populations with lower spike rates; (2) Responses decay exponentially due to noise and variable pre-stimulus spike rates; and (3) Shorter stimuli preferentially cause depressed post-activation spiking. These results allow us to propose mechanisms for the different LC responses observed across behavioral and task conditions, and to make explicit the role of baseline firing rates and the duration of task-related inputs in determining LC response.

A new method is proposed for increasing the sensitivity of a phase oscillator transducer by the use of positive phase feedback; the algorithm is described. Basic characteristics are given.

This study proposes a coupled system of phase oscillators for a single control of an urban traffic network. An area-wide control is given by a network of autonomous systems, which exchange local information without any central monitoring system. Each signal is modeled by a simple phase oscillator with a periodic motion, and pair-wise interactions with the adjacent signals control its offset to attain a desired value. The interaction strength depends on a traffic flow density between the corresponding pair of adjacent signals. The proposed control method is applicable to a rather general traffic flow, including a loop road, a one-way road, left- or right-turn traffic at a intersection, and any number of roads at an intersection. The results of computer simulations on several traffic flow patterns and conditions in a system of 2*2 signals are shown both with a macro model described by a flow density and with a micro model considering the dynamics of an individual car. Statistics such as average travel time show that, in most cases, effective control is achieved adaptively by the proposed method.

We suggest a definition for a type of chimera state that appears in networks of indistinguishable phase oscillators. Defining a "weak chimera" as a type of invariant set showing partial frequency synchronization, we show that this means they cannot appear in phase oscillator networks that are either globally coupled or too small. We exhibit various networks of four, six and ten indistinguishable oscillators where weak chimeras exist with various dynamics and stabilities. We examine the role of Kuramoto-Sakaguchi coupling in giving degenerate (neutrally stable) families of weak chimera states in these example networks., Comment: 9 figures

Cockroaches are remarkably stable runners, exhibiting rapid recovery from external perturbations. To uncover the mechanisms behind this important behavioral trait, we recorded leg kinematics of freely running animals in both undisturbed and perturbed trials. Functional coupling underlying inter-leg coordination was monitored before and during localized perturbations, which were applied to single legs via magnetic impulses. The resulting transient effects on all legs and the recovery times to normal pre-perturbation kinematics were studied. We estimated coupling architecture and strength by fitting experimental data to a six-leg-unit phase oscillator model. Using maximum-likelihood techniques, we found that a network with nearest-neighbor inter-leg coupling best fitted the data and that, although coupling strengths vary among preparations, the overall inputs entering each leg are approximately balanced and consistent. Simulations of models with different coupling strengths encountering perturbations suggest that the coupling schemes estimated from our experiments allow animals relatively fast and uniform recoveries from perturbations.

The human body is a spring-mass-damper system that oscillates up and down while moving. Wearable robots can add energy to the person at an appropriate time to enhance human moving performance and reduce the metabolic cost of performing cyclic tasks simultaneously. We have developed a phase oscillator to add assisted power at resonance so the system attains suitable energy to overcome the vertical resistance. The oscillator is robust to disturbances and initial conditions and allows wearable robots to enhance running, reduce metabolic cost, and increase hop height. Results show that when the external force is angled at 25 degrees, the phase oscillator can maintain robust bounded energy in a limit cycle to enhance human running performance.

Glacial-interglacial cycles are large variations in continental ice mass and greenhouse gases, which have dominated climate variability over the Quaternary. The dominant periodicity of the cycles is $\sim $40 kyr before the so-called middle Pleistocene transition between $\sim$1.2 and $\sim$0.7 Myr ago, and it is $\sim $100 kyr after the transition. In this paper, the dynamics of glacial-interglacial cycles are investigated using a phase oscillator model forced by the time-varying incoming solar radiation (insolation). We analyze the bifurcations of the system and show that strange nonchaotic attractors appear through nonsmooth saddle-node bifurcations of tori. The bifurcation analysis indicates that mode-locking is likely to occur for the 41 kyr glacial cycles but not likely for the 100 kyr glacial cycles. The sequence of mode-locked 41 kyr cycles is robust to small parameter changes. However, the sequence of 100 kyr glacial cycles can be sensitive to parameter changes when the system has a strange nonchaotic attractor., 25 pages, 9 figures

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with non-equal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.