The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed design of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitude death states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude unsynchronized domain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to death state are characterized using an average temporal interaction approximation, which agrees with the numerical results in temporal interaction. A first-order phase transition behavior may change into a second-order transition in spatial dynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possible abrupt first-order like transition is completely non-existent in the case of temporal dynamic interaction. Besides the study on periodic Stuart-Landau systems, we present results for paradigmatic chaotic model of Rössler oscillators and Mac-arthur ecological model.
We study the phase synchronization between collective rhythms of fully locked oscillator groups. For weakly interacting groups of two oscillators with global sinusoidal coupling, we analytically derive the collective phase coupling function, which determines the dynamics of the collective phase difference between the groups. We demonstrate that the groups can exhibit anti-phase collective synchronization in spite of microscopic in-phase external coupling and vice versa.
We theoretically investigate collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or anti-phase synchronization. We show tha…
We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct n…
Phase synchronization between collective oscillations exhibited by two weakly interacting groups of non-identical phase oscillators with internal and external global sinusoidal coupling of the groups is analyzed theoretically. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between…
Oscillators coupled in a network can synchronize with each other to yield a coherent population rhythm. If multiple such networks are coupled together, the question arises whether these rhythms will synchronize. We investigate the impact of noise on this synchronization for strong inhibitory pulse-coupling and find that increasing the noise can synchronize the population rhythms, even if the noisy inputs to different oscillators are completely uncorrelated. Reducing the system to a phenomenological iterated map we show that this synchronization of the rhythms arises from the noise-induced phas…
Systems of dynamical elements exhibiting spontaneous rhythms are found in various fields of science and engineering, including physics, chemistry, biology, physiology, and mechanical and electrical engineering. Such dynamical elements are often modeled as nonlinear limit-cycle oscillators. In this article, we briefly review phase reduction theory, which is a simple and powerful method for analyzing the synchronization properties of limit-cycle oscillators exhibiting rhythmic dynamics. Through phase reduction theory, we can systematically simplify the nonlinear multi-dimensional differential eq…
We explore the aging transition in a network of globally coupled Stuart-Landau oscillators under a discrete time-dependent coupling. In this coupling, the connections among the oscillators are turned ON and OFF in a systematic manner, having either a symmetric or an asymmetric time interval. We discover that depending upon the time period and duty cycle of the ON-OFF intervals, the aging region shrinks drastically in the parameter space, therefore promoting restoration of oscillatory dynamics from the aging. In the case of symmetric discrete coupling (where the ON-OFF intervals are equal), the…
Inhibitory circuits of relaxation oscillators are often-used models for the dynamics of biological networks. We present a qualitative and quantitative stability analysis of such a circuit constituted by three reciprocally coupled oscillators of a Fitzhugh-Nagumo type as nodes. Depending on inhibitory strengths, and parameters of individual oscillators, the circuit exhibits polyrhythmicity of up to five simultaneously stable rhythms. With methods of bifurcation analysis and phase reduction, we investigate qualitative changes in stability of these circuit rhythms for a wide range of parameters. …
Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the general…
In many animals, rhythmic motor activity is governed by neural limit cycle oscillations under the control of sensory feedback. In the fruit fly Drosophila melanogaster, the wingbeat rhythm is generated myogenically by stretch-activated muscles and hence independently from direct neural input. In this study, we explored if generation and cycle-by-cycle control of Drosophila's wingbeat are functionally separated, or if the steering muscles instead couple into the myogenic rhythm as a weak forcing of a limit cycle oscillator. We behaviourally tested tethered flying flies for characteristic proper…
Reverse quantum aging: Restoring rhythmicity from quantum aging state
Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan（巴德万大学物理系混沌与复杂系统研究实验室）；Department of Physics & Astronomy, Northwestern University（西北大学物理与天文学系）
A large variety of rhythms are observed in nature. Rhythms such as electroencephalogram signals in the brain can often be regarded as interacting. In this study, we investigate the dynamical properties of rhythmic systems in two populations of phase oscillators with different frequency distributions. We assume that the average frequency ratio between two populations closely approximates some small integer. Most importantly, we adopt a specific coupling function derived from phase reduction theory. Under some additional assumptions, the system of two populations of coupled phase oscillators red…
Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of identical rotators with higher-order coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminisc…
Employing both Bayesian statistics and the theory of nonlinear dynamics, we present a practically efficient method to extract a phase description of weakly coupled limit-cycle oscillators directly from time series observed in a rhythmic system. As a practical application, we numerically demonstrate that this method can retrieve all the interaction functions from the fluctuating rhythmic neuronal activity exhibited by a network of asymmetrically coupled neurons. This method can be regarded as a type of statistical phase reduction method that requires no detailed modeling, and as such, it is a v…
A universal approach is proposed for suppression of collective synchrony in a large population of interacting rhythmic units. We demonstrate that provided that the internal coupling is weak, stabilization of overall oscillations with vanishing stimulation leads to desynchronization in a large ensemble of coupled oscillators, without altering significantly the essential nature of each constituent oscillator. We expect our findings to be a starting point for the issue of destroying undesired synchronization, e. g. desynchronization techniques for deep brain stimulation for neurological diseases …
Biological rhythms are generated by pacemaker organs, such as the heart pacemaker organ (the sinoatrial node) and the master clock of the circadian rhythms (the suprachiasmatic nucleus), which are composed of a network of autonomously oscillatory cells. Such biological rhythms have notable periodicity despite the internal and external noise present in each cell. Previous experimental studies indicate that the regularity of oscillatory dynamics is enhanced when noisy oscillators interact and become synchronized. This effect, called the collective enhancement of temporal precision, has been stud…
The purpose of this paper is trifold -- to serve as an instructive resource and a reference catalog for biologically plausible modeling with i) conductance-based models, coupled with ii) strength-varying slow synapse models, culminating in iii) two canonical pair-wise rhythm-generating networks. We document the properties of basic network components: cell models and synaptic models, which are prerequisites for proper network assembly. Using the slow-fast decomposition we present a detailed analysis of the cellular dynamics including a discussion of the most relevant bifurcations. Several appro…
We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically …
We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule. We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks. We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a differentiation in the activation probabilities of the links. This, in turn, yields the formation of hubs associated to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can be used…
Spontaneous low-frequency oscillations on the order of several hertz are the drivers of many crucial processes in nature. From bacterial swimming to mammal gaits, the conversion of static energy inputs into slowly oscillating electrical and mechanical power is key to the autonomy of organisms across scales. However, the fabrication of slow artificial oscillators at micrometre scales remains a major roadblock towards the development of fully-autonomous microrobots. Here, we report the emergence of a low-frequency relaxation oscillator from a simple collective of active microparticles interactin…
The preBötzinger Complex, the mammalian inspiratory rhythm generator, encodes inspiratory time as motor pattern. Spike synchronization throughout this sparsely connected network generates inspiratory bursts albeit with variable latencies after preinspiratory activity onset in each breathing cycle. Using preBötC rhythmogenic microcircuit minimal models, we examined the variability in probability and latency to burst, mimicking experiments. Among various physiologically plausible graphs of 1000 point neurons with experimentally determined neuronal and synaptic parameters, directed Erdős-Rényi gr…
The revival of oscillation and maintaining rhythmicity in a network of coupled oscillators offer an open challenge to researchers as the cessation of oscillation often leads to a fatal system degradation and an irrecoverable malfunctioning in many physical, biological and physiological systems. Recently a general technique of restoration of rhythmicity in diffusively coupled networks of nonlinear oscillators has been proposed in [Zou et al. Nature Commun. 6:7709, 2015], where it is shown that a proper feedback parameter that controls the rate of diffusion can effectively revive oscillation fro…
The mammalian circadian system comprises a network of cell-autonomous oscillators, spanning from the central clock in the brain to peripheral clocks in other organs. These clocks are tightly coordinated to orchestrate rhythmic physiological and behavioral functions. Dysregulation of these rhythms is a hallmark of aging, yet it remains unclear how age-related changes lead to more easily disrupted circadian rhythms. Using a two-population model of coupled oscillators that integrates the central clock and the peripheral clocks, we derive simple mean-field equations that can capture many aspects o…
Entrainment of randomly coupled oscillator networks by periodic external forcing applied to a subset of elements is numerically and analytically investigated. For a large class of interaction functions, we find that the entrainment window with a tongue shape becomes exponentially narrow for networks with higher hierarchical organization. However, the entrainment is significantly facilitated if the networks are directionally biased, i.e., closer to the feedforward networks. Furthermore, we show that the networks with high entrainment ability can be constructed by evolutionary optimization proce…
Synchronization of network-coupled dynamical units is important to a variety of natural and engineered processes including circadian rhythms, cardiac function, neural processing, and power grids. Despite this ubiquity, it remains poorly understood how complex network structures and heterogeneous local dynamics combine to either promote or inhibit synchronization. Moreover, for most real-world applications it is impossible to obtain the exact specifications of the system, and there is a lack of theory for how uncertainty affects synchronization. We address this open problem by studying the Sync…
Soft-matter systems when driven out-of-equilibrium often give rise to structures that usually lie in-between the macroscopic scale of the material and microscopic scale of its constituents. In this paper we review three such systems, the two-dimensional square-lattice Ising model, the Kuramoto model and the Rayleigh-Bénard convection system which when driven out-of-equilibrium give rise to emergent spatio-temporal order through self-organization. A common feature of these systems is that the entities that self-organize are coupled to one another in some way, either through local interactions o…
We investigate different emergent dynamics namely oscillation quenching and revival of oscillation in a global network of identical oscillators coupled with diffusive (positive) delay coupling as it is perturbed by symmetry breaking localized repulsive delayed interaction. Starting from the oscillatory states (OS) we systematically identify three types of transition phenomena in the parameter space: (1) The system may reach inhomogeneous steady states (IHSS) from the homogeneous steady state (HSS) sometimes called as the transition from amplitude death (AD) to oscillation death (OD) state i.e.…
Physical and emergent nonpairwise interactions in oscillator networks: from higher-order phase reduction to coupling design
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（RIKEN跨学科理论与数学科学中心(iTHEMS)）；Department of Systems and Control Engineering, Institute of Science Tokyo（东京科学大学系统控制工程学院）；Research Center for Autonomous Systems Materialogy, Institute of Science Tokyo（东京科学大学自主系统材料学研究中心）；Department of Mathematics, Vrije Universiteit Amsterdam（阿姆斯特丹自由大学数学系）；Institute for Advanced Study, Technical University of Munich（慕尼黑工业大学高等研究院）；Department of Mathematics, University of Exeter（埃克塞特大学数学系）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
Synchronization among rhythmic elements is modeled by coupled phase-oscillators each of which has the so-called natural frequency. A symmetric natural frequency distribution induces a continuous or discontinuous synchronization transition from the nonsynchronized state, for instance. It has been numerically reported that asymmetry in the natural frequency distribution brings new types of bifurcation diagram having, in the order parameter, oscillation or a discontinuous jump which emerges from a partially synchronized state. We propose a theoretical classification method of five types of bifurc…
Despite their significant functional roles, beta-band oscillations are least understood. Synchronization in neuronal networks have attracted much attention in recent years with the main focus on transition type. Whether one obtains explosive transition or a continuous transition is an important feature of the neuronal network which can depend on network structure as well as synaptic types. In this study we consider the effect of synaptic interaction (electrical and chemical) as well as structural connectivity on synchronization transition in network models of Izhikevich neurons which spike reg…
Neural synchronization is believed to be critical for many brain functions. It frequently exhibits temporal variability, but it is not known if this variability has a specific temporal patterning. This study explores these synchronization/desynchronization patterns. We employ recently developed techniques to analyze the fine temporal structure of phase-locking to study the temporal patterning of synchrony of the human brain rhythms. We study neural oscillations recorded by EEG in $α$ and $β$ frequency bands in healthy human subjects at rest and during the execution of a task. While the phase-l…
We report an interesting symmetry-breaking transition in coupled identical oscillators, namely the continuous transition from homogeneous to inhomogeneous limit cycle oscillations. The observed transition is the oscillatory analog of the Turing-type symmetry-breaking transition from amplitude death (i.e., stable homogeneous steady state) to oscillation death (i.e., stable inhomogeneous steady state). This novel transition occurs in the parametric zone of occurrence of rhythmogenesis and oscillation death as a consequence of the presence of local filtering in the coupling path. We consider para…
In this paper we present a systematic, data-driven approach to discovering "bespoke" coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical…
When we look at the world around us, we see complex physical systems and emergent phenomena. Emergence occurs when a system is observed to have properties that its parts do not have on their own. These properties or behaviors emerge only when the parts interact in a wider whole. Examples of emergence can vary from the synchronization of pendulum clocks hanging on the same wall to the phenomenon of life as an emergent property of chemistry. One of the most complex systems that exist in nature is the human brain. It contains on average 100 to 200 billion neurons and about 100 trillion synapses c…
We extract data-driven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupled-agent system. They play the role of the independent spatial variables in this PDE (as opposed to the dependent, possibly also data-driven, state variables). This leads to an alternative description of the dynamics, local in these emergent coordinates, thus facilitating an alte…
Restoration of oscillation from an oscillation suppressed state in coupled oscillators is an important topic of research and has been studied widely in recent years. However, the same in the quantum regime has not been explored yet. Recent works established that under certain coupling conditions coupled quantum oscillators are susceptible to suppression of oscillations, such as amplitude death and oscillation death. In this paper, for the first time we demonstrate that quantum oscillation suppression states can be revoked and rhythmogenesis can be established in coupled quantum oscillators by …
