Yoji Kawamura

Phase synchronization between collective rhythms of globally coupled oscillator groups: Noiseless nonidentical case.

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. 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 the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.

Impact of hierarchical modular structure on ranking of individual nodes in directed networks

Many systems, ranging from biological and engineering systems to social systems, can be modeled as directed networks, with links representing directed interaction between two nodes. To assess the importance of a node in a directed network, various centrality measures based on different criteria have been proposed. However, calculating the centrality of a node is often difficult because of the overwhelming size of the network or because the information held about the network is incomplete. Thus, developing an approximation method for estimating centrality measures is needed. In this study, we focus on modular networks; many real-world networks are composed of modules, where connection is dense within a module and sparse across different modules. We show that ranking-type centrality measures, including the PageRank, can be efficiently estimated once the modular structure of a network is extracted. We develop an analytical method to evaluate the centrality of nodes by combining the local property (i.e. indegree and outdegree of nodes) and the global property (i.e. centrality of modules). The proposed method is corroborated by real data. Our results provide a linkage between the ranking-type centrality values of modules and those of individual nodes. They also reveal the hierarchical structure of

Structure of cell networks critically determines oscillation regularity.

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 studied theoretically using particular assumptions. In this study, we propose a general theoretical framework that enables us to understand the dependence of temporal precision on network parameters including size, connectivity, and coupling intensity; this effect has been poorly understood to date. Our framework is based on a phase oscillator model that is applicable to general oscillator networks with any coupling mechanism if coupling and noise are sufficiently weak. In particular, we can manage general directed and weighted networks. We quantify the precision of the activity of a single cell and the mean activity of an arbitrary subset of cells. We find that, in general undirected networks, the standard deviation of cycle-to-cycle periods scales with the system size N as 1/N, but only up to a certain system size N(⁎) that depends on network parameters. Enhancement of temporal precision is ineffective when N>N(⁎). We provide an example in which temporal precision considerably improves with increasing N while the level of synchrony remains almost constant; temporal precision and synchrony are independent dynamical properties. We also reveal the advantage of long-range interactions among cells to temporal precision.

Collective fluctuations in networks of noisy components

Collective dynamics result from interactions among noisy dynamical components. Examples include heartbeats, circadian rhythms and various pattern formations. Because of noise in each component, collective dynamics inevitably involve fluctuations, which may crucially affect the functioning of the system. However, the relation between the fluctuations in isolated individual components and those in collective dynamics is not clear. Here, we study a linear dynamical system of networked components subjected to independent Gaussian noise and analytically show that the connectivity of networks determines the intensity of fluctuations in the collective dynamics. Remarkably, in general directed networks including scale-free networks, the fluctuations decrease more slowly with system size than the standard law stated by the central limit theorem. They even remain finite for a large system size when global directionality of the network exists. Moreover, such non-trivial behavior appears even in undirected networks when nonlinear dynamical systems are considered. We demonstrate it with a coupled oscillator system.

Phase synchronization between collective rhythms of globally coupled oscillator groups: Noisy identical case

We theoretically investigate the 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 antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.

Phase-Reduction Approach to Synchronization of Spatiotemporal Rhythms in Reaction-Diffusion Systems

Reaction-diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. These rhythmic dynamics can be considered limit cycles of reaction-diffusion systems. However, the conventional phase-reduction theory, which provides a simple unified framework for analyzing synchronization properties of limit-cycle oscillators subjected to weak forcing, has mostly been restricted to low-dimensional dynamical systems. Here, we develop a phase-reduction theory for stable limit-cycle solutions of infinite-dimensional reaction-diffusion systems. By generalizing the notion of isochrons to functional space, the phase sensitivity function - a fundamental quantity for phase reduction - is derived. For illustration, several rhythmic dynamics of the FitzHugh-Nagumo model of excitable media are considered. Nontrivial phase response properties and synchronization dynamics are revealed, reflecting their complex spatiotemporal organization. Our theory will provide a general basis for the analysis and control of spatiotemporal rhythms in various reaction-diffusion systems.

Phase synchronization between collective rhythms of fully locked oscillator groups

A system of coupled oscillators can exhibit a rich variety of dynamical behaviors. When we investigate the dynamical properties of the system, we first analyze individual oscillators and the microscopic interactions between them. However, the structure of a coupled oscillator system is often hierarchical, so that the collective behaviors of the system cannot be fully clarified by simply analyzing each element of the system. For example, we found that two weakly interacting groups of coupled oscillators can exhibit anti-phase collective synchronization between the groups even though all microscopic interactions are in-phase coupling. This counter-intuitive phenomenon can occur even when the number of oscillators belonging to each group is only two, that is, when the total number of oscillators is only four. In this paper, we clarify the mechanism underlying this counter-intuitive phenomenon for two weakly interacting groups of two oscillators with global sinusoidal coupling.

Collective phase dynamics of globally coupled oscillators: Noise-induced anti-phase synchronization ✩

Abstract 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 numerical simulations of the three models.

Analysis of relative influence of nodes in directed networks

Many complex networks are described by directed links; in such networks, a link represents, for example, the control of one node over the other node or unidirectional information flows. Some centrality measures are used to determine the relative importance of nodes specifically in directed networks. We analyze such a centrality measure called the influence. The influence represents the importance of nodes in various dynamics such as synchronization, evolutionary dynamics, random walk, and social dynamics. We analytically calculate the influence in various networks, including directed multipartite networks and a directed version of the Watts-Strogatz small-world network. The global properties of networks such as hierarchy and position of shortcuts rather than local properties of the nodes, such as the degree, are shown to be the chief determinants of the influence of nodes in many cases. The developed method is also applicable to the calculation of the PAGERANK. We also numerically show that in a coupled oscillator system, the threshold for entrainment by a pacemaker is low when the pacemaker is placed on influential nodes. For a type of random network, the analytically derived threshold is approximately equal to the inverse of the influence. We numerically show that this relationship also holds true in a random scale-free network and a neural network.

Collective phase description of oscillatory convection.

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.