Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators.

The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for anti-phase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Phase reduction approach to elastohydrodynamic synchronization of beating flagella.

We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in infinite-dimensional dynamical systems, namely, the limit-cycle solutions to partial differential equations representing beating flagella. We derive the phase sensitivity function, which quantifies the phase response of a beating flagellum to weak perturbations applied at each point and at each time. Using the phase sensitivity function, we analyze the phase synchronization between a pair of beating flagella through hydrodynamic interactions at a low Reynolds number.

Phase reduction and synchronization of a network of coupled dynamical elements exhibiting collective oscillations.

A general phase reduction method for a network of coupled dynamical elements exhibiting collective oscillations, which is applicable to arbitrary networks of heterogeneous dynamical elements, is developed. A set of coupled adjoint equations for phase sensitivity functions, which characterize the phase response of the collective oscillation to small perturbations applied to individual elements, is derived. Using the phase sensitivity functions, collective oscillation of the network under weak perturbation can be described approximately by a one-dimensional phase equation. As an example, mutual synchronization between a pair of collectively oscillating networks of excitable and oscillatory FitzHugh-Nagumo elements with random coupling is studied.

Collective phase reduction of globally coupled noisy dynamical elements.

We formulate a theory for the collective phase reduction of globally coupled noisy dynamical elements exhibiting macroscopic rhythms. We first transform the Langevin-type equation that represents a group of globally coupled noisy dynamical elements into the corresponding nonlinear Fokker-Planck equation and then develop the phase reduction method for limit-cycle solutions to the nonlinear Fokker-Planck equation. The theory enables us to describe the collective dynamics of a group of globally coupled noisy dynamical elements by a single degree of freedom called the collective phase. As long as the group collectively exhibits macroscopic rhythms, the theory is applicable even when the coupling and noise are strong; it is also independent of the assumption that each element of the group is a self-sustained oscillator. We also provide a simple and accurate numerical algorithm for the collective phase description method and numerically illustrate the theory using a group of globally coupled noisy FitzHugh-Nagumo elements.

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling.

We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on the overall coupling intensity and the stationary phase difference. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems.

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Optimization of noise-induced synchronization of oscillator networks.

We investigate common-noise-induced synchronization between two identical networks of coupled phase oscillators exhibiting fully locked collective oscillations. Using the collective phase description method for fully locked oscillators, we demonstrate that two noninteracting networks of coupled phase oscillators can exhibit in-phase synchronization between the networks when driven by weak common noise. We derive the Lyapunov exponent characterizing the relaxation time for synchronization and develop a method of obtaining the optimal input pattern of common noise to achieve fast synchronization. We illustrate the theory using three representative networks with heterogeneous, global, and local coupling. The theoretical results are validated by direct numerical simulations.

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.

From the Kuramoto-Sakaguchi model to the Kuramoto-Sivashinsky equation.

We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type phase model via the Ott-Antonsen-type complex amplitude equation and demonstrate heterogeneity-induced collective-phase turbulence in nonlocally coupled individual-phase oscillators.

Noise-induced synchronization of oscillatory convection and its optimization.

We investigate common-noise-induced phase synchronization between uncoupled identical Hele-Shaw cells exhibiting oscillatory convection. Using the phase description method for oscillatory convection, we demonstrate that the uncoupled systems of oscillatory Hele-Shaw convection can exhibit in-phase synchronization when driven by weak common noise. We derive the Lyapunov exponent determining the relaxation time for the synchronization, and develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization. The theoretical results are confirmed by direct numerical simulations.

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.

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 phase description of globally coupled excitable elements.

We develop a theory of collective phase description for globally coupled noisy excitable elements exhibiting macroscopic oscillations. Collective phase equations describing macroscopic rhythms of the system are derived from Langevin-type equations of globally coupled active rotators via a nonlinear Fokker-Planck equation. The theory is an extension of the conventional phase reduction method for ordinary limit cycles to limit-cycle solutions in infinite-dimensional dynamical systems, such as the time-periodic solutions to nonlinear Fokker-Planck equations representing macroscopic rhythms. We demonstrate that the type of the collective phase sensitivity function near the onset of collective oscillations crucially depends on the type of the bifurcation, namely, it is type I for the saddle-node bifurcation and type II for the Hopf bifurcation.

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 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.

Collective-phase description of coupled oscillators with general network structure.

We develop a collective-phase description for a population of nonidentical limit-cycle oscillators with any network structure undergoing fully phase-locked collective oscillations. The whole network dynamics can be described by a single collective-phase variable. We derive a general formula for the collective-phase sensitivity, which quantifies the phase response of the whole network to weak external perturbations applied to the constituent oscillators. Moreover, we consider weakly interacting multiple networks and develop an effective phase coupling description for them. Several examples are given to illustrate our theory.

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 sensitivity.

The collective phase response to a macroscopic external perturbation of a population of interacting nonlinear elements exhibiting collective oscillations is formulated for the case of globally coupled oscillators. The macroscopic phase sensitivity is derived from the microscopic phase sensitivity of the constituent oscillators by a two-step phase reduction. We apply this result to quantify the stability of the macroscopic common-noise-induced synchronization of two uncoupled populations of oscillators undergoing coherent collective oscillations.

Hole structures in nonlocally coupled noisy phase oscillators.

We demonstrate that a system of nonlocally coupled noisy phase oscillators can collectively exhibit a hole structure, which manifests itself in the spatial phase distribution of the oscillators. The phase model is described by a nonlinear Fokker-Planck equation, which can be reduced to the complex Ginzburg-Landau equation near the Hopf bifurcation point of the uniform solution. By numerical simulations, we show that the hole structure clearly appears in the space-dependent order parameter, which corresponds to the Nozaki-Bekki hole solution of the complex Ginzburg-Landau equation.

Chimera Ising walls in forced nonlocally coupled oscillators.

Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal structure called a chimera, where the system splits into two groups of oscillators with sharp boundaries, one of which is phase locked and the other phase randomized. Two examples of chimera states are known: the first one appears in a ring of phase oscillators, and the second is associated with two-dimensional rotating spiral waves. In this paper, we report yet another example of the chimera state that is associated with the so-called Ising walls in one-dimensional spatially extended systems. This chimera state is exhibited by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.