Multistable attractors in a network of phase oscillators with three-body interaction

Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in asymptotic state can vary continuously within some range depending on the initial phase pattern.Comment: 5 pages, 3 figure

Higher order approximation of isochrons

Phase reduction is a commonly used techinque for analyzing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart-Landau oscillator to illustrate the method in order 2

Universal behavior in populations composed of excitable and self-oscillatory elements

We study the robustness of self-sustained oscillatory activity in a globally coupled ensemble of excitable and oscillatory units. The critical balance to achieve collective self-sustained oscillations is analytically established. We also report a universal scaling function for the ensemble's mean frequency. Our results extend the framework of the `Aging Transition' [Phys. Rev. Lett. 93, 104101 (2004)] including a broad class of dynamical systems potentially relevant in biology.Comment: 4 pages; Changed titl

Pulse-coupled resonate-and-fire models

We analyze two pulse-coupled resonate-and-fire neurons. Numerical simulation reveals that an anti-phase state is an attractor of this model. We can analytically explain the stability of anti-phase states by means of a return map of firing times, which we propose in this paper. The resultant stability condition turns out to be quite simple. The phase diagram based on our theory shows that there are two types of anti-phase states. One of these cannot be seen in coupled integrate-and-fire models and is peculiar to resonate-and-fire models. The results of our theory coincide with those of numerical simulations.Comment: 15 pages, 8 figure

Clustering and Synchronization of Oscillator Networks

Using a recently described technique for manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks with Poisson and scale-free degree distributions. For both types of network, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, scale-free networks show an additional, advanced transition instead of a single synchronization threshold. This cluster-enhanced synchronization of hubs may be relevant to the brain with its scale-free and highly clustered structure.Comment: Submitted to Phys. Rev.

Collective dynamical response of coupled oscillators with any network structure

We formulate a reduction theory that describes the response of an oscillator network as a whole to external forcing applied nonuniformly to its constituent oscillators. The phase description of multiple oscillator networks coupled weakly is also developed. General formulae for the collective phase sensitivity and the effective phase coupling between the oscillator networks are found. Our theory is applicable to a wide variety of oscillator networks undergoing frequency synchronization. Any network structure can systematically be treated. A few examples are given to illustrate our theory.Comment: 4 pages, 2 figure

Gauge Theory for the Rate Equations: Electrodynamics on a Network

Systems of coupled rate equations are ubiquitous in many areas of science, for example in the description of electronic transport through quantum dots and molecules. They can be understood as a continuity equation expressing the conservation of probability. It is shown that this conservation law can be implemented by constructing a gauge theory akin to classical electrodynamics on the network of possible states described by the rate equations. The properties of this gauge theory are analyzed. It turns out that the network is maximally connected with respect to the electromagnetic fields even if the allowed transitions form a sparse network. It is found that the numbers of degrees of freedom of the electric and magnetic fields are equal. The results shed light on the structure of classical abelian gauge theory beyond the particular motivation in terms of rate equations.Comment: 4 pages, 2 figures included, v2: minor revision, as publishe