Phase dynamics of coupled oscillators reconstructed from data

Estimation of Kramers-Moyal coefficients at low sampling rates

An optimization procedure for the estimation of Kramers-Moyal coefficients from stationary, one-dimensional, Markovian time series data is presented. The method takes advantage of a recently reported approach that allows one to calculate exact finite sampling interval effects by solving the adjoint Fokker-Planck equation. Therefore, it is well suited for the analysis of sparsely sampled time series. The optimization can be performed either by making a parametric ansatz for drift and diffusion functions or parameter free. We demonstrate the power of the method in several numerical examples with synthetic time series.

Detecting the harmonics of oscillations with time-variable frequencies

A method is introduced for the spectral analysis of complex noisy signals containing several frequency components. It enables components that are independent to be distinguished from the harmonics of nonsinusoidal oscillatory processes of lower frequency. The method is based on mutual information and surrogate testing combined with the wavelet transform, and it is applicable to relatively short time series containing frequencies that are time variable. Where the fundamental frequency and harmonics of a process can be identified, the characteristic shape of the corresponding oscillation can be determined, enabling adaptive filtering to remove other components and nonoscillatory noise from the signal. Thus the total bandwidth of the signal can be correctly partitioned and the power associated with each component then can be quantified more accurately. The method is first demonstrated on numerical examples. It is then used to identify the higher harmonics of oscillations in human skin blood flow, both spontaneous and associated with periodic iontophoresis of a vasodilatory agent. The method should be equally relevant to all situations where signals of comparable complexity are encountered, including applications in astrophysics, engineering, and electrical circuits, as well as in other areas of physiology and biology.

Phase synchronization of three locally coupled chaotic electrochemical oscillators: enhanced phase diffusion and identification of indirect coupling

Experiments are carried out with three locally coupled phase coherent chaotic electrochemical oscillators (A-B-C) in nickel dissolution in sulfuric acid. As the interaction strength is increased among the electrodes, an onset of synchronization is observed where the frequencies become identical and the phase differences are bounded. The precision of the period of the oscillators is characterized by phase diffusion coefficients from phases and phase differences. The transition to synchronization with increase of coupling strength was found to be accompanied by enhanced phase fluctuations that cause the precision of the oscillations to deteriorate. A parallel synchrony analysis showed that the direct (between A and B and B and C) and the indirect (between A and C) couplings can be correctly identified with the use of a partial phase synchrony index; therefore, the network topology can be deduced from dynamical measurements. Numerical simulations with a locally coupled model for electrochemical chaos confirm the presence of enhanced phase fluctuations close to the transition to synchronization and the usefulness of the partial phase synchrony index for differentiation of direct from indirect interactions in a small network of oscillators.

Phase Coupling Estimation from Multivariate Phase Statistics

Coupled oscillators are prevalent throughout the physical world. Dynamical system formulations of weakly coupled oscillator systems have proven effective at capturing the properties of real-world systems and are compelling models of neural systems. However, these formulations usually deal with the forward problem: simulating a system from known coupling parameters. Here we provide a solution to the inverse problem: determining the coupling parameters from measurements. Starting from the dynamic equations of a system of symmetrically coupled phase oscillators, given by a nonlinear Langevin equation, we derive the corresponding equilibrium distribution. This formulation leads us to the maximum entropy distribution that captures pairwise phase relationships. To solve the inverse problem for this distribution, we derive a closed-form solution for estimating the phase coupling parameters from observed phase statistics. Through simulations, we show that the algorithm performs well in high dimensions (d = 100) and in cases with limited data (as few as 100 samples per dimension). In addition, we derive a regularized solution to the estimation and show that the resulting procedure improves performance when only a limited amount of data is available. Because the distribution serves as the unique maximum entropy solution for pairwise phase statistics, phase coupling estimation can be broadly applied in any situation where phase measurements are made. Under the physical interpretation, the model may be used for inferring coupling relationships within cortical networks.

Phase resetting of collective rhythm in ensembles of oscillators

Phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations. We consider globally coupled ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of ensemble evolution to derive the analytical phase resetting equations. We show the final phase reset value to be composed of two parts: an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium. Analytical, semianalytical and numerical approximations of the final phase resetting curve are constructed. We support our findings with extensive numerical evidence involving identical and nonidentical oscillators. The validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit.

Effective phase dynamics of noise-induced oscillations in excitable systems

We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.

Detecting couplings between interacting oscillators with time-varying basic frequencies: instantaneous wavelet bispectrum and information theoretic approach

In the natural world, the properties of interacting oscillatory systems are not constant, but evolve or fluctuating continuously in time. Thus, the basic frequencies of the interacting oscillators are time varying, which makes the system analysis complex. For studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory. We show how these methods uncover the interacting properties and reveal the nature, strength, and direction of coupling. Wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase-time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver-response relationships in complex systems. We generate bivariate time-series numerically to mimic typical situations that occur in real measured data, apply both methods to the same time-series and discuss the results. The approach is applicable quite generally to any system of coupled nonlinear oscillators.

Synchronized Current Oscillations of Formic Acid Electro-oxidation in a Microchip-based Dual-Electrode Flow Cell

We investigate the oscillatory electro-oxidation of formic acid on platinum in a microchip-based dual-electrode cell with microfluidic flow control. The main dynamical features of current oscillations on single Pt electrode that had been observed in macro-cells are reproduced in the microfabricated electrochemical cell. In dual-electrode configuration nearly in-phase synchronized current oscillations occur when the reference/counter electrodes are placed far away from the microelectrodes. The synchronization disappears with close reference/counter electrode placements. We show that the cause for synchronization is weak albeit important, bidirectional electrical coupling between the electrodes; therefore the unidirectional mass transfer interactions are negligible. The experimental design enables the investigation of the dynamical behavior in micro-electrode arrays with well-defined control of flow of the electrolyte in a manner where the size and spacing of the electrodes can be easily varied.

Synchronization of sound sources

Sound generation and interaction are highly complex, nonlinear, and self-organized. Nearly 150 years ago Rayleigh raised the following problem: two nearby organ pipes of different fundamental frequencies sound together almost inaudibly with identical pitch. This effect is now understood qualitatively by modern synchronization theory M. Abel et al. [J. Acoust. Soc. Am. 119, 2467 (2006)10.1121/1.2170441]. For a detailed investigation, we substituted one pipe by an electric speaker. We observe that even minute driving signals force the pipe to synchronization, thus yielding three decades of synchronization-the largest range ever measured to our knowledge. Furthermore, a mutual silencing of the pipe is found, which can be explained by self-organized oscillations, of use for novel methods of noise abatement. Finally, we develop a reconstruction method which yields a perfect quantitative match of experiment and theory.

Estimating the phase of synchronized oscillators

The state of a collection of phase-locked oscillators is determined by a single phase variable or cyclic coordinate. This paper presents a computational method, Phaser, for estimating the phase of phase-locked oscillators from limited amounts of multivariate data in the presence of noise and measurement errors. Measurements are assumed to be a collection of multidimensional time series. Each series consists of several cycles of the same or similar systems. The oscillators within each system are not assumed to be identical. Using measurements of the noise covariance for the multivariate input, data from the individual oscillators in the system are combined to reduce the variance of phase estimates for the whole system. The efficacy of the algorithm is demonstrated on experimental and model data from biomechanics of cockroach running and on simulated oscillators with varying levels of noise.