# Inferring phase equations from multivariate time series.

@article{Tokuda2007InferringPE, title={Inferring phase equations from multivariate time series.}, author={Isao T. Tokuda and Swati Jain and Istv{\'a}n Zolt{\'a}n Kiss and John L. Hudson}, journal={Physical review letters}, year={2007}, volume={99 6}, pages={ 064101 } }

An approach is presented for extracting phase equations from multivariate time series data recorded from a network of weakly coupled limit cycle oscillators. Our aim is to estimate important properties of the phase equations including natural frequencies and interaction functions between the oscillators. Our approach requires the measurement of an experimental observable of the oscillators; in contrast with previous methods it does not require measurements in isolated single or two-oscillator…

## 63 Citations

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Application to prototypical models such as two interacting predator-prey systems in both limit cycle and chaotic regimes demonstrates the capability of detecting the anomalous structure from only a few sets of time series.

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### Phase Coupling Estimation from Multivariate Phase Statistics

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- 2010

This work derives a closed-form solution for estimating the phase coupling parameters from observed phase statistics and derives a regularized solution to the estimation and shows that the resulting procedure improves performance when only a limited amount of data is available.

### TOWARDS A PROTOCOL FOR ADAPTIVE DYNAMICAL BAYESIAN INFERENCE: CASE OF LIMIT-CYCLE OSCILLATORS

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The results of the evaluation of the introduced algorithm on a second system of coupled oscillators limit-cycle Poincaré oscillators in the presence of noise confirmed the relevance of the proposed algorithm for improved model inference, which allows for a deeper understanding of the interactions described by the coupling functions of the dynamical systems.

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We propose a method for estimating the asymptotic phase and amplitude functions of limit-cycle oscillators using observed time series data without prior knowledge of their dynamical equations. The…

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Estimators for the strengths of couplings are suggested which are based on modeling the observed oscillations with a set of stochastic phase oscillators and easily interpreted from a physical viewpoint to reveal an architecture of coupling reliably from a relatively short time series.

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- PhysicsPhilosophical Transactions of the Royal Society A
- 2019

The present paper reviews the above method in a way comprehensive to domain-scientists other than physics and presents applications of the method to detection of the network connectivity, inference of the phase sensitivity function, and approximation of the interaction among phase-coherent chaotic oscillators and experimental data from a forced Van der Pol electric circuit.

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### A dynamical systems approach for estimating phase interactions between rhythms of different frequencies from experimental data

- PhysicsPLoS Comput. Biol.
- 2018

An estimation method is proposed to identify the phase oscillator model from real data of cross-frequency synchronized activities and can estimate the coupling function governing the properties of synchronization.

## References

SHOWING 1-10 OF 31 REFERENCES

### Learning phase synchronization from nonsynchronized chaotic regimes.

- Physics, Computer SciencePhysical review letters
- 2002

A novel modeling approach for reconstruction of the global behavior of coupled chaotic systems from bivariate time series is presented and it is shown that the technique enables the recovery of the synchronization diagram from only three data sets.

### Predicting mutual entrainment of oscillators with experiment-based phase models.

- PhysicsPhysical review letters
- 2005

The experiment-based model properly describes in-phase and antiphase mutual entrainment with positive and negative interactions in small sets as well as dynamical clustering in populations of oscillators.

### Determination of a coupling function in multicoupled oscillators.

- PhysicsPhysical review letters
- 2006

A new method to determine a coupling function in a phase model is theoretically derived for coupled self-sustained oscillators and applied to Belousov-Zhabotinsky (BZ) oscillators. The synchronous…

### Emerging Coherence in a Population of Chemical Oscillators

- PhysicsScience
- 2002

Experiments on populations of chemical oscillators and a 25-year-old theory of Kuramoto that predicts that global coupling in a set of smooth limit-cycle oscillators with different frequencies produces a phase transition in which some of the elements synchronize are reported.

### Efficient estimation of phase-resetting curves in real neurons and its significance for neural-network modeling.

- BiologyPhysical review letters
- 2005

A simple method is presented for estimating the phase-resetting curve of real neurons to simplify the complex dynamics of a single neuron to a phase model and illustrate how to infer the existence of coherent network activity from the estimated PRC.

### Synchronization - A Universal Concept in Nonlinear Sciences

- PhysicsCambridge Nonlinear Science Series
- 2001

This work discusseschronization of complex dynamics by external forces, which involves synchronization of self-sustained oscillators and their phase, and its applications in oscillatory media and complex systems.

### Entrainment of randomly coupled oscillator networks by a pacemaker.

- PhysicsPhysical review letters
- 2004

It is found that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker, so that only shallow networks can thus exhibit frequency locking to thepacemaker.

### Fitting ordinary differential equations to chaotic data.

- MathematicsPhysical review. A, Atomic, molecular, and optical physics
- 1992

It is claimed that the problem of estimating parameters in systems of ordinary differential equations which give rise to chaotic time series is naturally tackled by boundary value problem methods and Lyapunov exponents can be computed accurately from time series much shorter than those required by previous methods.

### Pacemaker Synchronization of Electrically Coupled Rabbit Sinoatrial Node Cells

- Biology, PhysicsThe Journal of general physiology
- 1998

It is demonstrated that, at low coupling conductances, mutual pacemaker synchronization results mainly from the phase-resetting effects of the action potential of one cell on the depolarization phase of the other, and at high coupling conductance, the tonic, diastolic interactions become more important.

### Synchronization of eukaryotic cells by periodic forcing.

- Biology, PhysicsPhysical review letters
- 2006

We study a cell population described by a minimal mathematical model of the eukaryotic cell cycle subject to periodic forcing that simultaneously perturbs the dynamics of the cell cycle engine and…