A practical method for estimating coupling functions in complex dynamical systems

TL;DR

This paper reviews and extends a practical method for estimating coupling functions in complex dynamical systems, enabling inference of network topology and interactions from empirical data across various systems.

Contribution

It broadens the applicability of a phase-based inference method to diverse systems, including chaotic oscillators and experimental circuits, making it accessible to more scientists.

Findings

Successfully detects network connectivity from empirical data.

Accurately infers phase sensitivity functions.

Reconstructs interactions in chaotic and experimental systems.

Abstract

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett.99, 064101) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community.

Full text

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1 Ritsumeikan University, Japan $\Rightarrow$ 1 Department of Mechanical Engineering, Ritsumeikan University, Kusatsu, Japan

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2 Faculty of Information Studies in Novo Mesto, Slovenia $\Rightarrow$ 2 Complex Systems and Data Science Lab, Faculty of Information Studies in Novo Mesto, Novo Mesto, Slovenia

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network’s dynamics $\Rightarrow$ network dynamics

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e.g. current $\Rightarrow$ e.g., current

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the nonlinear $\Rightarrow$ these nonlinear

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$\sum_{k=1}^{N}\frac{{\partial}f_{i}}{{\partial}{\theta}_{k}}\frac{{\partial}{\phi}_{k}}{{\partial}p_{j}}$ $\Rightarrow$ $\sum_{k=1}^{N}\frac{{\partial}f_{i}}{{\partial}{\phi}_{k}}\frac{{\partial}{\phi}_{k}}{{\partial}p_{j}}$

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oscillator, ${\omega}_{i}$ gives $\Rightarrow$ oscillator and ${\omega}_{i}$ gives

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i.e. ${\omega}_{i}$ $\Rightarrow$ i.e., ${\omega}_{i}$

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determine $\Rightarrow$ determines

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${\int}_{0}^{2{\pi}}\bm{Z}({\theta}_{i})\bm{G}({\theta}_{i},{\theta}_{k})d{\theta}_{k}$ $\Rightarrow$ ${\int}_{0}^{2{\pi}}\bm{Z}({\theta}_{i}+{\theta}^{\prime})\bm{G}({\theta}_{i}+{\theta}^{\prime},{\theta}_{k}+{\theta}^{\prime})d{\theta}^{\prime}$

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via following $\Rightarrow$ via the following

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with the specific $\Rightarrow$ with this specific

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Please set the equal signs of (2.5) and (2.6) to locate horizontally at the same position.

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significantly. $\Rightarrow$ significantly (Figure 1d).

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$\tilde{T}_{3,1},\tilde{T}_{4,1}=0$ $\Rightarrow$ $\tilde{T}_{3,1},\tilde{T}_{4,1}$

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connectivity. Figure 2d, on the other hand, shows $\Rightarrow$ connectivity. (start a new line) Figure 2d shows

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oscillators, because $\Rightarrow$ oscillators, since

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${\int}_{0}^{2{\pi}}Z({\theta})G({\theta}+{\psi})d{\psi}$ $\Rightarrow$ ${\int}_{0}^{2{\pi}}Z({\psi})G({\theta}+{\psi})d{\psi}$

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remove ”and”

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is weak. $\Rightarrow$ is weak in equation (4.1).

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Please make the size of the table smaller.

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$V=6$ V $\Rightarrow$ $V=0.6$ V

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they desynchronized, $\Rightarrow$ they are desynchronized with each other,

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We have an experimental data generated from electric circuit. We would like to make these data available via a public repository, as soon as the manuscript is accepted for publication. Until then, we would like to provide the data upon individual request. $\Rightarrow$ Experimental data generated from electric circuit are available in Dryad dataset (https://doi.org/10.5061/dryad.z34tmpg80).