ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Khorev V. S., Grubov V. V., Badarin A. A. Mathematical model and dynamical analysis of the human equilibrium seeking training. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 3, pp. 409-420. DOI: 10.18500/0869-6632-2021-29-3-409-420

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Mathematical model and dynamical analysis of the human equilibrium seeking training

Khorev Vladimir Sergeevich, Innopolis University
Grubov Vadim Valerevich, Immanuel Kant Baltic Federal University
Badarin Artem Aleksandrovich, Immanuel Kant Baltic Federal University

The purpose of this work is to determine the ability of the partial directed coherence method to identify the
directed interaction between nonlinear systems correctly in presence of nonlinear couplings between systems, as well as
in the case when the measured signals are generated by objects of high dimension. The another purpose is to determine
the dependence of the coupling estimation results on the parameters: series length, sampling rate, model dimension and the
coupling architecture. Methods. In this paper, the possibilities and limitations of the frequency-resolved approach (partial
directed coherence) to describe the couplings between high-dimensional time series are investigated. Surrogate time series
constructed by permutation of realization are used to determine the significance of the results. Results. Coupling architecture
in ensembles of small-dimensional oscillators can be correctly identified for linear and nonlinear systems connected in case of
both linear and nonlinear coupling. For complex composite signals, when each measured time series is the sum of the signals
of many individual oscillators, the technique is not specific enough, revealing non-existent connections, and it is not sensitive
enough, missing the existing ones. Conclusion. The criteria for applying the partial directed coherence method to different
signals are formulated. The measure does not show indirect couplings at sufficient series length, sampling rate and model
dimension in contrast to the pairwise methods of Granger causality and transfer entropy. The measure works well for noisy
time series. The method allows to study the connectivity in an ensemble of an arbitrary number of oscillators. The method
allows to determine at what frequencies the interaction occurs. The partial directed coherence method gives acceptable results
for series of length 80 and more characteristic periods in comparison with the Granger causality method, for which the
efficiency is declared already at 4–16 characteristic periods.

This work has been supported by the program supporting Russian leading scientific schools (Grant No. NSh-2594.2020.2)
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