Mathematical model and dynamical analysis of the human equilibrium seeking training

Abstract

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.

Keywords

• partical directed coherence
• coupling
• nonlinearity
• nonlinear systems
• nonlinear coupling
• high–dimensional system