Effect of nonlinearity on coupling estimations between oscillators using partial directed coherence approach

The purpose of this work is to determine the ability of the partial directed coherence method to identify directed interactions between nonlinear systems correctly in presence of nonlinear couplings between systems, as well as in the case of measured signals generated by objects of high dimension. The other purpose is to determine the dependence of the coupling estimation results on the parameters: series length, sampling rate, model dimension and coupling architecture. Methods. Ensembles composed of four differently coupled oscillators and dynamical mesoscale model of epilepsy are considered as test systems. 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 in both cases of linear and nonlinear coupling. For complex composite signals, when each measured time series is the sum of signals from many individual oscillators, the technique is not specific enough, revealing non-existent connections, and it is not sensitive enough, missing the existing ones. Outcomes. 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 like Granger causality or transfer entropy. The measure works well for noisy time series. The method allows to study connectivity in an ensemble of 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 of 80 and more characteristic periods in comparison with the Granger causality method, for which the efficiency is declared already at 4–16 characteristic periods.