*4.1. Neural Mass Models Results*

The experiments described in Section 3.1 are intended to assess whether the phasebased connectivity measures considered in this study correctly detect the direction of interaction between two time series of known oscillatory properties. Figure 5 presents the results obtained from such experiments. Namely, column **A** shows the connectivity values obtained for different levels of coupling strength, column **B** compares the connectivities estimated for ideal signals with those of signals contaminated with noise and mixing, and column **C** displays the results obtained for bidirectional narrowband couplings. The rows in Figure 5 correspond to each of the phase-based connectivity measures studied. The first row contains average PSI values computed on the frequency range between 2 Hz and 60 Hz, while rows two to five display average net connectivity values for TE*<sup>θ</sup> κα*, TE*<sup>θ</sup> KSG*, TE*<sup>θ</sup> Sym*, and GC*θ*, respectively. Circled values indicate statistically significant connectivities at a particular frequency, according to a permutation test based on randomized surrogate trials. The test identifies connectivity values that are, on average, significantly different from those expected for that connectivity measure applied to non-interacting signals. For the three experiments involving simulated data from NMMs, we use the PSI as a comparison standard, since it is a robust and well-stablished measure of linear directed interactions defined in terms of phase relations [12,13]. Therefore, it is suited to analyze the coupled, mildly nonlinear time series generated by NMMs.

Regarding the first experiment, which modifies the coupling strength between the simulated signals, the obtained results (Figure 5, column **A**) show that all the measures studied satisfactorily detect the coupling direction of the simulated data. Note that since we set the NMMs to generate unidirectional interactions from **x** to **y**, and because of the way we defined Δ*λ*, then all net connectivity values for the simulated coupled signals should be positive. The same is true for the PSI(**x** → **y**). On the other hand, only the PSI, TE*<sup>θ</sup> κα*, and GC*<sup>θ</sup>* fulfill the criteria for an overall description of the phase-based interactions present in the data. First, we observe higher net connectivity values at higher coupling strengths, that is to say, stronger interactions lead to larger connectivity estimates. Second, for each coupling strength, there are higher net connectivity values around the frequencies corresponding to the main oscillatory components of the time series generated by the NMMs, in this case, oscillations in the range between 8 Hz and 20 Hz. Thirdly, there are statistically significant results for all the coupling strengths explored, except for noninteracting time series (*C*<sup>12</sup> = 0). TE*<sup>θ</sup> KSG* does not capture statistically significant interactions for a coupling coefficient value of 0.2, indicating a lower sensitivity to weak couplings. While TE*<sup>θ</sup> Sym* exhibits a very distorted connectivity profile when compared with the PSI. In addition, it has much larger standard deviations for all the coupling strengths considered.

The second experiment assesses the robustness of our proposal to realistic levels of noise and signal mixing, two sources of signal degradation that can lead to spurious connectivity results. In electrophysiological signals, such as EEG, signal mixing arises as a result of field spread, while noise is the result of technical and physiological artifacts [9,57,58]. The results in Figure 5, column **B**, show that PSI, TE*<sup>θ</sup> κα*, and GC*<sup>θ</sup>* capture statistically significant interactions in the frequencies of interest for both the ideal (no noise or signal mixing) and realistic conditions. The smaller connectivity values for the data contaminated with noise and signal mixing, as compared with the ideal signals, are mostly explained by the reduction in asymmetry between the driving and driven signals caused by

mixing [8]. On the contrary, we observe that neither TE*<sup>θ</sup> Sym* nor TE*<sup>θ</sup> KSG* produce statistically significant results under the realistic scenario, indicating that those estimators are less robust to signal degradation.

**Figure 5.** Obtained results for the experiments performed using simulated data from NMMs. Column (**A**) shows the average connectivity values obtained for different levels of coupling strength. Column (**B**) presents the average connectivity values estimated for ideal signals and for signals contaminated with noise and signal mixing. Column (**C**) displays the average connectivity values obtained for bidirectional narrowband couplings. The rows correspond to each of the net phase-based effective connectivity estimation approaches considered for the aforementioned experiments. Circled values indicate statistically significant results at a Bonferroni-corrected alpha level of 3.3 <sup>×</sup> <sup>10</sup><sup>−</sup>4, according to a permutation test based on randomized surrogate trials.

Finally, the third experiment aims to evaluate how our proposal deals with bidirectional interactions of localized frequency content. Because of our experimental setup, the obtained results should exhibit a positive deflection around 10 Hz in order to capture the directed interaction from **x** to **y** and a negative deflection around 40 Hz to represent the directed interaction from **y** to **x**. Figure 5, column **C**, shows that both PSI and TE*<sup>θ</sup> κα* successfully detect the change in the direction of interaction in localized frequency bands, with statistically significant connectivity values around the frequencies of interest. However, under this scenario, TE*<sup>θ</sup> κα* is less frequency specific for high-frequency interactions than PSI, with statistically significant connections present for a large range of frequency values around 40 Hz. This is probably due to the filtering step involved in the estimation of TE*<sup>θ</sup> κα*, while PSI is directly defined on the data spectra. Additionally, TE*<sup>θ</sup> KSG* and TE*<sup>θ</sup> Sym* fail to produce any significant results, while GC*<sup>θ</sup>* shows a statistically significant, non-existing coupling from **y** to **x** for frequencies under 10 Hz. Note that, ultimately, the permutation test indicates whether the connectivity values obtained are unlikely to be the result of chance and not whether they correctly capture the directed interactions present in the data. In this case, the statistically significant results mean that GC*<sup>θ</sup>* consistently found a directed interaction from **y** to **x** in the range mentioned before.

The results discussed above indicate that the proposed phase TE estimator is able to detect directed interactions between time series resembling electrophysiological data for different levels of coupling strength, under the presence of noise and signal mixing and for bidirectional narrowband couplings. Furthermore, they show that it is competitive with well-established approaches for phase-based net connectivity estimation, such as PSI, in the case of weakly nonlinear signals. Lastly, our results also show that commonly used single-trial TE estimators, such TE*KSG* and TE*Sym*, are ill-suited to measure directed interactions between instantaneous phase time series.
