## 4 Results and Discussion  
### Neural Mass Models Results  
The experiments described in Section 3.1 are intended to assess whether the phase-based 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 \(\text{TE}^{\theta}_{\text{exa}}\), \(\text{TE}^{\theta}_{\text{KSG}}\), \(\text{TE}^{\theta}_{\text{sym}}\), and \(\text{GC}^{\theta}\), 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 \(\Delta\lambda\), then all net connectivity values for the simulated coupled signals should be positive. The same is true for the \(\text{PSI}(\textbf{x}\rightarrow\textbf{y})\). On the other hand, only the PSI, \(\text{TE}^{\theta}_{\text{exa}}\), and \(\text{GC}^{\theta}\) 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 non-interacting time series (\(C_{12}=0\)). \(\text{TE}^{\theta}_{\text{KSG}}\) does not capture statistically significant interactions for a coupling coefficient value of 0.2, indicating a lower sensitivity to weak couplings. While \(\text{TE}^{\theta}_{\text{Sym}}\) 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, \(\text{TE}^{\theta}_{\text{exa}}\), and \(\text{GC}^{\theta}\) capture statistically significant interactions in the frequencies of interest for both the ideal (no noise or signal mixing) and realistic conditions.