## 4 Results and Discussion
### Neural Mass Models Results
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 \(\text{TE}^{\theta}_{Sym}\) nor \(\text{TE}^{\theta}_{KSG}\) 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 \(3.3\times 10^{-4}\), 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 \(\mathbf{x}\) to \(\mathbf{y}\) and a negative deflection around 40 Hz to represent the directed interaction from \(\mathbf{y}\) to \(\mathbf{x}\). Figure 5, column \(\mathbf{C}\), shows that both PSI and TE\({}^{\theta}_{\kappa\alpha}\) 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\({}^{\theta}_{\kappa\alpha}\) 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\({}^{\theta}_{\kappa\alpha}\), while PSI is directly defined on the data spectra. Additionally, TE\({}^{\theta}_{KSG}\) and TE\({}^{\theta}_{Sym}\) fail to produce any significant results, while GC\({}^{\theta}\) shows a statistically significant, non-existing coupling from \(\mathbf{y}\) to \(\mathbf{x}\) 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\({}^{\theta}\) consistently found a directed interaction from \(\mathbf{y}\) to \(\mathbf{x}\) 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.