## 4 Results and Discussion  
### Simulated Data  
Figure 4 presents the results obtained through the proposed \(\mathrm{TE}_{\kappa\kappa}^{\theta\theta^{c}}\) approach for the simulated data in the case where \(\mathrm{SNR}=3\). Figure 4A shows the average values obtained for \(\mathrm{TE}_{\kappa\kappa}^{\theta\theta^{c}}(\mathbf{x}_{\eta}^{w}\to \mathbf{y}_{\eta}^{w},f_{l},f_{h})\), with \(f_{l}\) and \(f_{h}\) varying in the range from 3 Hz to 45 Hz, in 3 Hz steps. That is to say, it shows the TE values that were computed following our proposal, assuming that the oscillation phases in \(\mathbf{x}_{\eta}^{w}\) drove the amplitude envelopes of the oscillations in \(\mathbf{y}_{\eta}^{w}\). Similarly, Figure 4B shows the average obtained \(\mathrm{TE}_{\kappa\kappa}^{\theta\theta^{c}}(\mathbf{y}_{\eta}^{w}\to \mathbf{x}_{\eta}^{w},f_{l},f_{h})\) values, where the underlying assumption is that oscillation phases in \(\mathbf{y}_{\eta}^{w}\) are causal to the amplitude envelopes of the oscillations in \(\mathbf{x}_{\eta}^{w}\). Figure 4C,D display the results returned by the permutation tests carried out over the TE data that were estimated for all the simulated trials, and whose average values are displayed in Figure 4A,B, respectively (statistically significant results are indicated in white). The results show that the proposed approach captures strong and statistically significant phase-amplitude-directed interactions around the target frequencies used to generate the simulated data. Statistically significant results were only obtained when TE was estimated from the phase of \(\mathbf{x}_{\eta}^{w}\) in the \(\theta\) band, at around 6 Hz, to the amplitude of \(\mathbf{y}_{\eta}^{w}\) in the \(\beta\) band, at around 24 Hz, and not when estimated assuming causality in the opposite direction.  
Figure 5 shows the results of the permutation tests carried out on the connectivity values estimated for all trials from the simulated data using, from top to bottom, CFD, \(\mathrm{TE}_{\kappa\kappa}^{\theta\theta}\) (conventional approach to estimate directed phase-amplitude through TE, using  
the kernel-based Renyi's \(\pi\) TE estimator proposed in [16] ), and \(\text{TE}_{\text{xx}}^{\theta\%}\), under the three modeled noise conditions, from left to right, low (\(\text{SNR}=6\)), moderate (\(\text{SNR}=3\)) and high (\(\text{SNR}=1\)) noise levels. In this case, all connectivity measures were obtained assuming the correct direction of causality, from \(\mathbf{\chi}_{\mathbf{q}}^{w}\) to \(\mathbf{\chi}_{\mathbf{q}}^{w}\). As before, statistically significant results are displayed in white. For the low and moderate noise levels, the three connectivity estimation methods successfully capture statistically significant, phase-amplitude directed interactions around the target frequencies. Note that the CFD and \(\text{TE}_{\text{xx}}^{\theta\%}\) display significant results on narrower frequency ranges around the interacting frequencies that were actually present in the data than \(\text{TE}_{\text{xx}}^{\theta\%}\). This is likely to be associated with the additional filtering stages involved in the computation of \(\text{TE}_{\text{xx}}^{\theta\%}\) and, in the case of CFD, its high-frequency specificity is probably linked to its robustness against false positives [11]. However, unlike \(\text{TE}_{\text{xx}}^{\theta\%}\), both CFD and \(\text{TE}_{\text{xx}}^{\theta\%}\) failed to capture any statistically significant interactions for the high-noise scenario. Therefore, our proposal exhibits higher robustness to the noise present in the data. Thus, from the results presented in Figures 4 and 5, we can argue that the proposed \(\text{TE}_{\text{xx}}^{\theta\%}\) approach allows for directed phase-amplitude interactions to be uncovered, capturing their strength and direction, even in the presence of high levels of noise and a confounding factor such as signal-mixing due to volume conduction.