## 2 Methods
### Phase Transfer Entropy and Directed Phase-Amplitude Interactions  
We begin by noting that the conventional approach to estimate directed phase-amplitude interactions though TE, as expressed by Equation (3), implies the computation of TE from data of different properties, a phase time series \(\mathbf{\theta}^{x}\), which represents a circular variable, and a smooth, continuous amplitude envelope \(\xi^{y}\). In this work, we reformulate the problem of directed phase-amplitude interaction detection using TE as a phase TE-estimation task [21]. We do this by applying the idea behind the CFD: obtaining the directionality of interactions between phase and amplitude time series can be redefined as estimating the direction of interaction between two-phase time series. We hypothesize that such a change can improve the robustness of phase-amplitude TE estimates to signal degradation by noise and volume conduction effects. Given two time series \(\mathbf{x}\) and \(\mathbf{y}\), we want to estimate the TE from the phase of \(\mathbf{x}\) at a frequency of \(f_{l}\) to the amplitude envelope of \(\mathbf{y}\) at a frequency of \(f_{h}\). As before, \(\mathbf{\theta}^{x}\in[-\pi,\pi]_{l=1}^{T}\) and \(\xi^{y}\in\mathbb{R}^{T}\) are the corresponding phase and amplitude time series, obtained at the adequate frequencies. However, before TE computation, we obtain a complex representation of \(\xi^{y}\) at \(f_{l}\), \(\mathbf{s}_{\xi}(f_{l})=\xi^{c}e^{i\mathbf{\theta}^{c}}\in\mathbb{C}^{T}\), where \(\mathbf{\theta}^{c},\in[-\pi,\pi]_{l=1}^{T}\) is an instantaneous phase time series, and \(\xi^{c}\in\mathbb{R}^{T}\) is an amplitude envelope (See Figure 1B). Next, we define:  
\[TE^{\theta\xi}(\mathbf{x}\rightarrow\mathbf{y},f_{l},f_{h})=H_{S}\Big{(}\mathbf{ \theta}_{t-1}^{c,d\zeta},\mathbf{\theta}_{t-u}^{c,dx}\Big{)}-H_{S}\Big{(}\mathbf{ \theta}_{t}^{c},\mathbf{\theta}_{t-1}^{c,dc},\mathbf{\theta}_{t-u}^{c,dx}\Big{)}+H_{S }\Big{(}\mathbf{\theta}_{t}^{c},\mathbf{\theta}_{t-1}^{c,dc}\Big{)}-H_{S}\Big{(}\mathbf{ \theta}_{t-1}^{c,dc}\Big{)}, \tag{5}\]  
where \(\mathbf{\theta}_{t}^{c,dx}\) and \(\mathbf{\theta}_{t}^{c,dc}\) are time-embedded versions of \(\mathbf{\theta}^{x}\) and \(\mathbf{\theta}^{c}\). Figure 1: (**A**) The conventional approach to capture directed phase-amplitude interactions through TE consists of estimating TE from the instantaneous phase and amplitude time series that were extracted at frequencies \(f_{l}\) and \(f_{h}\), respectively. (**B**) Our approach implies a further step, comprising the extraction of the phase of the amplitude time series at frequency \(f_{l}\), to reformulate the problem as a phase TE estimation task. deviations of \(0.1A\) and \(0.2T\), respectively. Then, we concatenated the \(d\) segments to obtain a continuous signal of varying amplitude, as depicted in Figure 2B, \(\mathbf{x}^{\prime}\in\mathbb{R}^{N}=\left[\mathbf{x}_{1}^{\prime},\mathbf{x}_{2 }^{\prime},\ldots,\mathbf{x}_{d}^{\prime}\right]\), and with a power spectrum peaking at around \(f_{l}\). Next, we generate a signal oscillating at \(f_{h}\), whose amplitude is a function of \(\mathbf{x}^{\prime}\), by defining:  
\[\mathbf{y}^{\prime}=\zeta\bigg{(}1-\frac{1}{1+\exp(-a(\mathbf{x}^{\prime}-c ))}\bigg{)}(sin(2\pi f_{h}\mathbf{t})+1), \tag{11}\]  
where \(\mathbf{t}=\{0,dt,2 dt,\ldots,(N-1)dt\}\), \(\zeta=\sqrt{f_{l}/f_{h}}\), \(c=0.6\), and \(a=10\) (see Figure 2C). The constant \(c\) represents a threshold value, so that when \(\mathbf{x}^{\prime}<c\), the amplitude of \(\mathbf{y}^{\prime}\) increases, \(a\) controls the steepness of that increase. Next, we imposed a directionality of interaction from \(\mathbf{x}^{\prime}\) to \(\mathbf{y}^{\prime}\) by time-shifting \(\mathbf{y}^{\prime}\) by \(\Delta t\) seconds \(\mathbf{y}_{\Delta t}^{\prime}=\mathbf{y}^{\prime}(t+\Delta t)\). Afterward, we construct two pairs of auxiliary signals following the steps described above. From one pair, we select the signal with low-frequency components, \(\mathbf{x}^{\prime\prime}\).