## 2 Methods  
### Phase Transfer Entropy  
Transfer entropy (TE) is a Wiener-causal measure of directed interactions between two dynamical systems [14; 15]. Given two time series \(\mathbf{x}=\{x_{t}\}_{t=1}^{T}\) and \(\mathbf{y}=\{y_{t}\}_{t=1}^{T}\), with \(t\in\mathbb{N}\) a discrete time index, \(T\in\mathbb{N}\), the TE from \(\mathbf{x}\) to \(\mathbf{y}\) estimates whether the ability to predict the future of \(\mathbf{y}\) improves by considering the past of both \(\mathbf{x}\) and \(\mathbf{y}\), as compared to the case when only the past of \(\mathbf{y}\) is considered. Formally, TE can be defined as:  
\[TE(\mathbf{x}\rightarrow\mathbf{y})=\sum_{y_{t},\mathbf{y}_{t-1}^{d_{x}}, \mathbf{x}_{t-u}^{dx}}p\left(y_{t},\mathbf{y}_{t-1}^{dy},\mathbf{x}_{t-u}^{ dx}\right)\text{log}\left(\frac{p\left(y_{t}|\mathbf{y}_{t-1}^{dy},\mathbf{x}_{t-u}^{ dx}\right)}{p\left(y_{t}|\mathbf{y}_{t-1}^{dy}\right)}\right), \tag{1}\]  
where \(\mathbf{x}_{t}^{dx},\mathbf{y}_{t}^{dy}\in\mathbb{R}^{D\times d}\) are time embedded versions of \(\mathbf{x}\) and \(\mathbf{y}\), \(D=T-(\tau(d-1))\) with \(d\), \(\tau\in\mathbb{N}\) the embedding dimension and delay, respectively; \(u\in\mathbb{N}\) represents the interaction delay between the driving and the driven systems, and \(p(\cdot)\) indicates a probability density function [30] (Henceforth, the summation symbol is to be interpreted in an extended way, that is to say, as a summation or an integral depending on whether the variable is discrete or continuous). Regarding the time embeddings, we have that \(\mathbf{x}_{t}^{d}=(x(t),x(t-\tau),x(t-2\tau),\ldots,x(t-(d-1)\tau))\)[31; 32]. Furthermore, using the definition of Shannon entropy, \(H_{S}(X)=-\sum_{x}p(x)\text{log}(p(x))\), where \(X\) is a discrete random variable (\(x\in X\)), we can also express Equation (1) as:  
\[TE(\mathbf{x}\rightarrow\mathbf{y})=H_{S}\Big{(}\mathbf{y}_{t-1}^{dy}, \mathbf{x}_{t-u}^{dx}\Big{)}-H_{S}\Big{(}y_{t},\mathbf{y}_{t-1}^{dy},\mathbf{ x}_{t-u}^{dx}\Big{)}+H_{S}\Big{(}y_{t},\mathbf{y}_{t-1}^{dy}\Big{)}-H_{S} \Big{(}\mathbf{y}_{t-1}^{dy}\Big{)}. \tag{2}\]  
where \(H_{S}(\cdot,\cdot)\), and \(H_{S}(\cdot)\) stand for joint and marginal entropies.  
In phase TE, the time series \(\mathbf{x}\) and \(\mathbf{y}\) are replaced by instantaneous phase time series \(\theta^{x}(f)\in[-\pi,\pi]_{t=1}^{T}\) and \(\theta^{y}(f)\in[-\pi,\pi]_{t=1}^{T}\), obtained from \(\mathbf{s}_{x}=\xi_{x}e^{i\theta^{x}(f)}\in\mathbb{C}^{T}\) and \(\mathbf{s}_{y}=\xi_{y}e^{i\theta^{y}(f)}\in\mathbb{C}^{T}\), which contain the complex-filtered values of \(\mathbf{x}\) and \(\mathbf{y}\) at frequency \(f\), respectively, and with \(\xi_{x},\xi_{y}\in\mathbb{R}^{T}\) the amplitude envelopes of the filtered time series [8]. Thus, we have that  
\[TE^{\theta}(\mathbf{x}\rightarrow\mathbf{y},f)=H_{S}\Big{(}\theta_{t-1}^{y,dy },\mathbf{\theta}_{t-u}^{x,dx}\Big{)}-H_{S}\Big{(}\theta_{t}^{y},\mathbf{ \theta}_{t-1}^{y,dy},\mathbf{\theta}_{t-u}^{dx}\Big{)}+H_{S}\Big{(}\theta_{t} ^{y},\mathbf{\theta}_{t-1}^{y,dy}\Big{)}-H_{S}\Big{(}\mathbf{\theta}_{t-1}^{ y,dy}\Big{)}, \tag{3}\]  
where \(\mathbf{\theta}_{t}^{xx}\) and \(\mathbf{\theta}_{t}^{y,dy}\) are time embedded versions of \(\theta^{x}\) and \(\mathbf{\theta}^{y}\). Note that for the sake of notation simplicity we have dropped the explicit dependency of the phase time series on \(f\).