*2.1. Phase Transfer Entropy*

Transfer entropy (TE) is a Wiener-causal measure of directed interactions between two dynamical systems [14,15]. Given two time series **<sup>x</sup>** = {*xt*}*<sup>T</sup> <sup>t</sup>*=<sup>1</sup> and **<sup>y</sup>** = {*yt*}*<sup>T</sup> <sup>t</sup>*=1, with *<sup>t</sup>* <sup>∈</sup> <sup>N</sup> a discrete time index, *<sup>T</sup>* <sup>∈</sup> <sup>N</sup>, the TE from **<sup>x</sup>** to **<sup>y</sup>** estimates whether the ability to predict the future of **y** improves by considering the past of both **x** and **y**, as compared to the case when only the past of **y** is considered. Formally, TE can be defined as:

$$TE(\mathbf{x}\rightarrow\mathbf{y}) = \sum\_{y\_t, \mathbf{y}\_{t-1}^{dy}, \mathbf{x}\_{t-u}^{dx}} p\left(y\_t, \mathbf{y}\_{t-1}^{dy}, \mathbf{x}\_{t-u}^{dx}\right) \log\left(\frac{p\left(y\_t|\mathbf{y}\_{t-1}^{dy}, \mathbf{x}\_{t-u}^{dx}\right)}{p\left(y\_t|\mathbf{y}\_{t-1}^{dy}\right)}\right),\tag{1}$$

where **x***dx <sup>t</sup>* , **<sup>y</sup>***dy <sup>t</sup>* <sup>∈</sup> <sup>R</sup>*D*×*<sup>d</sup>* are time embedded versions of **<sup>x</sup>** and **<sup>y</sup>**, *<sup>D</sup>* <sup>=</sup> *<sup>T</sup>* <sup>−</sup> (*τ*(*<sup>d</sup>* <sup>−</sup> <sup>1</sup>)) with *<sup>d</sup>*, *<sup>τ</sup>* <sup>∈</sup> <sup>N</sup> the embedding dimension and delay, respectively; *<sup>u</sup>* <sup>∈</sup> <sup>N</sup> represents the interaction delay between the driving and the driven systems, and *p*(·) 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 **x***<sup>d</sup> <sup>t</sup>* = (*x*(*t*), *x*(*t* − *τ*), *x*(*t* − 2*τ*), ... , *x*(*t* − (*d* − 1)*τ*)) [31,32]. Furthermore, using the definition of Shannon entropy, *HS*(*X*) = − ∑*<sup>x</sup> p*(*x*)log(*p*(*x*)), where *X* is a discrete random variable (*x* ∈ *X*), we can also express Equation (1) as:

$$TE(\mathbf{x}\rightarrow\mathbf{y}) = H\_S\left(\mathbf{y}\_{t-1}^{dy}, \mathbf{x}\_{t-u}^{dx}\right) - H\_S\left(y\_t, \mathbf{y}\_{t-1}^{dy}, \mathbf{x}\_{t-u}^{dx}\right) + H\_S\left(y\_t, \mathbf{y}\_{t-1}^{dy}\right) - H\_S\left(\mathbf{y}\_{t-1}^{dy}\right). \tag{2}$$

where *HS*(·, ·), and *HS*(·) stand for joint and marginal entropies.

In phase TE, the time series **x** and **y** are replaced by instantaneous phase time series *<sup>θ</sup>x*(*f*) ∈ [−*π*, *<sup>π</sup>*] *T <sup>t</sup>*=<sup>1</sup> and *<sup>θ</sup>y*(*f*) ∈ [−*π*, *<sup>π</sup>*] *T <sup>t</sup>*=1, obtained from **<sup>s</sup>***<sup>x</sup>* <sup>=</sup> *<sup>ς</sup>xeiθx*(*f*) <sup>∈</sup> <sup>C</sup>*<sup>T</sup>* and **<sup>s</sup>***<sup>y</sup>* <sup>=</sup> *<sup>ς</sup>yeiθy*(*f*) <sup>∈</sup> <sup>C</sup>*T*, which contain the complex-filtered values of **<sup>x</sup>** and **<sup>y</sup>** at frequency *<sup>f</sup>* , respectively, and with *<sup>ς</sup>x*, *<sup>ς</sup><sup>y</sup>* <sup>∈</sup> <sup>R</sup>*<sup>T</sup>* the amplitude envelopes of the filtered time series [8]. Thus, we have that

$$TE^{\theta}(\mathbf{x}\rightarrow\mathbf{y},f) = H\_{\mathbb{S}}\Big(\boldsymbol{\theta}\_{t-1}^{y,dy},\boldsymbol{\theta}\_{t-u}^{x,dx}\Big) - H\_{\mathbb{S}}\Big(\boldsymbol{\theta}\_{t}^{y},\boldsymbol{\theta}\_{t-1}^{y,dy},\boldsymbol{\theta}\_{t-u}^{x,dx}\Big) + H\_{\mathbb{S}}\Big(\boldsymbol{\theta}\_{t}^{y},\boldsymbol{\theta}\_{t-1}^{y,dy}\Big) - H\_{\mathbb{S}}\Big(\boldsymbol{\theta}\_{t-1}^{y,dy}\Big),\text{ (3)}$$

where *θx*,*dx <sup>t</sup>* and *θ y*,*dy <sup>t</sup>* are time embedded versions of *<sup>θ</sup><sup>x</sup>* and *<sup>θ</sup>y*. Note that for the sake of notation simplicity we have dropped the explicit dependency of the phase time series on *f* .
