*3.3. Parameter Selection*

We used in-house Python implementations of the algorithms for all the connectivity measures studied (The TE*<sup>θ</sup> κα* implementation is available at https://github.com/ide270 4/Kernel\_Phase\_Transfer\_Entropy, accessed on 13 July 2021), except for TE*<sup>θ</sup> KSG*. In that case, we used the implementation provided by the open access toolbox TRENTOOL, a TE estimation and analysis toolbox for Matlab [34].

Regarding the selection of parameters involved in the different effective connectivity estimation methods, we proceeded as follows: For the TE methods, we estimated all parameters from the real-valued time series, i.e., before extracting the phase time series. The embedding delay *τ* was set to 1 autocorrelation time (ACT), as proposed in [31]. The embedding dimension *d* was selected from the range *d* = {1, 2, ... , 10} using Cao's criterion [34,53]. Note that for any signal pair, the embedding parameters selected are those of the driven or target time series, i.e., to estimate TE(**x** → **y**) we use for both time series the embedding parameters found for **y**. The interaction delay *u* was set as the value generating the largest TE from ranges that varied depending on the experiment: *u* = {1, 2, ... , 10} for the NMMs, *u* = {1, 4, ... , 25} for the MI data, and *u* = {50, 60, ... , 250} for the WM data. Note that the meaning of *u* in terms of the time delay of the directed interaction between the driving and driven systems is associated with the sampling frequency, e.g., *u* = {1, 2, . . . , 10} for data sampled at 250 Hz translates to a time range between 4 ms and 40 ms. For TE*<sup>θ</sup> κα* we select a value of *α* = 2, which is neutral to weighting, a convenient choice when there is no previous knowledge about the values of the *α* parameter better suited for a particular application [10,28]. In addition, as kernel function, we employ an RBF kernel with Euclidean distance (see Equation (13)). The bandwidth *σ* was set in each case as the median distance of the data [54]. For TE*<sup>θ</sup> KSG* the Theiler correction window and the number of neighbors were left at their default values in TRENTOOL, 4 and 1 ACT, respectively [34]. For the GC methods the order of the autoregressive models *o* was selected from the range *o* = {1, 3, ... , 9} using Akaike information criterion [55,56]. Furthermore, in order to estimate the PSI we employed a sliding window 5 frequency bins long (3 bins long for the WM data), centered on the frequency of interest. Finally, for all the connectivity methods involving the extraction of phase time series through Morlet wavelets, we varied the parameter *m* (see Equation (10)) from 3 to 10 in a logarithmic scale, according to the selected frequency of the filter.

## **4. Results and Discussion**
