## 2 Methods
### Working Memory Data
#### 3.2.3 Classification Setup  
Our aim is to set up a subject-dependent classification system (one classifier per subject) that allows the different levels of the change detection task, to be differentiated using relevant directed phase-amplitude interactions captured through the proposed TE\({}_{xa}^{\theta\phi^{c}}\) approach as inputs. Feature Extraction  
Let \(\Psi=\{\textbf{X}_{n}\in\mathbb{R}^{C\times M}\}_{n=1}^{N}\) be the EEG set holding \(N\) trials from the WM dataset, recorded from a single subject, with \(C\) as the number of channels and \(M\) as the number of samples of each trial. Let \(\{l_{n}\}_{n=1}^{N}\) be a set whose \(n\)-th element corresponds to a label assigned to the \(n\)-th trial \(\textbf{X}_{n}\). The elements of \(l_{n}\) can take the values of 1, 2, and 3 to indicate low-, medium-, and high-memory loads, respectively. Our aim is to predict the label \(l_{n}\) from the transfer entropy features \(\left(TE_{xa}^{\theta\phi^{c}}\right)\) that capture the directed phase-amplitude interactions present in \(\textbf{X}_{n}\). Let \(\lambda(\textbf{x}_{c}\rightarrow\textbf{x}_{c^{\prime}},f_{c},f_{c^{\prime}})\) be a TE measure between the phase of channel \(\textbf{x}_{c}\) at frequency \(f_{c}\), and the amplitude envelope of \(\textbf{x}_{c^{\prime}}\) at frequency \(f_{c^{\prime}}\), as defined in Equation (6). For all pairwise combinations of channels in \(\textbf{X}_{n}\), computing \(\lambda(\textbf{x}_{c}\rightarrow\textbf{x}_{c^{\prime}},f_{c^{\prime}}f_{c^{ \prime}})\) yields a connectivity matrix \(\textbf{\Lambda}(f_{c^{\prime}},f_{c^{\prime}})\in\mathbb{R}^{C\times C}\). When \(c=c^{\prime}\), then \(\lambda(\textbf{x}_{c^{\prime}},f_{c^{\prime}}f_{c^{\prime}})=0\). The values of \(f_{c},f_{c^{\prime}}\) vary in the range from 4 Hz to 18 Hz, in 2 Hz steps, since activity in that frequency range has been associated with interactions between different brain regions during WM [6]. Then, three bandwidths are defined \(\Delta f\in\{\theta\in[4-6],\alpha\in[8-12],\beta_{l}\in[14-18]\}\) Hz, and the matrices \(\textbf{\Lambda}(f_{c},f_{c^{\prime}})\) are averaged within each pairwise combination of bandwidths (\(\theta-\alpha,\theta-\beta_{l}\), etc.). After that, each of the averaged matrices are normalized to the range \([0,1]\), and stacked together. Therefore, each trial is characterized by a connectivity matrix \(\textbf{\Lambda}^{\prime}\in\mathbb{R}^{C\times C\times 6}\). For the \(N\) trials, a set of connectivity matrices \(\{\textbf{\Lambda}^{\prime}_{n}\in\mathbb{R}^{C\times C\times 6}\}_{n=1}^{N}\) is used. Then, by applying vector concatenation to \(\textbf{\Lambda}^{\prime}_{n}\) a vector \(\boldsymbol{\phi}_{n}\in\mathbb{R}^{1\times(C\times C\times 6)}\) is obtained. Finally, the \(N\) vectors \(\boldsymbol{\phi}_{n}\) are stacked together in order to obtain a single bi-dimensional  
Figure 3: (**A**) Graphical representation of the change detection task for WM. (**B**) EEG channels selected from the montage used for the acquisition of the WM database. matrix \(\mathbf{\Phi}\in\mathbb{R}^{N\times p}\), \(P=C\times C\times 6\), that characterizes \(\Psi\) in terms of phase-amplitude TE measures. In practice, for each subject, the training data \(\mathbf{\Phi}\) have the following dimensions: \(N\) rows, which correspond to the number of trials of the task correctly performed by the subject (\(N\leq 96\)), and \(P=C\times(C-1)\times 6=5952\) features, where \(C-1\) accounts for the fact that the TE of a time series is not defined, and, therefore, the main diagonals of the \(\mathbf{\Lambda}_{n}^{\prime}\) matrices are empty (or all zeros, as in our case). Feature Selection and Classification  
In order to identify the TE features relevant to the discrimination between the cognitive load levels in the WM database, we rely on a relevance analysis methodology based on centered kernel alignment (CKA). CKA quantifies the similarity between two sample spaces by comparing two kernel functions that characterize each space [40].