## 1 Introduction  
Neural oscillations are observed in the mammalian brain at different temporal and spatial scales [1]. Oscillations in specific frequency bands are present in distinct neural networks, and their interactions have been linked to fundamental cognitive processes such as attention and memory [2, 3] and to information processing at large [4]. Three properties characterize such oscillations: amplitude, frequency, and phase, the latter referring to the position of a signal within an oscillation cycle [5]. Oscillation amplitudes are related to neural synchrony expansion in a local assembly, while the relationships between the phases of neural oscillations, such as phase synchronization, are involved in the coordination of anatomically distributed processing [6]. Moreover, from a functional perspective, phase synchronization and amplitude correlations are independent phenomena [7], hence the interest in studying phase-based interactions independently from other spectral relationships. Additionally, phase relationships are linked to neural synchronization and information flow within networks of connected neural assemblies [8]. Therefore, a measure that aims to capture phase-based interactions among signals from distributed brain regions should ideally include a description of the direction of interaction. A fitting framework for such measure is that of brain effective connectivity [9]. Effective brain connectivity, also known as directed functional connectivity, measures the influence that a neural assembly has over another one, establishing a direction for their interaction by estimating statistical causation from their signals [10]. Directed interactions between oscillations of similar frequency can be captured through measures such as Geweke-Granger causality statistics, partially directed coherence, and directed transfer function [9; 11]. However, since these metrics depend on both amplitude and phase signal components, they do not identify phase-specific information flow [8]. The phase slope index (PSI), introduced in [12], measures the direction of coupling between oscillations from the slope of their phases; still, it only captures linear phase relationships [13]. In this context arises the concept of phase transfer entropy, a phase-specific nonlinear directed connectivity measure introduced in [8]. Transfer entropy (TE) is an information-theoretic quantity, based on [PERSON]'s definition of causality, that estimates the directed interaction, or information flow, between two dynamical systems [14; 15]. In [8], the authors first extract instantaneous phase time series by complex filtering the signals of interest in a particular frequency, since a signal's phase is only physically meaningful when its spectrum is narrow-banded [16]. Such filtering-based approach has also been explored to obtain phase-specific versions of other information-theoretic metrics such as permutation entropy and time-delayed mutual information [7; 16]. Then, the authors compute TE from the obtained phase time series. Nonetheless, since conventional TE estimators are not well suited for periodical variables, in [8] phase TE estimates are obtained through a binning approach performed over multiple trials simultaneously, in a procedure termed trial collapsing. Phase TE has found multiple applications in neuroscience, such as gaining insight into reduced levels of consciousness by evaluating brain connectivity [17], analyzing resting-state networks [18], and assessing brain connectivity changes in children diagnosed with attention deficit hyperactivity disorder following neurofeedback training [19]. It has even been used to detect fluctuations in financial markets data [20]. Nonetheless, phase TE, estimated as in [8], cannot be employed as a characterization strategy for brain-computer interfaces (BCI) since they require features extracted on an independent trial basis, i.e., each trial must be associated with a set of features.