## 1 Introduction  
Biological neural systems exhibit rhythmic activity across many scales, from the spiking activity of individual neurons to the electric potentials generated by large populations of neurons, measurable from the scalp in the form of electroencephalographic (EEG) recordings [1]. Interactions between oscillations of different frequencies, known as cross-frequency couplings (CFCs), have been hypothesized to be directly related to the integration of distributed information in the brain [2] by regulating and synchronizing multi-scale communication within and across neural ensembles [3]. The most widely studied instance of CFC is the modulation of the amplitude envelope of high-frequency oscillations by the phase evolution of low-frequency activity, known as phase-amplitudecoupling (PAC) [2; 3; 4]. These phase-amplitude interactions seem to be linked to normal and pathological brain processes in different mammalian species, including humans [5], and they were locally and interregionally observed across a wide range of cognitive tasks [6]. Theoretically, PAC allows for information transfer from large-scale brain networks associated with low-frequency oscillations to local, fast cortical processing areas exhibiting high-frequency activity [7]. Phase-amplitude interactions are commonly assessed using electrophysiological data through metrics of statistical dependency, such as the modulation index, mean vector length, and variations in the concept of mutual information [8; 9; 10]. However, they are unable to capture the directionality and delay of phase-amplitude interactions, quantities that are intrinsic to the concept of information being sent from one neural assembly to another [1; 5; 11]. A natural solution to this limitation, within the framework of information theory, would be to assess PACs using transfer entropy [3; 9; 11; 12; 13]. Transfer entropy (TE) is a model-free connectivity measure that estimates the directed interaction, or information flow, between two dynamical systems [14; 15]. It is specially well-suited to exploratory analysis in neuroscience because of its ability to detect unknown non-linear interactions [16; 17]. However, as a model-free information-theoretic measure, it does not capture the details of how the information transfer is carried out [1]. Standard TE will not reveal anything about the spectral characteristics of the interactions it detects. The most common approach explored in the literature to assess directed phase-amplitude interactions through TE involves bandpass-filtering the signals in the target frequency bands before the extraction of the instantaneous phase and amplitude time series, which are then fed to a TE analysis [5; 18]. Nonetheless, it has been argued that filtering before TE computation negatively affects the results of TE [19; 20], leading to false positives, and that it may not have the desired frequency-specific effects [1]. In this work, we reframe the problem of estimating directed phase-amplitude interactions through TE as the computation of TE between two instantaneous phase time series, known as phase TE [21], by borrowing the underlying premise of the cross-frequency directionality (CFD), a linear connectivity measure capable of estimating the PAC direction [11]. We previously addressed the problem of phase TE estimation from single trial data [22] and hypothesized that, for directed PAC, the proposed approach can correctly identify the interacting frequencies, as well as the direction of interaction, while being robust to common factors that degrade the performance of connectivity estimation methods, such as the presence of high levels of noise and volume conduction effects [23]. To test our proposal, we use a simulation model that allows generating synthetic data with unidirectionally phase-amplitude couplings at two target frequencies [11], and real EEG data from a change detection task with several difficulty levels designed to study visuospatial working memory [24; 25].