## 1 Introduction
Effective connectivity measures, such as phase TE, can be used to assess the induced physiological variations in the brain occurring during BCI tasks [21]. Discriminative information may be hidden in the dynamical interactions among spatially separated brain regions that characterization methods commonly employed in BCI are not able detect [22]. This information could be relevant to address issues such as the inefficiency problem in some BCI systems [23]. In that context, authors in [6] applied a binning strategy to estimate single-trial phase TE to set up classification systems for visual attention. Nonetheless, binning estimators for single trial-based estimation of information-theoretic measures exhibit systematic bias [8]. Furthermore, spectrally resolved TE estimation methods that can obtain single-trial TE estimates have been recently proposed in the literature [24; 25]. Yet, phase TE is conceptually different from them [25], as they are not phase-specific metrics. Here, we propose a novel methodology to estimate TE between single pairs of instantaneous phase time series. Our approach combines the kernel-based TE estimator we introduced in [10], with phase time series obtained by convolving neural signals with a Morlet Wavelet. The kernel-based TE estimator expresses TE as a linear combination of Renyi's entropy measures of order \(\alpha\)[26; 27] and then approximates them through functionals defined on positive definite and infinitely divisible kernel matrices [28]. Its most important property is that it sidesteps the need to obtain the probability distributionsunderlying the data. Instead, the estimator computes TE directly from kernel matrices that, in turn, capture the similarity relations among data. It is robust to varying noise levels and data sizes and to the presence of multiple interaction delays in a network [10]. In this work, we hypothesize that the above-described estimator could overcome the hurdles other single-trial TE estimators face when obtaining TE values from instantaneous phase time series since it would not have to explicitly obtain probability distributions from circular variables [8]. Additionally, since our primary motivation to introduce a robust phase TE estimation methodology is the use of such measures in the context of BCI applications, we also explore a relevance analysis strategy based on centered kernel alignment (CKA) [29]. The CKA-based analysis allows us to identify the set of pairwise channel connectivities relevant to discriminate between specific conditions, favoring the neurophysiological interpretation of our results and providing an option to avoid carrying out all to all channel connectivity estimations in practical BCI systems based on phase TE. We employ simulated and real-world EEG data to test the introduced effective connectivity measure. The simulated data are obtained from neural mass models, mathematical models of neural mechanisms that generate time series with oscillatory behavior similar to electrophysiological signals. Obtained results for such data show that the proposed kernel-based phase TE estimation method successfully detects the direction of interaction imposed by the model. Indeed, it detects statistically significant connections in the frequency bands of interest, even for weak couplings and narrowband bidirectional interactions. It also displays robustness to realistic levels of noise and signal mixing. Regarding the EEG data, we consider two databases containing signals recorded under two different cognitive paradigms, consisting of motor imagery tasks and a change detection task designed to study working memory. Attained classification results demonstrate that our approach is competitive compared to real-valued and phase-based directed connectivity measures. Thus, this proposal extends the approach described in [10] by introducing a measure that captures directed interactions between the phases of oscillations at specific frequencies.