## 5 Conclusions  
In this work, we proposed a novel approach to estimate directed phase-amplitude interactions through TE. The central idea behind our proposal is to recast the problem of detecting directed PAC as that of estimating directed interactions between phase time series. Doing this allowed us to employ a single-trial, kernel-based phase TE-estimator to assess the cross-frequency interactions of interest. We tested the performance of our proposal on synthetic data containing directed phase-amplitude interactions and on an EEG database obtained under a WM paradigm. The obtained results for the synthetic data show that our approach successfully detects the direction of interaction and the interacting frequencies, while being more robust to noise than alternative methods. Additionally, for the WM data, our proposal revealed discriminant, directed, phase-amplitude interactions associated with the different cognitive loads of the task.  
In future work, we will attempt to optimize the strategies for selecting the parameters involved in the filtering and time-embedding stages of the proposed approach and the kernel-based TE estimator.  
Conceptualization, I.D.L.P.P., V.G.-O. and A.A.-M.; methodology, I.D.L.P.P., V.G.-O., and A.A.-M.; software, I.D.L.P.P., A.A.-M. and D.C.-P.; validation, I.D.L.P.P., A.A.-M. and D.C.-P.; formal analysis, I.D.L.P.P., V.G.-O. and A.O.-G.; investigation, I.D.L.P.P., V.G.-O. and A.A.-M.; resources, I.D.L.P.P., A.A.-M. and A.O.-G.; data curation, I.D.L.P.P. and V.G.-O.; writing--original draft preparation, I.D.L.P.P., V.G.-O. and A.A.-M.; writing--review and editing, A.A.-M. and D.C.-P.; visualization, I.D.L.P.P.; supervision, A.O.-G. and A.A.-M.; project administration, A.O.-G.; funding acquisition, I.D.L.P.P., V.G.-O. and A.A.-M. All authors have read and agreed to the published version of the manuscript  
Authors [PERSON] and [PERSON] were supported by the program \"Doctorado Nacional en Empresa--Convoctoria 758 de 2016\", funded by Minicencias. [PERSON] thanks the project _Prototip de interfaz cerebro-computador multimodal para la detecion de patrones relevantes relacionados con trastornos de impulsividad_-HERMES 50835, funded by Universidad Nacional de Colombia.  
Institutional Review Board StatementIn this study, we use a public-access EEG database introduced in a previously published work, and made freely available by its authors [24]. We did not collect any data from human participants ourselves.  
Informed Consent StatementThis study uses an anonymized public database introduced in a previously published work by another group [24].  
Data Availability StatementThe database used in this study is public and can be found at the following link: database from brain activity during visual working memory [[https://data.mendeley.com/datasets/j2v7](https://data.mendeley.com/datasets/j2v7) bchdy/2]([https://data.mendeley.com/datasets/j2v7](https://data.mendeley.com/datasets/j2v7) bchdy/2) (accessed on 4 September 2021).  
Conflicts of InterestThe authors declare that this research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.