Transfer entropy (TE) is a model-free effective connectivity measure based on information theory. It has been increasingly used in neuroscience because of its ability to detect unknown non-linear interactions, which makes it well suited for exploratory brain effective connectivity analyses. Like all information theoretic quantities, TE is defined regarding the probability distributions of the system under study, which in practice are unknown and must be estimated from data. Commonly used methods for TE estimation rely on a local approximation of the probability distributions from nearest neighbor distances, or on symbolization schemes that then allow the probabilities to be estimated from the symbols' relative frequencies. However, probability estimation is a challenging problem, and avoiding this intermediate step in TE computation is desirable. In this work, we propose a novel TE estimator using functionals defined on positive definite and infinitely divisible kernels matrices that approximate Renyi's entropy measures of order α. Our data-driven approach estimates TE directly from data, sidestepping the need for probability distribution estimation. Also, the proposed estimator encompasses the well-known definition of TE as a sum of Shannon entropies in the limiting case when α → 1. We tested our proposal on a simulation framework consisting of two linear models, based on autoregressive approaches and a linear coupling function, respectively, and on the public electroencephalogram (EEG) database BCI Competition IV, obtained under a motor imagery paradigm. For the synthetic data, the proposed kernel-based TE estimation method satisfactorily identifies the causal interactions present in the data. Also, it displays robustness to varying noise levels and data sizes, and to the presence of multiple interaction delays in the same connected network. Obtained results for the motor imagery task show that our approach codes discriminant spatiotemporal patterns for the left and right-hand motor imagination tasks, with classification performances that compare favorably to the state-of-the-art.

Neural oscillations are present in the brain at different spatial and temporal scales, and they are linked to several cognitive functions. Furthermore, the information carried by their phases is fundamental for the coordination of anatomically distributed processing in the brain. The concept of phase transfer entropy refers to an information theory-based measure of directed connectivity among neural oscillations that allows studying such distributed processes. Phase TE is commonly obtained from probability estimations carried out over data from multiple trials, which bars its use as a characterization strategy in brain–computer interfaces. In this work, we propose a novel methodology to estimate TE between single pairs of instantaneous phase time series. Our approach combines a kernel-based TE estimator defined in terms of Renyi’s α entropy, which sidesteps the need for probability distribution computation with phase time series obtained by complex filtering the neural signals. Besides, a kernel-alignment-based relevance analysis is added to highlight relevant features from effective connectivity-based representation supporting further classification stages in EEG-based brain–computer interface systems. Our proposal is tested on simulated coupled data and two publicly available databases containing EEG signals recorded under motor imagery and visual working memory paradigms. Attained results demonstrate how the introduced effective connectivity succeeds in detecting the interactions present in the data for the former, with statistically significant results around the frequencies of interest. It also reflects differences in coupling strength, is robust to realistic noise and signal mixing levels, and captures bidirectional interactions of localized frequency content. Obtained results for the motor imagery and working memory databases show that our approach, combined with the relevance analysis strategy, codes discriminant spatial and frequency-dependent patterns for the different conditions in each experimental paradigm, with classification performances that do well in comparison with those of alternative methods of similar nature.

Adjusting the stimulation parameters is a challenge in deep brain stimulation (DBS) therapy due to the vast number of different configurations available. As a result, systems based on the visualization of the volume of tissue activated (VTA) produced by a particular stimulation setting have been developed. However, the medical specialist still has to search, by trial and error, for a DBS set-up that generates the desired VTA. Therefore, our goal is developing a DBS parameter tuning strategy for current clinical devices that allows defining a target VTA under biophysically viable constraints. We propose a machine learning approach that allows estimating the DBS parameter values for a given VTA, which comprises two main stages: i) A K-nearest neighbors-based deformation to define a target VTA preserving biophysically viable constraints. ii) A parameter estimation stage that consists of a data projection using metric learning to highlight relevant VTA properties, and a regression/classification algorithm to estimate the DBS parameters that generate the target VTA. Our methodology allows setting a biophysically compliant target VTA and accurately predicts the required configuration of stimulation parameters. Also, the performance of our approach is stable for both isotropic and anisotropic tissue conductivities. Furthermore, the computational cost of the trained system is acceptable for real-world implementations.

Cross-frequency interactions, a form of oscillatory neural activity, are thought to play an essential role in the integration of distributed information in the brain. Indeed, phase-amplitude interactions are believed to allow for the transfer of information from large-scale brain networks, oscillating at low frequencies, to local, rapidly oscillating neural assemblies. A promising approach to estimating such interactions is the use of transfer entropy (TE), a non-linear, information-theory-based effective connectivity measure. The conventional method involves feeding instantaneous phase and amplitude time series, extracted at the target frequencies, to a TE estimator. In this work, we propose that the problem of directed phase-amplitude interaction detection is recast as a phase TE estimation problem, under the hypothesis that estimating TE from data of the same nature, i.e., two phase time series, will improve the robustness to the common confounding factors that affect connectivity measures, such as the presence of high noise levels. We implement our proposal using a kernel-based TE estimator, defined in terms of Renyi’s α entropy, which has successfully been used to compute single-trial phase TE. We tested our approach on the synthetic data generated through a simulation model capable of producing a time series with directed phase-amplitude interactions at two given frequencies, and on EEG data from a cognitive task designed to activate working memory, a memory system whose underpinning mechanisms are thought to include phase–amplitude couplings. Our proposal detected statistically significant interactions between the simulated signals at the desired frequencies for the synthetic data, identifying the correct direction of the interaction. It also displayed higher robustness to noise than the alternative methods. The results attained for the working memory data showed that the proposed approach codes connectivity patterns based on directed phase–amplitude interactions, that allow for the different cognitive load levels of the working memory task to be differentiated.

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