1. Introduction
From the neuroscience perspective, each physiological or cognitive process will produce a particular pattern of electrical interactions linking neurons from different brain regions. Therefore, the electrical response related to the interactions between neuron assemblies allows studying the brain function. Under such a perspective, the neuroscientists have found brain electrical activity (BEA) patterns associated with motor functions, cognitive processes, and neuropathologies [
1]. However, the wide range of possible mental conditions hampers the BEA analysis and poses very challenging tasks [
2]. In particular, capturing BEA by placing a set of electrodes over the scalp, known as electroencephalography (EEG), gathers and amplifies currents reflected in the brain cortex from all the possible brain sources, yielding a mixture of latent activity sources at each channel. For discovering such a latent activity, the literature considers different types of EEG analyses ranging from time and spectral domain processing [
3,
4], through connectivity measures between channels [
5], to the complex network analysis [
6].
Due to the non-stationarity behavior of EEG data, the classical temporal analyses cannot decode differences among several mental states or conditions. Besides, several studies demonstrate that the variations in the EEG oscillatory patterns play a fundamental role in the maintenance of brain functions and the identification of different neural conditions [
7]. Hence, spectral decomposition approaches look for relevant information at the brain rhythms (namely alpha, beta, theta, delta, and gamma). For instance, performing working memory or creative tasks evokes discriminative oscillatory patterns [
8,
9]. In addition, neuropathologies, such as Alzheimer’s disease, cause abnormal cortical neural synchronization at resting-state rhythms [
10]. The general spectral analysis procedure consists of a channel-wise frequency band splitting, followed by an identification of relevant time intervals, usually supervised by a specialist. The subsequent stages compute descriptors from the time-frequency representation. The power spectral density (PSD) and spectral entropy are among the most considered descriptors due to their straightforward interpretation [
11,
12]. Although EEG frequency analysis has proven to extract useful information for brain function understanding, the channel-wise approach still lacks the interpretability of several cognitive processes because each channel only holds a reflected version of BEA from a neural assembly [
13]. Besides, the low spatial resolution of EEG restricts the extracted information about the behavior of some brain regions involved in high-level cognitive or physiological processes [
4].
In recent years, connectivity analysis techniques account for channel interdependencies to enhance EEG spatial resolution through functional relationships captured in the BEA. Metrics from several domains attempt to quantify different properties of the BEA. Specifically, the coherence (COH), phase value index (PVI), and phase-locking value (PLV) capture pair-wise channel dependencies in the frequency domain. Moreover, the latter two gained attention due to their non-linear capability for unraveling latent connectivity patterns, which have proven valuable for applications, such as motor imagery and emotion recognition [
14,
15]. Nevertheless, the selection of the connectivity measure is not entirely straightforward. Their dependence on an estimated cross-spectrum and subject data variability yield to low generalization capability in a wide range of scenarios [
16,
17].
The above issues demand reliable brain connectivity approaches to automatically identify the relevant spectral information from the inherent EEG uncertainty. In this regard, a probabilistic modeling framework, such as Gaussian Processes (GPs), along with an appropriate covariance function possesses the capability for characterizing the latent processes from BEA data [
18,
19]. Moreover, the extension to vector-valued processes or multi-output GPs (MOGPs) adjusts the probabilistic model to map inputs into a multidimensional output space as an EEG channel array [
20,
21]. Former GP applications to EEG analysis in stress detection and cognitive stimulus recognition demonstrated the potential of GPs for BEA data modeling [
22,
23]. Furthermore, a recently proposed covariance function, known as the multi-output spectral mixture (MOSM) analyzes dependencies with spectral information for multidimensional output processes [
24]. The proposal novelty relies on the PSD design, ensuring the frequency constraints for real-valued signals. Additionally, the inverse Fourier transform of the MOSM covariance results in a temporal kernel with positive definiteness conditions that are harder to accomplish during kernel design.
In this work, we propose the extension of MOGPs with an MOSM kernel for BEA discrimination, termed MOSM-GP. To this end, we learn an MOSM-GP for each EEG trial in a training set to model and quantify channel relationship patterns associated with particular EEG conditions. Then, we implement a likelihood measure to label new trial data into a specific class. The proposed framework of discriminative MOSM-GP, termed DMOSM-GP, is tested in two publicly available EEG datasets acquired under emotional [
25] and motor imagery [
26] conditions. The attained results show that the likelihood measure of testing data on the trained MOSM-GPs corresponds to a reliable discriminative index.
The manuscript organization is as follows:
Section 2.2 describes the theoretical background of MOSM-GPs and introduces the proposed framework.
Section 3 presents the attained results on both EEG modeling and classification performance. Finally,
Section 4 concludes the works with the most relevant findings.
4. Discussion and Concluding Remarks
Brain information processing is a complex task that is not yet entirely mapped and understood. Despite the previous knowledge about brain regions interactions in motor or emotional means, works that allow improving the interpretability of results in different BEA scenarios would lead to the development of more precise frameworks for analysis, diagnosis, and treatment of mental pathologies, among other tasks. In this work, we proposed a framework for discriminate BEA using raw data with the support of a spectral kernel that identifies relationships between channels on behalf of a probabilistic methodology of multi-output Gaussian process. One of the essential remarks of this framework is the capability of learning EEG connectivity patterns by estimating raw data spectral components without a feature extraction stage. Further, this proposal of generative models working directly with EEG data has the advantage of adjusting the model relying on the optimization of kernel hyperparameters just from the channels information in a data-driven framework.
The results presented in
Section 3 evidenced that introducing the MOSM kernel to MOGPs becomes a reliable tool for BEA modeling, due to the spectral designing properties. It is well known that EEG channels share complex frequency relationships that can be exploited using the design of a covariance function in that particular domain. Regarding this property, the posterior distribution over the data measured by the MAE in
Figure 3 shows an adequate adjustment of the model on the original data. It also allows us to conclude that the inclusion of more spectral components into the covariance function benefits the model adjusting to the data.
Moreover, the identification of the spectral relationships between channels, performed by the MOSM kernel, allows gaining a better understanding of the latent functional connectivity between brain regions. As
Figure 6 evidenced, the MOSM-GP identifies representative frequency bands for the cognitive process. The positive linear relationships are grouped among channels from the same hemisphere, with strong specific dependencies between channels, like
, and
. The negative relationships are explained by lower PSD amplitudes at
,
, and
for the left-hand MI task, and
for the right-hand MI task. All these interactions quantified by the MOSM kernel in terms of higher or lower values of the PSD can be directly related with the activations of neural cells in different regions of the brain related with emotions (amygdala, hippocampus, and frontal cortex) or related with motor activities (primary motor cortex and posterior parietal cortex). Nevertheless, further studies must be completed from the evidence of these relationships to an accurate source reconstruction before determining the specific brain region of the acquired neural activity.
Finally, the discriminative results regarding the emotional and motor imagery conditions conclude that probabilistic models can be efficiently employed as a classification tool for EEG data. In this case, the probability distribution of tested data against the trained models directly becomes a classification algorithm by following a direct comparison of the mean likelihood value between the models from two classes. Despite lacking a feature extraction stage, the proposed DMOSM-GP produces discriminative information from the data. Moreover, the total accuracy of the subject-dependent and condition-dependent tests is comparable with state-of-art works, as
Table 5 and
Table 6 illustrate.
One of the drawbacks of this framework is the computational complexity of training a considerable number of MOSM-GP models. In addition, increasing the number of MOSM kernel mixtures and the size of BEA data will derive into an exponential growth of the training time. Further improvements of this methodology will be directed towards using lighter versions of probabilistic models, such as the sparse GPs aiming at solving more difficult supervised learning tasks from EEG data as multilabel classification or regression.