Geometric Neural Network based on Phase Space for BCI-EEG decoding

TL;DR

This work targets reliable motor-imagery EEG decoding in BCIs under data- and electrode-limited constraints. It introduces Phase-SPDNet, which fuses Takens-based phase-space reconstruction with SPDNet on the SPD manifold, enabling richer dynamic representations from as few as three electrodes. The model leverages an Augmented Covariance Method to create phase-augmented SPD features and trains a geometry-aware network using Riemannian optimization, achieving state-of-the-art performance across MOABB datasets while maintaining a compact parameter footprint. The approach also provides interpretability through GradCam++ visualizations, highlighting both diagonal and off-diagonal SPD contributions and offering practical benefits for real-world BCI deployments with improved comfort and efficiency.

Abstract

Objective: The integration of Deep Learning (DL) algorithms on brain signal analysis is still in its nascent stages compared to their success in fields like Computer Vision. This is particularly true for BCI, where the brain activity is decoded to control external devices without requiring muscle control. Electroencephalography (EEG) is a widely adopted choice for designing BCI systems due to its non-invasive and cost-effective nature and excellent temporal resolution. Still, it comes at the expense of limited training data, poor signal-to-noise, and a large variability across and within-subject recordings. Finally, setting up a BCI system with many electrodes takes a long time, hindering the widespread adoption of reliable DL architectures in BCIs outside research laboratories. To improve adoption, we need to improve user comfort using, for instance, reliable algorithms that operate with few electrodes. Approach: Our research aims to develop a DL algorithm that delivers effective results with a limited number of electrodes. Taking advantage of the Augmented Covariance Method and the framework of SPDNet, we propose the Phase-SPDNet architecture and analyze its performance and the interpretability of the results. The evaluation is conducted on 5-fold cross-validation, using only three electrodes positioned above the Motor Cortex. The methodology was tested on nearly 100 subjects from several open-source datasets using the Mother Of All BCI Benchmark (MOABB) framework. Main results: The results of our Phase-SPDNet demonstrate that the augmented approach combined with the SPDNet significantly outperforms all the current state-of-the-art DL architecture in MI decoding. Significance: This new architecture is explainable and with a low number of trainable parameters.

Figures (9)

• Figure 1: Representation of the augmented procedure. The original signal comprises three electrodes, represented by the plot in blue, while the red plot represents the augmented dataset using embedding parameters $\tau = 5$ and $\psi=2$.
• Figure 2: Overview of our approach.a. Phase space reconstruction for the EEG signal. For each trial from $\mathbf{X}$, we apply the function $f_{delay}$ to reconstruct the phase space. During each evaluation fold, we adopted the MDOP algorithm or optimized search to estimate suitable function parameters for embedding dimension $\psi$ and delay $\tau$. These hyperparameters are marked in red in the figure. For illustrative purposes, this figure considers 3 electrodes located over the motor cortex with parameters $\psi=2$ and $\tau=10$. b. Conversion of the phase space time-series with $f_{\mathbb{C}\text{ov}}$ for the covariance representation of the SPD, represented in green. With the parameter considered, the covariance is a square matrix of dimension 6. c. The covariance feature space is then used to train the Symmetric Positive Definite Neural Network with BiMap, ReEig, LogEig, and MLP layers. Once the representation returns to the Euclidean space (red), after the LogEig, we adjust a fully connected layer to predict the associate label $y$ for each trial. The light blue color represents the layer that possesses trainable parameters.
• Figure 3: Results for Right vs Left-hand classification, using Within-Session evaluation. Plot (a) provides the relative improvement of the AUC-ROC in percentage of the method considered with respect to the standard SPDNet of the different pipelines considered. Plot (b) shows a combined meta-analysis (over all datasets) of the different pipelines. It shows the significance of the algorithm on the y-axis being better than the one on the x-axis. The gray level represents the significance level of the ROC-AUC difference in terms of t-values. We only show significant interactions ($p < 0.05$). Plots (c), (d), and (e) show the meta-analysis of Phase-SPDNet $_{OPT}$ against SPDNet, ShallowNet, and DynSpat+ShallowNet, respectively. We show the standardized mean differences of p-values computed as a one-tailed Wilcoxon signed-rank test for the hypothesis given in the plot title. The gray bar denotes the $95\%$ interval. * stands for $p < 0.05$, ** for $p < 0.01$, and *** for $p < 0.001$.
• Figure 4: Most and least responsive 5 subjects of Cho2017. (a) Plot showing the box plot of the five most responsive subjects (3, 14, 35, 41, 43) of Cho2017. We see that the augmentation procedure consistently leads to notable improvements in the classification performance for all the SPD estimators considered. (b) Plot showing the box plot of the least responsive five subjects (2, 7, 29, 34, 50) of Cho2017. We see that the augmentation procedure increases the performance for the covariance feature, while the improvement is less clear for some FC estimators. In the plot, the names Ins and Imn stand respectively for Instantaneous Coherence and Imaginary Coherence.
• Figure 5: Interpretability parallel between GradCam++ and t-test for right versus left-hand classification for subject 35 of Cho2017. Plot (a) is the GradCam++ saliency map obtained at the level of the ReEig layer, for Phase-SPDNet $_{OPT}$ with a target left hand. Plot (b) is a zoom in the region corresponding to standard Covariance. Plot (c) the GradCam++ saliency map obtained at the level of the ReEig layer, for SPDNet with a target left hand. This discrepancy highlights the added value of our method in identifying crucial inter-channel relationships that might be overlooked by traditional covariance analysis.