Learning Cortico-Muscular Dependence through Orthonormal Decomposition of Density Ratios

TL;DR

The paper tackles the challenge of modeling cortico-muscular connectivity beyond traditional linear measures by learning an orthonormal decomposition of the density ratio between EEG and EMG signals. It introduces FMCA-T, a matrix-trace variant of Functional Maximal Correlation that jointly trains two neural nets to estimate the top eigenvalues and eigenfunctions, enabling context-aware representations captured by $\rho(X,Y)\approx \sum_{k=1}^K \sigma_k^{1/2}\,\phi_k(X)\psi_k(Y)$. The method further localizes dependence into channel- and temporal-level density ratios to reveal spatio-temporal activation patterns and movement/subject information without labeled data. Experimentally, FMCA-T shows robustness to nonstationary noise and delays, yields eigenfunctions that discriminate movements and subjects, and produces spatial maps consistent with known brain activations, outperforming several baselines in inter- and cross-subject classification. The work suggests strong potential for neuroscience analysis and brain-machine interfaces, while recognizing limitations due to dataset size and proposing future expansion to larger multimodal datasets.

Abstract

The cortico-spinal neural pathway is fundamental for motor control and movement execution, and in humans it is typically studied using concurrent electroencephalography (EEG) and electromyography (EMG) recordings. However, current approaches for capturing high-level and contextual connectivity between these recordings have important limitations. Here, we present a novel application of statistical dependence estimators based on orthonormal decomposition of density ratios to model the relationship between cortical and muscle oscillations. Our method extends from traditional scalar-valued measures by learning eigenvalues, eigenfunctions, and projection spaces of density ratios from realizations of the signal, addressing the interpretability, scalability, and local temporal dependence of cortico-muscular connectivity. We experimentally demonstrate that eigenfunctions learned from cortico-muscular connectivity can accurately classify movements and subjects. Moreover, they reveal channel and temporal dependencies that confirm the activation of specific EEG channels during movement. Our code is available at https://github.com/bohu615/corticomuscular-eigen-encoder.

Figures (16)

• Figure 1: The cortico-muscular pathway allows brain-muscle communication with coherent cortical and peripheral oscillations. This paper models this connectivity through the statistical dependence between their concurrent recordings of EEG and EMG.
• Figure 2: Diagram for learning cortico-muscular dependence by decomposing density ratios: (a) Network $\boldsymbol{\mathit{f}}_\theta$ is applied to EEG $\boldsymbol{\mathit{X}}_{1:T}$ and $\boldsymbol{\mathit{g}}_\omega$ to EMG $\boldsymbol{\mathit{Y}}_{1:T}$ to minimize a matrix trace cost. (b) EEG-EMG pairs are sampled from a joint distribution, from which a density ratio $\rho(X,Y)$ is defined and considered a positive definite function. Its linear operator has a spectral decomposition of eigenfunctions $\{\boldsymbol{\mathit{\phi}}$, $\boldsymbol{\mathit{\psi}}\}$ and eigenvalues $\{\sigma_k\}$. The networks provably approximate the dominant eigenvalues and eigenfunctions of this decomposition with network outputs $\{\widehat{\boldsymbol{\mathit{f}}_\theta}$, $\widehat{\boldsymbol{\mathit{g}}_\omega}\}$, and SVD results $\{\lambda_k\}$. Eigenvalues here measure multivariate statistical dependence; eigenfunctions are optimal feature projectors. (c) After training, the eigenfunctions, specifically those from EEG, form a projection space containing contextual information for motor control and participant identification. (d) To provide channel activation and activity synchronization for cortico-muscular analysis, we compute density ratios between channel-level $\boldsymbol{\mathit{Z}}_c$ and temporal-level features $\boldsymbol{\mathit{Z}}_{c, s}$ against global features $\boldsymbol{\mathit{Z}}_F$ to quantify channel-level and temporal-level dependencies.
• Figure 3: Density ratios from FMCA-T are robust to various noise types: (a) stationary white Gaussian noise, (b) nonstationary Gaussian noise, (c) nonstationary pink noise, and (d) random delays. FMCA-T proves the most robust estimations across all noise types and outperforms all linear and nonlinear baselines. Note that as delays increase, estimations using CC produce negative values given the opposite phase between the paired sinusoids.
• Figure 4: Visualizing eigenfunctions and density ratios in EEG-EMG fusion with FMCA-T: (a) t-SNE of EEG's eigenfunctions for a single subject (SUB1) show nine clusters specific to three movements (MOV1$\sim$MOV3) across three sessions. (b) t-SNE of EEG's eigenfunctions for reaching movement (MOV1) across 10 subjects (SUB1$\sim$SUB10) shows clusters specific to subjects, where each color is a subject. (c) Density ratios and (d) their mean and std of each cluster (C1$\sim$C9) demonstrate intra-cluster consistency and inter-cluster separability. (e-h) Comparison of baseline measures, where only MINE is comparable but with higher variance and instability.
• Figure 5: Comparison of classification accuracies: supervised, self-supervised, and our EEG-EMG dependence learning. FMCA-T's eigenfunctions, trained with trace cost without labels, are optimal feature projectors for EEG. EMG is not required for testing, but only used for training.

Theorems & Definitions (4)

• Lemma 1
• proof
• Lemma 2
• proof