## INTRODUCTION

<figure><img src="image_1.png" /></figure>

Brain-computer interfaces based on EEG often use pairwise features such as spectral power and imaginary coherence (Imcoh), which capture only bivariate local connections. We instead apply topological data analysis (TDA) to summarize global network structure and higher-order interactions, and test whether these features distinguish motor imagery (MI) from rest.

## PIPELINE

### 1. From EEG to Point Cloud via bivariate ImCoh

<figure><img src="image_2.png" /><figcaption>Figure 1. Raw EEG, β-band imaginary coherence (64×64), and the 3-D Laplacian spectral embedding of channels (single MI run).</figcaption></figure>

### 2. Alpha Complex Filtration and Persistent Homology

<figure><img src="image_3.png" /><figcaption>Figure 2. Filtration and persistent homology of the embedded point cloud.</figcaption></figure>

### 3. Homological Features

<figure><img src="image_4.png" /><figcaption>Figure 3: Persistence diagram and distributions of $H_0/H_1$ total persistence.</figcaption></figure>

## METHOD

We analyze MI vs Rest EEG on a per-subject basis. For each run, imaginary coherence (ImCoh) is computed between all channel pairs and averaged in the β band (12–26 Hz) to yield a symmetric connectivity matrix. The graph is mapped to a point cloud via Laplacian spectral embedding (3-D). We then build an α-complex on these points and compute persistent homology up to $H_2$ (GUDHI¹). From the diagrams we extract compact, interpretable features—$H_0$, $H_1$ component count and $H_0$, $H_1$ total persistence. As a baseline we compare against node-strength features. For classification, all features are z-scored and modeled with Linear Discriminant Analysis (LDA), evaluated with 3-fold stratified cross-validation and a 20% hold-out split.

We use the BRACCIO motor-imagery EEG dataset² (64-channel, 500 Hz) with three sessions per subject—each comprising six ≈10-min runs alternating MI and Rest.

For homology dimension $k$, let the persistence diagram be

$$D_k = \{(b_i', d_i)\}_{i=1}^{n_k}$$

$$H_{k\_count} = n_k = |D_k|$$

Count

Total persistence

$$H_{k\_tot\_pers} = \sum_{i=1}^{n_k} \max(0, \sqrt{d_i} - \sqrt{b_i})$$

## RESULTS

<figure><img src="image_5.png" /><figcaption>Figure 4: Subject-wise LDA — Topological features vs Node strength. Across high- and low-performing subjects, topological persistence features outperform node strength in LDA classification (0.71 vs 0.58; 0.65 vs 0.55).</figcaption></figure>

## CONCLUSION

Topological summaries of β-band connectivity ($H_0/H_1$ total persistence and count) add complementary information beyond node strength, yielding modest but consistent LDA gains (good subject 0.71 vs 0.58). This points to higher-order network geometry helping separate MI from Rest. Still, interpretability needs sharpening—do these invariants mainly track strongly connected components, or capture task-specific reconfiguration with distinct meaning? Next, we'll validate on the full cohort with nested CV, examine other bands, and run ablations/controls to link $H_0/H_1$ to neurophysiology and refine subject-specific calibration, and test alternative embeddings.

## REFERENCES

1. The GUDHI Project. (2025). GUDHI User and Reference Manual (v3.11.0). GUDHI Editorial Board. https://gudhi.inria.fr/doc/3.11.0/

2. Venot, T., Desbois A., Corsi M. C., Hugueville L., Saint-Bauzel L., & De Vico Fallani F. (2024). Intentional binding for noninvasive BCI control. Journal of Neural Engineering, 21(4), 046026. https://doi.org/10.1088/1741-2552/ada628c