LEGEND: Lorentzian electro-modal graph encoder for neural decoding for SCI rehabilitation

Spinal cord injury (SCI) severs voluntary motor control between cortical intent (electroencephalography, EEG, above the lesion) and peripheral muscle activity (electromyography, EMG, below it), leaving spinal circuit signals (electrospinography, ESG) as the critical intermediate. A viable neural bypass must jointly model all three tiers of the motor hierarchy. We introduce LEGEND (Lorentzian Electro-modal Graph Encoder for Neural Decoding), a two-stage architecture that encodes EEG/ESG/EMG in Lorentz hyperboloid space via a shared TriModalLorentzNet, connects the 51 channel nodes through a signed tri-layer Phase-Locking Value (PLV) graph, and refines embeddings with a class-prototype-conditioned HyperbolicGraphAttentionHead. On the Steele dataset Steele et al. (2023) [1], the only public simultaneous tri-modal motor corpus under strict leave-one-subject-out (LOSO) evaluation, LEGEND achieves 56.51%±12.27%, a +23.4 pp gain over EEGNet (33.11%) and +22.5 pp over ShallowConvNet (33.97%) (p<0.01, Wilcoxon), while outperforming three recent graph-neural-network baselines adapted to the same tri-modal 51-channel input: BrainTopoGCN (35.84%), EEG-GLT-Net (30.63%), and SAMGCN (44.82%), all p≤0.007. An Input-GradCAM extractor identifies class-discriminative EEG→ESG→EMG functional chains (mean pairwise top-5 Jaccard = 0.207). Evaluating the EEG encoder alone on standard motor-imagery benchmarks: we provide the strict cross-subject LOSO evaluation on BCI-IV-2a (9 subjects, 4-class motor imagery, MI), where LEGEND achieves 42.9%±7.0%, outperforming EEGNet-LOSO (35.1%), ShallowConvNet-LOSO (37.8%), and all three graph baselines reimplemented under the same protocol: BrainTopoGCN (38.8%), EEG-GLT-Net (25.2%), and SAMGCN (28.9%). On PhysioNet (109 subjects, binary MI), 80.50%±10.48% LOSO, competitive with EEG-Conformer (81.3%) despite the stricter protocol. Taken together, these results demonstrate that Lorentzian hyperbolic representations are broadly beneficial across EEG-only and tri-modal brain-computer interface (BCI) paradigms, and most importantly provide the validated computational foundation for a neural bypass targeting spinal cord injury rehabilitation.