State-Space Modeling of Real-Time Visual Biofeedback and Late-Stance Belt-Speed Modulation for Quantifying Error- and Use-Dependent Learning of Gait Propulsion in Individuals Post-Stroke

This study developed an exploratory, model-constrained state-space model to characterize how real-time visual biofeedback (VB) and late-stance, phase-specific belt modulation (propulsion-facilitating, PF) were associated with improvements in affected-limb propulsion following stroke. Twelve individuals with chronic stroke participated in a single-session experiment conducted on the Adaptive Propulsion Enhancement eXperience (APEX) system, which utilizes an instrumented split-belt treadmill. The system continuously displays the affected-limb anterior ground reaction force (AGRF) while modulating belt speed during late stance. The experimental session consisted of three sequential phases: baseline, combined VB and PF training, and post-training assessment. Step-to-step AGRFs were modeled as the sum of an error-based learning (EBL) state with separate VB-related and PF-related model input channels, a use-dependent learning (UDL) state, and a direct same-step PF feed-through component. Model parameters and latent states were estimated using maximum likelihood with an extended Kalman filter and Rauch-Tung-Striebel smoother. The model reconstructed propulsion trajectories with high in-sample fidelity within the fitted session (R² = 0.976 ± 0.004), whereas temporal hold-out and post-training prediction analyses indicated limited extrapolative performance. Therefore, the decomposition should be interpreted as a model-constrained, within-session descriptive analysis rather than as evidence of robust predictive generalizability. In this model-constrained decomposition, propulsion gains were approximately distributed across EBL-related processes (~56%), UDL-related processes (~43%), and direct same-step PF feed-through (~1%). Within EBL, the VB-related and PF-related model channels contributed comparably (25% and 31%), and both adaptive states were consistent with very slow decay within the imposed identifiable range, rather than providing precise estimates of long-duration retention. These results suggest that, within the model-constrained decomposition and despite limited extrapolative performance on unseen data, propulsion enhancement in the fitted session was more closely aligned with adaptive components than with direct same-step PF feed-through. This exploratory, proof-of-concept framework provides interpretable patient-level parameters as hypothesis-generating descriptors of within-session propulsion learning and may serve as a foundation for future mechanism-informed, personalized post-stroke gait rehabilitation strategies that require prospective validation.