From Observed Viability to Internal Predictive Approximation: A Single-Subject Latent-Space Analysis of Gait Dynamics Under Occlusal Constraint

Technical Summary: From Observed Viability to Internal Predictive Approximation

Problem Statement
Adaptive biomechanical systems, such as human gait, present a challenge where observable performance (e.g., spatiotemporal variables) does not uniquely identify the underlying latent organization. Previous work established that similar observable gait can arise from distinct latent configurations and that longitudinal viability (system stability over time) cannot be inferred solely from instantaneous performance. While a prior "Level 4" framework successfully assessed longitudinal viability retrospectively by measuring centroid displacements in a latent space (M1 to M2), it remained purely descriptive. It could not approximate the transformation before observing the final state. The central problem addressed in this study is whether the observed M1–M2 latent transformations, which characterize longitudinal viability, can be internally approximated within a predictive latent-space framework using a single-subject dataset.

Methodology
The study employs an exploratory, retrospective, single-subject design involving a participant with Parkinson's disease. Data were collected over two sessions (M1 and M2) separated by eleven weeks, during which the participant underwent a structured sensorimotor intervention.

• Data Acquisition: Gait was recorded using instrumented insoles under six distinct "occlusal observational probes": Neutral Natural Occlusion (ONL), Wide Open-Mouth Disengagement (OBL), Strong Voluntary Clenching (OSL), and three Vertical Dimension of Occlusion (VDO) variations (OC2.5, OC3, OC3P). These probes were applied only during measurement, serving as discrete constraints rather than continuous causal drivers of longitudinal evolution.
• Latent Representation: Principal Component Analysis (PCA) was used to map high-dimensional biomechanical variables into a low-dimensional latent space (PC1–PC2). Two projections were generated: one aligned with the previous Level 4 analysis (using ONL, OC2.5, OC3) and an extended projection including all six probes.
• Predictive Model: A simplified supervised machine-learning model (a feed-forward neural network with two hidden layers of 16 neurons) was trained to approximate the M1–M2 transformation directly in the latent space.
  • Inputs: M1 latent coordinates, encoded occlusal probe descriptors (representing mechanical constraints like dental contact, VDO increase, and protrusion), and a transition indicator.
  • Output: Predicted M2 latent coordinates.
  • Objective: Minimize the mean squared error between observed and predicted M2 centroids.
• Evaluation Strategy: The study utilized three internal evaluation levels to test approximation without claiming generalizability:
  1. Full-dataset approximation: Training and testing on the complete set of conditions.
  2. Held-out M2: Withholding a subset of M2 observations during training.
  3. Leave-condition-out: Training on five probes and evaluating on the sixth (unseen) probe.

Key Contributions

1. Introduction of Level 5 Analysis: The paper defines a new analytical level focused on "internal predictive approximation." This shifts the framework from retrospective observation of viability to a model-based approximation of viability-related latent trajectories within a single subject.
2. Methodological Distinction: It rigorously distinguishes between "observational constraints" (occlusal probes applied during measurement) and "longitudinal system evolution" (the combined effect of time, intervention, and spontaneous variability between sessions). The model approximates the latter while conditioning on the former.
3. Proof-of-Concept for Latent Trajectory Approximation: The study demonstrates that a simplified neural network can approximate the observed M1–M2 displacement in latent space and preserve the relative hierarchy of displacements identified in previous retrospective analyses.
4. Descriptor-Based Encoding: To enable leave-condition-out testing, the authors developed a descriptor-based encoding for occlusal probes (e.g., presence of contact, magnitude of VDO increase) rather than simple one-hot categorical encoding, allowing the model to generalize to unseen constraint combinations within the same subject.

Results

• Core Analysis (Level 4 Alignment): In the analysis restricted to ONL, OC2.5, and OC3, the model successfully preserved the displacement hierarchy observed in the previous Level 4 study: $d_{OC3} < d_{ONL} < d_{OC2.5}$. The predicted centroid displacements closely matched the observed values (e.g., predicted $d_{OC3} = 5.32$ vs. observed $5.35$), with low centroid errors (0.01–0.06).
• Extended Six-Probe Analysis: The model approximated the M1–M2 transformation for all six probes, preserving the global structure of the exploratory ordering. OSL showed the lowest displacement, while OBL and OC2.5 showed the highest. OC3 and OC3P remained closely grouped.
• Internal Robustness:
  • Held-out M2: The model maintained centroid-level approximation ability on unseen M2 observations within the same subject, though pointwise errors varied by condition.
  • Leave-condition-out: The model could approximate the transformation of a withheld probe, though with higher centroid errors (0.40–1.08) compared to full-dataset fitting, indicating the increased difficulty of approximating unseen constraints.
• Within-Session Organization: The analysis revealed that while OC3 remained close to the ONL reference in both sessions, OC2.5 remained more distant, suggesting that mechanical proximity (VDO increase) does not necessarily imply latent functional equivalence.

Significance and Claims
The paper explicitly frames its findings as a methodological bridge rather than a clinical solution.

• No Clinical Prediction: The authors state clearly that the study does not establish clinical predictive validity, causal occlusal effects, therapeutic superiority, or generalizable patient-level prediction. The term "predictive" is used strictly in a restricted methodological sense: the internal approximation of observed transformations within a single-subject dataset.
• Proof-of-Concept: The primary significance is demonstrating that viability-related latent trajectories, previously only observable retrospectively, can be internally approximated by a simplified model. This suggests that the "structure" of longitudinal viability is representable in a latent space.
• Path to Level 6: The work is positioned as a necessary prerequisite for a future "Level 6" framework. Level 5 tests whether trajectories are approximable in a single subject; Level 6 would be required to test whether these approximations are generalizable across multiple individuals to support clinically operational prediction.

In summary, the study validates a computational framework for moving from "observed longitudinal viability" to "internal predictive approximation" of latent trajectories, establishing a proof-of-concept that such transformations can be modeled without claiming immediate clinical applicability or causal inference.