A Multi-Scale Engineering Framework for Personalized Neuromodulation in Parkinson's Disease: Integrating Control Theory, Bayesian Inference, and Neural Dynamics

This conceptual-methodological paper introduces the Multi-Scale Adaptive Neuromodulation Engineering (MANE) framework, an integration of stochastic neural dynamics, nonlinear control theory, and time-varying Bayesian optimization for personalized closed-loop deep brain stimulation (DBS) in Parkinson's disease (PD). The framework is developed as a complete mathematical and computational pipeline rather than as a clinically validated device: it couples (i) a stochastic extension of the FitzHugh-Nagumo (FHN) neuron model with a fully derived supercritical Hopf-bifurcation analysis (first Lyapunov coefficient ℓ1 = -1/ω < 0), (ii) a proportional-integral-derivative (PID) feedback law with an explicit stochastic Lyapunov stability proof and closed-form gain conditions (Kp > b/c + σ²/2, Kd > 1/c), and (iii) a spatio-temporal Gaussian-process (GP) surrogate for online gain adaptation with a sublinear cumulative-regret bound RT = O(√(T log T)). Every numerical result reported below is produced by the self-contained, seed-fixed Python pipeline reproduced in the manuscript and is directly reproducible from the information given in this manuscript. Over 1,000 Monte Carlo replicates of a five-unit mean-field network, closed-loop PID control reduces relative beta-band (13 to 30 Hz) spectral power by 44.0% (Cohen's d = 1.26), suppresses the Kuramoto synchrony order parameter from 0.995 to 0.388 (61.0% reduction, d = 15.14), and reduces a synthetic composite index (SCI) by 20.3% (all p < 0.001, two-sided paired t-test, Bonferroni-corrected for three comparisons). A single-neuron demonstration under an identical noise realization shows a 59.9% reduction in peak membrane-potential deviation under closed-loop PID relative to the uncontrolled trajectory. Global (Saltelli) Sobol sensitivity analysis identifies noise intensity σ as the dominant source of output variance (first-order S1 = 0.848, total-order ST = 0.869), with the excitability, recovery, and timescale parameters a, b, c contributing only marginally. All model parameters are qualitatively informed by publicly reported summary statistics from the REMAP, PPMI, and OpenNeuro ds002778 datasets; no patient-level longitudinal clinical validation was performed, and we explicitly do not report a clinical correlation (e.g., against UPDRS trajectories) that the present pipeline is not equipped to support. The circuit-scale STN-GPe network (Layer 2) is directly simulated, not only stated analytically, producing a genuine beta-band (approximately 17 Hz) oscillator suppressed by 99.2% under closed-loop PID (n = 30, p = 4.2 × 10⁻³⁰); the Bayesian optimization layer is exercised as an actual online sequential decision loop, improving on fixed-gain control by roughly half on a composite beta-power-and-synchrony cost; a dedicated multi-decade scaling study of the regret bound's information-gain term γT finds it grows essentially linearly (not logarithmically) in the tested regime; a Milstein-scheme strong-convergence study empirically confirms the numerical integrator's stability regime; and the PID controller is benchmarked against sham and threshold-triggered aDBS baselines on the same network, showing a synchrony-suppression advantage (40.7% vs. 0.14% Kuramoto-R reduction) associated with the Lyapunov stability certificate. Dedicated sections address the scientific and technical risks of the framework and lay out a falsifiable, staged roadmap toward experimental and clinical validation. The complete simulation code, with fixed random seeds, is provided in full (two listings, covering the core and extended pipelines) for independent verification and reuse.