Early Diagnosis of Parkinson via Transient Beta Frequencies in a Delayed Van der Pol Model
Biological motor circuits are shaped by pathway-specific delays that generate history-dependent oscillations and short-lived transients not captured by memoryless models. We develop a delayed Van der Pol in which the delay ratio r acts as a control parameter for Hopf bifurcation and internal resonance and we pair it with an auditable, short-window signal-processing pipeline tailored to early beta activity. The pipeline combines Short-Time Fourier Transform (STFT), Continuous Wavelet Transform (CWT) ridge tracking, kernel-density estimates (KDE) of the dominant frequency, and algorithmic beta-burst detection under fixed, shared parameters. Transient burden is quantified by a metric triplet with a software-implementable attenuation rule and complemented by an early-detection index-the Transient Persistence Time. Across methods, we observe a monotonic attenuation of short-lived beta transients as r approaches a critical band. A reliable observation window of 0 - 0.35 s captures compact early packets at low r, progressive ridge stabilization in CWT, KDE narrowing over time, and near-elimination of bursts for near critical r, consistent with a transition to sustained narrowband rhythm. Clinically, these measures enable stage inference, a bedside Move-Stop protocol for rapid readout, and actionable policies for adaptive DBS and medication titration that minimize transient burden without inducing rigid locking. The fixed parameterization and audit-ready outputs support reproducibility and multi-center deployment. This framework advances data-driven, precision neuromodulation by targeting early transients before pathological synchrony becomes entrenched.