Structure-Preserving Koopman Model Predictive Control for Closed-Loop Stabilisation of Memristive Neural Dynamics

Why It Matters

The advance offers a mathematically rigorous, computationally efficient path to real‑time closed‑loop neurostimulation, potentially improving therapeutic outcomes for disorders driven by pathological neural oscillations. Its robustness and speed make it viable for next‑generation adaptive deep‑brain stimulation devices.

Key Takeaways

• Lie‑derivative dictionary yields 13‑dimensional Koopman lift
• MPC achieves RMSE 0.091 and 0.47 s settling time
• 100 % stabilization under input saturation
• Robust to 20 % model parameter mismatch
• Outperforms PID, LQR, sliding‑mode, NN‑MPC, DEPC

Pulse Analysis

Closed‑loop neurostimulation faces a dual challenge: controllers must react within milliseconds while handling the strong nonlinearities of excitable neural membranes. Traditional linear designs struggle to capture the rich dynamics of biophysical models such as the Hindmarsh‑Rose neuron, especially when memristive effects introduce electromagnetic induction and chaotic bursting. By leveraging the Koopman operator—a linear representation of nonlinear systems—researchers can embed these complex dynamics into a tractable predictive control framework, opening the door to faster, more reliable therapeutic interventions.

The core innovation lies in a structure‑preserving dictionary built from iterated Lie derivatives of the neuron’s governing equations. This analytical approach replaces ad‑hoc basis selection with a compact 13‑dimensional observable space that faithfully mirrors the polynomial structure of the underlying dynamics. A truncation‑error theorem quantifies the approximation fidelity, while a stability corollary links dictionary accuracy to the closed‑loop convergence region. Extended dynamic mode decomposition with control then extracts a lifted linear predictor, which feeds a constrained receding‑horizon MPC that respects input saturation and safety limits.

Benchmarking against five conventional controllers—including PID, LQR, sliding‑mode, neural‑network MPC, and data‑enabled predictive control—demonstrates the Koopman‑MPC’s superior performance: the lowest root‑mean‑square tracking error (0.091), the quickest settling time (0.47 seconds), and a perfect 100 % stabilization rate even under saturated inputs. Moreover, the controller maintains stability with up to 20 % mismatches in model parameters, underscoring its robustness. These results suggest a viable pathway for deploying adaptive deep‑brain stimulation systems that can dynamically counteract pathological oscillations, potentially transforming treatment protocols for Parkinson’s disease, essential tremor, and other neurological disorders.