Structure-preserving Koopman model predictive control for closed-loop stabilisation of memristive neural dynamics

Closed-loop neurostimulation for pathological neural oscillations requires controllers that are fast, constraint-aware, and robust to the strong nonlinearities inherent in excitable neural membranes. We develop a Koopman operator framework for model predictive control of a four-state memristive Hindmarsh–Rose neuron, a biophysical model that captures electromagnetic induction, coexisting attractors, and chaotic bursting. The key methodological contribution is a structure-preserving dictionary design based on iterated Lie derivatives of the governing vector field: instead of choosing basis functions ad hoc, the dictionary is constructed analytically from the polynomial structure of the neural dynamics, yielding a compact 13-dimensional observable space. A truncation error theorem quantifies the approximation quality of the finite-dimensional lifting, and a stability corollary connects dictionary fidelity to the closed-loop convergence neighbourhood. Extended dynamic mode decomposition with control identifies a lifted linear predictor that is embedded in a constrained receding-horizon controller with nonlinear re-lifting. A dictionary ablation study across four designs confirms that the physics-informed Lie-closure basis matches a generic mixed dictionary of six times its dimension. Monte Carlo benchmarks against five baseline controllers—including PID, LQR, sliding mode control, neural network MPC, and data-enabled predictive control—demonstrate the lowest tracking error (RMSE = 0.091), fastest settling (0.47 s), and 100% stabilisation rate under input saturation, with robustness confirmed under 20% parameter mismatch. These results establish a principled, computationally tractable route from biophysical neural models to real-time closed-loop control, with implications for the design of next-generation adaptive neurostimulation systems.