A computationally efficient adaptive phase response curve estimator for real-time closed-loop neuromodulation

OBJECTIVE: Accurate phase response curve (PRC) models are essential for closed-loop neuromodulation, yet biological non-stationarity-driven by medication, sleep cycles, or plasticity-limits the efficacy of traditional offline identification. An adaptive, computationally efficient PRC estimation algorithm is proposed for real-time, online identification in embedded devices. Approach: A parametric Fourier series model approximates the PRC, and coefficients are updated after each stimulus using a recursive Least Mean Squares (LMS) rule driven by prediction error. Spectral weighting enforces smoothness, while a power-law adaptive learning-rate schedule balances rapid initial acquisition with high-precision refinement. The framework was evaluated in a stochastic theta neuron model and a reduced Hodgkin Huxely model. Main results: Robust convergence was achieved in a stochastic setting. Analysis of the speed-precision trade-off identified an adaptive learning-rate schedule that is near-optimal for a given sample size. The Fourier representation also yields an instantaneous smooth derivative, enabling real-time selection of stimulation phases for synchronization or desynchronization without numerical smoothing or historical buffering. Significance: The algorithm requires only basic arithmetic, making it well suited to resource-constrained implantable pulse generators. Continuous adaptation allows tracking of non-stationary dynamics and supports personalized closed-loop neuromodulation without repeated offline recalibration. .