Physics-Informed Diffusion Surrogate Modeling of Stochastic Hodgkin–Huxley Dynamics

Stochastic conductance-based neuron models define distributions over voltage trajectories; estimating spike statistics from them requires many costly Monte Carlo trials. We present DiffHH, a conditional score-based diffusion surrogate for the stochastic (Fox–Lu) Hodgkin–Huxley system, trained with a vanishing physics-residual penalty. An elementary observation motivates the class: a diffuse trajectory law puts every single-path surrogate at total-variation distance one, so surrogates must be distributional (which does not by itself favor diffusion). On a five-dimensional parameter grid, DiffHH matches held-out in-grid inter-spike-interval (ISI) distributions to $0.31 \pm 0.04$ ms Wasserstein-1, reproduces the noise-smoothed type-II firing onset, and recovers the $O(1/\sqrt{N})$ channel-noise scaling of ISI variability at a representative parameter setting, sampling $\sim 135\times$ faster than our unoptimized reference simulator. We frame this as a proof of concept: the penalty buys 35–55% in the low-data regime, out-of-grid error grows $\sim 4\times$, and the pipeline pays off only amortized.