Diffusion models map hidden parameter compensations in biological systems

A new paper by Ruilin Zhang, Louis Tao, and Zhuo-Cheng Xiao introduces a diffusion learning framework to uncover viable parameter manifolds in complex biological dynamical systems. The core idea is that many parameters can co-vary to produce the same observed behavior, forming geometric structures called viable sets — the inverse image of target dynamical behaviors under a parameter-to-feature map. The key insight is that the effective codimension of these manifolds depends not on the number of reported features but on the rank of the mapping at the target scale. Co-varying features lower the codimension, while poor conditioning or regime mixing degrade learnability. The authors train conditional score-based diffusion models on simulated parameter–feature pairs, using them as amortized samplers to explore the prior-weighted viable sets.

In the Lorenz system, scalar trajectory statistics generate thin viable sheets; two-feature conditioning localizes a transition-adjacent corridor. In the Izhikevich neuron model, four firing descriptors collapse to nearly a two-dimensional family of features, revealing distinct regular and irregular compensation geometries. For an ODE reduction of finite spiking networks, the framework uncovers excitatory–inhibitory compensation, timescale–coupling tradeoffs, and input-dependent viable manifolds across 4–12 parameter dimensions. This work reframes robustness and compensation as inverse geometry problems, providing a practical tool for sampling and visualizing hidden parameter dependencies in biological systems.