Diffusion learning reveals viable parameter manifolds and compensation geometry in biological dynamical systems

Abstract

Models of complex systems often have many parameters, yet are constrained by far fewer experimentally accessible observables: similar activity can emerge from coordinated parameter changes. We formalize these compatible parameter sets as \emph{viable parameter manifolds}: the inverse images of a system's target dynamical behaviors under a parameter-to-feature map. The relevant codimension is not the number of reported features, but the effective rank of that map at the target scale. Co-varying features lower the codimension, while poor conditioning, high curvature, or regime mixing degrade learnability. We train conditional score-based diffusion models on simulated parameter--feature pairs and use them as amortized samplers of prior-weighted viable sets. In the Lorenz system, scalar trajectory statistics generate thin viable sheets, and two-feature conditioning localizes a transition-adjacent corridor. In the Izhikevich neuron model, four firing descriptors lie close to a nearly two-dimensional family of features, and the learned inverse images reveal distinct regular and irregular compensation geometries. In a recent ODE reduction of finite spiking networks, the same framework reveals excitatory--inhibitory compensation, timescale--coupling tradeoffs, and input-dependent viable manifolds across 4--12 parameter dimensions. In this view, robustness, compensation, and hidden parameter dependencies are organized as inverse geometry, with diffusion models providing practical tools for sampling, visualizing, and interrogating that geometry.