Machine learning sparse reaction-diffusion models from stochastic dynamics and spatiotemporal patterns

In recent years, live-cell imaging has generated detailed spatiotemporal datasets of biochemical networks within cells. These networks often exhibit characteristics of spatially-distributed excitable systems, with propagating waves of signaling activity that govern processes such as cell migration, division, and other essential physiological functions. Traditionally, these reaction-diffusion systems have been modeled using stochastic partial differential equations incorporating spatial Langevin-type dynamics. Although these knowledge-based models have provided valuable insights, they are typically not directly inferred from experimental data. In this study, we introduce and apply two data-driven methodologies for learning the structure and parameters of spatial Langevin equation (SLE) models: 1) Kramers-Moyal regression, an adaptation of an established approach to fit microscale stochastic dynamics, and 2) wave features optimization, a novel approach to match macroscale spatiotemporal patterns. Unlike black-box neural network models that predict system behavior but provide no mechanistic insight, this approach estimates nonlinear model equations that directly relate to system structure and dynamics. As a proof-of-concept, we focus on simulation datasets derived from two stochastic reaction-diffusion models: one based on the FitzHugh-Nagumo equations (FHN model) and another on a biochemically adapted version of the FHN equations (FR model). Our results demonstrate that optimizing stochastic dynamics and wave patterns enables the accurate estimation of SLE model structure and parameters directly from 1D and 2D spatial data. With sparsity enforcement, we also identify novel sparse reaction-diffusion models for excitable systems. We show that this approach effectively approximates system behavior even when working with datasets similar to experimental conditions, with low temporal resolution and unobserved molecular components. By leveraging machine learning techniques for robust estimation of excitable reaction-diffusion models from spatiotemporal data, this work enhances our ability to model and understand the complex systems that regulate cell behavior.