Causal Field Theory: Mechanistic Interpretability for Spatio-Temporal Biological Systems

Arash Mehrjou, Bernhard Schölkopf

Abstract: Understanding how biological interventions propagate through tissue and affect cellular organization is fundamental to drug discovery and therapeutic design. We present Causal Field Theory (CFT), a mathematical framework that extends causal inference to continuous biological fields by modeling causal influence as a propagating field. The framework introduces the response kernel, which quantifies the causal effect of localized interventions such as drug delivery and genetic perturbations, and the causal flux, which measures integrated influence between tissue regions. We demonstrate that tissue-scale interventions can generate greater causal flux than the sum of equivalent cell-level perturbations when aligned with coherent propagation modes. Through numerical experiments on reaction-diffusion systems modeling biological processes, we illustrate how CFT captures causal structure in continuous fields, enabling mechanistic interpretability and principled analysis of interventions in biological systems. This framework addresses a critical gap in current approaches by providing explicit causal semantics for spatio-temporal biological data, enabling lab-in-the-loop systems that reason about intervention effects and guide experimental design.

Submission Number: 16