Quantifying System‑Environment Synergistic Information by Effective Information Decomposition

Living systems maintain structural and functional stability while adapting to environmental changes, a capability independent of specific system‑environment states. Existing frameworks, such as self‑organization theory and free energy principles, cannot measure system‑environment interaction at the causal level. In this article, we propose a new causal indicator, Flexibility, to measure a system’s ability to respond to its environment. We construct this indicator based on information theory and interventional operations from causal inference, which implies the indicator depends only on the dynamical causal mechanism. We show this indicator satisfies the axiom system of the partial information decomposition (PID) framework and decomposes into two components, Expansiveness and Introversion, which correspond to different strategic tendencies for environmental adaptation. This decomposition reveals that Flexibility depends on the entanglement between system‑environment variables and noise magnitude. Through experiments on cellular automata (CA), random Boolean networks, and real gene regulatory networks (GRNs), we validate that the indicator identifies the most complex and computationally capable CA (Langton’s parameter at 0.5), while demonstrating that feedback loops carrying important biological functions in GRNs exhibit the highest flexibility. We also find that flexibility peaks at a moderate level of dynamical noise. Furthermore, we combine this framework with machine learning techniques to demonstrate its applicability when the underlying dynamics are unknown.