Stochastic processes with multiple temporal scales: timescale separation and information

Abstract

Complex systems are often characterized by the interplay of multiple interconnected dynamical processes operating across a range of temporal scales. This phenomenon is widespread in both biological and artificial scenarios, making it crucial to understand how such multiscale dynamics influence the overall functioning and behavior of these systems. Here, we present a general timescale separation approach that is valid for any set of stochastic differential equations coupled across multiple timescales. We show two alternative derivations, one based on an iterative procedure, and the other grounded on an a priori expansion in relevant small parameters associated with faster timescales. We provide an explicit expression for the conditional structure of the joint probability distribution of the whole set of processes that is solely determined by the multiscale interactions. This result has important consequences in determining how information is generated in the system and how it propagates across different timescales. To demonstrate that our findings are valid independently of the specific dynamical model, we test them against four different (linear and non-linear) dynamics. Then, we focus on the scenario in which two degrees of freedom are reciprocally coupled across their timescales. In this case, we show that the statistics of the fastest dynamics is captured by an effective self-interaction term in the slowest one, creating a regulatory loop. Finally, a connection with analogous previous results obtained for discrete-state systems with higher-order interactions is drawn, elucidating similarities and differences in the physical interpretation of the system. The presented framework might open the avenue for a clearer understanding of the interplay between the underlying multiscale structure and emergent functional behavior of biological and artificial systems.