Unravelling parametric variation of multiscale dynamics from data

Complex systems operate at the interplay between structure and variation - a phenomenon observed across scales from proteins and cells, to animal behavior, to the ecosystem. Revealing and studying the underlying parameters controlling variation in these systems remains a challenge as their dynamics is highly non-linear, varies across multiple timescales, and we can only access short and partial observations. We propose to overcome these challenges through a multiscale data-driven framework. Firstly, we capture dynamics across timescales from partial experimental observations using transfer operators. Secondly, we utilize a multiscale distance metric between operators that captures both fine and coarse dynamical differences. Using a non-linear embedding that accounts for the finite size of the data, we then reveal relevant dimensions of variation between dynamics, and robustly cluster together dynamics representing similar parameter regimes. Through applications to real world, out-of equilibrium phenomena of bacterial chemotaxis and larval zebrafish navigation, we showcase how underlying internal control variables and strong environmental fluctuations provide structure to inter-individual variation. Altogether, our approach discovers parametric variation from individual trajectories, bridging scales from internal parameters to structured variation in observations of complex phenomena.