Adaptive, locally linear models of complex dynamics

Significance Natural phenomena are teeming with temporal complexity, but such dynamics, however fascinating, offer substantial obstacles to quantitative understanding. We introduce a general method based on the simple idea that even complicated time series are locally linear. Our analysis transforms dynamical data into a parameterized space of linear models, and we detail a hierarchical clustering of this space into dynamical categories. The linear models reveal fine-scaled, interpretable states in the posture behavior and global brain activity of the nematode Caenorhabditis elegans. Furthermore, we find that the population of stable and unstable oscillations suggests a near-critical dynamics across both brains and behavior. We expect our approach to be widely applicable. The dynamics of complex systems generally include high-dimensional, nonstationary, and nonlinear behavior, all of which pose fundamental challenges to quantitative understanding. To address these difficulties, we detail an approach based on local linear models within windows determined adaptively from data. While the dynamics within each window are simple, consisting of exponential decay, growth, and oscillations, the collection of local parameters across all windows provides a principled characterization of the full time series. To explore the resulting model space, we develop a likelihood-based hierarchical clustering, and we examine the eigenvalues of the linear dynamics. We demonstrate our analysis with the Lorenz system undergoing stable spiral dynamics and in the standard chaotic regime. Applied to the posture dynamics of the nematode Caenorhabditis elegans, our approach identifies fine-grained behavioral states and model dynamics which fluctuate about an instability boundary, and we detail a bifurcation in a transition from forward to backward crawling. We analyze whole-brain imaging in C. elegans and show that global brain dynamics is damped away from the instability boundary by a decrease in oxygen concentration. We provide additional evidence for such near-critical dynamics from the analysis of electrocorticography in monkey and the imaging of a neural population from mouse visual cortex at single-cell resolution.