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SignificanceNatural 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.

An important model system for understanding genes, neurons and behavior, the nematode worm C. elegans naturally moves through a variety of complex postures, for which estimation from video data is challenging. We introduce an open-source Python package, WormPose, for 2D pose estimation in C. elegans, including self-occluded, coiled shapes. We leverage advances in machine vision afforded from convolutional neural networks and introduce a synthetic yet realistic generative model for images of worm posture, thus avoiding the need for human-labeled training. WormPose is effective and adaptable for imaging conditions across worm tracking efforts. We quantify pose estimation using synthetic data as well as N2 and mutant worms in on-food conditions. We further demonstrate WormPose by analyzing long (∼ 8 hour), fast-sampled (∼ 30 Hz) recordings of on-food N2 worms to provide a posture-scale analysis of roaming/dwelling behaviors.

Isolating slower dynamics from fast fluctuations has proven remarkably powerful, but how do we proceed from partial observations of dynamical systems for which we lack underlying equations? Here, we construct maximally predictive states by concatenating measurements in time, partitioning the resulting sequences using maximum entropy, and choosing the sequence length to maximize short-time predictive information. Transitions between these states yield a simple approximation of the transfer operator, which we use to reveal timescale separation and long-lived collective modes through the operator spectrum. Applicable to both deterministic and stochastic processes, we illustrate our approach through partial observations of the Lorenz system and the stochastic dynamics of a particle in a double-well potential. We use our transfer operator approach to provide a new estimator of the Kolmogorov–Sinai entropy, which we demonstrate in discrete and continuous-time systems, as well as the movement behavior of the nematode worm C. elegans.