Using timescale as a state coordinate reveals the metastable geometry of behavior

Animal behavior unfolds across many timescales, from fast movement patterns to slow changes in internal states such as hunger, arousal, and circadian phase. These slow variables are rarely measured directly and must instead be inferred from their effects on the faster movements that can be observed. Here we propose treating timescale itself as an explicit coordinate of the state representation, constructing a time-frequency state space where fast movements and slow modulations appear simultaneously. We find that slow modes emerge as linear arms radiating from a stationary-weighted hub in the leading non-trivial eigenvectors of the transfer operator, with one arm per metastable basin across three systems of increasing complexity. In a synthetic system, the framework recovers a hidden bistable driver across nearly three decades of dwell time, while a fixed-timescale analysis of the same trajectory finds no separable slow modes. In nematode locomotion, it reproduces the canonical run-pirouette organization. In freely moving fruit flies, where fast leg kinematics are orders of magnitude faster than the behavioral states they compose, the multi-timescale operator identifies four metastable behavioral basins directly from the postural time series, without first decomposing into a sequence of stereotyped actions. We further find that these basins exhibit a broad, heavy-tailed distribution of residence times. Treating timescale as a state coordinate thus exposes a predictable geometric form for the slow organization of behavior, providing a general route for extracting collective modes from partially observed biological time series without first organizing the dynamics into discrete events.