Multivariate oscillatory signals from complex systems often exhibit non-stationary dynamics and metastable regime structure, making dynamical interpretation challenging. We introduce a ``dynamical microscope'' framework that converts multichannel signals into circular phase--amplitude features, learns a data-driven latent trajectory representation with an autoencoder, and quantifies dynamical regimes through trajectory geometry and flow field metrics. Using a coupled Stuart--Landau oscillator network with topology-switching as ground-truth validation, we demonstrate that the framework recovers differences in dynamical laws even when regimes occupy overlapping regions of state space. Group differences can be expressed as changes in latent trajectory speed, path geometry, and flow organization on a shared manifold, rather than requiring discrete state separation. Speed and explored variance show strong regime discriminability ($η^2 > 0.5$), while some metrics (e.g., tortuosity) capture trajectory geometry orthogonal to topology contrasts. The framework provides a principled approach for analyzing regime structure in multivariate time series from neural, physiological, or physical systems.
1.1. Motivation: multivariate signals as dynamical systems Complex systems-whether neural, physiological, or physical-often produce multivariate oscillatory signals that are highly dynamic and metastable. Unlike responses time-locked to external events, ongoing activity exhibits continual fluctuations and short-lived patterns of coordination among components. Subsystems spontaneously couple and decouple in time, reflecting a balance between integration and segregation (Tognoli and Kelso, 2014). Metastability implies that the system does not reside in a single stationary state but transitions among a repertoire of semi-stable configurations across multiple time scales (Rabinovich et al., 2008;Tognoli and Kelso, 2014). These observations reinforce that ongoing activity is inherently non-stationary and structured in time.
Characterizing such dynamics is challenging because continuous recordings lack external markers that naturally segment the signal. Segment choice is often arbitrary, and stationarity assumptions are frequently violated. Traditional stationary summaries (e.g., a single spectrum or connectivity matrix over a long epoch) can obscure transient structure. These issues motivate approaches that treat multivariate signals as continuous trajectories evolving in a high-dimensional state space, where the objects of interest are not isolated states but the geometry and flow of trajectories over time.
Importantly, for such signals the “trajectory” should not be understood as a single clean curve in latent space. A more appropriate picture is a stochastic flow on a lowdimensional manifold: at each moment, the latent state undergoes structured drift (local flow) together with substantial variability, yielding an evolving probability density over states. As a result, single short realizations can appear as “snake-like” paths, while pooling longer time spans or multiple realizations naturally produces dense point clouds with metastable basins (high-occupancy regions) and preferred directions of motion (flow fields). In this sense, trajectory-centric (Lagrangian) and density/fieldcentric (Eulerian) views are complementary descriptions of the same underlying dynamics.
Many frameworks for analyzing dynamics reduce continuous activity to a sequence of discrete “states.” In microstate analysis, each time point is assigned to one of a few prototypical patterns, and comparisons across conditions focus on metrics such as fractional occupancy and mean duration (Michel and Koenig, 2018). Similarly, hidden Markov model approaches represent activity as prob-arXiv:2602.11054v1 [q-bio.NC] 11 Feb 2026 abilistic switching among latent states with characteristic patterns and dwell times (Baker et al., 2014;Vidaurre et al., 2018). Such state-based methods are useful for summarizing recurring configurations, but a primary focus on state occupancy can miss differences that are expressed in the motion through state space.
A complementary hypothesis is that conditions or regimes may share a common manifold (i.e., occupy broadly similar regions of state space) yet differ systematically in their trajectory laws: local flow structure, velocities, path geometry, recurrence, and exploration versus confinement. Theoretical work links altered dynamical conditions to changes in metastability and temporal exploration rather than wholesale changes in the set of accessible configurations (Cavanna et al., 2018;Deco et al., 2017). This motivates shifting emphasis from “which states are visited” to “how trajectories evolve,” and from static labels to dynamical descriptions of motion on learned coordinates.
Our framework is designed to complement, not replace, discrete-state methods. Some quantities we computesuch as occupancy entropy and dwell statistics-have direct analogues in microstate and HMM analyses (e.g., state entropy, transition rates, mean durations). We include these for continuity and interpretability, not as claims of novelty.
The novel contribution lies in continuous-trajectory quantities that do not reduce cleanly to discrete-state summaries:
• Path tortuosity: a geometric functional of the continuous trajectory (curvature and zig-zag structure) that captures within-region motion, not just boundary crossings between states.
• Speed variability (CV): local kinematics variability that quantifies how intermittently fast or slow the system moves, beyond simple “more or fewer switches.”
• Flow field geometry: spatially resolved drift structure showing where and how trajectories move, not just which regions are occupied.
These quantities are not directly recoverable from standard discrete-state summaries without additional assumptions, and require explicit continuous-trajectory representations.
We adopt a trajectory-centric dynamical-systems view and introduce a framework designed as a “dynamical microscope” rather than a classifier. Our contributions are:
• A circular phase-amplitude encoding of multichannel signals that
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