A Dynamical Microscope for Multivariate Oscillatory Signals: Validating Regime Recovery on Shared Manifolds

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.

💡 Research Summary

The paper introduces a “dynamical microscope” framework designed to uncover and quantify regime structure in high‑dimensional, non‑stationary oscillatory signals. The authors begin by converting each channel of a multivariate time series into a circular representation consisting of instantaneous phase and amplitude, obtained via the analytic signal (Hilbert transform). This step preserves the intrinsic periodicity of the data while providing a compact, rotation‑invariant feature set that is amenable to nonlinear learning.

Next, the phase‑amplitude vectors are fed into a deep autoencoder. The encoder compresses the high‑dimensional sequence into a low‑dimensional latent trajectory (typically two or three dimensions), while the decoder reconstructs the original sequence, enforcing a reconstruction loss. To ensure that the latent space respects temporal continuity, the authors incorporate recurrent layers (e.g., LSTM) or time‑conditional regularization, so that the learned manifold reflects the true flow of the underlying dynamical system rather than arbitrary static embeddings. As a result, each trial or recording is represented as a continuous curve on a shared latent manifold, enabling direct visual inspection and quantitative analysis.

Once the latent trajectories are obtained, the framework extracts several complementary metrics:

1. Speed – the time derivative of the latent coordinates, capturing how quickly the system moves through state space. Regime transitions typically manifest as abrupt changes in speed.
2. Explored variance – the trace of the covariance matrix of latent points, measuring the spatial extent of the trajectory. Different regimes often occupy distinct volumes even when they overlap in the original observation space.
3. Tortuosity (or curvature) – a geometric measure of how winding the trajectory is, providing information orthogonal to speed and variance.
4. Flow field – a vector‑field estimate derived from local velocity vectors, visualizing the overall directionality and organization of dynamics on the manifold.

To validate the approach, the authors construct a synthetic benchmark using a network of coupled Stuart‑Landau oscillators. The network’s coupling topology is switched periodically, thereby imposing two distinct dynamical laws on the same set of oscillator variables. Because the underlying state variables (phases and amplitudes) can occupy overlapping regions for both topologies, conventional clustering would fail to separate the regimes. The autoencoder nevertheless learns a common manifold, and the regime‑specific metrics reveal clear differences: speed and explored variance achieve high discriminability (η² > 0.5), while tortuosity captures geometry that is independent of the topology contrast. Flow‑field visualizations further illustrate that the direction of motion on the manifold reorganizes dramatically at each topology switch.

The key insight is that regime differences need not be expressed as discrete, non‑overlapping clusters; instead, they can be encoded as variations in trajectory speed, spatial spread, curvature, and vector‑field organization on a shared manifold. This perspective allows researchers to detect subtle dynamical changes even when the raw signals appear statistically indistinguishable.

The authors argue that the dynamical microscope is broadly applicable to neural recordings (EEG, LFP, calcium imaging), physiological signals (heart‑rate variability, respiration), and physical systems where multivariate oscillations arise. By providing a principled pipeline—from phase‑amplitude extraction through latent trajectory learning to multi‑metric regime quantification—the framework offers a powerful alternative to traditional state‑space clustering, enabling more nuanced interpretations of complex, non‑stationary time series.