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

TL;DR

This work addresses the challenge of interpreting non-stationary, metastable multivariate oscillatory signals by introducing a dynamical microscope that reframes analysis in terms of continuous trajectory laws on a learned manifold. It combines circular phase–amplitude encoding, an autoencoder-based latent trajectory representation, and flow- and geometry-based metrics to quantify regime structure beyond discrete state occupancy. Validation with a topology-switching Stuart–Landau oscillator network shows that regime differences expressed through coupling topology can be recovered even when regimes occupy overlapping state-space regions, with speed and explored variance achieving strong discriminability ($\eta^2 > 0.5$) and tortuosity capturing orthogonal geometric information. The framework yields a principled, trajectory-centric tool applicable to neural, physiological, and physical data, with open-source code enabling replication and extension, and highlights the complementary roles of Lagrangian trajectories and Eulerian flow descriptions in non-stationary dynamics.

Abstract

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.

Figures (6)

• Figure 1: Workflow schematic of the dynamical microscope framework. Multichannel oscillatory signals are transformed into circular phase--amplitude features, encoded into a latent trajectory via autoencoder, and analyzed using flow field estimation and trajectory metrics. The depicted "trajectory" is schematic; analyses emphasize both time-ordered realizations and ensemble summaries (densities and flow fields) obtained by pooling across time and/or realizations.
• Figure 2: Raw observations from coupled oscillator simulation. Top: multichannel time series (selected channels) showing regime switches as vertical dashed lines. Middle: power spectral density (mean across channels) with characteristic oscillator peak. Bottom: Hilbert amplitude envelope (channel 0) showing amplitude modulations across regimes.
• Figure 3: Dynamical microscope analysis of coupled Stuart--Landau oscillator network. (A) Ground-truth regime sequence: four coupling topologies (global, cluster, sparse, ring) cycling every 10 s for 160 s total. (B) UMAP-embedded latent trajectories colored by regime; regimes show substantial spatial overlap but visible clustering. (C) Density and flow field visualization showing local drift patterns across the shared manifold. (D) Normalized flow metrics by regime, demonstrating topology-dependent differences in speed, speed variability, tortuosity, and explored variance.
• Figure 4: Regime-specific flow fields. Each panel shows the density (background) and flow field (arrows) for trajectories within a single coupling topology. Despite substantial spatial overlap, regimes exhibit distinct local flow patterns. Divergence and curl values are annotated for each regime.
• Figure 5: Regime discriminability: per-window metric distributions. Violin plots show the distribution of speed, variance, and tortuosity across non-overlapping windows for each regime. Effect sizes ($\eta^2$) and significance levels are annotated. Speed and variance show clear regime separation with large effect sizes, while tortuosity distributions overlap substantially.