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

TL;DR

This paper introduces a novel 'dynamical microscope' framework that transforms multivariate oscillatory signals into features, learns latent trajectories, and quantifies regimes through geometric and flow metrics, enabling regime detection even in overlapping state spaces.

Contribution

The framework uniquely combines phase-amplitude features, autoencoder-based latent space learning, and trajectory geometry analysis to identify dynamical regimes without requiring discrete state separation.

Findings

Successfully recovers differences in dynamical laws in simulated oscillator networks.

Regime differences are captured by changes in trajectory speed, geometry, and flow organization.

Metrics like speed and variance show high discriminability ($^2 > 0.5$).

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 ($\eta^2 > 0.5$), 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.