FlowPrint: A Dynamical Microscope for Multivariate Oscillatory Signals

Abstract

Traditional methods for analyzing multivariate oscillatory signals often reduce dynamics to discrete states (microstates, HMM states) and compare occupancy statistics. We present FlowPrint, a complementary approach that treats the signal as a continuous trajectory evolving on a learned manifold and characterizes regimes by how the system moves, not just where it visits.

Using coupled Stuart-Landau oscillators with switchable coupling topologies as ground truth, we validate that flow-based metrics—speed, tortuosity, explored variance, and kinetic energy—can reliably discriminate between dynamical regimes on shared latent manifolds. Our framework provides a principled way to characterize continuous dynamics in EEG and other multivariate time series without reducing them to discrete states.

Interactive: Coupling Topologies

Explore how different network topologies create distinct dynamical regimes. Click on each topology to see its structure and characteristic dynamics.

Select Topology

Cluster 1

Cluster 2

Cluster 3

Method Overview

FlowPrint characterizes dynamical regimes through a four-stage pipeline:

Phase Extraction

Hilbert transform extracts instantaneous phase and amplitude from bandpass-filtered signals

Latent Encoding

Convolutional autoencoder compresses phase dynamics to low-dimensional trajectory

Flow Estimation

UMAP embedding + velocity field estimation reveals regime-specific flow patterns

Flow Characterization

Compute flow metrics (speed, tortuosity, variance, kinetic energy) as regime discriminators

Key Findings

High Discriminability

Speed and explored variance achieve η² > 0.5 effect sizes, reliably separating all four topology regimes.

Distinct Flow Fields

Each topology produces characteristic flow patterns on the shared UMAP manifold, enabling visual regime identification.

Kinetic Energy Signatures

Intermittency of kinetic energy ||v||² differs systematically across regimes, providing an additional discriminative marker.

Ground Truth Validation

Stuart-Landau oscillator simulation provides known regime labels for rigorous method validation.

Results Gallery

Flow fields by regime — **Flow Fields.** Regime-specific velocity patterns on the UMAP embedding reveal distinct dynamical signatures.

Discriminability analysis — **Discriminability.** ANOVA with η² effect sizes shows speed and explored variance best separate regimes.

Kinetic energy analysis — **Kinetic Energy.** ||v||² distributions and intermittency patterns differ systematically across topologies.

Electrode timeseries — **Raw Observations.** Simulated 30-channel EEG-like signals with regime transitions marked.

Learn More

Explore the paper through an AI-generated podcast discussion and video presentation, created using NotebookLM based on the paper and repository.

Podcast Discussion

AI-generated conversation exploring the key concepts of the dynamical microscope framework.

Video Presentation

Visual overview of the methodology and key results from the paper.

Generated with Google NotebookLM based on the paper and code repository.

Citation

@article{furman2026dynamical,
  title={A Dynamical Microscope for Multivariate Oscillatory Signals:
         Validating Regime Recovery on Shared Manifolds},
  author={Furman, Łukasz and Minati, Ludovico and Duch, Włodzisław},
  journal={TBA},
  year={2026}
}