Decomposing force fields as flows on graphs reconstructed from stochastic trajectories

TL;DR

This paper introduces a novel graph-based method to decompose stochastic trajectories into reversible and irreversible components, providing new insights into biological systems like blood cells and heartbeats.

Contribution

It presents a data-driven approach combining graph signal processing and Helmholtz-Hodge decomposition to analyze stochastic dynamics from trajectories.

Findings

Successfully applied to biological data such as red-blood cell flickering and heartbeat patterns.

Differentiated irreversible currents in healthy versus diseased states.

Validated on solvable and nonlinear systems.

Abstract

Disentangling irreversible and reversible forces from random fluctuations is a challenging problem in the analysis of stochastic trajectories measured from real-world dynamical systems. We present an approach to approximate the dynamics of a stationary Langevin process as a discrete-state Markov process evolving over a graph-representation of phase-space, reconstructed from stochastic trajectories. Next, we utilise the analogy of the Helmholtz-Hodge decomposition of an edge-flow on a contractible simplicial complex with the associated decomposition of a stochastic process into its irreversible and reversible parts. This allows us to decompose our reconstructed flow and to differentiate between the irreversible currents and reversible gradient flows underlying the stochastic trajectories. We validate our approach on a range of solvable and nonlinear systems and apply it to derive insight into the dynamics of flickering red-blood cells and healthy and arrhythmic heartbeats. In particular, we capture the difference in irreversible circulating currents between healthy and passive cells and healthy and arrhythmic heartbeats. Our method breaks new ground at the interface of data-driven approaches to stochastic dynamics and graph signal processing, with the potential for further applications in the analysis of biological experiments and physiological recordings. Finally, it prompts future analysis of the convergence of the Helmholtz-Hodge decomposition in discrete and continuous spaces.