Action Functional as an Early Warning Indicator in the Space of Probability Measures via Schrödinger Bridge

TL;DR

The paper develops an early warning framework for tipping in stochastic systems by reframing transition paths as probability-measure dynamics through the Schrödinger bridge. By linking the Onsager–Machlup action functional to entropy-regularized optimal transport, it defines a density-based indicator that signals approaching critical transitions and validates it on Morris–Lecar neuron models and Alzheimer's disease imaging data. The approach yields transition-path densities, demonstrates forward–backward SDE convergence, and provides a practical tool for detecting abrupt brain-state changes before clinical symptoms arise. This has potential implications for predictive neuroscience and the early intervention of neurodegenerative diseases, with a principled, geometry-aware transport perspective underpinning the analysis.

Abstract

Critical transitions and tipping phenomena between two meta-stable states in stochastic dynamical systems are a significant scientific issue. In this work, we expand the methodology of identifying the most probable transition pathway between two meta-stable states with Onsager-Machlup action functional, to investigate the evolutionary transition dynamics between two meta-stable invariant sets with Schrödinger bridge. In contrast to existing methodologies such as statistical analysis, bifurcation theory, information theory, statistical physics, topology, and graph theory for early warning indicators, we introduce a novel framework on Early Warning Signals (EWS) within the realm of probability measures that align with the entropy production rate (EPR). To validate our framework, we apply it to the Morris-Lecar model and investigate the transition dynamics between a meta-stable state and a stable invariant set (the limit cycle or homoclinic orbit) under various conditions. Additionally, we analyze real Alzheimer's data from the Alzheimer's Disease Neuroimaging Initiative database to explore EWS indicating the transition from healthy to pre-AD states. This framework not only expands the transition pathway to encompass measures between two specified densities on invariant sets, but also demonstrates the potential of our early warning indicators for complex diseases.

Figures (11)

• Figure 1: Phase portrait of the Morris-Lecar model. (A) The stable node and stable limit cycle of the Morris-Lecar model with bifurcation parameter $I = 92 \,\,\mu A/cm^2$. (B) The stable node, unstable spiral, and stable homoclinic cycle of the Morris-Lecar model with bifurcation parameter $I = 37 \,\,\mu A/cm^2$.
• Figure 2: Transition path dynamics between a stable state and invariant manifold in Morris-Lecar system. Upper row: evolutionary pathways when terminal time T = 20 and noise strength g = 0.3, from forward SDE (A) and backward SDE (B). Bottom row: evolutionary pathways when time T = 20 and noise strength g = 0.5, from forward SDE (C) and evolutionary pathways from backward SDE (D).
• Figure 3: The evolutionary density of transition paths from the stable node to the stable limit cycle with T=20,N=200, g=0.3.
• Figure 4: The evolutionary density of transition paths from the stable node to the stable homoclinic cycle with T=20, N=200,g=0.3.
• Figure 5: Terminal density concentrations of the transition from the stable node to the stable limit cycle, with (A) $\mathrm{T} = 3$ under various noise density g. (B) $\mathrm{g} = 1$ under various terminal time T.

Theorems & Definitions (2)

• Theorem 2.1
• Definition 3.1