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In neuroscience, scientists have some hypotheses on the brain's critical dynamics, which means the brain may stay in a phase transition state between ordered and disordered activities. Some tipping points, as a past of no return, can be critical for neural diseases. Therefore, a key question is how the critical brain hypothesis relates to pathological conditions compared with normal brain functionality. Action functional between two meta-stable states in stochastic dynamical systems is a good tool to study the critical transition and tipping. Here we extend the conventional Onsager-Machlup action functional of finding the most probable transition pathway to be looking for the evolutionary transition dynamics between two meta-stable invariant sets. Hence a rich theory from Schr\"odinger Bridge and Optimal Transport is brought in to solve this problem. Furthermore, different from various early warning indicators via statistics, bifurcation theory, information theory, statistical physics, topology, graph theory, we propose a novel viewpoint of early warning indicators in the space of probability measure, which facilitates us to build indicators based on action functional. To validate our framework, we apply this methodology to a Morris-Lecar model introduced to analyze the transition dynamics between a meta-stable state and the homo-clinic orbit. Besides, the real Alzheimer's data from ADNI database is also investigated to study the early warning signals of transition from healthy to pre-AD states. This framework not only extends the transition path to be pathway measures between two given densities on invariant sets but also shows potential ability for early warning indicators or biomarkers in complex diseases.
Zhang et al. (Fri,) studied this question.