Kubo-Martin-Schwinger states of Path-structured Flow in Directed Brain Synaptic Networks

TL;DR

This paper models the brain's synaptic network using algebraic quantum systems derived from graph C*-algebras, revealing how neuronal influence propagates and identifying key hub neurons in C. elegans based on thermodynamic properties.

Contribution

It introduces a novel application of KMS states in graph C*-algebras to analyze neural connectivity and identifies functional hubs in C. elegans through thermodynamic analysis.

Findings

Neurolocomotor neurons are primary flow hubs at entropy peaks.

KMS states describe stationary distributions of influence propagation.

Topological embedding influences neuronal centrality.

Abstract

The brain's synaptic network, characterized by parallel connections and feedback loops, drives interaction pathways between neurons through a large system with infinitely many degrees of freedom. This system is best modeled by the graph C*-algebra of the underlying directed graph, the Toeplitz-Cuntz-Krieger (TCK) algebra, which captures the diversity of path-structured flow connectivity. Equipped with the gauge action, the TCK algebra defines an {\em algebraic quantum system}, and here we demonstrate that its thermodynamic properties provide a natural framework for describing the dynamic mappings of potential flow pathways within the network. Specifically, the KMS states of this system represent the stationary distributions of a non-Markovian stochastic process with memory decay, capturing how influence propagates along exponentially weighted paths, and yield global statistical measures of neuronal interactions. Applied to the {\em C. elegans} synaptic network, our framework reveals that neurolocomotor neurons emerge as the primary hubs of incoming path-structured flow at inverse temperatures where the entropy of KMS states peaks. This finding aligns with experimental evidence of the foundational role of locomotion in {\em C. elegans} behavior, suggesting that functional centrality may arise from the topological embedding of neurons rather than solely from local physiological properties. Our results highlight the potential of algebraic quantum methods and graph algebras to uncover patterns of functional organization in complex systems and neuroscience.