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

TL;DR

This work addresses how to rigorously describe path-structured flow in directed neural networks by employing graph Toeplitz-Cuntz-Krieger (TCK) algebras and Kubo-Martin-Schwinger (KMS) states. The authors construct an algebraic quantum system $(\\mathcal{O},\\alpha)$ whose KMS states yield stationary distributions over neuronal interactions, capturing non-Markovian flow with exponentially decaying memory via the matrix $[\\mathbf{x}^{\\bullet|\\beta}]$ and pure-state profiles $\\mathbf{x}^{j|\\beta}$. They establish a phase-transition framework at $\\beta_c=\log r$ with symmetry-breaking behavior and define flow-entropy measures that quantify selectivity, applying the theory to the C. elegans connectome to reveal that locomotion-related neurons act as functional hubs in a specific temperature range. The results suggest that functional centrality can emerge from topological embedding rather than solely local physiology, and illustrate how algebraic quantum methods can illuminate global organizational patterns in neuroscience, with potential extensions to multiplex networks.

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.

Figures (7)

• Figure 1: The Hilbert space $\mathcal{H}$ is spanned by scaled paths $\delta_\gamma$. The projection $Q_j$ selects the subspace $\mathcal{H}_j$ of paths ending at neuron $j$. The operator $S_a$ extends a path $\delta_\gamma$ ending at neuron $j$ to $\delta_{a\gamma}$ reaching neuron $i$, while $S_a^*$ extracts the portion of flow flow to $i$ that passes through $j$.
• Figure 2: Schematic illustration of a directed network and its KMS matrices. (a) A network represented by a directed multigraph $G$ with $N=6$ nodes and 18 edges, including multiple parallel edges and self-loops. (b) Connectivity matrices defined by the KMS state matrices of $G$. The column representing the pure KMS state $\operatorname{{\bf x}}^{6|\beta}$ is the unit vector $(0,0,0,0,0,1)$ for all values of $\beta$, as node $6$ has no outgoing edges
• Figure 3: Average path-structured flow dynamics from touch receptor neurons. At low temperatures (high $\beta$), $C$ interacts only with direct neighbors; higher temperatures enable long-range interactions, mainly targeting motor neurons. Colors: green (sensory), yellow (inter), purple (motor). Self-interactions omitted.
• Figure 4: Comparison between weighted structural connectivity of C. elegans directed synaptic network and its KMS matrix at inverse temperature $\beta=10.7394$ (about 2.5 times the critical value $\beta_c = 4.295$).
• Figure 5: Binarized directed networks of (a) synaptic connectivity and (b) KMS matrix $[\operatorname{{\bf x}}^{\bullet|\beta}]$ for $4.38 \le\beta \le 4.6$. Node size reflects incoming connections. Locomotor neurons, despite few incoming synapses, dominate incoming flow at high temperature.