A fundamental paradigm in neuroscience is that cognitive functions -- such as perception, learning, memory, and locomotion -- are governed by the brain's structural organization. Yet, the theoretical principles explaining how the physical architecture of the nervous system shapes its function remain elusive. Here, we combine concepts from quantum statistical mechanics and graph C*-algebras to introduce a theoretical framework where functional states of a structural connectome emerge as thermal equilibrium states of the underlying directed network. These equilibrium states, defined from the Kubo-Martin-Schwinger states formalism (KMS states), quantify the relative contribution of each neuron to the information flow within the connectome. Using the prototypical connectome of the nematode {\em Caenorhabditis elegans}, we provide a comprehensive description of these KMS states, explore their functional implications, and establish the predicted functional network based on the nervous system's anatomical connectivity. Ultimately, we present a model for identifying the potential functional states of a detailed structural connectome and for conceptualizing the structure-function relationship.