Brain functions emerge as thermal equilibrium states of the connectome

A fundamental idea in neuroscience is that cognitive functions—such as perception, learning, memory, and locomotion—are shaped and constrained by the brain's structural organization. Despite significant progress in mapping and analyzing structural connectomes, the principles linking the brain's physical architecture to its functional capabilities remain elusive. Here, we introduce an algebraic quantum model to bridge this theoretical gap, offering insights into the relationship between the connectome and emergent brain functions while connecting structural data to functional predictions. Using the well-mapped anatomical and extrasynaptic connectomes, we demonstrate that brain functions, defined as functional networks of a neural system, emerge as thermal equilibrium states of an algebraic quantum system derived from the graph algebra of the underlying directed multigraph. Specifically, these equilibrium states, characterized by the Kubo-Martin-Schwinger formalism, reveal how individual neurons contribute to functional network formation. Our model illuminates the structure-function relationship in neural circuits through two key features: (1) a functional connectome that delineates topologically driven neuronal interactions and (2) an integration capacity index that quantifies how effectively neurons coordinate and modulate diverse information flows. Together these features provide a statistical and mechanistic account of information flow and reveal how the network topology of the connectome predicts cognition and complex behaviors.