  It is well established that the brain spontaneously traverses through a very
large number of states. Nevertheless, despite its relevance to understanding
brain function, a formal description of this phenomenon is still lacking. To
this end, we introduce a machine learning based method allowing for the
determination of the probabilities of all possible states at a given
coarse-graining, from which all the thermodynamics can be derived. This is a
challenge not unique to the brain, since similar problems are at the heart of
the statistical mechanics of complex systems. This paper uncovers a linear
scaling of the entropies and energies of the brain states, a behaviour first
conjectured by Hagedorn to be typical at the limiting temperature in which
ordinary matter disintegrates into quark matter. Equivalently, this establishes
the existence of a Zipf law scaling underlying the appearance of a wide range
of brain states. Based on our estimation of the density of states for large
scale functional magnetic resonance imaging (fMRI) human brain recordings, we
observe that the brain operates asymptotically at the Hagedorn temperature. The
presented approach is not only relevant to brain function but should be
applicable for a wide variety of complex systems.
