Linear scaling of entropy versus energy in human brain activity, the Hagedorn temperature, and the Zipf law

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, as well as the density of states, 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 provides a rigorous demonstration of the linear scaling of the entropies and energies of the brain states, a behavior first conjectured by Hagedorn to be typical at the limiting temperature at which ordinary matter disintegrates into quark matter. Equivalently, this establishes the thermodynamic origin of the 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 to a wide variety of complex systems.