On the linear scaling of entropy vs. energy in human brain activity, the Hagedorn temperature and the Zipf law

TL;DR

This study introduces a machine learning method to analyze brain state probabilities, revealing a linear entropy-energy relationship and Zipf law scaling, indicating the brain operates near a Hagedorn temperature, with implications for complex systems.

Contribution

It presents a novel machine learning approach to derive thermodynamics of brain states, uncovering linear entropy-energy scaling and Zipf law behavior in brain activity.

Findings

Linear scaling of entropy and energy in brain states

Brain operates near the Hagedorn temperature

Zipf law underlies the distribution of brain states

Abstract

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.