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

ArXiv Pub Date : 2024-12-11
Dante R Chialvo, Romuald A Janik
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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.

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人脑活动的统计热力学、哈格多恩温度和齐普夫定律。
众所周知,大脑会自发地穿越大量的状态。然而,尽管它与理解大脑功能息息相关,但目前仍缺乏对这一现象的正式描述。为此,我们引入了一种基于机器学习的方法,这种方法可以确定给定粗粒度下所有可能状态的概率,并由此推导出所有热力学。这并不是大脑所独有的挑战,因为类似的问题也是复杂系统统计力学的核心。本文揭示了大脑状态的熵和能量的线性缩放,这种行为最早由海格多恩(Hagedorn)推测,在普通物质解体为夸克物质的极限温度下具有典型性。同样,这也证明了齐普夫定律的存在,它是各种脑状态出现的基础。根据我们对大规模功能磁共振成像(fMRI)人脑记录的状态密度的估计,我们观察到大脑在哈格多恩温度下近似运行。所提出的方法不仅与大脑功能相关,而且应适用于各种复杂系统。
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