Statistical thermodynamics of the 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, 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.

Beehive scale-free emergent dynamics.

It has been repeatedly reported that the collective dynamics of social insects exhibit universal emergent properties similar to other complex systems. In this note, we study a previously published data set in which the positions of thousands of honeybees in a hive are individually tracked over multiple days. The results show that the hive dynamics exhibit long-range spatial and temporal correlations in the occupancy density fluctuations, despite the characteristic short-range bees' mutual interactions. The variations in the occupancy unveil a non-monotonic function between density and bees' flow, reminiscent of the car traffic dynamic near a jamming transition at which the system performance is optimized to achieve the highest possible throughput. Overall, these results suggest that the beehive collective dynamics are self-adjusted towards a point near its optimal density.

Mean-field solution of the neural dynamics in a Greenberg-Hastings model with excitatory and inhibitory units.

We present a mean-field solution of the dynamics of a Greenberg-Hastings neural network with both excitatory and inhibitory units. We analyze the dynamical phase transitions that appear in the stationary state as the model parameters are varied. Analytical solutions are compared with numerical simulations of the microscopic model defined on a fully connected network. We found that the stationary state of this system exhibits a first-order dynamical phase transition (with the associated hysteresis) when the fraction of inhibitory units f is smaller than some critical value f_{t}≲1/2, even for a finite system. Moreover, any solution for f<1/2 can be mapped to a solution for purely excitatory systems (f=0). In finite systems, when the system is dominated by inhibition (f>f_{t}), the first-order transition is replaced by a pseudocritical one, namely a continuous crossover between regions of low and high activity that resembles the finite size behavior of a continuous phase transition order parameter. However, in the thermodynamic limit (i.e., infinite-system-size limit), we found that f_{t}→1/2 and the activity for the inhibition dominated case (f≥f_{t}) becomes negligible for any value of the parameters, while the first-order transition between low- and high-activity phases for f

Trial-by-trial variability in cortical responses exhibits scaling of spatial correlations predicted from critical dynamics.

In the mammalian cortex, even simple sensory inputs or movements activate many neurons, with each neuron responding variably to repeated stimuli-a phenomenon known as trial-by-trial variability. Understanding the spatial patterns and dynamics of this variability is challenging. Using cellular 2-photon imaging, we study visual and auditory responses in the primary cortices of awake mice. We focus on how individual neurons' responses differed from the overall population. We find consistent spatial correlations in these differences that are unique to each trial and linearly scale with the cortical area observed, a characteristic of critical dynamics as confirmed in our neuronal simulations. Using chronic multi-electrode recordings, we observe similar scaling in the prefrontal and premotor cortex of non-human primates during self-initiated and visually cued motor tasks. These results suggest that trial-by-trial variability, rather than being random noise, reflects a critical, fluctuation-dominated state in the cortex, supporting the brain's efficiency in processing information.