A Statistical Mechanics Approach to Brain Activity

Neuronal avalanches are spatiotemporal cascades of spontaneous neuronal firing activity found in cortical tissue in vitro and in vivo of several animals, whose size and duration distributions follow power laws. This state of the dynamics is shown to optimize dynamic range and information transfer, and suggests that cortical networks operate near a critical point. Here, we demonstrate that critical avalanche dynamics necessarily induce power-law scaling of neuronal firing rates with brain size, an allometric scaling observed in several species. Using analytical calculations, we express the allometric exponent as a function of the exponents of the avalanche size and duration distribution, and explain that this relation holds for any system with avalanches exhibiting finite-size scaling. Next, using the so-called Maximum Entropy Method, we infer generalized Ising Hamiltonians consistent with time-averaged features of numerical and experimental neuronal firing time series at and off criticality, mapping the neuronal network dynamics into an analogous thermodynamic framework, offering an alternative approach to study critical signatures in neuronal data. We apply this method for the first time to data from an integrate-and-fire model, with threshold firing, refractory periods, and short- and long-term synaptic plasticity, enabling a controlled setting to study neuronal avalanche dynamics as opposed to experiments. The inferred Hamiltonians are poised near a spin-glass–like phase boundary and exhibit parameter disorder that depends on the underlying dynamical state of the datasets. These inferred models are able to correctly identify subcritical and non-subcritical signatures of the datasets, but are less suited to distinguish critical from supercritical signatures. These results help clarify both the utility and the limitations of Maximum Entropy Modeling to describe neuronal dynamics.