The brain keeps its overall dynamics in a corridor of intermediate activity and it has been a long standing question what possible mechanism could achieve this task. Mechanisms from the field of statistical physics have long been suggesting that this homeostasis of brain activity could occur even without a central regulator, via self-organization on the level of neurons and their interactions, alone. Such physical mechanisms from the class of self-organized criticality exhibit characteristic dynamical signatures, similar to seismic activity related to earthquakes. Measurements of cortex rest activity showed first signs of dynamical signatures potentially pointing to self-organized critical dynamics in the brain. Indeed, recent more accurate measurements allowed for a detailed comparison with scaling theory of non-equilibrium critical phenomena, proving the existence of criticality in cortex dynamics. We here compare this new evaluation of cortex activity data to the predictions of the earliest physics spin model of self-organized critical neural networks. We find that the model matches with the recent experimental data and its interpretation in terms of dynamical signatures for criticality in the brain. The combination of signatures for criticality, power law distributions of avalanche sizes and durations, as well as a specific scaling relationship between anomalous exponents, defines a universality class characteristic of the particular critical phenomenon observed in the neural experiments. Thus the model is a candidate for a minimal model of a self-organized critical adaptive network for the universality class of neural criticality. As a prototype model, it provides the background for models that may include more biological details, yet share the same universality class characteristic of the homeostasis of activity in the brain.

Spin models of neural networks and genetic networks are considered elegant as they are accessible to statistical mechanics tools for spin glasses and magnetic systems. However, the conventional choice of variables in spin systems may cause problems in some models when parameter choices are unrealistic from a biological perspective. Obviously, this may limit the role of a model as a template model for biological systems. Perhaps less obviously, also ensembles of random networks are affected and may exhibit different critical properties. We consider here a prototypical network model that is biologically plausible in its local mechanisms. We study a discrete dynamical network with two characteristic properties: Nodes with binary states 0 and 1, and a modified threshold function with Θ(0)(0)=0. We explore the critical properties of random networks of such nodes and find a critical connectivity K(c)=2.0 with activity vanishing at the critical point. Finally, we observe that the present model allows a more natural implementation of recent models of budding yeast and fission yeast cell-cycle control networks.

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