Criticality and avalanches in neural networks

Experimental work, both in vitro and in vivo, reveals the occurrence of neural avalanches with an inverse power law distribution in size and time duration. These properties are interpreted as an evident manifestation of criticality, thereby suggesting that the brain is an operating near criticality complex system: an attractive theoretical perspective that according to Gerhard Werner may help to shed light on the origin of consciousness. However, a recent experimental observation shows no clear evidence for power-law scaling in awake and sleeping brain of mammals, casting doubts on the assumption that the brain works at criticality. This article rests on a model proposed by our group in earlier publications to generate neural avalanches with the time duration and size distribution matching the experimental results on neural networks. We now refine the analysis of the time distance between consecutive firing bursts and observe the deviation of the corresponding distribution from the Poisson statistics, as the system moves from the non-cooperative to the cooperative regime. In other words, we make the assumption that the genuine signature of criticality may emerge from temporal complexity rather than from the size and time duration of avalanches. We argue that the Mittag-Leffler (ML) exponential function is a satisfactory indicator of temporal complexity, namely of the occurrence of non-Poisson and renewal events. The assumption that the onset of criticality corresponds to the birth of renewal non-Poisson events establishes a neat distinction between the ML function and the power law avalanches generating regime. We find that temporal complexity uncontaminated by periodicity occurs in a narrow range of values of the control parameter, in correspondence of which the information transfer from one network to another becomes maximal. We argue that if this enhancement of information transport is interpreted as a signature of criticality, then the power law avalanches are a manifestation of supercriticality rather than criticality. This leads us to conclude that the recent experiment showing no evidence of power-law avalanches in the brain of mammals would not conflict with the hypothesis that the brain works at criticality.