Neural Dynamics: Criticality, Cooperation, Avalanches and Entrainment between Complex Networks

The discovery of avalanches in neural systems [1] has aroused substantial interest among neurophysiologists and, more generally, among researchers in complex networks [2] as well. The main purpose of this chapter is to provide evidence in support of the hypothesis that the phenomenon of neural avalanches [1] is generated by the same cooperative properties as those responsible for a surprising effect that we call cooperation-induced synchronization, illustrated in [3]. The phenomenon of neural entrainment [4] is another manifestation of the same cooperative property. We also address the important issue of the connection between neural avalanches and criticality. Avalanches are thought to be a manifestation of criticality, and especially self-organized criticality [5, 6]. At the same time, criticality is accompanied by long-range correlation [5] and a plausible model for neural dynamics is expected to account for the astounding interaction between agents separated by relatively large distances. General agreement exists in the literature that brain function rests on these crucial properties, and the phase transition theory for physical phenomena [7] is thought to afford the most important theoretical direction for further research work on this subject. In this theory criticality emerges at a specific single value of a control parameter, designated by the symbol K c. In this chapter we illustrate a theoretical model generating avalanches, long-range correlation and entrainment, as a form of cooperation-induced synchronization, over a wide range of values of the control parameter K, thereby suggesting that the form of criticality within the brain is not the ordinary criticality of physical phase transitions but is instead the extended criticality recently introduced by Longo and co-workers [8] to explain biological processes. Cooperation is the common effort of the elements of a network for their mutual benefit. We use the term cooperation in the same loose sense as that adopted, for instance, by Montroll [9] to shed light on the equilibrium condition realized by the interacting spins of the Ising model. Although the term cooperation, frequently used in this chapter, does not imply a network's cognition, we follow the conceptual