Awaking and Sleeping a Complex Network

A network with local dynamics of logistic type is considered. We implement a mean-field multiplicative coupling among first-neighbor nodes. When the coupling parameter is small the dynamics is dissipated and there is no activity: the network is {\it turned off}. For a critical value of the coupling a non-null stable synchronized state, which represents a {\it turned on} network, emerges. This global bifurcation is independent of the network topology. We characterize the bistability of the system by studying how to perform the transition, which now is topology dependent, from the active state to that with no activity, for the particular case of a scale free network. This could be a naive model for the {\it wakening} and {\it sleeping} of a brain-like system.