Sensitivity, Itinerancy and Chaos in Partly–Synchronized Weighted Networks

We present exact results, as well as some illustrative Monte Carlo simulations, concerning a stochastic network with weighted connections in which the fraction of nodes that are dynamically synchronized, ρ ∈ [0, 1] , is a parameter. This allows one to describe from single–node kinetics (ρ → 0) to simultaneous updating of all the variables at each time unit (ρ → 1). An example of the former limit is the well–known sequential updating of spins in kinetic magnetic models whereas the latter limit is common for updating complex cellular automata. The emergent behavior changes dramatically as ρ is varied. For small values of ρ, we observe relaxation towards one of the attractors and a great sensibility to external stimuli and, for ρ ≥ ρc, itinerancy as in heteroclinic paths among attractors; tuning ρ in this regime, the oscillations with time may abruptly change from regular to chaotic and vice versa. We show how these observations, which may be relevant concerning computational strategies, closely resemble some actual situations related to both searching and states of attention in the brain. PACS: 02.50.Ey; 05.45.Gg; 05.70.Ln; 87.18.Sn; 89.20.-a