Sensitivity, Itinerancy and Chaos in Partly-Synchronized Weighted Networks

AbstractWe present exact results, as well as some illustrative Monte Carlosimulations, concerning a stochastic network with weighted connec-tions in which the fraction of nodes that are dynamically synchronized,ρ ∈ [0,1], is a parameter. This allows one to describe from single–nodekinetics (ρ → 0) to simultaneous updating of all the variables at eachtime unit (ρ → 1). An example of the former limit is the well–knownsequential updating of spins in kinetic magnetic models whereas thelatter limit is common for updating complex cellular automata. Theemergent behavior changes dramatically as ρ is varied. For small val-ues of ρ, we observe relaxation towards one of the attractors and agreat sensibility to external stimuli and, for ρ ≥ ρ c , itinerancy as inheteroclinic paths among attractors; tuning ρ in this regime, the os-cillations with time may abruptly change from regular to chaotic andvice versa. We show how these observations, which may be relevantconcerning computational strategies, closely resemble some actual sit-uations 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