  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 is a parameter. This
allows one to describe from single-node kinetics to simultaneous updating of
all the variables at each time unit. 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 the parameter is varied. For small values, we
observed relaxation towards one of the attractors and a great sensibility to
external stimuli, and for large synchronization, itinerancy as in heteroclinic
paths among attractors; tuning the parameter 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.
