  Recent experiments have shown that the spontaneous activity of young
dissociated neuronal cultures can be described as a process of highly
inhomogeneous nucleation and front propagation due to the localization of noise
activity, i.e., noise focusing. However, the basic understanding of the
mechanisms of noise build-up leading to the nucleation remain an open
fundamental problem. Here we present a minimal dynamical model called
stochastic quorum percolation that can account for the observed phenomena,
while providing a robust theoretical framework. The model reproduces the first
and second order phase--transitions of bursting dynamics and neuronal
avalanches respectively, and captures the profound effect metric correlations
in the network topology can have on the dynamics. The application of our
results to other systems such as in the propagation of infectious diseases and
of rumors is discussed.
