We study the stochastic version of the Wilson-Cowan model of neural dynamics, where the response function of neurons grows faster than linearly above the threshold, representing a cooperative effect of different synaptic inputs. The model shows a region of parameters where two attractive fixed points of the dynamics exist simultaneously, corresponding to lower and higher activity, and
the dynamics switches between them as observed in up and down states of cortical networks. Along with alternation of states, the model displays a bimodal distribution of the avalanches of activity, with a power law behaviour corresponding to the state of low activity, and a bump of very large
avalanches due to the high activity state. The bistability is due to the presence of a first order (discontinuous) transition in the phase diagram, and the observed critical behaviour is connected with the line where the low activity state becomes unstable (spinodal line).
See here a video, followed by a poster.