Susceptibility for extremely low external fluctuations and critical behavior of Greenberg-Hastings neuronal model

We consider the scaling behavior of the fluctuation susceptibility associated with the average activation in the Greenberg-Hastings neural network model and its relation to microscopic spontaneous activation. We found that, as the spontaneous activation probability tends to zero, a clear finite-size scaling behavior in the susceptibility emerges, characterized by critical exponents which follow already known scaling laws. This shows that the spontaneous activation probability plays the role of an external field conjugated to the order parameter of the dynamical activation transition. The roles of different kinds of activation mechanisms around the different dynamical phase transitions exhibited by the model are characterized numerically and using a mean-field approximation. However, while mean-field approximations predict directed percolation critical exponents, numerical simulations on Watts-Strogatz networks provide critical exponents estimations which do not seem to agree with any known universality class.