Mean-field solution of the neural dynamics in a Greenberg-Hastings model with excitatory and inhibitory units

We present a mean field solution of the dynamics of a Greenberg-Hastings neural network with both excitatory and inhibitory units. We analyse the dynamical phase transitions that appear in the stationary state as the model parameters are varied. Analytical solutions are compared with numerical simulations of the microscopic model defined on a fully connected network. We found that the stationary state of this system exhibits a first order dynamical phase transition (with the associated hysteresis) when the fraction of inhibitory units $f<f_t\le 1\mathrm{/}2$, even for a finite system. In finite systems, when $f>f_t$ the first order transition is replaced by a pseudo critical one, namely a continuous crossover between regions of low and high activity that resembles the finite size behaviour of a continuous phase transition order parameter. However, in the thermodynamic limit, we found that $f_t\to 1\mathrm{/}2$ and the activity for $f\ge f_t$ becomes negligible for any value of $T>0$, while the first order transition for $f<f_t$ remains.