Correlations Induced by Depressing Synapses in Critically Self-Organized Networks with Quenched Dynamics

In a recent work, mean-field analysis and computer simulations were employed to analyze critical self-organization in networks of excitable cellular automata where randomly chosen synapses in the network were depressed after each spike (the so-called annealed dynamics). Calculations agree with simulations of the annealed version, showing that the nominal \textit{branching ratio\/} $\sigma$ converges to unity in the thermodynamic limit, as expected of a self-organized critical system. However, the question remains whether the same results apply to the biological case where only the synapses of firing neurons are depressed (the so-called quenched dynamics). We show that simulations of the quenched model yield significant deviations from $\sigma=1$ due to spatial correlations. However, the model is shown to be critical, as the largest eigenvalue of the synaptic matrix approaches unity in the thermodynamic limit, that is, $\lambda_c = 1$ . We also study the finite size effects near the critical state as a function of the parameters of the synaptic dynamics.