  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.
