Differences between quenched and annealed neuronal networks with self-organized criticality

In a recent work, mean field analysis and computer simulations were employed to analyze critical self-organization in an annealed network of excitable cellular automata (SIRS) neuronal networks, where randomly chosen synapses are depressed after each neuron spike. Calculations agree with simulations of the annealed version, showing that the nominal branching ratio ($\sigma$) converges to the critical value $\sigma_c =1$ and fluctuations vanish in the thermodynamic limit, as expected of a self-organized critical system. However, the question remains whether the same results occurs to the quenched version of the model (which is biologically more plausible) where neighborhoods are fixed and only the acting synapses are depressed. We have seen that simulations of the quenched model yield a stationary value $\sigma(t \rightarrow \infty) = 1.105$ which is a significant deviation from $\sigma = 1$, due to spatio-temporal correlations produced by avalanches. However, the model is shown to be critical, as the largest eigenvalue $\lambda$ of the synaptic matrix is shown to approach $\lambda_c = 1$, with fluctuations vanishing in the thermodynamic limit. We also study the influence of the recovery and decay synaptic parameters in both types of models, as well the influence of the number of neighbors. As a future work, we intend to study the distribution of interspike intervals in this kind of neuronal networks.