Self-organized criticality and near criticality in neural networks J D Cowan, J Neuman, and W van Drongelen Abstract We show that an array of E -patches will self-organize around critical points of the directed percolation phase transition, and when driven by a weak stimu- lus will oscillate between UP and DOWN states each of which generates avalanches consistent with directed percolation. The array therefore exhibits self-organized crit- icality (SOC) and replicates the behavior of the original sandpile model of Bak et al. [1988]. We also show that an array of E /I patches will also self-organize to a weakly stable node located near the critical point of a directed percolation phase transition, so that fluctuations about the weakly stable node will also follow a power slope with a slope characteristic of directed percolation. We refer to this as self-organized near criticality (SONC). 1 Introduction Ideas about criticality in non-equilibrium dynamical systems have been around for at least fifty years or more. Criticality refers to the fact that nonlinear dynamical systems can have local equilibria that are marginally stable, so that small pertur- bations can drive the system away from the local equilibria towards one of several locally stable equilibria. In physical systems such marginally stable states manifest in several ways, in particular if the system is spatially as well as temporally orga- nized, then long-range correlations in both space and time can occur, and the statis- tics of the accompanying fluctuating activity becomes non-Gaussian, and in fact is self-similar in its structure, and therefore follows a Power Law. Bak et al. [1988] introduced a mechanism whereby such a dynamical system could self-organize to a marginally stable critical point, which they called self-organized criticality. Their paper immediately triggered an avalanche of papers on the topic, not the least of which was a connection with 1/ f or scale-free noise. However it was not until an- other paper appeared, by Gil and Sornette [1996], which greatly clarified the dy- Dept. of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637 1