  This paper contains an analysis of a simple neural network that exhibits
self-organized criticality. Such criticality follows from the combination of a
simple neural network with an excitatory feedback loop that generates
bistability, in combination with an anti-Hebbian synapse in its input pathway.
Using the methods of statistical field theory, we show how one can formulate
the stochastic dynamics of such a network as the action of a path integral,
which we then investigate using renormalization group methods. The results
indicate that the network exhibits hysteresis in switching back and forward
between its two stable states, each of which loses its stability at a
saddle-node bifurcation. The renormalization group analysis shows that the
fluctuations in the neighborhood of such bifurcations have the signature of
directed percolation. Thus the network states undergo the neural analog of a
phase transition in the universality class of directed percolation. The network
replicates precisely the behavior of the original sand-pile model of Bak, Tang
& Wiesenfeld.
