  We study the Witten-type topological field theory(W-TFT) of self-organized
criticality(SOC) for stochastic neural networks. The Parisi-Sourlas-Wu
quantization of general stochastic differential equations (SDEs) for neural
networks, the Becchi-Rouet-Stora-Tyutin(BRST)-symmetry of the diffusion system
and the relation between spontaneous breaking and instantons connecting steady
states of the SDEs, as well as the sufficient and necessary condition on
pseudo-supersymmetric stochastic neural networks are obtained. Suppose neuronal
avalanche is a mechanism of cortical information processing and storage
\cite{Beggs}\cite{Plenz1}\cite{Plenz2} and the model of stochastic neural
networks\cite{Dayan} is correct, as well as the SOC system can be looked upon
as a W-TFT with spontaneously broken BRST symmetry. Then we should recover the
neuronal avalanches and spontaneously broken BRST symmetry from the model of
stochastic neural networks. We find that, provided the divergence of drift
coefficients is small and non-constant, the model of stochastic neural networks
is BRST symmetric. That is, if the SOC of brain neural networks system can be
looked upon as a W-TFT with spontaneously broken BRST symmetry, then the
general model of stochastic neural networks which be extensively used in
neuroscience \cite{Dayan} is not enough to describe the SOC. On the other hand,
using the Fokker-Planck equation, we show the sufficient condition on diffusion
so that there exists a steady state probability distribution for the stochastic
neural networks. Rhythms of the firing rates of the neuronal networks arise
from the process, meanwhile some biological laws are conserved.
