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Witten-type topological field theory of self-organized criticality for stochastic neural networks

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.