Thouless–Anderson–Palmer equation for associative memory neural network with synaptic noise

We study the effects of temporal fluctuations in synaptic couplings on the properties of analog neural networks. Since no energy concept exists in networks with such couplings, the use of the replica method does not make sense. On the other hand, the self-consistent signal-to-noise analysis (SCSNA), which is an alternative to the replica method for deriving a set of order parameter equations, requires no energy concept and thus plays an important role for studying such networks. To apply the SCSNA to stochastic networks, it is necessary to define the deterministic networks equivalent to the original stochastic ones, which are given by the Thouless–Anderson–Palmer (TAP) equations. Therefore the TAP equation is of interest for studying the statistical properties of the networks with synaptic noise, while such study is very few. In this paper, we show the TAP equation together with a set of order parameter equations for such networks by using both the cavity method and the SCSNA.