Note: Moment approximations for Bernoulli neurons with sigmoidal nonlinearity

Consider a stochastic, binray, linear-nonlinear unit, with spiking output $s$, synaptic inputs $\mathbf{x}$, weights $\mathbf{w}$, and bias (threshold) $b$:

$$\begin{array}{}\text{(1)}& \begin{array}{rl}s& \sim \mathrm{Bernoulli}[p=\mathrm{\Phi }(a)]\\ a& ={\mathbf{w}}^{\mathrm{\top }}\mathbf{x}+b,\end{array}\end{array}$$

where $\mathrm{\Phi }(\cdot )$ is the cumulative distribution function of a standard normal distribution. Note that $\mathrm{\Phi }(\cdot )$ can be rescaled to closely approximate the logistic sigmoid if desired. Assuming the mean $\mu$ and covariance $\mathrm{\Sigma }$ of $\mathbf{x}$ are known, can we obtain the mean and covariance of $s$?