Computational Power of Neural Networks?
There are some references I could find: 
General-purpose computation with neural networks: A survey of complexity theoretic results, 2003 and Counting hierarchies: polynomial time and constant depth circuits, 1993. 

It appears that neural networks are considered threshold circuits; i.e. those circuits using MAJORITY gates. In (2) it is the case that a depth $d$ neural network has complexity $TC^0_d$ (here's a link to link to complexity zoo entry about $TC^0$). 

Since $TC^0$ containts $ACC^0$, which is $AC^0$ with arbitrary MOD gates, then $AC^0 \subset TC^0$. Also, it is mentioned in the zoo that such circuits with depth 3 are strictly more powerful than those of depth 2.

In On the computational power of sigmoid versus Boolean threshold circuits,1991 it is mentioned that for a constant depth $d$, Boolean and real-valued threshold circuits (with polynomially bounded weights) are essentially the same.