Analog circuits run at full-speed (no clock), use little power, and take up little space. An analog computer directly implements the mathematical function it represents as circuits instead of emulating it on a von Neumann-like machine. Long story short, they have issues that made people go digital. Yet, if you can use analog, you can get significant advantages. Example for math acceleration:

http://www.cisl.columbia.edu/grads/gcowan/vlsianalog.pdf

Brain is a bunch of components that are spread out 3D that operate like a mathematical function at slow speed. Mostly sounds analog. Results in us. So, a huge spread of analog components could get some results directly simulating something like that. Here's one of my favorites which is a wafer-scale, analog computer for neural networks.

www.kip.uni-heidelberg.de/Veroeffentlichungen/download.cgi/4713/ps/1856.pdf