In Li Deng's and Dong Yu's book on Deep Learning (March 2014) version they briefly relate Hierarchical Temporal Memory (HTM) to the Convolutional Neural Networks which are popular for Deep Learning.

http://research.microsoft.com/apps/pubs/default.aspx?id=2093...

It's worth noting that most people doing Deep Learning aren't trying to replicate the brain, but just want to do a better job at Machine Learning (ML) and Artificial Intelligence. Here's how I see it as someone working on Deep Learning; someone correct me if I'm wrong.

Deep Learning: Trying to do ML - yes Trying to replicate brain - no (for the most part)

Numenta (HTM/CLA): Trying to do ML - yes (not sure how much they succeed) Trying to replicate brain - yes, but (i) we don't know exactly how the brain works (ii) they make approximations

Projects like Nengo (http://nengo.ca/): Trying to do ML - no Trying to replicate brain - yes

I'm not very familiar with Nengo.

Edit: formatting