Five years ago I ran some queries on Google Scholar to see trends on the number of papers that mention particular phrase. The number of hits for each year was divided by the number of hits for "machine learning". Back then it looked like NN's started gaining in popularity with invention of back-propagation in 1980's, peaked in 1993 and went downhill from there.
Since then, there's been several major NN developments, involving deep learning and probabilistically founded versions so I decided to update the trend. I couldn't find a copy of scholar scraper script anymore, luckily Konstantin Tretjakov has maintained a working version and reran the query for me.
It looks like downward trend in 2000's was misleading because not all papers from that period have made it into index yet, and the actual recent trend is exponential growth!
One example of this "third wave" of Neural Network research is unsupervised feature learning. Here's what you get if you train a sparse auto-encoder on some natural scene images
What you get is pretty much a set of Gabor filters, but the cool thing is that you get them from your neural network rather than image processing expert