I'm surprised that Numenta [1] did not get a mention in the comments. As a mostly lay person (with moderate exposure to Computational Neuroscience) I quite like their approach even though it seems their results have slowed down a bit of late. I still haven't finished parsing BAMI [2], but seems very interesting.

[1] https://numenta.com/ [2] https://numenta.com/resources/biological-and-machine-intelli...