Alan Turing invented neural nets in a little-known paper entitled Intelligent Machinery (see [1]), in 1948. Since, the use of NNs has moved away decisively from inspiration by nature. I reckon, nature's last big win in AI were convolutional NNs: Kunihiko Fukushima's neocognitron was published in 1980, and inspired by 1950s work of Hubel and Wiesel [2]. Modern deep learning is largely an exercise in distributed systems: how can you feed stacks and stacks of tensor-cores and TPUs with floating point numbers, while minimising data movement (the real bottleneck of all computing)?
Not unlike, I think, how airplanes were originally inspired by birds, but nowadays the two have mostly parted ways, for solid technical reasons.
[1] http://www.alanturing.net/turing_archive/pages/Reference%20A...