>They're a multi-layered class of machine learning algorithms that essentially try to mimic how a human brain works...
This is a common misconception. The artificial "neurons" in a neural net are certainly inspired by biological neurons in nature, but the networks we construct out of them are very often deliberately un-like how biology constructs the soft-and-mushy version of the real world brain. For example many artificial deep neural nets are deliberately acyclic ("feedforward"), which isn't the case in the brain.
What we're generally aiming to mimic is the outcomes of the brain working - the learning part - rather than the underlying mechanisms or structure of the brain itself.
It's fair to say some classes of neural nets are biologically inspired in their structure, but the fundamental reality is we have as-good-as-no-idea how the brain is really functioning at the cellular level so we're really being inspired by the current-best models for how the brain might work. Models inspiring models. Turtles all the way down.