I work in the field, and while some models are based on biological structures/systems, there's a lot of fuzz about them being "based on biological foundations" that is now best avoided. Yes, it is true the model is based on them, but it's a model that only covers very little of the real complexity. So in a sense, it's naive to say "put a billion neurons in there and you'll get a rat brain" (as was publicized one time).

The effectiveness comes from their non-linear nature and their ability to "learn" (store knowledge in the weights, that is derived from the training process). And black magic, of course!