they can be continuously updated, assuming you re-run representative samples of the training set through them continuously. Unlike a mammal brain which preserves the function of neurons unless they activate in a situation which causes a training signal, deep nets have catastrophic forgetting because signals get scattered everywhere. If you had a model continuously learning about you in your pocket, without tons of cycles spent "remembering" old examples. In fact, this is a major stumbling block in standard training, sampling is a huge problem. If you just iterate through the training corpus, you'll have forgotten most of the english stuff by the time you finish with chinese or spanish. You have to constantly mix and balance training info due to this limitation.

The fundamental difference is that physical neurons have a discrete on/off activation, while digital "neurons" in a network are merely continuous differentiable operations. They also don't have a notion of "spike timining dependency" to avoid overwriting activations that weren't related to an outcome. There are things like reward-decay over time, but this applies to the signal at a very coarse level, updates are still scattered to almost the entire system with every training example.