I'm not a neurologist or a philosopher so I think this may not be something I can speak intelligently about, but it seems to me that the question doesn't really live in the function approximation, but in the gaps. Doing a single train of thought is the easy part. I think what you need is an "always online" feature to keep thinking and some way to be stateful (make memories) without constantly retraining the whole network or whole sets of layers.
This is, I think, the benefit of the parallelism and asynchrony of the "individual neuron" discussed in the book.