Bishop is talking bollocks but he's too fucking stupid (or hooked on Searle's deeply-flawed Chinese Room) to understand it. He could be replaced by a single layer neural net.
Our brains are neural nets. No magic involved. The fact that we have yet to manufacture an artificial neural net of similar complexity and organization doesn't mean that we cannot do so (thermal problems and/or light speed problems may mean it has to run a lot slower than a human brain, but that's a different matter).
Bishop is arguing that the small neural nets we make aren't up to the job therefore a big one won't be either. He is a fuckwit. Or religious (in my view, that's the same thing).
There is no magic involved in the way our brains work. No "soul" implanted by the Good Magic Sky Fairy at conception (why then and not gastrulation?). Mind is an emergent property of brain, not a bit of bolt-on magic stuff.
If you insist there is a soul, you need to explain a lot of things that are incompatible with soul theory. Such explanations require industrial strength ad hockery and much hand-waving, and even then are transparently stupid.
What incompatible things? Any of the effects of localized brain damage. Several are explored in Oliver Sacks' excellent The Man Who Mistook His Wife For A Hat.
One example: prosopagnosia. Damage to a small section of brain causes an inability to recognize faces. You can still see faces. You can still describe a face you see. If you're at all artistic, you can draw the face you're seeing (how well depends on how artistic you are, obviously). If you're very artistic you can see a face briefly then draw it from memory. But you can't recognize it. Not even if it's somebody very close to you. Not if it's your spouse, or your parent, or your child. Something most of us take for granted and consider to be an intrinsic part of our personality. Under soul theory, prosopagnosia cannot happen.
There are many similar examples of localized brain damage having specific effects upon things we consider an intrinsic part of our personality, our "soul." Then there are the effects of alcohol, drugs and anaesthetics. With soul theory, only local anaesthesia would be possible because your "soul" would remain awake even if your whole body was anasethetized. Under soul theory, alcohol and drugs might have pleasant effects upon the body (say a feeling of being stroked all over, or continuous orgasm) but no direct effect on the mind, because that is magic and survives even death intact.
So our minds are nothing but an emergent property of a large, complex neural net. When Bishop says that neural nets are incapable of doing what the human mind does, he is talking pure fucking bollocks. Like I said, he could be replaced with a single-layer neural net.