Say you randomly alter 1% of the molecules in the brain. Then almost every neuron would still recognizably be a neuron, and still have synapses that connected to the right things, and any concentration of neurotransmitter X would still recognizably be type X (rather than Y). There is no way I see for 1% random destruction to erase the person information-theoretically.

Would this be enough to keep thresholds for action potentials correct? I'm more familiar with neural nets for computational purposes than with actual neural architecture, but for neural nets this matters a lot. You can have wildly different behavior even with the same neurons connected to each other just by changing the potential levels. Learning behavior consists not just in constructing or removing connections but also in strengthening and weakening existing connections.

I don't know why you mention the concentrations of neurotransmitters since that's a fairly temporary thing which (as far as I'm aware) doesn't contain much in the way of actual data except about neurons which have fired very recently.