Because I hear this hypothesis come up from time to time--it's unlikely consciousness or long-term memory are maintained by e-fields generated in the brain, or that they require continuous electrical activity. Stark evidence against it come from ischemia studies. Ischemia usually accompanies serious underlying issues, so more controlled examples are easier to reason about: surgeons may use deep hypothermic circulatory arrest as protection for the brain in procedures with extended ischemia time requirements. In these cases, achieving electrocerebral silence (flat EEG) is one of the checklist items for the procedure. Clinical cases are worth reviewing for any hypothesis that suggests that long-term memory depends on continuous electrical activity* of neurons in the brain.
Since the events of ischemia, brain flatlining, and brain death are so closely linked in time, it's easy to conflate them. After a cold stop+start, the brain doesn't immediately jump back to normal function--there are a variety of processes that are worth studying better to get back to normal brain waves and brain function. Ischemia-related damage is often from a metabolic problems than from a discontinuity of electrical function. The reason for the cooling is to maintain the local energy reserves of the cells--when that is lost, the cells may have too much difficulty getting back to normal when blood is later reintroduced, and that's where you see brain damage or death. This kind of procedure is not without serious risk factors.
* Of course, electrical activity is needed for recall, but the point that I'm making is that the memory is later available for recall even after a period of discontinuity in the EEG.