At least they think they do. As they mentioned a typical materialist explanation that their maybe some undetectable brain activity in the brain that can account for these experiences without saying that consciousness can function without the brain. The problem is this is very improbable. As EEG expert Dr. John Greenfield has confirmed.
“It’s very unlikely that a hypoperfused brain [someone with no blood flow to the brain], with no evidence of electrical activity could generate NDEs. Human studies as well as animal studies have typically shown very little brain perfusion [blood flow] or glucose utilization when the EEG is flat. There are deep brain areas involved in generating memories that might still operate at some very reduced level during cardiac arrest, but of course any subcortically generated activity can’t be brought to consciousness without at least one functioning cerebral hemisphere. So even if there were some way that NDEs were generated during the hypoxic state [while the brain is shut off from oxygen], you would not experience them until reperfusion [blood flow] allowed you to dream them or wake up and talk about them”, Greenfield stated.
The philosophers of this book that is coming out about a naturalistic explanation for NDE's go on to say.
Physical explanations of NDEs are not only significantly more likely to be true than supernatural explanations, but that they are also capable of being deeply attractive and inspiring, Mitchell-Yellin says.
First of all I don't think the afterlife explanation is an supernatural explanation. It's an natural phenomenon in the sense that we can explore it.