This study found no significant changes in sleep architecture (time spent in various stages of sleep) between ME/CFS patients and controls but found significant changes in all heart rate variability measures between the two groups during sleep.
They also found reduced sleep efficiency and reduced total sleep time in the chronic fatigue syndrome patients. When they looked deeper, they found that patients who felt sleepier had a 'higher fractal scaling index',, which is believed to indicate increased sympathetic nervous system activity, in Stages 1, 2 and 3 sleep compared to controls.
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In my case, we noticed that each time I was sliding into relapse, I had nightmares about being chased by a dragon -- as the full article notes, "fight or flight activation".
We were never sure whether the nightmares occurred because of changes in brain chemistry (I was already relapsing) or whether the relapse started because I was only getting a couple hours of sleep before I woke up and was awake the rest of the night, so my immune system was being depleted by the lack of sleep. So we were still looking at a Chicken And The Egg question, but at least we had a little advance warning that my body was planning to give out soon.