Divergence of primitive sleep into REM and NREM states is thought to have occurred in the nocturnal Triassic ancestors of mammals as a natural accompaniment of the evolution of warm-bloodedness. As ambient temperatures during twilight portions of primitive sleep traversed these evolving ancestors' core temperature, mechanisms of thermoregulatory control that employ muscle contractions became superfluous. The resulting loss of need for such contractions during twilight sleep led to muscle atonia. With muscle tone absent, selection favored the persistence of the fast waves of nocturnal activity during twilight sleep. Stimulations by these waves reinforce motor circuits at the increasing temperatures of evolving warm-bloodedness without leading to sleep-disturbing muscle contractions. By these and related interlinked adaptations, twilight sleep evolved into REM sleep. The daytime period of sleep became NREM sleep. The evolution of NREM and REM sleep following this scenario has implications for sleep's maintenance processes for long-term memories. During NREM sleep, there is an unsynchronized, uncoordinated stimulation and reinforcement of individual distributed component circuits of consolidated memories by slow wave potentials, a process termed 'uncoordinated reinforcement'. The corresponding process during REM sleep is the coordinated stimulation and reinforcement of these circuits by fast wave potentials. This action temporally binds the individual component circuit outputs into fully formed memories, a process termed 'coordinated reinforcement'. Sequential uncoordinated and coordinated reinforcement, that is, NREM followed by REM sleep, emerges as the most effective mechanism of long-term memory maintenance in vertebrates. With the evolution of this two-stage mechanism of long-term memory maintenance, it became adaptive to partition sleep into several NREM-REM cycles, thereby achieving a more lengthy application of the cooperative sequential actions.