Take a bird-nap, not a cat nap.

A complete group of sleep characteristics (rapid-eye-movement sleep and slow-wave sleep as well as transition stages and quick spikes) has been found outside of mammals, in zebra finches, a surprising finding because birds lack a neocortex, the part of the mammalian brain thought necessary for such patterns. Low et al. suggest that ancestral characteristics of sleep evolved under selective pressures common to songbirds and mammals. This would fit with Tononi's suggestion that sleep is required for synaptic homeostasis and regenerations (a form of sleep is also observed in fruitflies). Here is their abstract:

A suite of complex electroencephalographic patterns of sleep occurs in mammals. In sleeping zebra finches, we observed slow wave sleep (SWS), rapid eye movement (REM) sleep, an intermediate sleep (IS) stage commonly occurring in, but not limited to, transitions between other stages, and high amplitude transients reminiscent of K-complexes. SWS density decreased whereas REM density increased throughout the night, with late-night characterized by substantially more REM than SWS, and relatively long bouts of REM. Birds share many features of sleep in common with mammals, but this collective suite of characteristics had not been known in any one species outside of mammals. We hypothesize that shared, ancestral characteristics of sleep in amniotes evolved under selective pressures common to songbirds and mammals, resulting in convergent characteristics of sleep.