YES, THEY DO - but not in the sense we understand sleep. Research conducted by James and Cottell into sleep patterns of insects (1983) showed that ants have a cyclical pattern of resting periods which each nest as a group observes, lasting around eight minutes in any 12-hour period. Although this means two such rest periods in any 24-hour period, only one of the rest periods bears any resemblance to what we would call sleep. Mandible and antennae activity is at a much lower level (usually up to 65 per cent lower) than during the other rest period in one 24-hour period, indicating a much deeper "resting" phase. Basing and McCluskey in 1986 used brain activity recorders on black, red, and soldier ants to determine whether the deeper resting period constituted actual "sleep". A steep decline in brain wave fluctuations supported the "sleep" hypothesis in black and red ants, but surprisingly showed a higher level of brain activity in soldier ants in a deep resting phase.
