In recent years, an increasing number of studies have demonstrated the critical importance of adequate sleep for human health. A broader biological implication of this is that sleep appears to be nearly universal in animals. Despite its importance, it is still unclear what sleep is for. Researchers are considering the idea of non-human animals dreaming during REM-like sleep because species act out waking-like actions in this condition.
Sleep spindles are visible in octopuses' sleep, just as they are in humans.
Furthermore, during the octopuses' peaceful sleep, the researchers discovered distinct brain waves. These closely mimic specific waveforms recorded in mammalian brains during non-REM sleep. These waves are referred to by scientists as sleep spindles. Scientists believe that these waves are important in octopus memory consolidation as they sleep. Dr. Tomoyuki Mano, the first co-author, employed a powerful microscope to discover that these sleep spindle-like waves occur in areas of the octopuses' brains associated with learning and memory. This implies a function similar to that of humans. The octopuses went into an active sleep phase that lasted about a minute every hour. Their brain activity closely reflected their awake state throughout this stage. This is similar to REM sleep in humans.

Okinawa Institute of Science and Technology (OIST) researchers, in collaboration with the University of Washington, have closely examined brain activity and skin patterning in octopuses (Octopus laqueus) during this active period of sleep and discovered that they closely resemble neural activity and skin patterning behavior seen when awake. In mammals, wake-like activity occurs during rapid eye movement (REM) sleep, which is when most dreams occur. In several senses, the REM sleep state is contradictory. It is characterized by increased metabolic activity and brainwave patterns more akin to wakefulness. At the same time, most skeletal muscles relax (the exception being the eye muscles).  Octopuses sleep in the same way as humans do but with a twist. The slumber bouts of these bright marine invertebrates are punctuated by bursts of frenetic activity. During sleep, the octopuses' eyes and arms quiver, their breathing becomes more rapid, and their skin flashes in a beautiful display of colors. The brain architecture of octopuses (and other cephalopods) differs dramatically from that of almost every other type of organism on the planet. They appear to be extremely intelligent, with extraordinary problem-solving abilities, and their brains have a few surprising similarities with humans. 
It's an intriguing discovery that could shed light on the evolution and function of sleep!