Lying awake in the dark praying for rest, or watching a driver struggling to stay alert at the wheel, you see that there is something frightening about the power of sleep. Once you become too conscious of its absence or presence, the balance has shifted and you have already lost. And, as Sleeping and Dreaming at the Wellcome Collection proves, there are enough extreme cases to give you pause for uneasy thought next time you slip under the covers.

Michael Corke, who suffered from fatal familial insomnia, a rare genetic disorder, went without sleep for six months before he died. Film footage from 1991 shows him, a conductor, mounting a stage in front of a school orchestra as slowly as a tin man in need of oil. After five months he was unable to talk, paralysed within his own wakeful body. Sometimes he appeared to be dozing, but EEG measurements showed no brain activity that could be classified as sleep.

Sleep isn’t necessary only for our physical wellbeing. “It is a common experience that a problem difficult at night is resolved in the morning after the committee of sleep has worked on it,” wrote John Steinbeck, who knew, like many authors and artists, how to harness sleep’s unruly power. The committee of Sleeping and Dreaming, led by the curator Michael Dorrmann, has not resolved the difficult questions of whether dreams have a purpose or whether there will ever be a way of overcoming sleep altogether – but its tirelessly thorough approach has created a show that, at its best, is fascinating and freakish.

The sound of heavy breathing, loud and slow, comes from Johannes Grebert and Philipp Stölzl’s haunting 2007 10-minute film Sleep Dream, creating an atmosphere that is at once comforting, intimate and threatening, like the black walls of the gallery space. As in the dream landscape of Luis Buñuel’s 1929 surrealist masterpiece Un chien andalou – in which the narrative moves through the rabbit-hole from bedroom to beach, from horror to humour – the show allows the viewer to make lucid but almost unconscious links between exhibits. Goya’s little creatures in one of the “Los Caprichos” etchings seem to terrorise their dreamer to the soundtrack of the “Devil’s Trill”, a sonata by Giuseppe Tartini inspired by the appearance in a dream of the devil playing a violin.

The Wellcome’s mission is to display art works alongside scientific material; indeed, the two become wonderfully, nightmarishly, blurred in this exhibition, so that, in its context, the documentary footage of Corke comes to resemble something out of a Gabriel García Márquez novel. When you are deprived of sleep, we hear in a sound recording of Karl Wilheim Frick, who was kidnapped from West Germany in 1955 and taken to East Germany to be interrogated, “you will do anything you are told. You feel as if you’re in the wrong film.” It is easy to experience real life as fictional narrative in this show – and vice versa. The front cover of a 1923 edition of Science and Invention advertises plans for “The Sleep Eliminator”, a piece of office equipment that uses an electrical current and an enhanced oxygen supply to eliminate the need for sleep. It looks rather useful until you consult your common sense.

Perhaps most successfully, this exhibition uses the distinction between art and science to make conspicuous the mystery surrounding dreams, suggesting how much more there is to learn – or perhaps how much can never be known. Dreams after all cannot be recorded by a machine or a graph, but only by the human voice or hand; Jane Gifford’s 2004 “Dream Paintings” are scientific records and childlike fantasies. Like Keats’s idea of negative capability, the show thrives on man’s capacity to be in “uncertainties, mysteries, doubts without any irritable reaching after fact and reason”.

Indeed, the main irritation in this show is when the hand of the curator tries to reach for an exhaustive approach. Interactive games featuring Trivial Pursuit-style questions have little substance – save the “How Tired Are You?” computerised test, which, if taken once at the beginning and once at the end of a visit, rewards us with the knowledge that the exhibition is, indeed, tiring. Towards the end of the show, cabinets of sedative pills or of alarm clocks, or information about avoiding jetlag, stand like dead-ends in a wonderfully tangled and infinitely sprouting spider-diagram.

During my visit, the cool, dark atmosphere and seemingly endless content proved too much for one man; he simply gave himself up to slumber and settled at the central table with his head resting on a catalogue. But even after a coffee and a brisk walk in the cold air, you will continue to make fresh connections between exhibits in your mind, even overnight.

Until March 9

Tel +44 020 7611 2222