By the end of planning her new exhibition, Caroline Fisher had come to an interesting conclusion. "It's even harder to exhibit rats than drugs," she says. The Home Office eventually granted her the necessary licences to exhibit a bottle of heroin, a ball of opium, some morphine, a selection of magic mushrooms, a peyote cactus, some hallucinogenic snuff and a variety of Victorian high-street pharmacy favourites including cocaine mouth lozenges and tincture of Indian cannabis – "as many drugs as we could get our hands on". But Health and Safety weren't having the rats. "We wanted to recreate a 7m-long Rat Park," Fisher sighs, referring to the classic 1970s Canadian experiment that showed opiate addiction in rodents was determined not by the drugs they took, but the living conditions they took them in.

Fisher is the co-curator of High Society: Mind-Altering Drugs in History and Culture at the Wellcome Collection in London, and offers a history of narcotics that feels fresh. After all, we hardly need another account of the Romantic poets getting carried away with hashish, or more woolly recollections from acid house revellers who outwitted the police on the M25 while going to Sunrise.

"I don't think anything similar has been done before," says Mike Jay, the exhibition's co-curator and author of an accompanying book. "There's always been two different discourses, the 'drug culture underground' one and a rather more straight-lens way of looking at it, from a medical or political view. It's the middle ground that feels interesting."

High Society strives to cover as much of this middle ground as possible. It spans from pre-2000 BC chillum-style pipes fashioned from puma bones, to mephedrone and other internet-distributed synthetic stimulants of the 21st century. Along the way it takes in kava drinking in the South Pacific, betel chewing in Papua New Guinea and cocaine snorting in Weimar Germany. Tea, coffee and sugar also feature (albeit in supporting roles) and there's plenty on the rise and fall of tobacco.

As such the exhibition is able to make its central premise: very few people live their lives without resorting to some sort of mind-altering substance. Taking drugs, it suggests, is "a universal impulse". "Drug cultures are endlessly varied, but drugs in general are more or less ubiquitous among our species," writes Jay. Later he quotes American anthropologist Donald Brown's celebrated work Human Universals, which lists "mood- or consciousness-altering techniques and/or substances" as one of the essential components of human culture, along with "music, conflict resolution, language and play". "The public perception is that drugs are this terrible thing that appeared with hippies in the 60s; that they're a modern disease," Jay says. "The historicality has been lost."

The curators are at pains to underline the mutability of culture and society, and how a drug's definition is determined by non-chemical factors such as intent behind its use, its method of administration and the social class of the user. (Nitrous oxide is a medicine when used by doctors, a drug when used for pleasure.) Even so a pattern soon establishes itself: a new mind-altering substance arrives accompanied by extravagant medical claims and counter-claims, gets enthusiastically taken up by sections of the public (usually the idle rich); then addiction and side-effects make themselves apparent over time.

"It was hard to designate drugs themselves as the problem when they were also being promoted to the public at large as the solution," writes Jay of the nurses, doctors and military officers who were treating local infections with morphine injections in the 1880s, ushering in the first "morphinomaniacs" in the process. Elsewhere the 18th-century botanist and pioneering drug cataloguer Carl Linnaeus frowned upon coffee – he felt it sapped vitality and brought on early senility – but endorsed tobacco as a means of fighting infection. In a tract published in Leipzig in 1707, we see early adopters of tea being reprimanded for "drinking themselves to death" in the mindless pursuit of fashion. Around the same time the British literary intelligentsia waxed lyrical on the benefits of rounding an evening off with a few pipes of opium, something they believed helped digestion, fortified against fever and improved performance in the bedroom. Only alcohol seems to have maintained a constant reputation, viewed as the boorish vice of the corrupt elite in Roman times, banned across much of the Islamic world and the subject of US prohibition in the 1920s.

Still, High Society remains morally neutral. There won't be any disclaimers. "We're not doing, 'Hey kids, drugs are good', so ultimately we don't need to do, 'Hey kids, drugs are bad,'" reasons Jay. "Since that's basically the entire popular discourse about drugs, it seems nice to get rid of both of them and take the subject on its own merit."

High Society has commissioned some interactive artworks to help convey the quixotic effects of drugs on mind and body in the sober medium of an exhibition space. Joshua White was the resident artist at New York's Fillmore East theatre during the late 60s. Using bottles of coloured liquids, hand-painted slides, lightbulbs on the end of sticks and clock faces, he projected his psychedelic "liquid light shows" on to live performances by Frank Zappa, Janis Joplin and Jefferson Airplane, among others. "Was my work best experienced on drugs? I would say so, yes," says White, who'll travel to the UK to install his new show at the Wellcome Collection. "Everybody had a different relationship with drugs back then, just as everybody in my parents' generation had a different relationship with alcohol. Some people had a nice buzz; some people threw up. We would hire speed freaks for our special projects – get them to stay up all night gluing jewels on to a ball."

There will also be a recreation of the "dreamachine", the light-emitting cylinder built by artist Brion Gysin and William Burroughs's "systems adviser" Ian Sommerville. "You look at it with your eyes shut in a dark room, and it supposedly recreates the hallucinatory experience," explains Fisher.

Other contemporary artwork includes the video piece Cannabis In the UK, of artist Mark Harris reading Baudelaire's Les Paradis Artificiels and Walter Benjamin's Hashish in Marseilles to cannabis plants ("I hope it won't be taken too seriously," says Harris. "I just thought, 'If you're going to read to plants to make them grow, what better than to read to cannabis plants something about the effects of the drug?'"), and photographer Mark Leffingwell's "collective intoxication" picture depicting 10,000 people gathered at the University of Colorado for a "smoke-in" to commemorate "420", an event observed across America every 20 April to promote the legalisation of marijuana.

If none of those do the trick, there are plenty of accounts from the history of self-experimentation. There's the study on nitrous oxide performed by 18th-century chemist Humphry Davy, who got fed up with testing the gas on rabbits, kittens and fish and took heroic quantities himself, reaching the less than empirical conclusion that "nothing exists but thoughts". There's the story of the family who discovered the liberty cap mushroom by accident: cooking some up for a morning broth they developed vertigo, visions and the overwhelming sensation they were dying, only to leave the house for help and forget why they had done so a few hundred metres later. (When a doctor did eventually reach them, the situation was scarcely improved by the family's eight-year-old, whose symptoms proved unique: bursting into raucous laughter every time his terrified parents opened their mouths.) And there's French psychiatrist Jacques-Joseph Moreau, who suggested that the low prevalence of insanity in the Arab world was down to a preference for cannabis over alcohol: testing his theory he swallowed three grams before dinner and found himself preparing to fight a duel with a bowl of candied fruit.

From more recent times there's a photograph of "father of MDMA" and sometime US Drug Enforcement Agency employee Alexander Shulgin. Shulgin's popularisation of ecstasy eventually gave rise to acid house, the last significant drug-led subculture. High Society largely steers clear of examining the hows and whys of such moments; in fact there's little on why we might be drawn towards illicit drugs in the first place. "I just think it's self-evident that people wouldn't take drugs if they didn't enjoy them," Jay shrugs.

The most recent UN figures put the illegal drug trade at $320bn (£200bn) a year – the third biggest international market on the planet, after arms and oil. "2011 is the 50th anniversary of the United Nations Single Convention on Narcotic Drugs," Jay says. "That's the 50th anniversary of global prohibition; they've been trying for 50 years to achieve that. What's so ironic is that 1961 was precisely the time when the drug counterculture formed; the point where policing started to fall apart with the surge in demand that was coming. Today our culture has become even more experimental: we regard it as a good thing to try something exotic and different, in a way that it just wasn't 50 years ago. So it's very hard to say, 'That's the way we are in culture. Oh – except for drugs, which have to be hived off.'"

Given that more people take more drugs than at any other time in history, you might wonder if they'll ever be part of a counterculture again. At a time when Keith Richards is a bestselling author off the back of his national treasure status as a chemical dustbin, Governor Arnold Schwarzenegger has taken steps to decriminalise marijuana possession in California and Prince Harry is found inhaling "hippy crack", it's difficult to see how drugs could be more mainstream. "I wouldn't be surprised if in five years, marijuana wasn't fully legalised all over the US," says Leffingwell. "Most people don't see it as any more harmful than having a beer."

Others suggest that the seeds of a new, drug-led counterculture are all around us. "I think smart drugs, things that boost your IQ such as Modafinil, could lend themselves to certain music," says Jay. "Very techy electronica."

To return to High Society's premise, then: the drugs we consume may change – from over-the-counter laudanum in Victorian times, to over-the-internet mephedrone today – but the human relationship with them remains strangely constant. "Nothing's changed," says White. "The form changes, the fickleness changes – but our cravings stay the same."

High Society: Mind-Altering Drugs in History and Culture is at the Wellcome Collection, 183 Euston Road, London NW1 2BE from 11 Nov to 27 Feb. wellcomecollection.org