Gransha, formerly Londonderry Lunatic Asylum, is the local mental hospital. They will have records of inmates but getting details depends on several things. If you wish to try to obtain any information from them you will have to put your request in writing (have gotten details from Downshire from 1950s by explaining I was interested in family medical details).
Some records (minute books, etc.) are deposited at PRONI, Belfast but patient records should still be at Gransha.