Mr Coulter had suffered addiction issues in recent years after swapping shifts with a colleague who later died as a result of a bomb blast.
Major Investigation Team detectives have now issued a photograph of Mr Coulter, along with a specific appeal for information.
The senior investigating officer in the case, Detective Chief Inspector Raymond Phelan, said: “Just after 12.30 pm on Monday afternoon, October 18, 2021, police received a report that the body of a deceased male had been found at a residential property in the Sandy Braes area.
“Police attended and arrested a man nearby on suspicion of murder who remains in custody.
“I am asking for help from the public to identify sightings and movements of Brian since last Monday, 11th October.
“I am particularly keen to hear from anyone who saw Brian, together with descriptions of anyone who was in his company through to Monday 18th October.
“If you have any information which you think could help, please contact detectives on 101 quoting reference 944 of 18/10/21.”
A report can be made using the online reporting form via: www.psni.police.uk/makeareport
Information can also be provided to Crimestoppers anonymously on 0800 555 111 or online at: www.crimestoppers-uk.org