It depends. It could be helpful to have them review the manuscript before you submit it. Or indeed if might be common in your field to invite the supervisor to coauthor. If neither of those things are relevant then you could go ahead and do it yourself. It is definitely sufficient to say it was your PhD work in the bio. You might want to acknowledge supervisors by name if they gave helpful feedback on the work.
But if it is the norm in your field/area/old department to coauthor with the supervisor, or if you feel they did contribute to producing that piece of work, then I think it would be good etiquette to contact them and let them know you are planning to submit it for publication.
Even if it's normal in your area to be the sole author on PhD work (I know it can be), it might still be beneficial to contact them and ask if they would comment on a draft or have suggestions (e.g., you are at an earlier stage and wondering which findings might make a good paper and could do with support). But that would be if you felt it was helpful.