Douglas, have you been able to double check with James' marriage cert. Daughter's birth here?
https://familysearch.org/pal:/MM9.1.1/FQSH-4QPHave you been able to trace any of the other siblings showing as living from mother Ann's 1855 death cert?
I've struggled to find the family in the 1861 census, after Ann's death. Have you been able to find them? James here, (sure you have this, but for background to the post):
James Bryden 39 miner b. Kirkoswald
Mary Bryden 43 Staymaker b. Dundee
Jemima M Bryden 7 b. Edinburgh
Address: 28 Tolbooth Wynd, Leith South
Possibly for James in 1861:
Mathew Park 40 Iron Stone Miner b. Muirkirk, Ayrshire
Isabella Park 36 b. Coylton, Ayrshire
George Park 5 nephew b. Muirkirk, Ayrshire
James Brydon 27 boarder, Iron Stone Miner b. Kirkmichael
Address: 18 Greenhill Road, Dalmellington
Monica