The Amsterdam page of www.barrow-lousada.org leads into a discussion of the early Baruch Louzadas there and to a special-purpose chart showing the early family once it left Livorno.
In accordance with one Jewish naming custom, we suggest Isaac's father was Abraham who led the family back into Judaism in Livorno. Isaac probably had brothers, one of whom may have been the Moses who died in London in 1677, whilst perhaps another brother stayed in Livorno.
birth suggested by James Greener