FO-uniform AC0 with some predicate
I am not completely sure what you are looking for, but the following might 
be interesting to you:


The idea that restricting numerical predicates in FO-formula corresponds to uniformity conditions is explicitly investigated, for example, in the paper
"FO(&lt;)-uniformity" by Behle and Lange.
The survey "Arithmetic, first-order logic, and counting quantifiers"
by Schweikardt provides i.a. an overview of known results about
the expressive power of different arithmetical predicates