The standard complexity classes such as P, NP are usually defined using Turing Machines. In finite model theory those classes can be defined via the classical first-order/second-order logics.

I am curious whether there exists an equivalent definition of such classes via $\mu$-recursive functions. What about register machines? lambda calculus? etc? In computability theory it is known that $\mu$-recursive functions, counter machines, lambda calculus and Turing Machines are all equivalent. (https://en.wikipedia.org/wiki/Computable_function)