Many complexity classes defined with Turing machines have definitions in terms of uniform circuits. For example, P can also be defined using uniform polynomial size circuits, and similarly BPP, NP, BQP, etc. can be defined with uniform circuits.
So is there a circuit-based definition of L?
An obvious idea would be to allow polynomial size circuits with some depth limitation, but this turn out to define the NC hierarchy.
I was thinking about this question a long time ago, but didn't find an answer. If I remember correctly, my motivation was to understand what the quantum analog of L would look like.