Separation of  space complexity classes: differeces between uniform class and nonuniform one as an analogy of circuit lower bounds project
Perhaps you'd be interested in switching networks. According to Potechin's Bounds on monotone switching networks for directed connectivity, one way to separate L from NL is to show that there is no polysize switching network for directed connectivity. There is in fact a (trivial) polysize switching-and-rectifier network for directed connectivity. The difference between the two models is that the former is undirected and the latter directed.

Potechin has been working on separating L and NL, but so far his method only works for monotone networks. See also his STOC 2012 paper with Siu Man Chan. (More papers are on the way.)