We already do (in a multitude of ways) when we sleep. There's research that indicates that sleep is used to clear away neurotoxic waste.[0][1] The role of deposits of tau and beta-amyloid proteins in Alzheimer's[2] and aging are also of particular interest. Other papers supporting the synaptic homeostasis hypothesis suggest that sleep is when synaptic strength is renormalized.[3][4] Basically, memories are reinforced (the signal) in relation to existing knowledge while daily noise is effectively pruned (among other processes).
The analogy is flawed, but the brain already has some pretty intricate garbage control methods. As we learn more about them, it'll open up new paths for clinical treatments for a host of problems.