  We investigate the slow time scales that arise from aging of the paths during
the process of path aggregation. This is studied using Monte-Carlo simulations
of a model aiming to describe the formation of fascicles of axons mediated by
contact axon-axon interactions. The growing axons are represented as
interacting directed random walks in two spatial dimensions. To mimic axonal
turnover, random walkers are injected and whole paths of individual walkers are
removed at specified rates. We identify several distinct time scales that
emerge from the system dynamics and can exceed the average axonal lifetime by
orders of magnitude. In the dynamical steady state, the position-dependent
distribution of fascicle sizes obeys a scaling law. We discuss our findings in
terms of an analytically tractable, effective model of fascicle dynamics.
