Ultime periodicity in chip-firing games

One can study variations of this discrete dynamical system : applying the rule sequentially (on exactly one vertex at each time step), the restriction to undirected graphs or other families of graphs, the extension to multi-graphs, etc. Such models are called chip-firing games (CFG). Some original definitions where given in [19] and [1], but the two references to this respect are [4] for the undirected case, and [5] for the directed case. On a finite graph with a finite number of chips, the trajectory from any configuration is ultimately periodic, whatever the update scheme is (parallel or sequential).