Remarks on bootstrap percolation in metric networks

We examine bootstrap percolation in d-dimensional, directed metric graphs in the context of recent measurements of firing dynamics in 2D neuronal cultures. There are two regimes, depending on the graph size N . Large metric graphs are ignited by the occurrence of critical nuclei, which initially occupy an infinitesimal fraction, f∗ → 0, of the graph and then explode throughout a finite fraction. Smaller metric graphs are effectively random in the sense that their ignition requires the initial ignition of a finite, unlocalized fraction of the graph, f∗ > 0. The crossover between the two regimes is at a size N∗ which scales exponentially with the connectivity range λ like N∗ ∼ expλd. The neuronal cultures are finite metric graphs of size N ≃ 105 − 106, which, for the parameters of the experiment, is effectively random since N ≪ N∗. This explains the seeming contradiction in the observed finite f∗ in these cultures. Finally, we discuss the dynamics of the firing front.