Remarks on Bootstrap Percolation in Metric Networks

TL;DR

This paper analyzes bootstrap percolation in high-dimensional metric networks, revealing size-dependent regimes and explaining experimental neuronal culture dynamics through theoretical insights on critical nuclei and ignition thresholds.

Contribution

It introduces a size-dependent framework for bootstrap percolation in metric graphs, connecting theoretical regimes to neuronal culture experiments.

Findings

Large graphs ignite via critical nuclei with infinitesimal initial activation.

Smaller graphs require finite initial activation for ignition.

Crossover size scales exponentially with connectivity range.

Abstract

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 \lambda like_* \sim \exp\lambda^d. The neuronal cultures are finite metric graphs of size N \simeq 10^5-10^6, 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.