 0925 Tsvi Tlusty, JeanPierre Eckmann
 Remarks on Bootstrap Percolation in Metric Networks
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Feb 19, 09

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Abstract. We examine bootstrap percolation in
ddimensional, 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 10^510^6, 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.
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