Remarks on bootstrap percolation in metric networks
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, flowast → 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, flowast > 0. The crossover between the two regimes is at a size Nlowast which scales exponentially with the connectivity range λ like Nlowast ~ exp λd. The neuronal cultures are finite metric graphs of size N sime 105 - 106, which, for the parameters of the experiment, is effectively random since N Lt Nlowast. This explains the seeming contradiction in the observed finite flowast in these cultures. Finally, we discuss the dynamics of the firing front.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- May 2009
- DOI:
- 10.1088/1751-8113/42/20/205004
- arXiv:
- arXiv:0902.3384
- Bibcode:
- 2009JPhA...42t5004T
- Keywords:
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- Condensed Matter - Statistical Mechanics;
- Physics - Biological Physics;
- Quantitative Biology - Neurons and Cognition
- E-Print:
- T Tlusty and J-P Eckmann 2009 J. Phys. A: Math. Theor. 42 205004