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We study propagation of avalanches in a certain excitable network. The modelis a particular case of the one introduced in [23], and is mathematicallyequivalent to an endemic variation of the Reed-Frost epidemic model introducedin [27]. Two types of heuristic approximation are frequently used for models ofthis type in applications, a branching process for avalanches of a small sizeat the beginning of the process and a deterministic dynamical system once theavalanche spreads to a significant fraction of a large network. In this paperwe prove several results concerning the exact relation between the avalanchemodel and these limits, including rates of convergence and rigorous bounds forcommon characteristics of the model.

Avalanches in an excitable network