Bootstrap Percolation with Inhibition

Hafsteinn Einarsson, Johannes Lengler, Konstantinos Panagiotou, Frank Mousset, Angelika Steger

Published 2014-10-13Version 1

A recurring phenomenon in bootstrap percolation theory is a type of `all-or-nothing' phenomenon: either the size of the starting set is so small that the process dies out almost immediately, or the process percolates (almost) completely. For many processes in physics and material sciences such a phenomenon is consistent with observations. Not so, in neurobiology. There external input (for example from sensory nerves) leads to a small initial activity that is then boosted by local connectivity. Nevertheless, the total activity never surpasses a certain level. Such a phenomenon is known as input normalization and is an important building block in many neuronal systems, including the human brain. In this work we provide a model of bootstrap percolation that does exhibit such a phenomenon: from a tiny starting level we percolate to a much larger level of activity that nevertheless is very far from complete percolation. To reach this goal we introduce and analyze bootstrap percolation on an Erdos-Renyi random graph model under the assumption that some vertices are inhibitory, i.e., they hinder percolation. We show that the standard, round-based percolation model does not show any stable normalizing effects. In contrast, if we move to a continuous time model in which every edge draws its transmission time randomly, then normalization is an automatic and intrinsic property of the process. Moreover, we find that random edge delays accelerate percolation dramatically, regardless of whether inhibition is present or not.