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Topological determinants of self-sustained activity in a simple model of excitable dynamics on graphs

Models of simple excitable dynamics on graphs are an efficient framework for studying the interplay between network topology and dynamics. This subject is a topic of practical relevance to diverse fields, ranging from neuroscience to engineering. Here we analyze how a single excitation propagates through a random network as a function of the excitation threshold, that is, the relative amount of activity in the neighborhood required for an excitation of a node. Using numerical simulations and analytical considerations, we can understand the onset of sustained activity as an interplay between topological cycle statistics and path statistics. Our findings are interpreted in the context of the theory of network reverberations in neural systems, which is a question of long-standing interest in computational neuroscience.