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The mechanisms responsible for contention of activity in systems representedby networks are crucial in various phenomena, as in diseases such as epilepsythat affects the neuronal networks, and for information dissemination in socialnetworks. The first models to account for contained activity includedtriggering and inhibition processes, but they cannot be applied to socialnetworks where inhibition is clearly absent. A recent model showed thatcontained activity can be achieved with no need of inhibition processesprovided that the network is subdivided in modules (communities). In thispaper, we introduce a new concept inspired in the Hebbian theory through whichactivity contention is reached by incorporating a dynamics based on a decayingactivity in a random walk mechanism preferential to the node activity. Uponselecting the decay coefficient within a proper range, we observed sustainedactivity in all the networks tested, viz. random, Barabasi-Albert andgeographical networks. The generality of this finding was confirmed by showingthat modularity is no longer needed if the dynamics based on theintegrate-and-fire dynamics incorporated the decay factor. Taken together,these results provide a proof of principle that persistent, restrained networkactivation might occur in the absence of any particular topological structure.This may be the reason why neuronal activity does not outspread to the entireneuronal network, even when no special topological organization exists.

A decaying factor accounts for contained activity in neuronal networks with no need of hierarchical or modular organization