  We study a mechanism of activity sustaining on networks inspired by a
well-known model of neuronal dynamics. Our primary focus is the emergence of
self-sustaining collective activity patterns, where no single node can stay
active by itself, but the activity provided initially is sustained within the
collective of interacting agents. In contrast to existing models of
self-sustaining activity that are caused by (long) loops present in the
network, here we focus on tree--like structures and examine activation
mechanisms that are due to temporal memory of the nodes. This approach is
motivated by applications in social media, where long network loops are rare or
absent. Our results suggest that under a weak behavioral noise, the nodes
robustly split into several clusters, with partial synchronization of nodes
within each cluster. We also study the randomly-weighted version of the models
where the nodes are allowed to change their connection strength (this can model
attention redistribution), and show that it does facilitate the self-sustained
activity.
