title: Role of long cycles in excitable dynamics on graphs.
abstract: Topological cycles in excitable networks can play an important role in maintaining the network activity. When properly activated, cycles act as dynamic pacemakers, sustaining the activity of the whole network. Most previous research has focused on the contributions of short cycles to network dynamics. Here, we identify the specific cycles that are used during different runs of activation in sparse random graphs, as a basis of characterizing the contribution of cycles of any length. Both simulation and a refined mean-field approach evidence a decrease in the cycle usage when the cycle length increases, reflecting a trade-off between long time for recovery after excitation and low vulnerability to out-of-phase external excitations. In spite of this statistical observation, we find that the successful usage of long cycles, though rare, has important functional consequences for sustaining network activity: The average cycle length is the main feature of the cycle length distribution that affects the average lifetime of activity in the network. Particularly, use of long, rather than short, cycles correlates with higher lifetime, and cutting shortcuts in long cycles tends to increase the average lifetime of the activity. Our findings, thus, emphasize the essential, previously underrated role of long cycles in sustaining network activity. On a more general level, the findings underline the importance of network topology, particularly cycle structure, for self-sustained network dynamics.
authors.affiliation: School of Engineering and Science, Jacobs University Bremen, D-28759 Bremen, Germany.
authors.forename: Guadalupe C
authors.initials: GC
authors.lastname: Garcia
authors.affiliation: LPTMC, CNRS UMR 7600, Université Pierre et Marie Curie, Sorbonne Universités, 4 place Jussieu, F-75252, Paris, France and IGMM, CNRS UMR 5535, Université de Montpellier, 1919 route de Mende, F-34293, Montpellier, France.
authors.forename: Annick
authors.initials: A
authors.lastname: Lesne
authors.affiliation: Department of Computational Neuroscience, University Medical Center Hamburg-Eppendorf, Hamburg University, D-20148 Hamburg, Germany and Department of Health Sciences, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: Claus C
authors.initials: CC
authors.lastname: Hilgetag
authors.affiliation: School of Engineering and Science, Jacobs University Bremen, D-28759 Bremen, Germany.
authors.forename: Marc-Thorsten
authors.initials: MT
authors.lastname: Hütt