Interplay between short and long time scales in adapting, periodically driven elastic flow networks

Existing theories of structural adaptation in biological flow networks are largely concerned with steady flows. However, biological networks are composed of elastic vessels, and many are driven by a pulsatile or periodic source, leading to spatiotemporal variations in the pressure and flow fields on short time-scales within each vessel. Here, we investigate the mathematical problem of how long-term adaptation in elastic networks is impacted by short-term pulsatile dynamics at the level of individual vessels. Using a a minimal one-loop network, we show that pulsatility gives rise to resonances that stabilize the loop for a much broader range of metabolic cost functions than predicted by existing theories. Our paper emphasizes the importance of correctly capturing the interplay of the short and long time-scales for a more realistic treatment of adaptation in periodically driven elastic flow networks.