Non-local transitions and ground state switching in the self-organization of vascular networks

TL;DR

This paper investigates how vascular networks self-organize into tree-like or cyclic structures, revealing a non-local transition between these states driven by fluctuations and bifurcations, with implications for understanding biological transport systems.

Contribution

It uncovers a non-local transition between tree-like and cyclic network states as cost-optimal solutions, expanding understanding of vascular network self-organization under fluctuating demands.

Findings

Cyclic structures emerge under large fluctuations in demand.

A non-local transition exists where tree-like and cyclic solutions exchange roles as ground states.

Noisy dynamics favor cyclic structures even when they are not cost-optimal.

Abstract

The model by Hu and Cai [Phys. Rev. Lett., Vol. 111(13) (2013)] describes the self-organization of vascular networks for transport of fluids from source to sinks. Diameters, and thereby conductances, of vessel segments evolve so as to minimize a cost functional E. The cost is the trade-off between the power required for pumping the fluid and the energy consumption for vessel maintenance. The model has been used to show emergence of cyclic structures in the presence of locally fluctuating demand, i.e. non-constant net flow at sink nodes. Under rapid and sufficiently large fluctuations, the dynamics exhibits bistability of tree-like and cyclic network structures. We compare these solutions in terms of the cost functional E. Close to the saddle-node bifurcation giving rise to the cyclic solutions, we find a parameter regime where the tree-like solution rather than the cyclic solution is cost-optimal. Thus we discover an additional, non-local transition where tree-like and cyclic solutions exchange their roles as minimum-cost (or ground) states. The findings hold both in a small system of one source and a few sinks and in an empirical vascular network with hundreds of sinks. In the small system, we further analyze the case of slower fluctuations, i.e., on the same time scale as network adaptation. We find that the noisy dynamics settles around the cyclic structures even when these structures are not cost-optimal.