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

The model by D. Hu and D. Cai [Phys. Rev. Lett. 111, 138701 (2013). doi:10.1103/PhysRevLett.111.138701] 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 the 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.