Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks

Transport networks are crucial for the functioning of natural and technological systems. We study a mathematical model of vascular network adaptation, where the network structure dynamically adjusts to changes in blood flow and pressure. The model is based on local feedback mechanisms that occur on different time scales in the mammalian vasculature. The cost exponent $\gamma$ tunes the vessel growth in the adaptation rule, and we test the hypothesis that the cost exponent is ...

Distribution networks -- from vasculature to urban transportation systems -- are prevalent in both the natural and consumer worlds. These systems are intrinsically physical in composition and are embedded into real space, properties that lead to constraints on their topological organization. In this study, we compare and contrast two types of biological distribution networks: mycelial fungi and the vasculature system on the surface of rodent brains. Both systems are alike in ...

Transport networks are crucial to the functioning of natural and technological systems. Nature features transport networks that are adaptive over a vast range of parameters, thus providing an impressive level of robustness in supply. Theoretical and experimental studies have found that real-world transport networks exhibit both tree-like motifs and cycles. When the network is subject to load fluctuations, the presence of cyclic motifs may help to reduce flow fluctuations and,...

Physarum polycephalum is an acellular slime mould that grows as a highly adaptive network of veins filled with protoplasm. As it forages, Physarum dynamically rearranges its network structure as a response to local stimuli information, optimising the connection between food sources. This high-level behaviour was already exploited to solve numerous optimisation problems. We develop a flow-based model for the adaptive network formation of Physarum, which solves some inconsisten...

A plethora of computational models have been developed in recent decades to account for the morphogenesis of complex biological fluid networks, such as capillary beds. Contemporary adaptation models are based on optimization schemes where networks react and adapt toward given flow patterns. Doing so, a system reduces dissipation and network volume, thereby altering its final form. Yet, recent numeric studies on network morphogenesis, incorporating uptake of metabolites by the...

The structure of networks that provide optimal transport properties has been investigated in a variety of contexts. While many different formulations of this problem have been considered, it is recurrently found that optimal networks are trees. It is shown here that this result is contingent on the assumption of a stationary flow through the network. When time variations or fluctuations are allowed for, a different class of optimal structures is found, which share the hierarc...

The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organisation of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialised membrane-localised proteins. Many venation models have been based on polarly localised efflux-mediator proteins of the PIN family. Here, we investigate a modeling framework for auxin transport with a positive feedbac...

The vascular network of leaves, comprising xylem and phloem, is a highly optimized system for the delivery of water, nutrients, and sugars. The design rules for these naturally occurring networks have been studied since the time of Leonardo da Vinci, who constructed a local rule for comparing the widths of in- and outgoing veins at branch points. Recently, physical models have been developed that seek to explain the full morphogenesis of leaf venial networks in which veins gr...

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three...

Recent work on self-organized remodeling of vasculature in slime-mold, leaf venation systems and vessel systems in vertebrates has put forward a plethora of potential adaptation mechanisms. All these share the underlying hypothesis of a flow-driven machinery, meant to alter rudimentary vessel networks in order to optimize the system's dissipation, flow uniformity, or more, with different versions of constraints. Nevertheless, the influence of environmental factors on the long...