Optimization dynamics and fluctuations in the self-organization of vascular networks

Konstantin Klemm, Erik Andreas Martens
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Abstract

The model by Hu and Cai [Phys. Rev. Lett., Vol. 111(13) (2013)1 ] 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. Further increase of fluctuation amplitude then leads to an additional transition at which the cyclic solution becomes optimal. The findings hold both in a small system of one source and two 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.
血管网络自组织中的优化动力学和波动性
Hu 和 Cai [Phys. Rev. Lett.该成本是泵送流体所需的功率与血管维护所消耗的能量之间的权衡。该模型已被用于显示局部波动需求(即下沉节点处的非恒定净流量)情况下出现的循环结构。在快速和足够大的波动条件下,动力学表现出树状和循环网络结构的稳定性。在接近产生循环解的鞍节点分叉处,我们发现了一个参数体系,在该体系中,树状解而不是循环解是成本最优的。这些发现在由一个源和两个汇组成的小型系统和由数百个汇组成的经验血管网络中都成立。在小型系统中,我们进一步分析了波动较慢的情况,即与网络适应的时间尺度相同。我们发现,即使循环结构不是成本最优的,噪声动态也会在这些结构周围稳定下来。
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