School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
Proc Natl Acad Sci U S A. 2013 May 7;110(19):7880-5. doi: 10.1073/pnas.1300753110. Epub 2013 Apr 22.
Spatially embedded complex networks, such as nervous systems, the Internet, and transportation networks, generally have nontrivial topological patterns of connections combined with nearly minimal wiring costs. However, the growth rules shaping these economical tradeoffs between cost and topology are not well understood. Here, we study the cellular nervous system of the nematode worm Caenorhabditis elegans, together with information on the birth times of neurons and on their spatial locations. We find that the growth of this network undergoes a transition from an accelerated to a constant increase in the number of links (synaptic connections) as a function of the number of nodes (neurons). The time of this phase transition coincides closely with the observed moment of hatching, when development switches metamorphically from oval to larval stages. We use graph analysis and generative modeling to show that the transition between different growth regimes, as well as its coincidence with the moment of hatching, may be explained by a dynamic economical model that incorporates a tradeoff between topology and cost that is continuously negotiated and renegotiated over developmental time. As the body of the animal progressively elongates, the cost of longer-distance connections is increasingly penalized. This growth process regenerates many aspects of the adult nervous system's organization, including the neuronal membership of anatomically predefined ganglia. We expect that similar economical principles may be found in the development of other biological or manmade spatially embedded complex systems.
空间嵌入的复杂网络，如神经系统、互联网和交通网络，通常具有复杂但非平凡的拓扑连接模式，同时布线成本也接近最低。然而，塑造这些成本与拓扑之间经济权衡的生长规则还没有得到很好的理解。在这里，我们研究了线虫秀丽隐杆线虫的细胞神经系统，以及神经元的出生时间及其空间位置的信息。我们发现，随着节点（神经元）数量的增加，网络的生长经历了一个从加速增长到恒定增长的转变。这个相变的时间与观察到的孵化时刻非常吻合，此时发育从椭圆形到幼虫阶段发生了变态。我们使用图分析和生成模型表明，不同生长阶段之间的转变，以及它与孵化时刻的吻合，可能可以用一个动态经济模型来解释，该模型将拓扑和成本之间的权衡进行了连续的协商和重新协商，这个过程贯穿了整个发育时间。随着动物身体的逐渐延长，远距离连接的成本会受到越来越多的惩罚。这个生长过程再生了成年神经系统组织的许多方面，包括解剖上预先定义的神经节的神经元成员。我们预计，类似的经济原则可能存在于其他生物或人为的空间嵌入复杂系统的发展中。
