Phase transition in the economically modeled growth of a cellular nervous system

Abstract

Spatially-embedded complex networks, such as nervous systems, the Internet and transportation networks, generally have non-trivial topological patterns of connections combined with nearly minimal wiring costs. However the growth rules shaping these economical trade-offs between cost and topology are not well understood. Here we study the cellular nervous system of the nematode worm C. 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 modelling to show that the transition between different growth regimes, as well as its coincidence with the moment of hatching, can be explained by a dynamic economical model which incorporates a trade-off between topology and cost that is continuously negotiated and re-negotiated over developmental time. As the body of the animal progressively elongates, the cost of longer distance connections is increasingly penalised. This growth process regenerates many aspects of the adult nervous system's organization, including the neuronal membership of anatomically pre-defined ganglia. We expect that similar economical principles can be found in the development of other biological or man-made spatially-embedded complex systems.