A mathematical model for adaptive vein formation during exploratory migration of Physarum polycephalum: routing while scouting

Exploring free space (scouting) efficiently is a non-trivial task for organisms of limited perception, such as the amoeboid Physarum polycephalum. However, the strategy behind its exploratory behaviour has not yet been characterised. In this organism, as the extension of the frontal part into free space is directly supported by the transport of body mass from behind, the formation of transport channels (routing) plays the main role in that strategy. Here, we study the organism's exploration by letting it expand through a corridor of constant width. When turning at a corner of the corridor, the organism constructed a main transport vein tracing a centre-in-centre line. We argue that this is efficient for mass transport due to its short length, and check this intuition with a new algorithm that can predict the main vein's position from the frontal tip's progression. We then present a numerical model that incorporates reaction-diffusion dynamics for the behaviour of the organism's growth front and current reinforcement dynamics for the formation of the vein network in its wake, as well as interactions between the two. The accuracy of the model is tested against the behaviour of the real organism and the importance of the interaction between growth tip dynamics and vein network development is analysed by studying variants of the model. We conclude by offering a biological interpretation of the well-known current reinforcement rule in the context of the natural exploratory behaviour of Physarum polycephalum.