Wavy synchronization in locomotion of train millipedes
The sophisticated control of many legs by millipedes and centipedes establish well-organized temporal structures such as wavy synchronization in their gate patterns. Quantification and mathematical modeling of such temporal patterns can contribute to the understandings of behavioral mechanisms in biological locomotion. In this paper, we investigated the temporal pattern and its mechanism in the locomotion of small millipedes, train millipedes (Parafontaria laminata armigera), by combining empirical data with a system of coupled phase oscillators. First, we performed behavioral experiments by using actual millipedes and characterized their locomotion patterns based on an order parameter capable of detecting wavy synchronization with a specific wavenumber k. Second, we proposed a phase oscillator model incorporating two phase shift parameters and estimated their suitable values by comparing numerical simulations of the model with the empirically derived wavenumber through the proposed order parameter. Consequently, we established a concise mathematical model reproducing not only the wavy synchronization of the train millipedes but also the other types of wavy synchronization. From the biological points of views, our result suggests that the asymmetric interaction between neighboring legs is a key factor in the sophisticated motion control of the millipedes, enabling them to maintain a specific wavenumber during forward walking.