Robust phase-waves in chains of half-center oscillators.

Many neuronal circuits driving coordinated locomotion are composed of chains of half-center oscillators (HCOs) of various lengths. The HCO is a common motif in central pattern generating circuits (CPGs); an HCO consists of two neurons, or two neuronal populations, connected by reciprocal inhibition. To maintain appropriate motor coordination for effective locomotion over a broad range of frequencies, chains of CPGs must produce approximately constant phase-differences in a robust manner. In this article, we study phase-locking in chains of nearest-neighbor coupled HCOs and examine how the circuit architecture can promote phase-constancy, i.e., inter-HCO phase-differences that are frequency-invariant. We use two models with different levels of abstraction: (1) a conductance-based model in which each neuron is modeled by the Morris-Lecar equations (the ML-HCO model); and (2) a coupled phase model in which the state of each HCO is captured by its phase (the phase-HCO model). We show that one of four phase-waves with inter-HCO phase-differences at approximately 0, 25, 50 or 75 % arises robustly as a result of the inter-HCO connection topology, and its robust existence is not affected by the number of HCOs in the chain, the difference in strength between the ascending and descending nearest-neighbor connections, or the number of nearest-neighbor connections. Our results show that the internal anti-phase structure of the HCO and an appropriate inter-HCO connection topology together can provide a mechanism for robust (i.e., frequency-independent) limb coordination in segmented animals, such as the 50 % interlimb phase-differences in the tripod gate of stick insects and cockroaches, and the 25 % interlimb phase-differences in crayfish and other long-tailed crustaceans during forward swimming.