A Minimal Network Model for Quadrapedal Locomotion based on Symmetry and Stability

Four-legged animals move with several distinct patterns of rhythmic leg movements, called gaits. Standard quadrapedal gaits include walk, pace, trot, bound, and gallop. Networks of coupled oscillators have been used to model the central pattern generators (CPGs) that produce these patterns. In these models, symmetric gaits are related to phase-locked states of the network possessing the same symmetries. Pioneer works by Golubitsky et al were based on symmetry analysis that gave conditions for the existence of these states. We show that models based on symmetry alone cannot generate a model circuit of practical use, i.e., a circuit people can actually install in a four-legged robot capable of moving with different gaits. A functioning network should possess not only enough symmetry to guarantee the existence of these solutions but a mechanism to segregate each one of them dynamically. Our new theory, based on the analysis of both the existence and stability of these phase-locked states, allows us to achieve both goals. We show that a minimal network of four identical neurons is capable of generating dynamically independent patterns for all standard quadrapedal gaits. A circuit is designed based on this theory using a realistic neuronal model and synaptic currents. Numerical simulations of this model circuit confirmed the analytical results.

(Others involved in part of this work: Drs. Yuqing Wang and Robert Miura)

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).