### Math Biology Seminar Abstracts

Wednesday November 13, 2002

## A Minimal Network Model for Quadrupedal Locomotion Based on Symmetry and Stabili
ty

Yue-Xian Li

Department of Mathematics, University of British Columbia

Abstract: 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)

For more information contact J. Keener, 1-6089

E-mail:
keener@math.utah.edu