Mathematics Colloquium
3:00 p.m.
Math Annex 1100
Yue Xian Li
UBC
A Minimal Network Model for Quadrapedal Locomotion based on Symmetry and Stability
Fourlegged 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 phaselocked 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 fourlegged 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
phaselocked 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).
