Central pattern generators for bipedal locomotion

Abstract

We use symmetry to study two central pattern generator (CPG) models for biped locomotion. The first one is a coupled four-cell network, proposed by Golubitsky, Stewart, Buono, and Collins, that models rhythms associated to legs. A classification based on symmetry shows that this network can produce periodic solutions with rhythms corresponding to the standard bipedal gaits of run, walk, hop, gallop, and skip, among others. Moreover, the four-cell model can produce two types of hop, two types of gallop, and three additional symmetry types of periodic solutions that have yet to be identified with the rhythms of known bipedal gaits. The second locomotor CPG network models interlimb coordination in bipeds (arms+legs). It is obtained by breaking the symmetry between fore and hind legs in an eight-cell CPG network for quadruped gaits, also proposed by Golubitsky et al. We match the rhythms of perturbed periodic solutions found in this eight-cell network with legs rhythms produced by the four-cell CPG model. We also compare patterns of oscillation of gaits of the eight-cell model with results on bipedal interlimb coordination in the literature, showing that the eight-cell model is a plausible network for modeling human interlimb coordination.We show numerical simulations of periodic solutions corresponding to the bipedal gaits in the two CPG models. These simulations use clamped Hodgkin-Huxley equations to model cell internal dynamics and partial linear coupling (where only the electrical potentials of different cells are coupled). We use synaptic coupling in the four-cell model and diffusive coupling in the eight-cell model.

Symmetry and order parameter dynamics of the human odometer

“…For experimental distances of 7.6, 15.2, and 22.9 m, Turvey et al (2009) confirmed Schwartz's (1999 finding that walk-walk and run-walk were equivalent M − R conditions (see also Isenhower et al 2012), but showed that it was a special case. They did so by manipulating the symmetry classes of bipedal gaits identified by Pinto and Golubitsky (2006).…”

“…1). Then, the bipedal network's symmetry group consists of transpositions ρ, τ , and ρτ that swap (or transpose) pairs (12) and (34), pairs (13) and (24), and pairs (14) and (23), respectively (Pinto 2007;Pinto and Golubitsky 2006).…”

“…A primary bipedal gait (walk, run, slow two-footed hop, fast two-footed hop) is one for which the spatio-temporal symmetries satisfy the dihedral group D 2 (see Pinto and Golubitsky 2006;Pinto 2007;Turvey et al 2009Turvey et al , 2012. All three transpositions are entailed.…”

“…All three transpositions are entailed. A secondary bipedal gait (skip, gallop-run, gallop-walk, hesitation-walk) is one for which the spatio-temporal symmetries approximate the cyclic group Z 2 (see Pinto and Golubitsky 2006;Pinto Turvey et al (2012) positive sign when R gait is D 2 and of negative sign when R gait is Z 2 ). In what follows, we identify a dynamical system befitting the foregoing characterization of human odometry and we report an experiment by which it can be evaluated.…”

“…Figure 4 is a schematic of the preceding and present research (identified by italics) on the human odometer. Group symmetry analysis of bipedal gait (Pinto and Golubitsky 2006) and principles of self-organizing systems, as expressed in order parameter dynamics (e.g., Frank 2005;Haken 1977Haken , 1983Kelso 1995), provide the general concepts and theoretical underpinning for the experimentation and modeling. The preceding research investigated odometry overground (precisely, the floor of a gymnasium).…”