A modular network for legged locomotion

Physica D: Nonlinear Phenomena

Volumn 115, Issue 1-2, 1998, Pages 56-72

  Golubitsky, Martin ^a   Stewart, Ian ^b   Buono, Pietro Luciano ^a   Collins, J J ^c  

^a University of Houston   (United States)

^b UNIVERSITY OF WARWICK   (United Kingdom)

^c Boston University   (United States)

Author keywords

Control pattern generator; Gaits; Hopf bifurcation; Symmetry

Indexed keywords

EID: 0002659841     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(97)00222-4     Document Type: Article

References (32)

1. R. McN. Alexander, Terrestrial locomotion, in: R. McN. Alexander, J.M. Goldspink (Eds.), Mechanics and Energetics of Animal Locomotion, Chapman and Hall, London, 1977, pp. 168-203.
2. A.H. Cohen, P.J. Holmes, R.H. Rand, The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: a mathematical model, J. Math. Biol. 13 (1982) 345-369.
3. J.J. Collins, C.C. Chow, A.C. Capela, T.T. Imhoff, Aperiodic stochastic resonance, Phys. Rev. E 54 (1996) 5575-5584.
4. J.J. Collins, S.A. Richmond, Hard-wired central pattern generators for quadrupedal locomotion, Biol. Cybern. 71 (1994) 375-385.
5. J.J. Collins, I. Stewart, Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion, J. Math. Biol. 30 (1992) 827-838.
6. J.J. Collins, I. Stewart, Coupled nonlinear oscillators and the symmetries of animal gaits, J. Nonlin. Sci. 3 (1993) 349-392.
7. J.J. Collins, I. Stewart, Hexapodal gaits and coupled nonlinear oscillator models, Biol. Cybern. 68 (1993) 287-298.
8. J.J. Collins, I. Stewart, A group-theoretic approach to rings of coupled biological oscillators, Biol. Cybern. 71 (1994) 95-103.
9. B. Dionne, M. Golubitsky, I. Stewart, Coupled cells with internal symmetry. Part I: wreath products, Nonlinearity 9 (1996) 559-574.
10. B. Dionne, M. Golubitsky, I. Stewart, Coupled cells with internal symmetry, Part II: direct products, Nonlinearity 9 (1996) 575-599.
11. I.R. Epstein, M. Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos 3 (1) (1993) 1-5.
12. G.B. Ermentrout, N. Kopell, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol. 29 (1991) 195-217.
13. P.P. Gambaryan, How Mammals Run: Anatomical Adaptations, Wiley, New York, 1974.
14. M. Golubitsky, I.N. Stewart, Hopf bifurcation in the presence of symmetry, Arch. Rational Mech. Anal. 87 (2) (1985) 107-165.
15. M. Golubitsky, I.N. Stewart, Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, in: M. Golubitsky and J. Guckenheimer (Eds.), Multiparameter Bifurcation Theory, Providence, RI, 1986, Contemporary Mathematics 56, 1986, pp. 131-173.
16. M. Golubitsky, I.N. Stewart, D.G. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol II, Applied Mathematical Sciences, vol. 69, Springer, New York, 1988.
17. S. Grillner, Locomotion in vertebrates: central mechanisms and reflex interaction, Physiol. Rev. 55 (1975) 247-304.
18. S. Grillner, Control of locomotion in bipeds, tetrapods and fish, in: V.B. Brooks (Ed.), Handbook of Physiology, Section 1: The Nervous System, Vol. II: Motor Control, American Physiological Society, Bethesda, 1981, pp. 1179-1236.
19. S. Grillner, Neurobiological bases of rhythmic motor acts in vertebrates, Science 228 (1985) 143-149.
20. S. Grillner, Neural networks for vertebrate locomotion, Sci. Am. 274 (1) (1996) 48-53.
21. N. Kopell, G.B. Ermentrout, Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39 (1986) 623-660.
22. N. Kopell, G.B. Ermentrout, Coupled oscillators and the design of central pattern generators, Math. Biosci. 89 (1988) 14-23.
23. N. Kopell, G.B. Ermentrout, Phase transitions and other phenomena in chains of oscillators, SIAM J. Appl. Math. 50 (1990) 1014-1052.
24. K.G. Pearson, Common principles of motor control in vertebrates and invertebrates, Annu. Rev. Neurosci. 16 (1993) 265-297.
25. R.H. Rand, A.H. Cohen, P.J. Holmes, Systems of coupled oscillators as models of central pattern generators, in: A.H. Cohen, S. Rossignol, S. Grillner (Eds.), Neural Control of Rhythmic Movements in Vertebrates, Wiley, New York, 1988, pp. 333-367.
26. G. Schöner, W.Y. Jiang, J.A.S. Kelso, A synergetic theory of quadrupedal gaits and gait transitions, J. Theor. Biol. 142 (1990) 359-391.
27. M.L. Shik, G.N. Orlovsky, Neurophysiology of locomotor automatism, Physiol. Rev. 56 (1976) 465-501.
28. S. Smale, A mathematical model of two cells via Turing's equation, in: J.D. Cowan (Ed.), Some Mathematical Questions in Biology V, Amer. Math. Soc. Lecture Notes on Mathematics in the Life Sciences, vol. 6, AMS, Providence, RI, 1974, pp. 15-26.
29. P.S.G. Stein, Motor systems, with specific reference to the control of locomotion, Annu. Rev. Neurosci. 1 (1978) 61-81.
30. T.L. Williams, K.A. Sigvardt, N. Kopell, G.B. Ermentrout, M.P. Remler, Forcing of coupled nonlinear oscillators: studies of intersegmental coordination in the lamprey locomotor central pattern generator, J. Neurophysiol. 64 (1990) 862-871.
31. H. Yuasa, M. Ito, Coordination of many oscillators and generation of locomotory patterns, Biol. Cybern. 63 (1990) 177-184.
32. H. Yuasa, M. Ito, Generation of locomotive patterns and self-organization, J. Robot. Mechatron. 4 (1992) 142-147.