Biological Cybernetics

, Volume 71, Issue 5, pp 375–385 | Cite as

Hard-wired central pattern generators for quadrupedal locomotion

  • J. J. Collins
  • S. A. Richmond


Animal locomotion is generated and controlled, in part, by a central pattern generator (CPG), which is an intraspinal network of neurons capable of producing rhythmic output. In the present work, it is demonstrated that a hard-wired CPG model, made up of four coupled nonlinear oscillators, can produce multiple phase-locked oscillation patterns that correspond to three common quadrupedal gaits — the walk, trot, and bound. Transitions between the different gaits are generated by varying the network's driving signal and/or by altering internal oscillator parameters. The above in numero results are obtained without changing the relative strengths or the polarities of the system's synaptic interconnections, i.e., the network maintains an invariant coupling architecture. It is also shown that the ability of the hard-wired CPG network to produce and switch between multiple gait patterns is a model-independent phenomenon, i.e., it does not depend upon the detailed dynamics of the component oscillators and/or the nature of the inter-oscillator coupling. Three different neuronal oscillator models — the Stein neuronal model, the Van der Pol oscillator, and the FitzHugh-Nagumo model -and two different coupling schemes are incorporated into the network without impeding its ability to produce the three quadrupedal gaits and the aforementioned gait transitions.


Oscillator Parameter Gait Pattern Central Pattern Generator Internal Oscillator Neuronal Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Afelt Z, Blaszczyk J, Dobrzecka C (1983) Speed control in animal locomotion: transitions between symmetrical and nonsymmetrical gaits in the dog. Acta Neurobiol Exp 43:235–250Google Scholar
  2. Alexander JC, Doedel EJ, Othmer HG (1989) Resonance and phaselocking in excitable systems. In: Othmer HG (ed) Some mathematical questions in biology: the dynamics of excitable media, Vol 21. American Mathematics Society, Providence, pp 1–37Google Scholar
  3. Alexander RMcN, Jayes AS (1983) A dynamic similarity hypothesis for the gaits of quadrupedal mammals. J Zool Lond 201:135–152Google Scholar
  4. Andersson O, Grillner S (1993) Peripheral control of the cat's step cycle. II. Entrainment of the central pattern generators for locomotion by sinusoidal hip movements during “fictive locomotion”. Acta Physiol Scand 118:229–239Google Scholar
  5. Arshavsky YI, Kots YM, Orlovsky GN, Rodionov IM, Shik ML (1965) Investigation of the biomechanics of running by the dog. Biophysics 10:737–746Google Scholar
  6. Ashwin P (1990) Symmetric chaos in systems of three and four forced oscillators. Nonlinearity 3:603–617Google Scholar
  7. Bässler U (1986) On the definition of central pattern generator and its sensory control. Biol Cybern 54:65–69Google Scholar
  8. Bay JS, Hemami H (1987) Modeling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans Biomed Eng 34:297–306Google Scholar
  9. Beer RD (1990) Intelligence as adaptive behavior: an experiment in computational neurothology. Academic Press, San DiegoGoogle Scholar
  10. Beer RD, Chiel HJ, Quinn RD, Espenschied KS, Larsson P (1992) A distributed neural network architecture for hexapod robot locomotion. Neural Comput 4:356–365Google Scholar
  11. Chiel HJ, Beer RD, Quinn RD, Espenschied KS (1992) Robustness of a distributed neural network controller for locomotion in a hexapod robot. IEEE Trans Robot Autom 8:293–303Google Scholar
  12. Collins JJ (1995) Gait transitions. In: Arbib MA (ed) The handbook of brain theory and neural networks. MIT Press, Cambridge, Ma. (in press)Google Scholar
  13. Collins JJ, Stewart IN (1992) Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion. J Math Biol 30:827–838Google Scholar
  14. Collins JJ, Stewart IN (1993a) Coupled nonlinear oscillators and the symmetries of animal gaits. J Nonlinear Sci 3:349–392Google Scholar
  15. Collins JJ, Stewart I (1993b) Hexapodal gaits and coupled nonlinear oscillator models. Biol Cybern 68:287–298Google Scholar
  16. Dagg AI (1973) Gaits in mammals. Mammal Rev 3:135–154Google Scholar
  17. Davis WJ, Kennedy D (1972) Command interneurons controlling swimmeret movements in the lobster. I. Types of effects on motoneurons. J Neurophysiol 35:1–12Google Scholar
  18. Delcomyn F (1980) Neural basis of rhythmic behavior in animals. Science 210:492–498Google Scholar
  19. Deuel NR, Lawrence LM (1987) Laterality in the gallop gait of horses. J Biomech 20:645–649Google Scholar
  20. Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New YorkGoogle Scholar
  21. Farley CT, Taylor CR (1991) A mechanical trigger for the trot-gallop transition in horses. Science 253:306–308Google Scholar
  22. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445–466Google Scholar
  23. Flaherty JE, Hoppensteadt FC (1978) Frequency entrainment of a forced van der Pol oscillator. Stud Appl Math 58:5–15Google Scholar
  24. Gambaryan P (1974) How mammals run: anatomical adaptations. Wiley, New YorkGoogle Scholar
  25. Getting PA (1989) Emerging principles governing the operation of neural networks. Annu Rev Neurosci 12:185–204Google Scholar
  26. Grillner S (1975) Locomotion in vertebrates: central mechanisms and reflex interaction. Physiol Rev 55:247–304Google Scholar
  27. Grillner S (1981) Control of locomotion in bipeds, tetrapods and fish. In: Brooks VB (ed) Handbook of physiology, Sect 1. The nervous system, Vol II. Motor control. American Physiological Society Bethesda, pp 1179–1236Google Scholar
  28. Grillner S (1985) Neurological bases of rhythmic motor acts in vertebrates. Science 228:143–149Google Scholar
  29. Grillner S, Wallén P (1985) Central pattern generators for locomotion, with special reference to vertebrates. Annu Rev Neurosci 8:233–261Google Scholar
  30. Harris-Warrick RM (1988) Chemical modulation of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 285–331Google Scholar
  31. Harris-Warrick RM, Marder E (1991) Modulation of neural networks for behavior. Annu Rev Neurosci 14:39–57Google Scholar
  32. Hildebrand M (1976) Analysis of tetrapod gaits: general considerations and symmetrical gaits. In: Herman RM, Grillner S, Stein PSG, Stuart DG (eds) Neural control of locomotion. Plenum Press, New York, pp 203–236Google Scholar
  33. Hildebrand M (1977) Analysis of asymmetrical gaits. J Mammal 58:131–156Google Scholar
  34. Hoyt DF, Taylor CR (1981) Gait and the energetics of locomotion in horses. Nature 292:239–240Google Scholar
  35. Jayes AS, Alexander RMcN (1978) Mechanics of locomotion of dogs (Canis familiaris) and sheep (Ovis aris). J Zool Lond 185:289–308Google Scholar
  36. Jeka JJ, Kelso JAS, Kiemel T (1993a) Pattern switching in human multilimb coordination dynamics. Bull Math Biol 55:829–845Google Scholar
  37. Jeka JJ, Kelso JAS, Kiemel T (1993b) Spontaneous transitions and symmetry: pattern dynamics in human four-limb coordination. Hum Mov Sci 12:627–651Google Scholar
  38. Kaczmarek LK, Levitan IB (1987) Neuromodulation. Oxford University Press, New YorkGoogle Scholar
  39. Kelso JAS, Jeka JJ (1992) Symmetry breaking dynamics of human multilimb coordination. J Exp Psychol [Hum Percept] 18:645–668Google Scholar
  40. Kleinfeld D, Raccuia-Behling F, Chiel HJ (1990) Circuits constructed from identified Aplysia neurons exhibit multiple patterns of persistent activity. Biophys J 57:697–715Google Scholar
  41. Kopell N (1988) Toward a theory of modelling central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 369–413Google Scholar
  42. Lennard PR, Stein PSG (1977) Swimming movements elicited by electrical stimulation of turtle spinal cord. I. Low-spinal and intact preparations. J Neurophysiol 40:768–778Google Scholar
  43. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070Google Scholar
  44. Pearson KG (1993) Common principles of motor control in vertebrates and invertebrates. Annu Rev Neurosci 16:265–297Google Scholar
  45. Pearson KG, Ramirez JM, Jiang W (1992) Entrainment of the locomotor rhythm by group Ib afferents from ankle extensor muscles in spinal cats. Exp Brain Res 90:557–566Google Scholar
  46. Pecora LM, Carroll TL (1991) Pseudoperiodic driving: eliminating multiple domains of attraction using chaos. Phys Rev Lett 67:945–948Google Scholar
  47. Rand RH, Cohen AH, Holmes PJ (1988) Systems of coupled oscillators as models of central pattern generators. In: Cohen AH, Rossignol S, Grillner S (eds) Neural control of rhythmic movements in vertebrates. Wiley, New York, pp 333–367Google Scholar
  48. Schöner G, Jiang WY, Kelso JAS (1990) A synergetic theory of quadrupedal gaits and gait transitions. J Theor Biol 142:359–391Google Scholar
  49. Sharp AA, Abbott LF, Marder E (1992) Artificial electrical synapses in oscillatory networks. J Neurophysiol 67:1691–1694Google Scholar
  50. Sharp AA, O'Neil MB, Abbott LF, Marder E (1993) Dynamic clamp: computer-generated conductances in real neurons. J Neurophysiol 69:992–995Google Scholar
  51. Shik ML, Orlovsky GN (1976) Neurophysiology of locomotor automatism. Physiol Rev 56:465–501Google Scholar
  52. Shik ML, Severin FV, Orlovsky GN (1966) Control of walking and running by means of electrical stimulation of the mid-brain. Biophysics 11:756–765Google Scholar
  53. Stafford FS, Barnwell GM (1985) Mathematical models of central pattern generators in locomotion. III. Interlimb model for the cat. J Motor Behav 17:60–76Google Scholar
  54. Stein PSG (1978) Motor systems, with specific reference to the control of locomotion. Annu Rev Neurosci 1:61–81Google Scholar
  55. Stein RB, Leung KV, Mangeron D, 385–01 MN (1974a) Improved neuronal models for studying neural networks. Kybernetik 15:1–9Google Scholar
  56. Stein RB, Leung KV, 385–02 MN, Williams DW (1974b) Properties of small neural networks. Kybernetik 14:223–230Google Scholar
  57. Syed NI, Bulloch AGM, Lukowiak K (1990) In vitro reconstruction of the respiratory central pattern generator of the mollusk Lymnaea. Science 250:282–285Google Scholar
  58. Taga G, Yamaguchi Y, Shimizu H (1991) Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol Cybern 65:147–159Google Scholar
  59. Taylor CR (1978) Why change gaits? Recruitment of muscles and muscle fibers as a function of speed and gait. Am Zool 18:153–161Google Scholar
  60. Willis JB (1980) On the interaction between spinal locomotor generators in quadrupeds. Brain Res Rev 2:171–204Google Scholar
  61. Winfree AT (1990) The geometry of biological time. Springer, Berlin Heidelberg New YorkGoogle Scholar
  62. Yuasa H, Ito M (1990) Coordination of many oscillators and generation of locomotory patterns. Biol Cybern 63:177–184Google Scholar
  63. Yuasa H, Ito M (1992) Generation of locomotive patterns and self-organization. J Robot Mechatron 4:142–147Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • J. J. Collins
    • 1
  • S. A. Richmond
    • 1
  1. 1.NeuroMuscular Research Center and Department of Biomedical EngineeringBoston UniversityBostonUSA

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