Skip to main content
Log in

Hard-wired central pattern generators for quadrupedal locomotion

  • Published:
Biological Cybernetics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Afelt Z, Blaszczyk J, Dobrzecka C (1983) Speed control in animal locomotion: transitions between symmetrical and nonsymmetrical gaits in the dog. Acta Neurobiol Exp 43:235–250

    Google Scholar 

  • 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 1–37

    Google Scholar 

  • Alexander RMcN, Jayes AS (1983) A dynamic similarity hypothesis for the gaits of quadrupedal mammals. J Zool Lond 201:135–152

    Google Scholar 

  • 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 118:229–239

    Google Scholar 

  • Arshavsky YI, Kots YM, Orlovsky GN, Rodionov IM, Shik ML (1965) Investigation of the biomechanics of running by the dog. Biophysics 10:737–746

    Google Scholar 

  • Ashwin P (1990) Symmetric chaos in systems of three and four forced oscillators. Nonlinearity 3:603–617

    Google Scholar 

  • Bässler U (1986) On the definition of central pattern generator and its sensory control. Biol Cybern 54:65–69

    Google Scholar 

  • Bay JS, Hemami H (1987) Modeling of a neural pattern generator with coupled nonlinear oscillators. IEEE Trans Biomed Eng 34:297–306

    Google Scholar 

  • Beer RD (1990) Intelligence as adaptive behavior: an experiment in computational neurothology. Academic Press, San Diego

    Google Scholar 

  • Beer RD, Chiel HJ, Quinn RD, Espenschied KS, Larsson P (1992) A distributed neural network architecture for hexapod robot locomotion. Neural Comput 4:356–365

    Google Scholar 

  • 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 8:293–303

    Google Scholar 

  • 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 

  • Collins JJ, Stewart IN (1992) Symmetry-breaking bifurcation: a possible mechanism for 2:1 frequency-locking in animal locomotion. J Math Biol 30:827–838

    Google Scholar 

  • Collins JJ, Stewart IN (1993a) Coupled nonlinear oscillators and the symmetries of animal gaits. J Nonlinear Sci 3:349–392

    Google Scholar 

  • Collins JJ, Stewart I (1993b) Hexapodal gaits and coupled nonlinear oscillator models. Biol Cybern 68:287–298

    Google Scholar 

  • Dagg AI (1973) Gaits in mammals. Mammal Rev 3:135–154

    Google Scholar 

  • Davis WJ, Kennedy D (1972) Command interneurons controlling swimmeret movements in the lobster. I. Types of effects on motoneurons. J Neurophysiol 35:1–12

    Google Scholar 

  • Delcomyn F (1980) Neural basis of rhythmic behavior in animals. Science 210:492–498

    Google Scholar 

  • Deuel NR, Lawrence LM (1987) Laterality in the gallop gait of horses. J Biomech 20:645–649

    Google Scholar 

  • Edelstein-Keshet L (1988) Mathematical models in biology. Random House, New York

    Google Scholar 

  • Farley CT, Taylor CR (1991) A mechanical trigger for the trot-gallop transition in horses. Science 253:306–308

    Google Scholar 

  • FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1:445–466

    Google Scholar 

  • Flaherty JE, Hoppensteadt FC (1978) Frequency entrainment of a forced van der Pol oscillator. Stud Appl Math 58:5–15

    Google Scholar 

  • Gambaryan P (1974) How mammals run: anatomical adaptations. Wiley, New York

    Google Scholar 

  • Getting PA (1989) Emerging principles governing the operation of neural networks. Annu Rev Neurosci 12:185–204

    Google Scholar 

  • Grillner S (1975) Locomotion in vertebrates: central mechanisms and reflex interaction. Physiol Rev 55:247–304

    Google Scholar 

  • 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 1179–1236

    Google Scholar 

  • Grillner S (1985) Neurological bases of rhythmic motor acts in vertebrates. Science 228:143–149

    Google Scholar 

  • Grillner S, Wallén P (1985) Central pattern generators for locomotion, with special reference to vertebrates. Annu Rev Neurosci 8:233–261

    Google Scholar 

  • 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 285–331

    Google Scholar 

  • Harris-Warrick RM, Marder E (1991) Modulation of neural networks for behavior. Annu Rev Neurosci 14:39–57

    Google Scholar 

  • 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 203–236

    Google Scholar 

  • Hildebrand M (1977) Analysis of asymmetrical gaits. J Mammal 58:131–156

    Google Scholar 

  • Hoyt DF, Taylor CR (1981) Gait and the energetics of locomotion in horses. Nature 292:239–240

    Google Scholar 

  • Jayes AS, Alexander RMcN (1978) Mechanics of locomotion of dogs (Canis familiaris) and sheep (Ovis aris). J Zool Lond 185:289–308

    Google Scholar 

  • Jeka JJ, Kelso JAS, Kiemel T (1993a) Pattern switching in human multilimb coordination dynamics. Bull Math Biol 55:829–845

    Google Scholar 

  • Jeka JJ, Kelso JAS, Kiemel T (1993b) Spontaneous transitions and symmetry: pattern dynamics in human four-limb coordination. Hum Mov Sci 12:627–651

    Google Scholar 

  • Kaczmarek LK, Levitan IB (1987) Neuromodulation. Oxford University Press, New York

    Google Scholar 

  • Kelso JAS, Jeka JJ (1992) Symmetry breaking dynamics of human multilimb coordination. J Exp Psychol [Hum Percept] 18:645–668

    Google Scholar 

  • Kleinfeld D, Raccuia-Behling F, Chiel HJ (1990) Circuits constructed from identified Aplysia neurons exhibit multiple patterns of persistent activity. Biophys J 57:697–715

    Google Scholar 

  • 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 369–413

    Google Scholar 

  • Lennard PR, Stein PSG (1977) Swimming movements elicited by electrical stimulation of turtle spinal cord. I. Low-spinal and intact preparations. J Neurophysiol 40:768–778

    Google Scholar 

  • Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070

    Google Scholar 

  • Pearson KG (1993) Common principles of motor control in vertebrates and invertebrates. Annu Rev Neurosci 16:265–297

    Google Scholar 

  • 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 90:557–566

    Google Scholar 

  • Pecora LM, Carroll TL (1991) Pseudoperiodic driving: eliminating multiple domains of attraction using chaos. Phys Rev Lett 67:945–948

    Google Scholar 

  • 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 333–367

    Google Scholar 

  • Schöner G, Jiang WY, Kelso JAS (1990) A synergetic theory of quadrupedal gaits and gait transitions. J Theor Biol 142:359–391

    Google Scholar 

  • Sharp AA, Abbott LF, Marder E (1992) Artificial electrical synapses in oscillatory networks. J Neurophysiol 67:1691–1694

    Google Scholar 

  • Sharp AA, O'Neil MB, Abbott LF, Marder E (1993) Dynamic clamp: computer-generated conductances in real neurons. J Neurophysiol 69:992–995

    Google Scholar 

  • Shik ML, Orlovsky GN (1976) Neurophysiology of locomotor automatism. Physiol Rev 56:465–501

    Google Scholar 

  • Shik ML, Severin FV, Orlovsky GN (1966) Control of walking and running by means of electrical stimulation of the mid-brain. Biophysics 11:756–765

    Google Scholar 

  • Stafford FS, Barnwell GM (1985) Mathematical models of central pattern generators in locomotion. III. Interlimb model for the cat. J Motor Behav 17:60–76

    Google Scholar 

  • Stein PSG (1978) Motor systems, with specific reference to the control of locomotion. Annu Rev Neurosci 1:61–81

    Google Scholar 

  • Stein RB, Leung KV, Mangeron D, 385–01 MN (1974a) Improved neuronal models for studying neural networks. Kybernetik 15:1–9

    Google Scholar 

  • Stein RB, Leung KV, 385–02 MN, Williams DW (1974b) Properties of small neural networks. Kybernetik 14:223–230

    Google Scholar 

  • Syed NI, Bulloch AGM, Lukowiak K (1990) In vitro reconstruction of the respiratory central pattern generator of the mollusk Lymnaea. Science 250:282–285

    Google Scholar 

  • Taga G, Yamaguchi Y, Shimizu H (1991) Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol Cybern 65:147–159

    Google Scholar 

  • Taylor CR (1978) Why change gaits? Recruitment of muscles and muscle fibers as a function of speed and gait. Am Zool 18:153–161

    Google Scholar 

  • Willis JB (1980) On the interaction between spinal locomotor generators in quadrupeds. Brain Res Rev 2:171–204

    Google Scholar 

  • Winfree AT (1990) The geometry of biological time. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Yuasa H, Ito M (1990) Coordination of many oscillators and generation of locomotory patterns. Biol Cybern 63:177–184

    Google Scholar 

  • Yuasa H, Ito M (1992) Generation of locomotive patterns and self-organization. J Robot Mechatron 4:142–147

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Collins, J.J., Richmond, S.A. Hard-wired central pattern generators for quadrupedal locomotion. Biol. Cybern. 71, 375–385 (1994). https://doi.org/10.1007/BF00198915

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00198915

Keywords

Navigation