Journal of Mathematical Biology

, Volume 13, Issue 3, pp 345–369 | Cite as

The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: A mathematical model

  • Avis H. Cohen
  • Philip J. Holmes
  • Richard H. Rand


We present a theoretical model which is used to explain the intersegmental coordination of the neural networks responsible for generating locomotion in the isolated spinal cord of lamprey.

A simplified mathematical model of a limit cycle oscillator is presented which consists of only a single dependent variable, the phase θ(t). By coupling N such oscillators together we are able to generate stable phase locked motions which correspond to traveling waves in the spinal cord, thus simulating “fictive swimming”. We are also able to generate irregular “drifting” motions which are compared to the experimental data obtained from cords with selective surgical lesions.

Key words

Locomotion Pattern generator Dynamical systems Oscillators 


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  1. Berkenblitt, M. B., Deliagina, T. H., Feldman, A. G., Gelfand, I. M., Orlovsky, G. N.: Generation of scratching. I. Activity of spinal interneurons during scratching. J. Neurophysiol. 41, 1040–1057 (1978)Google Scholar
  2. Carpenter, G. A.: Bursting phenomena in excitable membranes. SIAM J. Appl. Math. 36(2), 334–372 (1979)Google Scholar
  3. Chillingworth, D. R. J.: Differential topology with a view to applications. Pitman, 1976Google Scholar
  4. Cohen, A. H., Buchanan, J. T.: Activity of identified spinal neurons in lamprey during “fictive swimming”, Abstract No. 129.2, Soc. Neurosci., 1980Google Scholar
  5. Cohen, A. H., Wallén, P.: The neuronal correlate of locomotion in fish. “Fictive swimming” induced in an in vitro preparation of the lamprey spinal cord. Exp. Brain Res. 41, 11–18 (1980)Google Scholar
  6. Cohen, D. S., Neu, J. C.: Interacting oscillatory chemical reactors. Bifurcation theory and applications in the scientific disciplines. Gurel, O., Rössler, D. E. (eds.). Annals N. Y. Acad. Sci. 316, 332–337 (1979)Google Scholar
  7. Dismukes, R. K.: New concepts of molecular communication among neurons. Beh. Brain Sci. 2, 409–448 (1979)Google Scholar
  8. Ermentrout, G. B.: nm phase locking of weakly coupled oscillators. J. Math. Biol. 12, 327–342 (1981)Google Scholar
  9. Friesen, W. O., Stent, G. S.: Generation of a locomotory rhythm by a neural network with recurrent cyclic inhibition. Biol. Cyber. 28, 27–40 (1977)Google Scholar
  10. Getting, P. A., Lennard, P. R., Hume, R. I.: Central pattern generator mediating swimming in Tritonia. I. Identification and synaptic interactions. J. Neurophysiol. 44, 151–164 (1980)Google Scholar
  11. Glass, L., Young, R. E.: Structure and dynamics of neural network oscillators. Brain Res. 179, 207–218 (1979)Google Scholar
  12. Grasman, J., Jansen, M. J. W.: Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J. Math. Biol. 7, 171–197 (1979)Google Scholar
  13. Grillner, S.: On the generation of locomotion in the spinal dogfish. Exp. Brain Res. 20, 459–470 (1975)Google Scholar
  14. Grillner, S.: Locomotion in vertebrates: Central mechanisms and reflex interaction. Physiol. Rev. 55, 247–304 (1975)Google Scholar
  15. Grillner, S., Kashin, S.: On the generation and performance of swimming in fish. In: Neural control of locomotion. (Herman, R., Grillner, S., Stein, P., Stuart, D., eds). Vol. 18, pp. 181–202. New York: Plenum Press 1976Google Scholar
  16. Grillner, S., Zangger, P.: How detailed is the central pattern generator for locomotion? Brain Res. 88, 367–371 (1975)Google Scholar
  17. Holmes, P. J.: Phase locking and chaos in coupled limit cycle oscillators. Proc. Symp. on Recent Advances in Structural Dynamics, Southampton, England, 1980Google Scholar
  18. Keener, J. P., Hoppensteadt, F. C., Rinzel, J.: Integrate-and-fire models of nerve membrane response to oscillatory input. SIAM J. Appl. Math. (in press)Google Scholar
  19. Kristan, W. B.: Neural control of movement. Group report. In: Function and formation of neural systems (Stent, G. S., ed.), pp. 329–354. Berlin: Dahlem Konferenz, 1977Google Scholar
  20. Linkens, D. A.: Analytical solution of large numbers of mutually coupled nearly sinusoidal oscillators. IEEE Trans. Circuits and Systems, CAS-21(2), 294–300 (1974)Google Scholar
  21. Linkens, D. A.: Stability of entrainment conditions for a particular form of mutually coupled van der Pol oscillators. IEEE Trans. Circuits and Systems, CAS-23(2), 113–121 (1976)Google Scholar
  22. Neu, J. C.: Coupled chemical oscillators. SIAM J. Appl. Math. 37(2), 307–315 (1979)Google Scholar
  23. Neu, J. C.: The method of near-identity transformations and its applications. SIAM J. Appl. Math. 38(2), 189–208 (1980)Google Scholar
  24. Neu, J. C.: Large populations of coupled chemical oscillators. SIAM J. Appl. Math. 38(2), 305–316 (1980)Google Scholar
  25. Pavlidis, T.: Biological oscillators: Their mathematical analysis. Academic Press, 1973Google Scholar
  26. Pavlidis, T., Pinsker, H. M.: Oscillator theory and neurophysiology. Fed. Proc. 36, 2033–2035 (1977)Google Scholar
  27. Pinsker, H. M., Bell, J.: Phase plane description of endogenous neuronal oscillators in Aplysia. Biol. Cyber. (in press)Google Scholar
  28. Poon, M. L. T.: Induction of swimming in lamprey by L-DOPA and amino acids. J. Comp. Physiol. 136, 337–344 (1980)Google Scholar
  29. Poon, M., Friesen, W. D., Stent, G. S.: Neural control of swimming in the medicinal leech. V. Connections between the oscillatory interneurons and the motor neurons. J. Exp. Biol. 75, 45–63 (1978)Google Scholar
  30. Rand, R. H., Holmes, P. J.: Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Nonlinear Mechanics 15, 387–399 (1980)Google Scholar
  31. Rovainen, C. M.: Neurobiology of lampreys. Physiol. Rev. 59, 1007–1077 (1979)Google Scholar
  32. Russell, D. F., Wallén, P.: On the pattern generator for fictive swimming in the lamprey, Icthyomyzon unicuspis. Acta Physiol. Scand. (abst.) (1979)Google Scholar
  33. Selverston, A. I.: Are central pattern generators understandable? Beh. Brain Sci. 3, 535–571 (1980)Google Scholar
  34. Selverston, A. I., Russell, D. F., Miller, J. P., King, D. G.: The stomatogastric nervous system; structure and function of a small neural network. Prog. Neurobiol. 6, 1–75 (1976)Google Scholar
  35. Stein, P. S. G.: Mechanisms of interlimb phase control. In: Neural control of locomotion (Herman, R., Grillner, S., Stein, P., Stuart, D., eds.). Vol. 18, pp. 465–487. New York: Plenum Press 1976Google Scholar
  36. Tang, D., Selzer, M. E.: Projections of lamprey spinal neurons determined by the retrograde axonal transport of horseradish peroxidase. J. Comp. Neurol. 188, 629–646 (1979)Google Scholar
  37. Vidal, C., Viala, D., Buser, P.: Central locomotor programming in the rabbit. Brain Res. 168, 57–73 (1979)Google Scholar
  38. Wilson, D. M.: The central nervous control of flight in a locust. J. Exp. Biol. 38, 471–479 (1961)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Avis H. Cohen
    • 1
  • Philip J. Holmes
    • 2
  • Richard H. Rand
    • 2
  1. 1.Department of PhysiologySchool of Veterinary MedicineUSA
  2. 2.Department of Theoretical and Applied Mechanics and Center for Applied MathematicsCornell UniversityIthacaUSA

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