Journal of Mathematical Biology

, Volume 29, Issue 3, pp 195–217 | Cite as

Multiple pulse interactions and averaging in systems of coupled neural oscillators

  • G. B. Ermentrout
  • N. Kopell


Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.

Key words

Oscillations Neurons Averaging Neural circuits 


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  1. 1.
    Glass, L., Mackey, M.: From clocks to chaos: the rhythms of life. Princeton: Princeton University Press 1988Google Scholar
  2. 2.
    Kopell, N. Ermentrout, G. B.: Coupled oscillators and the design of central pattern generators. Math. Biosci. 90, 87–109 (1988)Google Scholar
  3. 3.
    Friesen, W. O., Poon, M., Stent, G.: Neuronal control of swimming in the medicinal leech IV. Identification of a network of oscillatory interneurons. J. Exp. Biol. 75, 25–43 (1978)Google Scholar
  4. 4.
    Kopell, N.: Toward a theory of modelling central pattern generators. In: Cohen, A. H., Rossignol, S., Grillner, S. (eds.), The neural control of rhythmic movements in vertebrates, pp. 369–413. New York: Wiley 1987Google Scholar
  5. 5.
    Ermentrout G. B., Rinzel, J. M.: Phase walkthrough in biological oscillators. Am. J. Physiol. 246, R602–606 (1983)Google Scholar
  6. 6.
    Schrieber, I., Marek, M.: Strange attractors in coupled reaction-diffusion cells. Physica 15d, 258–272 (1982)Google Scholar
  7. 7.
    Ermentrout, G. B., Kopell, N.: Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math., to appearGoogle Scholar
  8. 8.
    Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophysical J. 35, 193–213 (1981)Google Scholar
  9. 9.
    Wilson, H. R., Cowan J. D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972)Google Scholar
  10. 10.
    Ermentrout, G. B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15, 215–237 (1984)Google Scholar
  11. 11.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)Google Scholar
  12. 12.
    Sanders, J. A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. (Appl. Math. Sci., vol. 59) Springer: New York 1985Google Scholar
  13. 13.
    Ermentrout, G. B.: The behavior of rings of coupled oscillators. J. Math. Biol, 23. 55–74 (1986)Google Scholar
  14. 14.
    Aronson, D. G., Ermentrout, G. B., Kopell, N.: Amplitude response of coupled oscillators. Physica D, to appearGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • G. B. Ermentrout
    • 1
  • N. Kopell
    • 2
  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Department of MathematicsBoston UniversityBostonUSA

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