Journal of Mathematical Biology

, Volume 23, Issue 1, pp 55–74 | Cite as

The behavior of rings of coupled oscillators

  • G. B. Ermentrout
Article

Abstract

Coupled oscillators in a ring are studied using perturbation and numerical methods. Stability of waves with nearest neighbor weak coupling is shown for a class of simple oscillators. Linkens' [23] model for colorectal activity is analyzed and several stable modes are found. Stability of waves with general (non nearest neighbor coupling) is determined and comparisons to the nearest neighbor case are made. Approximate solutions to a ring with inhomogeneities are compared with numerical simulations.

Keywords

Oscillators Waves Perturbation theory 

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References

  1. 1.
    Alexander J.: Patterns at primary Hopf bifurcations of a plexus of identical oscillators. SIAM J Appl. Math., to appearGoogle Scholar
  2. 2.
    Amari, S. A.: Dynamics of pattern formation in lateral-inhibition type neutral fields. Biol. Cyb. 27, 77–87 (1977)Google Scholar
  3. 3.
    Auchmuty, J. F. G.: Bifurcating Waves. In: Gurel, O., Rössler, O. E. (eds.) Bifurcation Theory and Applications in Scientific Disciplines. Ann. N.Y. Acad. Sci. 316, 263–278 (1978)Google Scholar
  4. 4.
    Burnstock, G.: Structure of Smooth Muscle and Its Innervations. In: Arnold, E. (ed.) Smooth Muscle, pp. 1–69. London 1970Google Scholar
  5. 5.
    Buzano, E., Golubitsky, M.: Bifurcation on the hexagonal lattice and the planar Benard problem. Philos. Trans. R. Soc. Lond., 1983Google Scholar
  6. 6.
    Cohen, D. S., Hoppensteadt, F. C., Miura, R. M.: Slowly modulated oscillations in nonlinear diffusion processes. SIAM J. Appl. Math. 33, 217–229 (1977)Google Scholar
  7. 7.
    Ermentrout, G. B.: n∶m phaselocking of weakly coupled oscillators. J. Math. Biol. 12, 327–342 (1981)Google Scholar
  8. 8.
    Ermentrout, G. B.: Asymptotic behavior of stationary, homogeneous neutronal nets. In: Amari, S. A., Arbib, M. S. (ed.) Competition and Cooperation in Neural Nets. Lect. Notes Biomath. 45, pp. 57–70. Berlin Heidelberg New York: Springer 1982Google Scholar
  9. 9.
    Ermentrout, G. B., Cowan, J. D.: Temporal oscillations in neutral nets. J. Math. Biol. 7, 265–280 (1979a)Google Scholar
  10. 10.
    Ermentrout, G. B., Cowan, J. D.: A mathematical theory of visual hallucination patterns. Biol. Cyb. 34, 137–158 (1979b)Google Scholar
  11. 11.
    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
  12. 12.
    Ermentrout, G. B., Rinzel, J.: Waves in a simple, excitable or oscillatory, reaction-diffusion model. J. Math. Biol. 11, 269–294 (1981)Google Scholar
  13. 13.
    Gmitro, J.: Concentration patterns generated by reaction and diffusion. Univ. Minnesota, Ph.D. Thesis 1969Google Scholar
  14. 14.
    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
  15. 15.
    Hagan, P. S.: Sprial waves in reaction-diffusion equations. SIAM J. Appl. Math. 42 762–786 (1983)Google Scholar
  16. 16.
    Hagan, P. S., Cohen, M. S.: Diffusion-induced morphogenesis in the development of dictyostelium. J. Theor. Biol. 93, 881–908 (1981)Google Scholar
  17. 17.
    Kawato, M., Suzuki, R.: Two coupled neural oscillators as a model of the circadian pacemeker. J. Theoret. Biol. 86, 547–575 (1980)Google Scholar
  18. 18.
    Keener, J. M.: On the validity of the two-timing method for large times. SIAM J. Math. Anal. 8, 1067–1091 (1977)Google Scholar
  19. 19.
    Kopell, N., Howard, L. N.: Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 52, 291–328 (1973)Google Scholar
  20. 20.
    Linkens, D. A.: The stability of entrainment conditions for RLC coupled van der Pol oscillators used as a model for intestinal electrical rhythms. Bull. Math. Biol. 39, 359–372 (1977)Google Scholar
  21. 21.
    Linkens, D. A., Datardina, S.: Frequency entrainment of coupled Hodgkin-Huxley-type oscillators for modeling gastro-intestinal electrical activity. IEEE Trans. Biomed. Engn., BME-24, 362–365 (1977)Google Scholar
  22. 22.
    Linkens, D. A., Duthie, H. L., Brown, B. H.: Dynamic model of the human small intenstine. IFAC Int. Symposium on dynamics and controls in physiological systems. August 1973, 140Google Scholar
  23. 23.
    Linkens, D. A., Taylor, I., Duthie, H. L.: Mathematical modeling of the colorectal myoelectrical activity in humans. IEEE Trans. Biomed. Engn., BME-23 101–110 (1976)Google Scholar
  24. 24.
    Nelson, T. S., Becker, J. C.: Simulation of the electrical and mechanical gradient of the small intestine. Am. J. Physiol. 214, 749–757 (1968)Google Scholar
  25. 25.
    Neu, J.: Coupled chemical oscillators. SIAM J. Appl. Math. 37, 307–315, (1979)Google Scholar
  26. 26.
    Rand, R. H., Holmes, P. J.: Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Nonlinear Mech. 15, 387–399 (1980)Google Scholar
  27. 27.
    Sarna, S. K., Daniel, E. E., Kingma, Y. J.: Simulation of the electric-control activity of the stomach by an array of relaxation oscillators. Digestive Diseases 17. 299–310 (1972)Google Scholar
  28. 28.
    Schlüter, A., Lortz, D., Busse, F.: On the stability of steady finite amplitude convection. J. Fluid Mech. 23, 129–144 (1965)Google Scholar
  29. 29.
    Torre, V.: Synchronization of nonlinear biochemical oscillators coupled by diffusion. Biol. Cyb. 17 137–144 (1975)Google Scholar
  30. 30.
    Traub, R. D.: Simulation of intrinsic bursing in CA3 hippocampal neurons. Neurosci. 7, 1233–1242 (1982)Google Scholar
  31. 31.
    Traub, R. D., Wong, R. K. S.: Cellular mechanism of neuronal synchronization in epilepsy. Science 216, 745–747 (1982)Google Scholar
  32. 32.
    Winfree, A. T.: The geometry of biological time. Berlin Heidelberg New York: Springer 1980Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • G. B. Ermentrout
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of PittsburghPittsburghUSA

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