Abstract
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.
Similar content being viewed by others
References
Glass, L., Mackey, M.: From clocks to chaos: the rhythms of life. Princeton: Princeton University Press 1988
Kopell, N. Ermentrout, G. B.: Coupled oscillators and the design of central pattern generators. Math. Biosci. 90, 87–109 (1988)
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)
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 1987
Ermentrout G. B., Rinzel, J. M.: Phase walkthrough in biological oscillators. Am. J. Physiol. 246, R602–606 (1983)
Schrieber, I., Marek, M.: Strange attractors in coupled reaction-diffusion cells. Physica 15d, 258–272 (1982)
Ermentrout, G. B., Kopell, N.: Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math., to appear
Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophysical J. 35, 193–213 (1981)
Wilson, H. R., Cowan J. D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1–24 (1972)
Ermentrout, G. B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15, 215–237 (1984)
Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)
Sanders, J. A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. (Appl. Math. Sci., vol. 59) Springer: New York 1985
Ermentrout, G. B.: The behavior of rings of coupled oscillators. J. Math. Biol, 23. 55–74 (1986)
Aronson, D. G., Ermentrout, G. B., Kopell, N.: Amplitude response of coupled oscillators. Physica D, to appear
Author information
Authors and Affiliations
Additional information
Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University
Rights and permissions
About this article
Cite this article
Ermentrout, G.B., Kopell, N. Multiple pulse interactions and averaging in systems of coupled neural oscillators. J. Math. Biol. 29, 195–217 (1991). https://doi.org/10.1007/BF00160535
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00160535