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Multiple pulse interactions and averaging in systems of coupled neural oscillators. (English) Zbl 0718.92004
Summary: 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.

92C20 Neural biology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text: DOI
[1] Glass, L., Mackey, M.: From clocks to chaos: the rhythms of life. Princeton: Princeton University Press 1988 · Zbl 0705.92004
[2] Kopell, N. Ermentrout, G. B.: Coupled oscillators and the design of central pattern generators. Math. Biosci. 90, 87-109 (1988) · Zbl 0649.92009 · doi:10.1016/0025-5564(88)90059-4
[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)
[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 1987
[5] Ermentrout G. B., Rinzel, J. M.: Phase walkthrough in biological oscillators. Am. J. Physiol. 246, R602-606 (1983)
[6] Schrieber, I., Marek, M.: Strange attractors in coupled reaction-diffusion cells. Physica 15d, 258-272 (1982)
[7] Ermentrout, G. B., Kopell, N.: Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math., to appear · Zbl 0686.34033
[8] Morris, C., Lecar, H.: Voltage oscillations in the barnacle giant muscle fiber. Biophysical J. 35, 193-213 (1981) · doi:10.1016/S0006-3495(81)84782-0
[9] Wilson, H. R., Cowan J. D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12, 1-24 (1972) · doi:10.1016/S0006-3495(72)86068-5
[10] Ermentrout, G. B., Kopell, N.: Frequency plateaus in a chain of weakly coupled oscillators, I. SIAM J. Math. Anal. 15, 215-237 (1984) · Zbl 0558.34033 · doi:10.1137/0515019
[11] Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193-226 (1971) · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017
[12] Sanders, J. A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. (Appl. Math. Sci., vol. 59) Springer: New York 1985 · Zbl 0586.34040
[13] Ermentrout, G. B.: The behavior of rings of coupled oscillators. J. Math. Biol, 23. 55-74 (1986) · Zbl 0583.92002
[14] Aronson, D. G., Ermentrout, G. B., Kopell, N.: Amplitude response of coupled oscillators. Physica D, to appear · Zbl 0703.34047
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