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The nature of the coupling between segmental oscillators of the lamprey spinal generator for locomotion: a mathematical model. (English) Zbl 0476.92003

92B05 General biology and biomathematics
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37-XX Dynamical systems and ergodic theory
92Cxx Physiological, cellular and medical topics
Full Text: DOI
[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)
[2] Carpenter, G. A.: Bursting phenomena in excitable membranes. SIAM J. Appl. Math. 36(2), 334-372 (1979) · Zbl 0411.35050 · doi:10.1137/0136027
[3] Chillingworth, D. R. J.: Differential topology with a view to applications. Pitman, 1976 · Zbl 0336.58001
[4] Cohen, A. H., Buchanan, J. T.: Activity of identified spinal neurons in lamprey during ?fictive swimming?, Abstract No. 129.2, Soc. Neurosci., 1980
[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) · doi:10.1007/BF00236674
[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)
[7] Dismukes, R. K.: New concepts of molecular communication among neurons. Beh. Brain Sci. 2, 409-448 (1979)
[8] Ermentrout, G. B.: n?m phase locking of weakly coupled oscillators. J. Math. Biol. 12, 327-342 (1981) · Zbl 0476.92007 · doi:10.1007/BF00276920
[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) · doi:10.1007/BF00360911
[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)
[11] Glass, L., Young, R. E.: Structure and dynamics of neural network oscillators. Brain Res. 179, 207-218 (1979) · doi:10.1016/0006-8993(79)90439-6
[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) · Zbl 0398.92005
[13] Grillner, S.: On the generation of locomotion in the spinal dogfish. Exp. Brain Res. 20, 459-470 (1975)
[14] Grillner, S.: Locomotion in vertebrates: Central mechanisms and reflex interaction. Physiol. Rev. 55, 247-304 (1975) · doi:10.1104/pp.55.2.247
[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 1976
[16] Grillner, S., Zangger, P.: How detailed is the central pattern generator for locomotion? Brain Res. 88, 367-371 (1975) · doi:10.1016/0006-8993(75)90401-1
[17] Holmes, P. J.: Phase locking and chaos in coupled limit cycle oscillators. Proc. Symp. on Recent Advances in Structural Dynamics, Southampton, England, 1980
[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) · Zbl 0476.92010
[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, 1977
[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) · doi:10.1109/TCS.1974.1083848
[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) · doi:10.1109/TCS.1976.1084184
[22] Neu, J. C.: Coupled chemical oscillators. SIAM J. Appl. Math. 37(2), 307-315 (1979) · Zbl 0417.34063 · doi:10.1137/0137022
[23] Neu, J. C.: The method of near-identity transformations and its applications. SIAM J. Appl. Math. 38(2), 189-208 (1980) · Zbl 0446.70015 · doi:10.1137/0138017
[24] Neu, J. C.: Large populations of coupled chemical oscillators. SIAM J. Appl. Math. 38(2), 305-316 (1980) · Zbl 0442.92030 · doi:10.1137/0138026
[25] Pavlidis, T.: Biological oscillators: Their mathematical analysis. Academic Press, 1973
[26] Pavlidis, T., Pinsker, H. M.: Oscillator theory and neurophysiology. Fed. Proc. 36, 2033-2035 (1977)
[27] Pinsker, H. M., Bell, J.: Phase plane description of endogenous neuronal oscillators in Aplysia. Biol. Cyber. (in press)
[28] Poon, M. L. T.: Induction of swimming in lamprey by L-DOPA and amino acids. J. Comp. Physiol. 136, 337-344 (1980) · doi:10.1007/BF00657354
[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)
[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) · Zbl 0447.70028 · doi:10.1016/0020-7462(80)90024-4
[31] Rovainen, C. M.: Neurobiology of lampreys. Physiol. Rev. 59, 1007-1077 (1979)
[32] Russell, D. F., Wallén, P.: On the pattern generator for fictive swimming in the lamprey, Icthyomyzon unicuspis. Acta Physiol. Scand. (abst.) (1979)
[33] Selverston, A. I.: Are central pattern generators understandable? Beh. Brain Sci. 3, 535-571 (1980) · doi:10.1017/S0140525X00006580
[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) · doi:10.1016/0301-0082(76)90004-6
[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 1976
[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) · doi:10.1002/cne.901880408
[37] Vidal, C., Viala, D., Buser, P.: Central locomotor programming in the rabbit. Brain Res. 168, 57-73 (1979) · doi:10.1016/0006-8993(79)90128-8
[38] Wilson, D. M.: The central nervous control of flight in a locust. J. Exp. Biol. 38, 471-479 (1961)
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