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Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. (English) Zbl 0783.58053
Summary: We present two hypotheses on the mathematical mechanism underlying bursting dynamics in a class of differential systems: (1) that the transition from continuous firing of spikes to bursting is caused by a crisis which destabilizes a chaotic state of continuous spiking; and (2) that the bursting corresponds to a homoclinicity to this unstable chaotic state. These propositions are supported by a numerical test on the Hindmarsh-Rose model, a prototype of its kind. We conclude by a unified view for three types of complex multi-modal oscillations: homoclinic systems, bursting, and the Pomeau-Manneville intermittency.

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C23 Bifurcation theory for ordinary differential equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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[1] Alexander, J.C.; Cai, D.Y., J. math. biol., 29, 405, (1991)
[2] Chay, T.R.; Keizer, J., Biophys. J., 42, 181, (1983)
[3] Chay, T.R.; Rinzel, J., Biophys. J., 47, 357, (1985)
[4] Decroly, O.; Goldbeter, A., J. theor. biol., 124, 219, (1987)
[5] Doedel, E., Cong. num., 30, 265, (1981)
[6] Eckmann, J.-P.; Ruelle, D., Rev. mod. phys., 57, 617, (1985)
[7] Feigenbaum, M.J., J. stat. phys., 19, 25, (1978)
[8] Gaspard, P.; Wang, X.-J., J. stat. phys., 48, 151, (1987)
[9] Gavrilov, N.K.; Šil’nikov, L.P.; Gavrilov, N.K.; Šil’nikov, L.P., Math. USSR sb., Math. USSR sb., 19, 139, (1973)
[10] Grebogi, C.; Ott, E.; Yorke, J.A., Phys. rev. A, 36, 5365, (1987)
[11] Hindmarsh, J.L.; Rose, R.M., Proc. R. soc. lond. B, 221, 87, (1984)
[12] Kaas-Petersen, C., (), 183
[13] Li, T.-Y.; Yorke, J.A., Am. math. mon., 82, 985, (1975)
[14] Morris, C.; Lecar, H., Biophys. J., 35, 193, (1981)
[15] Newhouse, S., Proc. AMS symp. pure math., 14, 191, (1970)
[16] Newhouse, S., Topology, Publ. math. IHES, 50, 101, (1979)
[17] Ovsyannikov, I.M.; Šil’nikov, L.P., Math. USSR sb., 58, 557, (1987)
[18] Pomeau, Y.; Manneville, P., Commun. math. phys., 74, 189, (1980)
[19] Rinzel, J., (), 304
[20] Rinzel, J., (), 1578
[21] Rinzel, J.; Lee, Y.S., (), 19
[22] Rinzel, J.; Ermentrout, G.B., (), 135
[23] Robinson, C., Commun. math. phys., 90, 433, (1983)
[24] Selverston, A.I.; Moulin, M., Ann. rev. physiol., 47, 29, (1985)
[25] Šil’nikov, L.P., Soviet math. dokl., Math. USSR sb., 10, 91, (1970)
[26] Šil’nikov, L.P., Soviet math. dokl., 9, 624, (1968)
[27] Steriade, M.; Jones, E.G.; Llinás, R.R., Thalamic oscillations and signaling, (1990), Wiley New York
[28] Tedeschini-Lalli, L.; Yorke, J.A., Commun. math. phys., 106, 635, (1986)
[29] Terman, D., SIAM J. appl. math., 51, 1418, (1991)
[30] Terman, D., The transition from bursting to continuous spiking in excitable membrane models, (1991), Ohio State University at Columbus, preprint · Zbl 0900.92059
[31] van Strien, S.J., (), 316
[32] Wang, X.-J., Commun. math. phys., 131, 317, (1990)
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