Abstract
The dynamics of three-variable models of bursting are studied. It is shown that under certain conditions, the dynamics on the attractor can be essentially reduced to two dimensions. The salient dynamics on the attractor can thus be completely described by the return map of a section which is a logistic interval map. Two specific bursting models from the literature are shown to fit in the general framework which is developed. Bifurcation of the full system for one case in investigated and the dynamical behavior on the attractor is shown to depend on the position of a certain nullcline.
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Adams, W. B., Benson, J. A.: The generation and modulation of endogenous rhythmicity in the Aplysia bursting pacemaker neuron R15. In Noble, D., Bundle, T. L. (eds.)Progress in biophysics and molecular biology, vol. 46, pp. 1–49. Elmsford, NY: Pergamon Press, 1985
Alexander, J. C., Doedel, E. J., Othmer, H. G.: Resonance of phase-locking in excitable systems. In: Othmer, H. G. (ed.) Some mathematical questions in biology: the dynamics of excitable media. (Lect. Math. Life Sci., vol. 21, pp. 7–37.) Am. Math. Soc., Providence, RI
Alexander, D. C., Doedel, E. J., Othmer, H. G.: On the resonance structure in a forced excitable system. SIAM J. Appl. Math. 50, 1313–1418 (1990)
Alving, B. O.: Spontaneous activity in isolated somata of Aplysia pacemaker neurons. J. Gen. Physiol. 51, 29–45 (1968)
Argemi, J., Chagneux, H., Ducreux, C., Gola, M.: Qualitative study of a dynamical system for metrazol-induced paroxysmal depolarization shifts. Bull. Math. Biol. 46, 903–922 (1984)
Argemi, J., Gola, M., Chagneux, H.: Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells–I. Reduced systems. Bull. Math. Biol. 41, 665–686 (1979)
Argemi, J., Gola, M., Chagneux, H.: Qualitative analysis of a model generating long potential waves in Ba-treated nerve cells–II. Complete systems. Bull. Math. Biol. 42, 221–238 (1980)
Atwater, I., Dawson, M., Scott, A., Eddlestone, G., Rojas, E.: The nature of oscillatory behavior in electrical activity for the pancreatic β-cell. J. Hormone Metabol. Res., Suppl. 10, 100–107 (1980)
Baer, S. M., Tier, C.: An analysis of dendritic neuron model with an active membrane site. J. Math. Biol. 23, 137–161 (1986)
Chay, T. R.: Chaos in a three-variable model of an excitable cell, Phys. D 16, 233–242 (1985)
Chay, T. R.: Oscillations and chaos in the pancreatic β-cell. In: Othmer, H. G. (ed.) Nonlinear oscillations in biology and chemistry. (Lect. Notes Biomath., vol. 66, pp. 2–18.) Springer, Berlin-Heidelberg-New York, 1986
Chay, T. R., Keizer J.: Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42, 181–190 (1983)
Chay, T. R., Rinzel, J.: Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47, 357–366 (1985)
DeKepper, P., Rossi, A., Pacault, A.: Etude experimentale d'une reaction chimique periodique. Diagramme d'etat de la reaction de Belousov-Zhabotinskii. C.R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre C 283, 371–375 (1976)
Decroly, O., Goldbeter, A.: From simple to complex oscillatory behavior: Analysis of bursting in a multiply regulated biochemical system. J. Theor. Biol. 124, 219–250 (1987)
Doedel, E. J.: AUTO: A program for the automatic bifurcation and analysis of autonomous systems. Congr. Numer. 30, 265–284 (1987)
Ermentrout, G. B., Kopell N.: Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46, 233–253 (1986)
Fitzhugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1967)
Hindmarsh, J. L., Rose, R. M.: A model of neuronal bursting using three coupled first order differential equations. Philos. Trans. Roy. Soc. London Ser. B221, 87–102 (1984)
Honerkamp, J., Mutschler, C., Seitz, R.: Coupling of a slow and a fast oscillator can generate bursting. Bull. Math. Biol. 47, 1–21 (1985)
Hudson, J. L., Hart, M., Marinko, D.: An experimental study of multiple peak periodic and non-periodic oscillations in the Belousov-Zhabotinskii reaction. J. Chem. Phys. 71, 1601–1606 (1979)
Hudson, J. L., Lamba, P., Mankin, J. C.: Experiments on low-amplitude forcing of a chemical oscillator. J. Phys. Chem. 90, 3430–3434 (1986)
Janz, R. D., Vanacek, D. J., Field, R. J.: Composite double oscillation in a modified version of the oregonator model of the Belousov-Zhabotinsky reaction. J. Chem. Phys. 73, 3132–3138 (1980)
Johnson, D., Brown, T. H.: Mechanisms of neuronal burst generation. In: Schwartzkroin and H. V. Wheal (eds.) Electrophysiology of epilepsy, pp. 277–301. New York: Academic Press, 1984
Plant, R. E.: The effects of calcium on bursting neurons. J. Neurophys. 21, 217–237 (1978)
Plant, R. E.: Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biol. 11, 11–32 (1981)
Plant, R. E., Kim, M.: Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations. Biophys. J. 16, 227–244 (1976)
Rinzel, J.: Bursting oscillations in an excitable membrane model. In: Sleeman, B. D., Jarvis R. J. (eds.) Ordinary and partial differential equations proceedings of the eight conference held at Dundee, Scotland, June 25–29, 1984. (Lect. Notes Math., vol. 1151, pp. 304–316.) Springer, Berlin-Heidelberg-New York, 1985
Rinzel, J.: A formal classification of bursting mechanisms in excitable systems. In: Proceedings of the International Congress of Mathematics, vol. 2, pp. 1578–1593. American Mathematical Society, Providence, RI, 1986
Rinzel, J., Lee, Y. S.: On different mechanisms for membrane potential bursting. In: Othmar, H. G. (ed.) Nonlinear oscillations in biology and chemistry. (Lect Notes Biomath., vol. 66, pp. 19–33.) Springer, Berlin-Heidelberg-New York, 1986
Rinzel, J. Lee, Y. S.: Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25, 653–675 (1987)
Rinzel, J., Troy, W. C.: Bursting phenomena in a simplified Oregonator flow system model. J. Chem. Phys. 76, 1775–1789 (1982)
Rinzel, J., Troy, W. C.: A one-variable map analysis of bursting in the Belousov-Zhabotinskii reaction. Cont. Math. 17, 411–427 (1983)
Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes, Ohio State University, preprint (1989)
Traub, R. D.: Simulation of intrinsic bursting in CA3 hippocampal neurons. Neuroscience 7, 1233–1242 (1982)
Wong, R. K. S., Prince, D. A.: Participation of calcium spikes during intrinsic burst firing in hippocampal neurons. Brain Res. 159, 385–390 (1978)
Wong, R. K. S., Prince, D. A.: After potential generation in hippocampal pyramidal cells. J. Neurophysiol. 45, 89–97 (1981)
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Supported in part by N.S.F.
On leave at University of Maryland
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Alexander, J.C., Cai, DY. On the dynamics of bursting systems. J. Math. Biol. 29, 405–423 (1991). https://doi.org/10.1007/BF00160469
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DOI: https://doi.org/10.1007/BF00160469