Abstract
Burst activity is characterized by slowly alternating phases of near steady state behavior and trains of rapid spike-like oscillations; examples of bursting patterns are shown in Fig. 2. These two phases have been called the silent and active phases respectively [2], In the case of electrical activity of biological membrane systems the slow time scale of bursting is on the order of tens of seconds while the spikes have millisecond time scales. In our study of several specific models for burst activity we have identified a number of different mechanisms for burst generation (which are characteristic of classes of models). We will describe qualitatively some of these mechanisms by way of the schematic diagrams in Fig. 1.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, W. B., and J. A. Benson. 1985. The generation and modulation of endogenous rhythmieity in the Aplysia bursting pacemaker neurone R15. Prog. Biophys. Molec. Biol. 46;1–49.
Atwater, I., C. M. Dawson, A. Scott, G. Eddiestone, E. Rojas. 1980. The nature of the oscillatory behavior in electrical activity for pancreatic β-cell. J. of Hormone and Metabolic Res., Suppl. 10:100–107.
Atwater, I., B. Ribalet, and E. Rojas. 1979. Mouse pancreatic β-cells: tetra-ethylammonium blockage of the potassium permeability increase induced by depolarization. J. Physiol. 288:561–574.
Atwater, I., and J. Rinzel. The β-cell bursting pattern and intracellular calcium. In Ionic Channels in Cells and Model Systems (ed., R. Latorre). Plenum, New York, 1986.
Beigelman, P. M., B. Ribalet, and I. Atwater. 1977. Electrical activity of mouse pancreatic beta-cells II. Effects of glucose and arginine. J. Physiol., Paris 73:201–217.
Chay, T. R., and J. Keizer. 1983. Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42:181–190.
Chay, T. R., and J. Keizer. 1985. Theory of the effect of extracellular potassium en oscillations in the pancreatic β-cell. Biophys. J. 48:815–827.
Chay, T. R., and J. Rinzel. 1985. Bursting, beating, and chaos in an excitable membrane model. Biophys. J. 47:357–366.
Doedel, E. J. Software for Continuation Problems in Ordinary Differential Equations. Tech. Report, Applied Math. Dept., Cal Tech., 1986.
FitzHugh, R. 1961. Impulses and physiological states in models of nerve membrane. Biophys. J. 1:445–466.
Guttman, R., S. Lewis, and J. Rinzel. 1980. Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator. J. Physiol. 305:377–395.
Guttman, R., and R. Barnhill. 1970. Oscillation and repetitive firing in squid axons: Comparison of experiments with computations. J. Gen. Physiol. 55:104–118.
Hindmarsh, A. C. Ordinary differential equations systems solver. Report UCID-30001. Lawrence Livermore Lab. Livermore, CA, 1974.
Hindmarsh, J. L., and R. M. Rose. 1984. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B 221:87–102.
Hodgkin, A. L., and A. F. Huxley. 1952. A quantitative description of membrane current and its applicaton to conduction and excitation in nerve. J. Physiol. (Lond) 117:500–544.
Honorkamp, J., G. Mutschler, and R. Seitz. 1985. Coupling of a slow and a fast oscillator can generate bursting. Bull. Math. Biol. 47:1–21.
Hsu, I., and N. K. Kazarinoff. 1976. An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J. Math. Anal. App. 55:61–89.
Kopell, N., and G. B. Ermentrout. 1986. Subcellular oscillations and bursting. Math. Biosci. 78:265–291.
Martiel, J. L., and A. Goldbeter. Origin of bursting and biorhythmicity in a model for cyclic AMP oscillations in Dictyostelium cells. In Proceedings of Kyoto International Symposium on Mathematical Biology (ed., E. Teramoto). Springer, to appear.
Nagumo, J. S., S. Arimoto, and S. Yoshizawa. 1962. An active pulse transmission line simulating nerve axon. Proc. IRE. 50:2061–2070.
Plant, R. E. 1981. Bifurcation and resonance in a model for bursting nerve cells. J. Math. Biol. 11:15–32.
Rinzel, J. 1978. On repetitive activity in nerve. Federation Proc. 37:2793–2802.
Rinzel, J. Bursting oscillations in an excitable membrane model. In Ordinary and Partial Differential Equations (eds., B. D. Sleeman and R. J. Jarvis). Lecture Notes in Mathematics 1151, Springer-Verlag, New York, 1985.
Rinzel, J., and Y. S. Lee. On different mechanisms for membrane potential bursting. In Nonlinear Oscillations in Biology and Chemistry (ed., H. G. Othmer). Springer, New York, to appear.
Rinzel, J., and Y. S. Lee. Dissection of a model for neuronal parabolic bursting, preprint.
Rinzel, J., and I. B. Schwartz. 1984. One variable map prediction of Belousov-Zhabotinskii mixed mode oscillations. J. Chem. Phys. 80:5610–5615.
Rinzel, J., and W. C. Troy. 1982. Bursting phenomena in a simplified Oregonator flow system model. J. Chem. Phys. 76:1775–1789.
Siska, J., L. Kubinova, and I. Dvorak. Time hierarchy in systems with general attractors. In Proceedings of the Fourth International Conference on Mathematical Modeling (Zurich, 1983). Pergamon Press, to appear.
Tikhonov, A. N. 1952. Systems of differential equations containing a small parameter multiplying the highest derivatives. Mat. Sb. NS (31) 73:575–585. (in Russian)
Troy, W. C. 1974. Oscillation phenomena in nerve conduction equations. Doctoral thesis, State Univ. of New York at Buffalo.
Wasow, W. Asymptotic Expansions for Ordinary Differential Equations. Wiley-Interscience, New York, 1965, pps. 297–303.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rinzel, J. (1987). A Formal Classification of Bursting Mechanisms in Excitable Systems. In: Teramoto, E., Yumaguti, M. (eds) Mathematical Topics in Population Biology, Morphogenesis and Neurosciences. Lecture Notes in Biomathematics, vol 71. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93360-8_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-93360-8_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-17875-0
Online ISBN: 978-3-642-93360-8
eBook Packages: Springer Book Archive