Abstract
Pancreatic β-cells exhibit periodic bursting electrical activity (BEA) consisting of active and silent phases. The Sherman-Rinzel-Keizer (SRK) model of this phenomenon consists of three coupled first-order nonlinear differential equations which describe the dynamics of the membrane potential, the activation parameter for the voltage-gated potassium channel, and the intracellular calcium concentration. These equations are nondimensionalized and transformed into a Liénard differential equation coupled to a single first-order differential equation for the slowly changing nondimensional calcium concentration. Leading-order perturbation problems are derived for the silent and active phases of the BEA on slow and fast time scales. Numerical solutions of these leading-order problems are compared with those for the exact equation in their respective regions. The leading-order solution in the active phase has a limit cycle behavior with a slowly varying frequency. It is observed that the “damping term” in the Liénard equation is small numerically.
This is a preview of subscription content, log in via an institution to check access.
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
Atwater, I., Ribalet, B., Rojas, E. (1978a): Cyclic changes in potential and resistance of the β-cell membrane induced by glucose in islets of Langerhans from mouse. J. Physiology (Lond.). 278, 117–139.
Atwater, I., Ribalet, B., Rojas, E. (1978b): Mouse pancreatic β-cells: tetraethylammonium blockage of the potassium permeability increase induced by depolarization. J. Physiology (Lond.). 288, 561–574.
Atwater, I., Dawson, CM., Ribalet, B., Rojas, E. (1978c): Potassium permeability activated by intracellular calcium ion concentration in the pancreatic β-cell. J. Physiology (Lond.). 288, 575–588.
Atwater, I., Dawson, C.M., Scott, A., Eddlestone, G., Rojas, E. (1980): The nature of the oscillatory behaviour in electrical activity from pancreatic β-cell. Hormone and Metabolic Research, supp.10, 100–107.
Bender, C.M., Orszag, S.A. (1978): Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York.
Bourland, F.J., Haberman, R. (1990): Separatrix crossing: time-invariant potentials with dissipation. SIAM J. Appl. Math. 50, 1716–1744.
Chay, T.R., Keizer, J. (1983): Minimal model for membrane oscillations in the pancreatic β-cell. Biophys. J. 42, 181–189.
Chay, T.R., Keizer, J. (1985): Theory of the effect of extracellular potassium on oscillations in the pancreatic β-cell. Biophys. J. 48, 815–827.
Dean, P.M., Matthews, E.K. (1970a): Glucose-induced electrical activity in pancreatic islet cells. J. Physiology (Lond.). 210, 255–264.
Dean, P.M., Matthews, E.K. (1970b): Electrical activity in pancreatic islet cells: effects of ions. J. Physiology (Lond.). 210, 265–275.
Henquin, J.C., Meissner, H.P. (1984): Effects of theophylline and dibutyryl cyclic adenosine monophosphate on the membrane-potential of mouse pancreatic β-cells. J. Physiology (Lond.). 351, 595–612.
Hodgkin, A.L., Huxley, A.F. (1952): A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiology (Lond.). 117, 500–544.
Keizer, J. (1988): Electrical activity and insulin release in pancreatic beta cells. Math. Biosci. 90, 127–138.
Kevorkian, J., Cole, J.D. (1981): Perturbation Methods in Applied Mathematics. (Applied Mathematical Sciences, Vol. 34). Springer-Verlag, Berlin-Heidelberg-New York.
Kevorkian, J. (1987): Perturbation techniques for oscillatory systems with slowly varying coefficients. SIAM Rev. 29, 391–461.
Kuzmak, G.E. (1959): Asymptotic solutions of nonlinear second order differential equations with variable coefficients. P.M.M. 23, 730–744.
Luke, J.C (1966): A perturbation method for nonlinear dispersive wave problems. Proc. Roy. Soc, Ser. A. 292, 403–412.
Meissner, H.P., Preissler, M. (1980): Ionic mechanisms of the glucose-induced membrane potential changes in β-cells. Hormone and Metabolic Research, supp.10, 91–99.
Meissner, H.P., Schmelz, H. (1974): Membrane potential of beta-cells in pancreatic islets. Pfluegers Arch. 351, 195–206.
Minorsky, N. (1962): Nonlinear Oscillations. Robert E. Krieger Publishing, Malabar, Florida.
Ozawa, S., Sand, O. (1986): Electrophysiology of endocrine cells. Phys. Reviews, 66, 887–952.
Rinzel, J. (1987): A formal classification of bursting mechanisms in excitable systems. In: Teramato, E., Yamaguti, M. (eds.) Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences. Lecture Notes in Biomath. Vol. 71 267–281, Springer-Verlag, Berlin-Heidelberg-New York.
Rinzel, J. (1987): Dissection of a model for neuronal parabolic bursting. J. Math. Biol. 25, 653–675.
Rinzel, J., Lee, Y.S. (1986): On different mechanisms for membrane potential bursting. In: Othmer, H.G. (ed.) Nonlinear Oscillations in Biology and Chemistry, Lecture Notes Biomath. 66, 19–83, Springer-Verlag, Berlin-Heidelberg-New York.
Rorsman, P., Trube, G. (1986): Calcium and delayed potassium currents in mouse pancreatic β-cells under voltage-clamp conditions. J. Physiology. 374, 531–550.
Sherman, A., Rinzel, J., Keizer, J. (1988): Emergence of organized bursting in clusters of pancreatic β-cells by channel sharing. Biophys. J. 54, 411–425.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pernarowski, M., Miura, R.M., Kevorkian, J. (1991). The Sherman-Rinzel-Keizer Model for Bursting Electrical Activity in the Pancreatic β-Cell. In: Busenberg, S., Martelli, M. (eds) Differential Equations Models in Biology, Epidemiology and Ecology. Lecture Notes in Biomathematics, vol 92. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45692-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-45692-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54283-4
Online ISBN: 978-3-642-45692-3
eBook Packages: Springer Book Archive