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Diffusively coupled bursters: Effects of cell heterogeneity

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An Erratum to this article was published on 01 September 1999

Abstract

The interaction of a pair of weakly coupled biological bursters is examined. Bursting refers to oscillations in which an observable slowly alternates between phases of relative quiescence and rapid oscillatory behavior. The motivation for this work is to understand the role of electrical coupling in promoting the synchronization of bursting electrical activity (BEA) observed in the β-cells of the islet of Langerhans, which secrete insulin in response to glucose. By studying the coupled fast subsystem of a model of BEA, we focus on the interaction that occurs during the rapid oscillatory phase. Coupling is weak, diffusive and non-scalar. In addition, non-identical oscillators are permitted. Using perturbation methods with the assumption that the uncoupled oscillators are near a Hopf bifurcation, a reduced system of equations is obtained. A detailed bifurcation study of this reduced system reveals a variety of patterns but suggests that asymmetrically phase-locked solutions are the most typical. Finally, the results are applied to the unreduced full bursting system and used to predict the burst pattern for a pair of cells with a given coupling strength and degree of heterogeneity.

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References

  • Aronson, D. G., G. B. Ermentrout and N. Kopell (1990). Amplitude response of coupled oscillators. Physica D 41, 403–449.

    Article  MATH  MathSciNet  Google Scholar 

  • Bar-Eli, K. (1984). Coupling of chemical oscillators. J. Phys. Chem. 88, 3616–3622.

    Article  Google Scholar 

  • Bar-Eli, K. (1985). On the stability of coupled chemical oscillators. Physica D 14, 242–252.

    Article  MathSciNet  Google Scholar 

  • Bertram, R., P. Smolen, A. Sherman, D. Mears, I. Atwater, F. Martin and B. Soria (1995). A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic β-cells. Biophys. J. 68, 2323–2332.

    Google Scholar 

  • Doedel, E. (1981). AUTO: a program for the automatic bifurcation analysis of autonomous systems. Cong. Num. 30, 265–284.

    MATH  MathSciNet  Google Scholar 

  • Eddlestone, G. T., A. Goncalves, J. A. Bangham and E. Rojas (1984). Electrical coupling between cells in islets of Langerhans in mouse. J. Membr. Biol. 77, 1–14.

    Article  Google Scholar 

  • Ermentrout B. (1981a). Unpublished lecture notes.

  • Ermentrout, B. (1981b). n:m Phase-locking of weakly coupled oscillators. J. Math. Biology 12, 327–342.

    Article  MATH  MathSciNet  Google Scholar 

  • Ermentrout, B. and N. Kopell (1990). Oscillator death in systems of coupled neural oscillators. SIAM J. Appl. Math. 50, 125–146.

    Article  MATH  MathSciNet  Google Scholar 

  • Han, S. K., C. Kurrer and Y. Kuramoto (1995). Dephasing and bursting in coupled neural oscillators. Phys. Rev. Lett. 75, 3190–3193.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Kawato, M., M. Sokabe and R. Suzuki (1979). Synergism and antagonism of neurons caused by an electrical synapse. Biol. Cybernetics 34, 81–89.

    Article  MATH  MathSciNet  Google Scholar 

  • Keener, J. P. (1988). Principles of Applied Mathematics. Redwood City, CA: Addison-Wesley Publishing Company.

    MATH  Google Scholar 

  • Khibnik, A. I., Y. A. Kuznetsov, V. V. Levitin and E. V. Nikolaev (1993). Continuation techniques and interactive software for bifurcation analysis and interate maps. Physica D 62, 360–371.

    Article  MATH  MathSciNet  Google Scholar 

  • Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence. Berlin: Springer.

    MATH  Google Scholar 

  • Li Y.-X. (1990). Conditions for the occurrence of degeneracy between Turing and phase instabilities. Phys. Lett. A 147, 204–208.

    Google Scholar 

  • Manor, Y., J. Rinzel, I. Segev and Y. Yarom (1997). Low-amplitude oscillations in the inferior olive: a model based on electrical coupling of neurons with heterogeneous channel densities. J. Neurophysiol. 77, 2736–2752.

    Google Scholar 

  • Meda, P., I. Atwater, A. Bangham, L. Orci and E. Rojas (1984). The topography of electrical synchrony among β-cells in the mouse islet of Langerhans. Q. J. Exp. Physiol. 69, 719–735.

    Google Scholar 

  • Meda, P., R. M. Santos and I. Atwater (1986). Direct identification of electrophysiologically monitored cells within intact mouse islets of Langerhans. Diabetes 35, 232–236.

    Google Scholar 

  • Michaels, R. L., R. L. Sorenson, J. A. Parsons and J. D. Sheridan (1987). Prolactin enhances cell-to-cell communication among β-cells in pancreatic islets. Diabetes 36, 1098–1102.

    Google Scholar 

  • Morita, Y. (1986). A secondary bifurcation problem of weakly coupled oscillators with time delay. Japan. J. Appl. Math. 3, 223–247.

    Article  MATH  MathSciNet  Google Scholar 

  • Morita, Y. (1987). A periodic wave and its stability to a circular chain of weakly coupled oscillators. SIAM J. Math. Anal. 18, 1681–1698.

    Article  MATH  MathSciNet  Google Scholar 

  • Morris C. and H. Lecar (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213.

    Google Scholar 

  • Perez-Armendariz, M., D. C. Spray and M. V. L. Bennett (1991). Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells. Biophys. J. 59, 76–92.

    Google Scholar 

  • Pernarowski, M. (1998). Fast and slow subsystems for a continuum model of bursting activity. SIAM J. Appl. Math. In press.

  • Rinzel, J. (1985). Bursting oscillations in an excitable membrane model, in Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1151, B. D. Sleeman and R. J. Jarvis (Eds), New York: Springer, pp. 304–316.

    Google Scholar 

  • Sherman, A. (1994). Anti-phase, asymmetric and aperiodic oscillations in excitable cells-I. coupled bursters. Bull. Math. Biol. 56, 811–835.

    MATH  Google Scholar 

  • Sherman, A. (1995). Theoretical aspects of synchronized bursting in β-cells, in Pacemaker Activity and Intercellular Communication, J. D. Huizinga (Ed.), Boca Raton, FL: CRC Press, pp. 323–337.

    Google Scholar 

  • Sherman, A. (1996). Contributions of modeling to understanding stimulus-secretion coupling in pancreatic β-cells. Am. J. Physiol. 271, E362–E372.

    Google Scholar 

  • Sherman, A. and J. Rinzel (1991). Model for synchronization of pancreatic β-cells by gap junctions. Biophys. J. 59, 547–559.

    Google Scholar 

  • Sherman, A. and J. Rinzel (1992). Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc. Natl Acad. Sci. USA 89, 2471–2474.

    Article  Google Scholar 

  • Smolen, P. and J. Keizer (1992). Slow voltage inactivation of Ca2+ currents and bursting mechanisms for the mouse pancreatic beta-cell. Biophys. J. 127, 9–19.

    Google Scholar 

  • Smolen, P., J. Rinzel and A. Sherman (1993). Why pancreatic islets burst but single β cells do not: the heterogeneity hypothesis. Biophys. J. 64, 1668–1680.

    Article  Google Scholar 

  • Taylor, M. A. and I. G. Kevrekidis (1991). Some common dynamic features of coupled reacting systems. Physica D 51, 274–292.

    Article  MATH  MathSciNet  Google Scholar 

  • Taylor, M. A. and I. G. Kevrekidis (1991). Couple, double, toil and trouble: a computer assisted study of two coupled CSTRs. Chemical Engineering Science 48, 2129–2149.

    Article  Google Scholar 

  • Valdeolmillos, M., A. Gomis and J. V. Sanchez-Andres (1996). In vivo synchronous membrane potential oscillations in mouse pancreatic beta-cells: lack of co-ordination between islets. J. Physiol. 493, 9–18.

    Google Scholar 

  • Wang X.-J. and J. Rinzel (1995). Oscillatory and bursting properties of neurons, in The Handbook of Brain Theory and Neural Networks, M. A. Arbib (Ed.), Cambridge, MA: The MIT Press, 686–691.

    Google Scholar 

  • Yamaguchi Y. and H. Shimizu (1984). Theory of self-organization in the presence of native frequency distributions and external noise. Physica D 11, 212–226.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Research Branch, NIDDK, National Institutes of Health, Bethesda, MD, 20892, USA

    Gerda De Vries & Arthur Sherman

  2. 16203 S. 26th Place, Phoenix, AZ, 85048, USA

    Hsiu-Rong Zhu

Authors
  1. Gerda De Vries
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  2. Arthur Sherman
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  3. Hsiu-Rong Zhu
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Additional information

An erratum to this article is available at http://dx.doi.org/10.1006/bulm.1999.0125.

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De Vries, G., Sherman, A. & Zhu, HR. Diffusively coupled bursters: Effects of cell heterogeneity. Bull. Math. Biol. 60, 1167–1200 (1998). https://doi.org/10.1006/bulm.1998.0057

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  • DOI: https://doi.org/10.1006/bulm.1998.0057

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