American Institute of Mathematical Sciences

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September  2012, 17(6): 1969-1990. doi: 10.3934/dcdsb.2012.17.1969

Gap junctions and excitation patterns in continuum models of islets

Pranay Goel 1, and  James Sneyd 2,

1. 

Indian Institute of Science Education and Research, Pune, Maharashtra 411021, India
2. 

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

Received  October 2011 Revised  March 2012 Published  May 2012

We extend the development of homogenized models for excitable tissues coupled through "doughball" gap junctions. The analysis admits nonlinear Fickian fluxes in rather general ways and includes, in particular, calcium-gated conductance. The theory is motivated by an attempt to understand wave propagation and failure observed in the pancreatic islets of Langerhans. We reexamine, numerically, the role that gap junctional strength is generally thought to play in pattern formation in continuum models of islets.
Keywords: Homogenization, partial wave propagation., gap junctions, pancreatic islets, excited activity.
Mathematics Subject Classification: Primary: 74Q10, 35B27; Secondary: 92B2.
Citation: Pranay Goel, James Sneyd. Gap junctions and excitation patterns in continuum models of islets. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1969-1990. doi: 10.3934/dcdsb.2012.17.1969
References:
[1]

Gerda de Vries and Arthur Sherman, Beyond synchronization: Modulatory and emergent effects of coupling in square-wave bursting, in "Bursting: The Genesis of Rhythm in the Nervous System" (eds. Stephen Coombes and Paul C. Bressloff), World Scientific, London, (2005), 243-272.

[2]

J.-L. Auriault and H. I. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Int. J. Heat Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5.

[3]

R. K. Benninger, M. Zhang, W. S. Head, L. S. Satin and D. W. Piston, Gap junction coupling and calcium waves in the pancreatic islet, Biophys. J., 95 (2008), 5048-5061. doi: 10.1529/biophysj.108.140863.

[4]

R. Bertram, L. Satin, M. Zhang, P. Smolen and A. Sherman, Calcium and glycolysis mediate multiple bursting modes in pancreatic islets, Biophys. J., 87 (2004), 3074-3087. doi: 10.1529/biophysj.104.049262.

[5]

F. C. Brunicardi, J. Stagner, S. Bonner-Weir, H. Wayland, R. Kleinman, E. Livingston, P. Guth, M. Menger, R. McCuskey, M. Intaglietta, A. Charles, S. Ashley, A. Cheung, E. Ipp, S. Gilman, T. Howard and E. Passaro, Microcirculation of the islets of Langerhans, Long Beach Veterans Administration Regional Medical Education Center Symposium, Diabetes, 45 (1996), 385-392.

[6]

O. Cabrera, D. M. Berman, N. S. Kenyon, C. Ricordi, P. O. Berggren and A. Caicedo, The unique cytoarchitecture of human pancreatic islets has implications for islet cell function, Proc. Natl. Acad. Sci. U.S.A., 103 (2006), 2334-2339. doi: 10.1073/pnas.0510790103.

[7]

B. Ermentrout and K. Bar-Eli, Oscillation death, Scholarpedia, 3 (2008), 5371.

[8]

G. B. Ermentrout and D. H. Terman, "Mathematical Foundations of Neuroscience," Interdisciplinary Applied Mathematics, 35, Springer, New York, 2010.

[9]

P. Goel and A. Sherman, The geometry of bursting in the dual oscillator model of pancreatic beta-cells, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1664-1693.

[10]

P. Goel, A. Sherman and A. Friedman, Multiscale modeling of electrical and intracellular activity in the pancreas: The islet tridomain equations, Multiscale Modeling & Simulation, 7 (2009), 1609-1642.

[11]

P. Goel, J. Sneyd and A. Friedman, Homogenization of the cell cytoplasm: The calcium bidomain equations, Multiscale Modeling & Simulation, 5 (2006), 1045-1062.

[12]

P. E. Hand, B. E. Griffith and C. S. Peskin, Deriving macroscopic myocardial conductivities by homogenization of microscopic models, Bull. Math. Biol., 71 (2009), 1707-1726. doi: 10.1007/s11538-009-9421-y.

[13]

E. Hertzberg, ed., "Gap Junctions," Advances in Molecular and Cell Biology, Elsevier Science, 2000.

[14]

A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, Journal of Dynamics and Differential Equations, 22 (2010), 79-119. doi: 10.1007/s10884-010-9157-2.

[15]

H. J. Hupkes, D. Pelinovsky and B. Sandstede, Propagation failure in the discrete nagumo equation, Proceedings of the American Mathematical Society, 139 (2011), 3537-3551. doi: 10.1090/S0002-9939-2011-10757-3.

[16]

T. Kanno, S. O. Gopel, P. Rorsman and M. Wakui, Cellular function in multicellular system for hormone-secretion: Electrophysiological aspect of studies on alpha-, beta- and delta-cells of the pancreatic islet, Neurosci. Res., 42 (2002), 79-90. doi: 10.1016/S0168-0102(01)00318-2.

[17]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM Journal on Applied Mathematics, 47 (1987), 556-572. doi: 10.1137/0147038.

[18]

J. P. Keener, Diffusion induced oscillatory insulin secretion, Bull. Math. Biol., 63 (2001), 625-641. doi: 10.1006/bulm.2001.0235.

[19]

W. Krassowska and J. C. Neu, Effective boundary conditions for syncytial tissues, IEEE Trans Biomed Eng, 41 (1994), 143-150. doi: 10.1109/10.284925.

[20]

A. Lazrak and C. Peracchia, Gap junction gating sensitivity to physiological internal calcium regardless of pH in Novikoff hepatoma cells, Biophys. J., 65 (1993), 2002-2012. doi: 10.1016/S0006-3495(93)81242-6.

[21]

L. W. Maki and J. Keizer, Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells, Bull. Math. Biol., 57 (1995), 569-591. doi: 10.1007/BF02460784.

[22]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199.

[23]

G. P. and S. J., A comparison of effective conductivity in two models of gap junction coupling in tissues,, submitted.

[24]

C. Peracchia, Chemical gating of gap junction channels; roles of calcium, pH and calmodulin, Biochim. Biophys. Acta, 1662 (2004), 61-80.

[25]

M. Perez-Armendariz, C. Roy, D. C. Spray and M. V. Bennett, Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells, Biophys. J., 59 (1991), 76-92. doi: 10.1016/S0006-3495(91)82200-7.

[26]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001.

[27]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, "Methods in Neuronal Modeling," MIT Press, Cambridge, MA, USA, (1989), 135-169.

[28]

J. V. Rocheleau, G. M. Walker, W. S. Head, O. P. McGuinness and D. W. Piston, Microfluidic glucose stimulation reveals limited coordination of intracellular Ca2+ activity oscillations in pancreatic islets, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 12899-12903. doi: 10.1073/pnas.0405149101.

[29]

A. Sherman and J. Rinzel, Model for synchronization of pancreatic beta-cells by gap junction coupling, Biophys. J., 59 (1991), 547-559. doi: 10.1016/S0006-3495(91)82271-8.

[30]

C. L. Stokes and J. Rinzel, Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans, Biophys. J., 65 (1993), 597-607. doi: 10.1016/S0006-3495(93)81092-0.

[31]

K. Tsaneva-Atanasova, C. L. Zimliki, R. Bertram and A. Sherman, Diffusion of calcium and metabolites in pancreatic islets: Killing oscillations with a pitchfork, Biophys. J., 90 (2006), 3434-3446. doi: 10.1529/biophysj.105.078360.

show all references

References:
[1]

Gerda de Vries and Arthur Sherman, Beyond synchronization: Modulatory and emergent effects of coupling in square-wave bursting, in "Bursting: The Genesis of Rhythm in the Nervous System" (eds. Stephen Coombes and Paul C. Bressloff), World Scientific, London, (2005), 243-272.

[2]

J.-L. Auriault and H. I. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Int. J. Heat Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5.

[3]

R. K. Benninger, M. Zhang, W. S. Head, L. S. Satin and D. W. Piston, Gap junction coupling and calcium waves in the pancreatic islet, Biophys. J., 95 (2008), 5048-5061. doi: 10.1529/biophysj.108.140863.

[4]

R. Bertram, L. Satin, M. Zhang, P. Smolen and A. Sherman, Calcium and glycolysis mediate multiple bursting modes in pancreatic islets, Biophys. J., 87 (2004), 3074-3087. doi: 10.1529/biophysj.104.049262.

[5]

F. C. Brunicardi, J. Stagner, S. Bonner-Weir, H. Wayland, R. Kleinman, E. Livingston, P. Guth, M. Menger, R. McCuskey, M. Intaglietta, A. Charles, S. Ashley, A. Cheung, E. Ipp, S. Gilman, T. Howard and E. Passaro, Microcirculation of the islets of Langerhans, Long Beach Veterans Administration Regional Medical Education Center Symposium, Diabetes, 45 (1996), 385-392.

[6]

O. Cabrera, D. M. Berman, N. S. Kenyon, C. Ricordi, P. O. Berggren and A. Caicedo, The unique cytoarchitecture of human pancreatic islets has implications for islet cell function, Proc. Natl. Acad. Sci. U.S.A., 103 (2006), 2334-2339. doi: 10.1073/pnas.0510790103.

[7]

B. Ermentrout and K. Bar-Eli, Oscillation death, Scholarpedia, 3 (2008), 5371.

[8]

G. B. Ermentrout and D. H. Terman, "Mathematical Foundations of Neuroscience," Interdisciplinary Applied Mathematics, 35, Springer, New York, 2010.

[9]

P. Goel and A. Sherman, The geometry of bursting in the dual oscillator model of pancreatic beta-cells, SIAM Journal on Applied Dynamical Systems, 8 (2009), 1664-1693.

[10]

P. Goel, A. Sherman and A. Friedman, Multiscale modeling of electrical and intracellular activity in the pancreas: The islet tridomain equations, Multiscale Modeling & Simulation, 7 (2009), 1609-1642.

[11]

P. Goel, J. Sneyd and A. Friedman, Homogenization of the cell cytoplasm: The calcium bidomain equations, Multiscale Modeling & Simulation, 5 (2006), 1045-1062.

[12]

P. E. Hand, B. E. Griffith and C. S. Peskin, Deriving macroscopic myocardial conductivities by homogenization of microscopic models, Bull. Math. Biol., 71 (2009), 1707-1726. doi: 10.1007/s11538-009-9421-y.

[13]

E. Hertzberg, ed., "Gap Junctions," Advances in Molecular and Cell Biology, Elsevier Science, 2000.

[14]

A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, Journal of Dynamics and Differential Equations, 22 (2010), 79-119. doi: 10.1007/s10884-010-9157-2.

[15]

H. J. Hupkes, D. Pelinovsky and B. Sandstede, Propagation failure in the discrete nagumo equation, Proceedings of the American Mathematical Society, 139 (2011), 3537-3551. doi: 10.1090/S0002-9939-2011-10757-3.

[16]

T. Kanno, S. O. Gopel, P. Rorsman and M. Wakui, Cellular function in multicellular system for hormone-secretion: Electrophysiological aspect of studies on alpha-, beta- and delta-cells of the pancreatic islet, Neurosci. Res., 42 (2002), 79-90. doi: 10.1016/S0168-0102(01)00318-2.

[17]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM Journal on Applied Mathematics, 47 (1987), 556-572. doi: 10.1137/0147038.

[18]

J. P. Keener, Diffusion induced oscillatory insulin secretion, Bull. Math. Biol., 63 (2001), 625-641. doi: 10.1006/bulm.2001.0235.

[19]

W. Krassowska and J. C. Neu, Effective boundary conditions for syncytial tissues, IEEE Trans Biomed Eng, 41 (1994), 143-150. doi: 10.1109/10.284925.

[20]

A. Lazrak and C. Peracchia, Gap junction gating sensitivity to physiological internal calcium regardless of pH in Novikoff hepatoma cells, Biophys. J., 65 (1993), 2002-2012. doi: 10.1016/S0006-3495(93)81242-6.

[21]

L. W. Maki and J. Keizer, Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells, Bull. Math. Biol., 57 (1995), 569-591. doi: 10.1007/BF02460784.

[22]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Crit. Rev. Biomed. Eng., 21 (1993), 137-199.

[23]

G. P. and S. J., A comparison of effective conductivity in two models of gap junction coupling in tissues,, submitted.

[24]

C. Peracchia, Chemical gating of gap junction channels; roles of calcium, pH and calmodulin, Biochim. Biophys. Acta, 1662 (2004), 61-80.

[25]

M. Perez-Armendariz, C. Roy, D. C. Spray and M. V. Bennett, Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells, Biophys. J., 59 (1991), 76-92. doi: 10.1016/S0006-3495(91)82200-7.

[26]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge Nonlinear Science Series, 12, Cambridge University Press, Cambridge, 2001.

[27]

J. Rinzel and G. B. Ermentrout, Analysis of neural excitability and oscillations, "Methods in Neuronal Modeling," MIT Press, Cambridge, MA, USA, (1989), 135-169.

[28]

J. V. Rocheleau, G. M. Walker, W. S. Head, O. P. McGuinness and D. W. Piston, Microfluidic glucose stimulation reveals limited coordination of intracellular Ca2+ activity oscillations in pancreatic islets, Proc. Natl. Acad. Sci. U.S.A., 101 (2004), 12899-12903. doi: 10.1073/pnas.0405149101.

[29]

A. Sherman and J. Rinzel, Model for synchronization of pancreatic beta-cells by gap junction coupling, Biophys. J., 59 (1991), 547-559. doi: 10.1016/S0006-3495(91)82271-8.

[30]

C. L. Stokes and J. Rinzel, Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans, Biophys. J., 65 (1993), 597-607. doi: 10.1016/S0006-3495(93)81092-0.

[31]

K. Tsaneva-Atanasova, C. L. Zimliki, R. Bertram and A. Sherman, Diffusion of calcium and metabolites in pancreatic islets: Killing oscillations with a pitchfork, Biophys. J., 90 (2006), 3434-3446. doi: 10.1529/biophysj.105.078360.

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