September  2011, 16(2): 569-588. doi: 10.3934/dcdsb.2011.16.569

A reliability study of square wave bursting $\beta$-cells with noise

1. 

Department of Dynamics and Control, Beihang University, Beijing, 100191, China

2. 

Department of Mathematics, The University of Texas at Arlington, Arlington, TX 76019, United States, United States

3. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, United States

Received  May 2010 Revised  December 2010 Published  June 2011

Reliability of spike timing has been a hot topic recently. However reliability has not been considered for bursting behavior, as commonly observed in a variety of nerve and endocrine cells, including $\beta$-cells in intact pancreatic islets. In this paper, reliability of $\beta$-cells with noise is considered. A method to numerically study reliability of bursting cells is presented. Reliability of a single cell will decrease as noise level becomes larger. The reliability of networks of $\beta$-cells coupled by gap junctions or synaptic excitation is investigated. Simulations of the network of $\beta$-cells reveal that increasing noise level decreases the reliability. But the reliability of the network is higher than that of single cell. The effect of coupling strength on reliability is also investigated. Reliability will decrease when coupling strength is small and increase when coupling strength is large.
Citation: Jiaoyan Wang, Jianzhong Su, Humberto Perez Gonzalez, Jonathan Rubin. A reliability study of square wave bursting $\beta$-cells with noise. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 569-588. doi: 10.3934/dcdsb.2011.16.569
References:
[1]

J. C. Jonas, P. Gilon and J. C. Henquin, Temporal and quantitative correlation between insulin secretion and stably elevated or oscillatory cytoplasmic Ca2+ in mouse pancreatic $\beta$-cells,, Diabetes, 47 (1998), 1266. doi: 10.2337/diabetes.47.8.1266. Google Scholar

[2]

P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic $\beta$-cells under voltage clamp conditions,, J. Physiol., 374 (1986), 531. Google Scholar

[3]

P. M. Dean and E. K. Matthews, Glucose-induced electrical activity in pancreatic islet cells,, J. Physiol., 210 (1970), 255. Google Scholar

[4]

A. M. Scott, I. Atwater and E. Rojas, A method for the simultaneous measurement of insulin release and $\beta$-cell membrane potential in single mouse islets of Langerhans,, Diabetologia., 21 (1981), 470. doi: 10.1007/BF00257788. Google Scholar

[5]

F. M. Ashcroft and P. Rorsman, Electrophysiology of the pancreatic $\beta$-cell,, Prog. Biophys. Mol. Biol., 54 (1989), 87. doi: 10.1016/0079-6107(89)90013-8. Google Scholar

[6]

S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation,, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 495. doi: 10.3934/dcdsb.2010.14.495. Google Scholar

[7]

G. De Vries and A. Sherman, Channel sharing in pancreatic $\beta$-cell revisited: enhancement of emergent bursting by noise,, J. Theor. Biol., 207 (2000), 513. doi: 10.1006/jtbi.2000.2193. Google Scholar

[8]

M. I. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance,, Phys. D, 137 (2000), 333. doi: 10.1016/S0167-2789(99)00191-8. Google Scholar

[9]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Mod. Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223. Google Scholar

[10]

T. Wellens, V. Shatokhin and A. Buchleitner, Stochastic resonance,, Rep. Prog. Phys., 67 (2004), 45. doi: 10.1088/0034-4885/67/1/R02. Google Scholar

[11]

A. Bar-Even, J. Paulsson, N. Maheshri, M. Carmi, E. O'Shea, Y. Pilpel and N. Barkai, Noise in protein expression scales with natural protein abundance,, Nat Genet, 38 (2006), 636. doi: 10.1038/ng1807. Google Scholar

[12]

J. R. Newman, S. Ghaemmaghami, J. Ihmels, D. K. Breslow, M. Noble, J. L. Derisi and J. S. Weissman, Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise,, Nature, 441 (2006), 840. doi: 10.1038/nature04785. Google Scholar

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E. M. Ozbudak, M. Thattai, I. Kurtser, A. D. Grossman and A. van Oudenaarden, Regulation of noise in the expression of a single gene,, Nat Genet, 31 (2002), 69. doi: 10.1038/ng869. Google Scholar

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G. M. Suel, R. P. Kulkarni, J. Dworkin, J. Garcia-Ojalvo and M. B. Elowitz, Tunability and noise dependence in differentiation dynamics,, Science, 315 (2007), 1716. doi: 10.1126/science.1137455. Google Scholar

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M. Turcotte, J. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state,, PNAS, 105 (2008), 15732. doi: 10.1073/pnas.0806349105. Google Scholar

[16]

I. Atwater, L. Rosario and E. Rojas, Properties of the Ca-activated K+ channel in pancreatic beta-cells,, Cell Calcium, 4 (1983), 451. doi: 10.1016/0143-4160(83)90021-0. Google Scholar

[17]

T. R. Chay and H. S. Kang, Role of single-channel stochastic noise on bursting clusters of pancreatic $\beta$-cells,, Biophys. J., 54 (1988), 427. doi: 10.1016/S0006-3495(88)82976-X. Google Scholar

[18]

A. Sherman, J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic $\beta$-cells by channel sharing,, Biophys. J., 54 (1988), 411. doi: 10.1016/S0006-3495(88)82975-8. Google Scholar

[19]

M. Pedersen and M. Sørensen, The effect of noise on $\beta$-cell burst period,, SIAM J. Appl. Math., 67 (2007), 530. doi: 10.1137/060655663. Google Scholar

[20]

J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters,, Nonlinearity, 17 (2004), 133. doi: 10.1088/0951-7715/17/1/009. Google Scholar

[21]

Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons,, Science, 268 (1995), 1503. doi: 10.1126/science.7770778. Google Scholar

[22]

R. D. Kumbhani, M. J. Nolt and L. A. Palmer, Precision, reliability, and information-theoretic analysis of visual thalamocortical neurons,, J. Neurophysiol, 98 (2007), 2647. doi: 10.1152/jn.00900.2006. Google Scholar

[23]

S. Schreiber, J. M. Fellous, D. Whitmer, P. Tiesinga and T. J. Sejnowski, A new correlation-based measure of spike time reliability,, Neurocomputing, 52-54 (2003), 52. doi: 10.1016/S0925-2312(02)00838-X. Google Scholar

[24]

X. Hu, H. Jiang, C. G, C. Li and D. L. Sparks, Reliability of oculomotor command signals carried by individual neurons,, PNAS, 104 (2007), 8137. doi: 10.1073/pnas.0702799104. Google Scholar

[25]

M. Pernarowski, Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting,, SIAM J. Appl. Math., 54 (1994), 814. doi: 10.1137/S003613999223449X. Google Scholar

[26]

P. Fatt and B. Katz, Spontaneous subthreshold activity at motor nerve endings,, J Physiol., 117 (1952), 109. Google Scholar

[27]

B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Sysems: A Guide to XPPAUT for Researchers and Students,", Software Environ. Tools 14, (2002). Google Scholar

[28]

G. L. Gerstein and N. Y. Kiang, An approach to the quantitative analysis of electrophysiological data from single neurons,, Biophys. J., 1 (1960), 15. doi: 10.1016/S0006-3495(60)86872-5. Google Scholar

[29]

N. B. Rushforth, Behavioral and electrophysiological studies of hydra. I. Analysis of contraction pulse patterns,, Biol. Bull., 140 (1971), 255. doi: 10.2307/1540073. Google Scholar

[30]

H. Shimazaki and S. Shinomoto, A method for selecting the bin size of a time histogram,, Neural Computation, 19 (2007), 1503. doi: 10.1162/neco.2007.19.6.1503. Google Scholar

[31]

L. G. Nowak, M. V. Sanchez-vives and D. A. McCormick, Influence of low and high frequency inputs on spike timing in visual cortical neurons,, Cerebral Cortex, 7 (1997), 487. doi: 10.1093/cercor/7.6.487. Google Scholar

[32]

R. J. Butera, Jr., J. Rinzel and J. C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. populations of coupled pacemaker neurons,, J. Neurophysiol, 82 (1999), 398. Google Scholar

[33]

J. E. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.021917. Google Scholar

[34]

R. Rodriguez and H. C. Tuckwell, Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks,, Phys. Rev. E, 54 (1996), 5585. doi: 10.1103/PhysRevE.54.5585. Google Scholar

[35]

S. Tanabe and K. Pakdaman, Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.031911. Google Scholar

[36]

M. G. Pedersen, A comment on noise enhanced bursting in pancreatic beta-cells,, J. Theoret. Biol., 235 (2005), 1. doi: 10.1016/j.jtbi.2005.01.025. Google Scholar

[37]

M. Frey and E. Simiu, Noise-induced chaos and phase space flux,, Phys. D, 63 (1993), 321. doi: 10.1016/0167-2789(93)90114-G. Google Scholar

[38]

N. Berglund and B. Gentz, "Stochastic Dynamic Bifurcations and Excitability in: Stochastic Methods in Neuroscience,", Carlo Laing and Gabriel Lord (eds.), (2009). Google Scholar

show all references

References:
[1]

J. C. Jonas, P. Gilon and J. C. Henquin, Temporal and quantitative correlation between insulin secretion and stably elevated or oscillatory cytoplasmic Ca2+ in mouse pancreatic $\beta$-cells,, Diabetes, 47 (1998), 1266. doi: 10.2337/diabetes.47.8.1266. Google Scholar

[2]

P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic $\beta$-cells under voltage clamp conditions,, J. Physiol., 374 (1986), 531. Google Scholar

[3]

P. M. Dean and E. K. Matthews, Glucose-induced electrical activity in pancreatic islet cells,, J. Physiol., 210 (1970), 255. Google Scholar

[4]

A. M. Scott, I. Atwater and E. Rojas, A method for the simultaneous measurement of insulin release and $\beta$-cell membrane potential in single mouse islets of Langerhans,, Diabetologia., 21 (1981), 470. doi: 10.1007/BF00257788. Google Scholar

[5]

F. M. Ashcroft and P. Rorsman, Electrophysiology of the pancreatic $\beta$-cell,, Prog. Biophys. Mol. Biol., 54 (1989), 87. doi: 10.1016/0079-6107(89)90013-8. Google Scholar

[6]

S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation,, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 495. doi: 10.3934/dcdsb.2010.14.495. Google Scholar

[7]

G. De Vries and A. Sherman, Channel sharing in pancreatic $\beta$-cell revisited: enhancement of emergent bursting by noise,, J. Theor. Biol., 207 (2000), 513. doi: 10.1006/jtbi.2000.2193. Google Scholar

[8]

M. I. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance,, Phys. D, 137 (2000), 333. doi: 10.1016/S0167-2789(99)00191-8. Google Scholar

[9]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Mod. Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223. Google Scholar

[10]

T. Wellens, V. Shatokhin and A. Buchleitner, Stochastic resonance,, Rep. Prog. Phys., 67 (2004), 45. doi: 10.1088/0034-4885/67/1/R02. Google Scholar

[11]

A. Bar-Even, J. Paulsson, N. Maheshri, M. Carmi, E. O'Shea, Y. Pilpel and N. Barkai, Noise in protein expression scales with natural protein abundance,, Nat Genet, 38 (2006), 636. doi: 10.1038/ng1807. Google Scholar

[12]

J. R. Newman, S. Ghaemmaghami, J. Ihmels, D. K. Breslow, M. Noble, J. L. Derisi and J. S. Weissman, Single-cell proteomic analysis of S. cerevisiae reveals the architecture of biological noise,, Nature, 441 (2006), 840. doi: 10.1038/nature04785. Google Scholar

[13]

E. M. Ozbudak, M. Thattai, I. Kurtser, A. D. Grossman and A. van Oudenaarden, Regulation of noise in the expression of a single gene,, Nat Genet, 31 (2002), 69. doi: 10.1038/ng869. Google Scholar

[14]

G. M. Suel, R. P. Kulkarni, J. Dworkin, J. Garcia-Ojalvo and M. B. Elowitz, Tunability and noise dependence in differentiation dynamics,, Science, 315 (2007), 1716. doi: 10.1126/science.1137455. Google Scholar

[15]

M. Turcotte, J. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state,, PNAS, 105 (2008), 15732. doi: 10.1073/pnas.0806349105. Google Scholar

[16]

I. Atwater, L. Rosario and E. Rojas, Properties of the Ca-activated K+ channel in pancreatic beta-cells,, Cell Calcium, 4 (1983), 451. doi: 10.1016/0143-4160(83)90021-0. Google Scholar

[17]

T. R. Chay and H. S. Kang, Role of single-channel stochastic noise on bursting clusters of pancreatic $\beta$-cells,, Biophys. J., 54 (1988), 427. doi: 10.1016/S0006-3495(88)82976-X. Google Scholar

[18]

A. Sherman, J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic $\beta$-cells by channel sharing,, Biophys. J., 54 (1988), 411. doi: 10.1016/S0006-3495(88)82975-8. Google Scholar

[19]

M. Pedersen and M. Sørensen, The effect of noise on $\beta$-cell burst period,, SIAM J. Appl. Math., 67 (2007), 530. doi: 10.1137/060655663. Google Scholar

[20]

J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters,, Nonlinearity, 17 (2004), 133. doi: 10.1088/0951-7715/17/1/009. Google Scholar

[21]

Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons,, Science, 268 (1995), 1503. doi: 10.1126/science.7770778. Google Scholar

[22]

R. D. Kumbhani, M. J. Nolt and L. A. Palmer, Precision, reliability, and information-theoretic analysis of visual thalamocortical neurons,, J. Neurophysiol, 98 (2007), 2647. doi: 10.1152/jn.00900.2006. Google Scholar

[23]

S. Schreiber, J. M. Fellous, D. Whitmer, P. Tiesinga and T. J. Sejnowski, A new correlation-based measure of spike time reliability,, Neurocomputing, 52-54 (2003), 52. doi: 10.1016/S0925-2312(02)00838-X. Google Scholar

[24]

X. Hu, H. Jiang, C. G, C. Li and D. L. Sparks, Reliability of oculomotor command signals carried by individual neurons,, PNAS, 104 (2007), 8137. doi: 10.1073/pnas.0702799104. Google Scholar

[25]

M. Pernarowski, Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting,, SIAM J. Appl. Math., 54 (1994), 814. doi: 10.1137/S003613999223449X. Google Scholar

[26]

P. Fatt and B. Katz, Spontaneous subthreshold activity at motor nerve endings,, J Physiol., 117 (1952), 109. Google Scholar

[27]

B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Sysems: A Guide to XPPAUT for Researchers and Students,", Software Environ. Tools 14, (2002). Google Scholar

[28]

G. L. Gerstein and N. Y. Kiang, An approach to the quantitative analysis of electrophysiological data from single neurons,, Biophys. J., 1 (1960), 15. doi: 10.1016/S0006-3495(60)86872-5. Google Scholar

[29]

N. B. Rushforth, Behavioral and electrophysiological studies of hydra. I. Analysis of contraction pulse patterns,, Biol. Bull., 140 (1971), 255. doi: 10.2307/1540073. Google Scholar

[30]

H. Shimazaki and S. Shinomoto, A method for selecting the bin size of a time histogram,, Neural Computation, 19 (2007), 1503. doi: 10.1162/neco.2007.19.6.1503. Google Scholar

[31]

L. G. Nowak, M. V. Sanchez-vives and D. A. McCormick, Influence of low and high frequency inputs on spike timing in visual cortical neurons,, Cerebral Cortex, 7 (1997), 487. doi: 10.1093/cercor/7.6.487. Google Scholar

[32]

R. J. Butera, Jr., J. Rinzel and J. C. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. populations of coupled pacemaker neurons,, J. Neurophysiol, 82 (1999), 398. Google Scholar

[33]

J. E. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters,, Phys. Rev. E, 74 (2006). doi: 10.1103/PhysRevE.74.021917. Google Scholar

[34]

R. Rodriguez and H. C. Tuckwell, Statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks,, Phys. Rev. E, 54 (1996), 5585. doi: 10.1103/PhysRevE.54.5585. Google Scholar

[35]

S. Tanabe and K. Pakdaman, Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.031911. Google Scholar

[36]

M. G. Pedersen, A comment on noise enhanced bursting in pancreatic beta-cells,, J. Theoret. Biol., 235 (2005), 1. doi: 10.1016/j.jtbi.2005.01.025. Google Scholar

[37]

M. Frey and E. Simiu, Noise-induced chaos and phase space flux,, Phys. D, 63 (1993), 321. doi: 10.1016/0167-2789(93)90114-G. Google Scholar

[38]

N. Berglund and B. Gentz, "Stochastic Dynamic Bifurcations and Excitability in: Stochastic Methods in Neuroscience,", Carlo Laing and Gabriel Lord (eds.), (2009). Google Scholar

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