# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems - B, 2011, 16 (2) : 569-588. doi: 10.3934/dcdsb.2011.16.569
##### References:
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Rojas, Properties of the Ca-activated K+ channel in pancreatic beta-cells, Cell Calcium, 4 (1983), 451-461. doi: 10.1016/0143-4160(83)90021-0. [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), 427-435. doi: 10.1016/S0006-3495(88)82976-X. [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), 411-425. doi: 10.1016/S0006-3495(88)82975-8. [19] M. Pedersen and M. Sørensen, The effect of noise on $\beta$-cell burst period, SIAM J. Appl. Math., 67 (2007), 530-542. doi: 10.1137/060655663. [20] J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters, Nonlinearity, 17 (2004), 133-157. doi: 10.1088/0951-7715/17/1/009. [21] Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons, Science, 268 (1995), 1503-1506. doi: 10.1126/science.7770778. [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), 2647-2663. doi: 10.1152/jn.00900.2006. [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), 925-931. doi: 10.1016/S0925-2312(02)00838-X. [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), 8137-8142. doi: 10.1073/pnas.0702799104. [25] M. Pernarowski, Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), 814-832. doi: 10.1137/S003613999223449X. [26] P. Fatt and B. Katz, Spontaneous subthreshold activity at motor nerve endings, J Physiol., 117 (1952), 109-128. [27] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Sysems: A Guide to XPPAUT for Researchers and Students," Software Environ. Tools 14, SIAM, Philadelphia, 2002. [28] G. L. Gerstein and N. Y. Kiang, An approach to the quantitative analysis of electrophysiological data from single neurons, Biophys. J., 1 (1960), 15-28. doi: 10.1016/S0006-3495(60)86872-5. [29] N. B. Rushforth, Behavioral and electrophysiological studies of hydra. I. Analysis of contraction pulse patterns, Biol. Bull., 140 (1971), 255-273. doi: 10.2307/1540073. [30] H. Shimazaki and S. Shinomoto, A method for selecting the bin size of a time histogram, Neural Computation, 19 (2007), 1503-1527. doi: 10.1162/neco.2007.19.6.1503. [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), 487-501. doi: 10.1093/cercor/7.6.487. [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), 398-415. [33] J. E. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E, 74 (2006), 021917. doi: 10.1103/PhysRevE.74.021917. [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), 5585-5590. doi: 10.1103/PhysRevE.54.5585. [35] S. Tanabe and K. Pakdaman, Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations, Phys. Rev. E, 63 (2001), 031911. doi: 10.1103/PhysRevE.63.031911. [36] M. G. Pedersen, A comment on noise enhanced bursting in pancreatic beta-cells, J. Theoret. Biol., 235 (2005), 1-3. doi: 10.1016/j.jtbi.2005.01.025. [37] M. Frey and E. Simiu, Noise-induced chaos and phase space flux, Phys. 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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), 1266-1273. doi: 10.2337/diabetes.47.8.1266. [2] P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic $\beta$-cells under voltage clamp conditions, J. Physiol., 374 (1986), 531-550. [3] P. M. Dean and E. K. Matthews, Glucose-induced electrical activity in pancreatic islet cells, J. Physiol., 210 (1970), 255-264. [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), 470-475. doi: 10.1007/BF00257788. [5] F. M. Ashcroft and P. Rorsman, Electrophysiology of the pancreatic $\beta$-cell, Prog. Biophys. Mol. Biol., 54 (1989), 87-143. doi: 10.1016/0079-6107(89)90013-8. [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), 495-513. doi: 10.3934/dcdsb.2010.14.495. [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), 513-530. doi: 10.1006/jtbi.2000.2193. [8] M. I. Freidlin, Quasi-deterministic approximation, metastability and stochastic resonance, Phys. D, 137 (2000), 333-352. doi: 10.1016/S0167-2789(99)00191-8. [9] L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance, Rev. Mod. Phys., 70 (1998), 223-287. doi: 10.1103/RevModPhys.70.223. [10] T. Wellens, V. Shatokhin and A. Buchleitner, Stochastic resonance, Rep. Prog. Phys., 67 (2004), 45-105. doi: 10.1088/0034-4885/67/1/R02. [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), 636-643. doi: 10.1038/ng1807. [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), 840-846. doi: 10.1038/nature04785. [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), 69-73. doi: 10.1038/ng869. [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), 1716-1719. doi: 10.1126/science.1137455. [15] M. Turcotte, J. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state, PNAS, 105 (2008), 15732-15737. doi: 10.1073/pnas.0806349105. [16] I. Atwater, L. Rosario and E. Rojas, Properties of the Ca-activated K+ channel in pancreatic beta-cells, Cell Calcium, 4 (1983), 451-461. doi: 10.1016/0143-4160(83)90021-0. [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), 427-435. doi: 10.1016/S0006-3495(88)82976-X. [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), 411-425. doi: 10.1016/S0006-3495(88)82975-8. [19] M. Pedersen and M. Sørensen, The effect of noise on $\beta$-cell burst period, SIAM J. Appl. Math., 67 (2007), 530-542. doi: 10.1137/060655663. [20] J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters, Nonlinearity, 17 (2004), 133-157. doi: 10.1088/0951-7715/17/1/009. [21] Z. F. Mainen and T. J. Sejnowski, Reliability of spike timing in neocortical neurons, Science, 268 (1995), 1503-1506. doi: 10.1126/science.7770778. [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), 2647-2663. doi: 10.1152/jn.00900.2006. [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), 925-931. doi: 10.1016/S0925-2312(02)00838-X. [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), 8137-8142. doi: 10.1073/pnas.0702799104. [25] M. Pernarowski, Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), 814-832. doi: 10.1137/S003613999223449X. [26] P. Fatt and B. Katz, Spontaneous subthreshold activity at motor nerve endings, J Physiol., 117 (1952), 109-128. [27] B. Ermentrout, "Simulating, Analyzing, and Animating Dynamical Sysems: A Guide to XPPAUT for Researchers and Students," Software Environ. Tools 14, SIAM, Philadelphia, 2002. [28] G. L. Gerstein and N. Y. Kiang, An approach to the quantitative analysis of electrophysiological data from single neurons, Biophys. J., 1 (1960), 15-28. doi: 10.1016/S0006-3495(60)86872-5. [29] N. B. Rushforth, Behavioral and electrophysiological studies of hydra. I. Analysis of contraction pulse patterns, Biol. Bull., 140 (1971), 255-273. doi: 10.2307/1540073. [30] H. Shimazaki and S. Shinomoto, A method for selecting the bin size of a time histogram, Neural Computation, 19 (2007), 1503-1527. doi: 10.1162/neco.2007.19.6.1503. [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), 487-501. doi: 10.1093/cercor/7.6.487. [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), 398-415. [33] J. E. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Phys. Rev. E, 74 (2006), 021917. doi: 10.1103/PhysRevE.74.021917. [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), 5585-5590. doi: 10.1103/PhysRevE.54.5585. [35] S. Tanabe and K. Pakdaman, Dynamics of moments of FitzHugh-Nagumo neuronal models and stochastic bifurcations, Phys. Rev. E, 63 (2001), 031911. doi: 10.1103/PhysRevE.63.031911. [36] M. G. Pedersen, A comment on noise enhanced bursting in pancreatic beta-cells, J. Theoret. Biol., 235 (2005), 1-3. doi: 10.1016/j.jtbi.2005.01.025. [37] M. Frey and E. Simiu, Noise-induced chaos and phase space flux, Phys. D, 63 (1993), 321-340. doi: 10.1016/0167-2789(93)90114-G. [38] N. Berglund and B. Gentz, "Stochastic Dynamic Bifurcations and Excitability in: Stochastic Methods in Neuroscience," Carlo Laing and Gabriel Lord (eds.), Oxford University Press, 2009.
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