2010, 7(4): 793-807. doi: 10.3934/mbe.2010.7.793

Minimal state models for ionic channels involved in glucagon secretion

1. 

Dept. Ciencias Básicas, Universidad Autónoma Metropolitana Azcapotzalco, México D.F., 02200, Mexico

2. 

Dept. Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Santander, 39005, Spain

3. 

Instituto de Bioingeniería, Universidad Miguel Hernández, Elche, 03202, Spain

Received  April 2010 Revised  August 2010 Published  October 2010

Pancreatic alpha cells synthesize and release glucagon. This hormone along with insulin, preserves blood glucose levels within a physiological range. During low glucose levels, alpha cells exhibit electrical activity related to glucagon secretion. In this paper, we introduce minimal state models for those ionic channels involved in this electrical activity in mice alpha cells. For estimation of model parameters, we use Monte Carlo algorithms to fit steady-state channel currents. Then, we simulate dynamic ionic currents following experimental protocols. Our aims are 1) To understand the individual ionic channel functioning and modulation that could affect glucagon secretion, and 2) To simulate ionic currents actually measured in voltage-clamp alpha-cell experiments in mice. Our estimations indicate that alpha cells are highly permeable to sodium and potassium which mainly manage action potentials. We have also found that our estimated N-type calcium channel population and density in alpha cells is in good agreement to those reported for L-type calcium channels in beta cells. This finding is strongly relevant since both, L-type and N-type calcium channels, play a main role in insulin and glucagon secretion, respectively.
Citation: Virginia González-Vélez, Amparo Gil, Iván Quesada. Minimal state models for ionic channels involved in glucagon secretion. Mathematical Biosciences & Engineering, 2010, 7 (4) : 793-807. doi: 10.3934/mbe.2010.7.793
References:
[1]

G. C. Amberg, S. D. Koh, Y. Imaizumi, S. Ohya and K. M. Sanders, A-type potassium currents in smooth muscle,, Am. J. Physiol. Cell Physiol., 284 (2003), 583. Google Scholar

[2]

S. Barg, J. Galvanovskis, S. O. Göpel, P. Rorsman and L. Eliasson, Tight coupling between electrical activity and exocytosis in mouse glucagon-secreting $\alpha$-cells,, Diabetes, 49 (2000), 1500. doi: doi:10.2337/diabetes.49.9.1500. Google Scholar

[3]

S. Barg, X. Ma, L. Eliasson, et al., Fast exocytosis with few ca channels in insulin-secreting mouse pancreatic b cells,, Biophys. J., 81 (2001), 3308. doi: doi:10.1016/S0006-3495(01)75964-4. Google Scholar

[4]

K. C. Bittner and D. A. Hanck, The relationship between single-channel and whole-cell conductance in the t-type $ca^{2+}$ channel $ca_v3.1$,, Biophys. J., 95 (2008), 931. doi: doi:10.1529/biophysj.107.128124. Google Scholar

[5]

M. Brissova, M. J. Fowler, W. E. Nicholson, A. Chu, B. Hirshberg, D. M. Harlan and A. C. Powers, Assessment of human pancreatic islet architecture and composition by laser scanning confocal microscopy,, J. Histochem. Cytochem., 53 (2005), 1087. doi: doi:10.1369/jhc.5C6684.2005. Google Scholar

[6]

R. C. Cannon and G. D'Alessandro, The ion channel inverse problem: Neuroinformatics meets biophysics,, PLoS Computational Biology, 2 (2006), 862. doi: doi:10.1371/journal.pcbi.0020091. Google Scholar

[7]

P. M. Diderichsen and S. O. Göpel, Modelling the electrical activity of pancreatic $\alpha$-cells based on experimental data from intact mouse islets,, J. Biol. Phys., 32 (2006), 209. doi: doi:10.1007/s10867-006-9013-0. Google Scholar

[8]

C. P. Fall, E. S. Marland, J. M. Wagner and J. J. Tyson, "Computational Cell Biology,", 1st ed., (2002). Google Scholar

[9]

Z.-P. Feng, J. Hamid, C. Doering, S. E. Jarvis, G. M. Bosey, E. Bourinet, T. P. Snutch and G. W. Zamponi, Amino acid residues outside of the pore region contribute to n-type calcium channel permeation,, J. Biol. Chem., 276 (2001), 5726. doi: doi:10.1074/jbc.C000791200. Google Scholar

[10]

A. Gil and V. González-Vélez, Exocytotic dynamics and calcium cooperativity effects in the calyx of held synapse: A modelling study,, J. Comput. Neurosci., 28 (2010), 65. doi: doi:10.1007/s10827-009-0187-x. Google Scholar

[11]

V. González-Vélez and H. González-Vélez, Parallel stochastic simulation of macroscopic calcium currents,, J. Bioinf. Comput. Biol., 5 (2007), 755. doi: doi:10.1142/S0219720007002679. Google Scholar

[12]

S. O. Göpel, T. Kanno, S. Barg, X-G. Weng, J. Gromada and P. Rorsman, Regulation of glucagon release in mouse $\alpha$-cells by $k_{ATP}$ channels and inactivation of ttx-sensitive $na^{+}$ channels,, J. Physiol., 528 (2000), 509. doi: doi:10.1111/j.1469-7793.2000.00509.x. Google Scholar

[13]

J. Gromada, K. Bokvist, W.-G. Ding, S. Barg, K. Buschard, E. Renström and P. Rorsman, Adrenaline stimulates glucagon secretion in pancreatic a-cells by increasing the $ca^{2+}$ current and the number of granules close to the l-type $ca^{2+}$ channels,, J. Gen. Physiol., 110 (1997), 217. doi: doi:10.1085/jgp.110.3.217. Google Scholar

[14]

J. Gromada, I. Franklin and C. B. Wollheim, $alpha$-cells of the endocrine pancreas: 35 years of research but the enigma remains,, Endocrine Rev. 28 (2007), 28 (2007), 84. doi: doi:10.1210/er.2006-0007. Google Scholar

[15]

M. Gurkiewicz and A. Korngreen, A numerical approach to ion channel modelling using whole-cell voltage-clamp recordings and a genetic algorithm,, PLoS Computational Biology, 3 (2007), 1633. doi: doi:10.1371/journal.pcbi.0030169. Google Scholar

[16]

H. Kasai and E. Neher, Dihydropyridine-sensitive and omega-conotoxin-sensitive calcium channels in a mammalian neuroblastoma-glioma cell line,, J. Physiol., 448 (1992), 161. Google Scholar

[17]

J. Klingauf and E. Neher, Modeling buffered ca$^{2+}$ diffusion near the membrane: Implications for secretion in neuroendocrine cells,, Biophys. J., 72 (1997), 674. doi: doi:10.1016/S0006-3495(97)78704-6. Google Scholar

[18]

Y. M. Leung, I. Ahmed, L. Sheu, R. G. Tsushima, N. E. Diamant and H. Y. Gaisano, Two populations of pancreatic islet $\alpha$-cells displaying distinct $ca^{2+}$ channel properties,, Biochem. Biophys. Res. Comm., 345 (2006), 340. doi: doi:10.1016/j.bbrc.2006.04.066. Google Scholar

[19]

Y. M. Leung, I. Ahmed, L. Sheu, R. G. Tsushima, N. E. Diamant, M. Hara and H. Y. Gaisano, Electrophysiological characterization of pancreatic islet cells in the mouse insulin promoter-green fluorescent protein mouse,, Endocrinology, 146 (2005), 4766. doi: doi:10.1210/en.2005-0803. Google Scholar

[20]

P. E. MacDonald, Y. Z. De Marinis, R. Ramracheya, A. Salehi, X. Ma, P. R. V. Johnson, R. Cox, L. Eliasson and P. Rorsman, A $k_{ATP}$ channel-dependent pathway within $\alpha$ cells regulates glucagon release from both rodent and human islets of langerhans,, PLoS Biology, 5 (2007), 1236. doi: doi:10.1371/journal.pbio.0050143. Google Scholar

[21]

M. E. Meyer-Hermann, The electrophysiology of the $\beta$-cell based on single transmembrane protein characteristics,, Biophys. J., 93 (2007), 2952. doi: doi:10.1529/biophysj.107.106096. Google Scholar

[22]

L. S. Milescu, G. Akk and F. Sachs, Maximum likelihood estimation of ion channel kinetics from macroscopic currents,, Biophys. J., 88 (2005), 2494. doi: doi:10.1529/biophysj.104.053256. Google Scholar

[23]

F. Qin, Principles of single-channel kinetic analysis,, Meth. Mol. Biol., 288 (2007), 253. doi: doi:10.1007/978-1-59745-529-9_17. Google Scholar

[24]

I. Quesada, E. Tudurí, C. Ripoll and A. Nadal, Physiology of the pancreatic $\alpha$-cell and glucagon secretion: Role in glucose homeostasis and diabetes,, J. Endocrin., 199 (2008), 5. doi: doi:10.1677/JOE-08-0290. Google Scholar

[25]

M. S. P. Sansom, F. G. Ball, C. J. Kerry, R. McGee, R. L. Ramsey and P. N. R. Usherwood, Markov, fractal, diffusion, and related models of ion channel gating,, Biophys. J., 56 (1989), 1229. doi: doi:10.1016/S0006-3495(89)82770-5. Google Scholar

[26]

J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters,, Biophys. J., 79 (2000), 1771. doi: doi:10.1016/S0006-3495(00)76429-0. Google Scholar

[27]

J. R. Serrano, E. Pérez-Reyes and S. W. Jones, State-dependent inactivation of the $\alpha$1g t-type calcium channel,, J. Gen. Physiol., 114 (1999), 185. doi: doi:10.1085/jgp.114.2.185. Google Scholar

[28]

M. F. Sheets, B. E. Scanley, D. A. Hanck, J. C. Makielski and H. A. Fozzard, Open sodium channel properties of single canine cardiac purkinje cells,, Biophys. J., 52 (1987), 13. doi: doi:10.1016/S0006-3495(87)83183-1. Google Scholar

[29]

M. Slucca, J. S. Harmon, E. A. Oseid, J. Bryan and R. P. Robertson, Atp-sensitive $k^+$ channel mediates the zinc switch-off signal for glucagon response during glucose deprivation,, Diabetes, 59 (2010), 128. doi: doi:10.2337/db09-1098. Google Scholar

[30]

D. O. Smith, J. L. Rosenheimer and R. E. Kalil, Delayed rectifier and a-type potassium channels associated with $k_v2.1$ and $k_v4.3$ expression in embryonic rat neural progenitor cells,, PLoS One, 3 (2008), 1. doi: doi:10.1371/journal.pone.0001604. Google Scholar

[31]

A. F. Strassberg and L. J. DeFelice, Limitations of the Hodgkin-Huxley formalism: Effects of single channel kinetics on transmembrane voltage dynamics,, Neural Comp., 5 (1993), 843. doi: doi:10.1162/neco.1993.5.6.843. Google Scholar

[32]

N. A. Tamarina, A. Kuznetsov, L. E. Fridlyand and L. H. Philipson, Delayed-rectifier ($k_v2.1$) regulation of pancreatic $\beta$-cell calcium responses to glucose: Inhibitor specificity and modeling,, Am. J. Physiol. Endocrinol. Metab., 289 (2005). doi: doi:10.1152/ajpendo.00054.2005. Google Scholar

[33]

T. I. Tóth and V. Crunelli, Estimation of the activation and kinetic properties of $i_{Na}$ and $i_k$ from the time course of the action potential,, J. Neurosci. Meth., 111 (2001), 111. doi: doi:10.1016/S0165-0270(01)00433-2. Google Scholar

[34]

E. Tudurí, E. Filiputti, E. M. Carneiro and I. Quesada, Inhibition of $ca^{2+}$ signaling and glucagon secretion in mouse pancreatic $\alpha$-cells by extracellular atp and purinergic receptors,, Am. J. Physiol. Endocrinol. Metab., 294 (2008). doi: doi:10.1152/ajpendo.00641.2007. Google Scholar

[35]

C. A. Vandenberg and F. Bezanilla, A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon,, Biophys. J., 60 (1991), 1511. doi: doi:10.1016/S0006-3495(91)82186-5. Google Scholar

[36]

S. Vignali, V. Leiss, R. Karl, F. Hofmann and A. Welling, Characterization of voltage-dependent sodium and calcium channels in mouse pancreatic a- and b-cells,, J. Physiol., 572 (2006), 691. Google Scholar

[37]

J. Villanueva, C. J. Torregrosa-Hetland, A. Gil, V. González-Vélez, J. Segura, S. Viniegra and L. M. Gutiérrez, The organization of the secretory machinery in neuroendocrine chromaffin cells as a major factor to model exocytosis,, HFSP Journal, 4 (2010), 85. doi: doi:10.2976/1.3338707. Google Scholar

[38]

S. Wang, V. E. Bondarenko, Y. Qu, M. J. Morales, R. L. Rasmusson and H. C. Strauss, Activation properties of $k_v4.3$ channels: Time, voltage and $[k^+]_o$ dependence,, J. Physiol., 557 (2004), 705. doi: doi:10.1113/jphysiol.2003.058578. Google Scholar

[39]

A. R. Willms, Neurofit: Software for fitting hodgkin-huxley models to voltage-clamp data,, J. Neurosci. Meth., 121 (2002), 139. doi: doi:10.1016/S0165-0270(02)00227-3. Google Scholar

[40]

A. R. Willms, D. J. Baro, R. M. Harris-Warrick and J. Guckenheimer, An improved parameter estimation method for hodgkin-huxley models,, J. Comp. Neurosci., 6 (1999), 145. doi: doi:10.1023/A:1008880518515. Google Scholar

[41]

G. Zamponi, "Voltage-gated Calcium Channels,", 1st ed., (2005). Google Scholar

show all references

References:
[1]

G. C. Amberg, S. D. Koh, Y. Imaizumi, S. Ohya and K. M. Sanders, A-type potassium currents in smooth muscle,, Am. J. Physiol. Cell Physiol., 284 (2003), 583. Google Scholar

[2]

S. Barg, J. Galvanovskis, S. O. Göpel, P. Rorsman and L. Eliasson, Tight coupling between electrical activity and exocytosis in mouse glucagon-secreting $\alpha$-cells,, Diabetes, 49 (2000), 1500. doi: doi:10.2337/diabetes.49.9.1500. Google Scholar

[3]

S. Barg, X. Ma, L. Eliasson, et al., Fast exocytosis with few ca channels in insulin-secreting mouse pancreatic b cells,, Biophys. J., 81 (2001), 3308. doi: doi:10.1016/S0006-3495(01)75964-4. Google Scholar

[4]

K. C. Bittner and D. A. Hanck, The relationship between single-channel and whole-cell conductance in the t-type $ca^{2+}$ channel $ca_v3.1$,, Biophys. J., 95 (2008), 931. doi: doi:10.1529/biophysj.107.128124. Google Scholar

[5]

M. Brissova, M. J. Fowler, W. E. Nicholson, A. Chu, B. Hirshberg, D. M. Harlan and A. C. Powers, Assessment of human pancreatic islet architecture and composition by laser scanning confocal microscopy,, J. Histochem. Cytochem., 53 (2005), 1087. doi: doi:10.1369/jhc.5C6684.2005. Google Scholar

[6]

R. C. Cannon and G. D'Alessandro, The ion channel inverse problem: Neuroinformatics meets biophysics,, PLoS Computational Biology, 2 (2006), 862. doi: doi:10.1371/journal.pcbi.0020091. Google Scholar

[7]

P. M. Diderichsen and S. O. Göpel, Modelling the electrical activity of pancreatic $\alpha$-cells based on experimental data from intact mouse islets,, J. Biol. Phys., 32 (2006), 209. doi: doi:10.1007/s10867-006-9013-0. Google Scholar

[8]

C. P. Fall, E. S. Marland, J. M. Wagner and J. J. Tyson, "Computational Cell Biology,", 1st ed., (2002). Google Scholar

[9]

Z.-P. Feng, J. Hamid, C. Doering, S. E. Jarvis, G. M. Bosey, E. Bourinet, T. P. Snutch and G. W. Zamponi, Amino acid residues outside of the pore region contribute to n-type calcium channel permeation,, J. Biol. Chem., 276 (2001), 5726. doi: doi:10.1074/jbc.C000791200. Google Scholar

[10]

A. Gil and V. González-Vélez, Exocytotic dynamics and calcium cooperativity effects in the calyx of held synapse: A modelling study,, J. Comput. Neurosci., 28 (2010), 65. doi: doi:10.1007/s10827-009-0187-x. Google Scholar

[11]

V. González-Vélez and H. González-Vélez, Parallel stochastic simulation of macroscopic calcium currents,, J. Bioinf. Comput. Biol., 5 (2007), 755. doi: doi:10.1142/S0219720007002679. Google Scholar

[12]

S. O. Göpel, T. Kanno, S. Barg, X-G. Weng, J. Gromada and P. Rorsman, Regulation of glucagon release in mouse $\alpha$-cells by $k_{ATP}$ channels and inactivation of ttx-sensitive $na^{+}$ channels,, J. Physiol., 528 (2000), 509. doi: doi:10.1111/j.1469-7793.2000.00509.x. Google Scholar

[13]

J. Gromada, K. Bokvist, W.-G. Ding, S. Barg, K. Buschard, E. Renström and P. Rorsman, Adrenaline stimulates glucagon secretion in pancreatic a-cells by increasing the $ca^{2+}$ current and the number of granules close to the l-type $ca^{2+}$ channels,, J. Gen. Physiol., 110 (1997), 217. doi: doi:10.1085/jgp.110.3.217. Google Scholar

[14]

J. Gromada, I. Franklin and C. B. Wollheim, $alpha$-cells of the endocrine pancreas: 35 years of research but the enigma remains,, Endocrine Rev. 28 (2007), 28 (2007), 84. doi: doi:10.1210/er.2006-0007. Google Scholar

[15]

M. Gurkiewicz and A. Korngreen, A numerical approach to ion channel modelling using whole-cell voltage-clamp recordings and a genetic algorithm,, PLoS Computational Biology, 3 (2007), 1633. doi: doi:10.1371/journal.pcbi.0030169. Google Scholar

[16]

H. Kasai and E. Neher, Dihydropyridine-sensitive and omega-conotoxin-sensitive calcium channels in a mammalian neuroblastoma-glioma cell line,, J. Physiol., 448 (1992), 161. Google Scholar

[17]

J. Klingauf and E. Neher, Modeling buffered ca$^{2+}$ diffusion near the membrane: Implications for secretion in neuroendocrine cells,, Biophys. J., 72 (1997), 674. doi: doi:10.1016/S0006-3495(97)78704-6. Google Scholar

[18]

Y. M. Leung, I. Ahmed, L. Sheu, R. G. Tsushima, N. E. Diamant and H. Y. Gaisano, Two populations of pancreatic islet $\alpha$-cells displaying distinct $ca^{2+}$ channel properties,, Biochem. Biophys. Res. Comm., 345 (2006), 340. doi: doi:10.1016/j.bbrc.2006.04.066. Google Scholar

[19]

Y. M. Leung, I. Ahmed, L. Sheu, R. G. Tsushima, N. E. Diamant, M. Hara and H. Y. Gaisano, Electrophysiological characterization of pancreatic islet cells in the mouse insulin promoter-green fluorescent protein mouse,, Endocrinology, 146 (2005), 4766. doi: doi:10.1210/en.2005-0803. Google Scholar

[20]

P. E. MacDonald, Y. Z. De Marinis, R. Ramracheya, A. Salehi, X. Ma, P. R. V. Johnson, R. Cox, L. Eliasson and P. Rorsman, A $k_{ATP}$ channel-dependent pathway within $\alpha$ cells regulates glucagon release from both rodent and human islets of langerhans,, PLoS Biology, 5 (2007), 1236. doi: doi:10.1371/journal.pbio.0050143. Google Scholar

[21]

M. E. Meyer-Hermann, The electrophysiology of the $\beta$-cell based on single transmembrane protein characteristics,, Biophys. J., 93 (2007), 2952. doi: doi:10.1529/biophysj.107.106096. Google Scholar

[22]

L. S. Milescu, G. Akk and F. Sachs, Maximum likelihood estimation of ion channel kinetics from macroscopic currents,, Biophys. J., 88 (2005), 2494. doi: doi:10.1529/biophysj.104.053256. Google Scholar

[23]

F. Qin, Principles of single-channel kinetic analysis,, Meth. Mol. Biol., 288 (2007), 253. doi: doi:10.1007/978-1-59745-529-9_17. Google Scholar

[24]

I. Quesada, E. Tudurí, C. Ripoll and A. Nadal, Physiology of the pancreatic $\alpha$-cell and glucagon secretion: Role in glucose homeostasis and diabetes,, J. Endocrin., 199 (2008), 5. doi: doi:10.1677/JOE-08-0290. Google Scholar

[25]

M. S. P. Sansom, F. G. Ball, C. J. Kerry, R. McGee, R. L. Ramsey and P. N. R. Usherwood, Markov, fractal, diffusion, and related models of ion channel gating,, Biophys. J., 56 (1989), 1229. doi: doi:10.1016/S0006-3495(89)82770-5. Google Scholar

[26]

J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters,, Biophys. J., 79 (2000), 1771. doi: doi:10.1016/S0006-3495(00)76429-0. Google Scholar

[27]

J. R. Serrano, E. Pérez-Reyes and S. W. Jones, State-dependent inactivation of the $\alpha$1g t-type calcium channel,, J. Gen. Physiol., 114 (1999), 185. doi: doi:10.1085/jgp.114.2.185. Google Scholar

[28]

M. F. Sheets, B. E. Scanley, D. A. Hanck, J. C. Makielski and H. A. Fozzard, Open sodium channel properties of single canine cardiac purkinje cells,, Biophys. J., 52 (1987), 13. doi: doi:10.1016/S0006-3495(87)83183-1. Google Scholar

[29]

M. Slucca, J. S. Harmon, E. A. Oseid, J. Bryan and R. P. Robertson, Atp-sensitive $k^+$ channel mediates the zinc switch-off signal for glucagon response during glucose deprivation,, Diabetes, 59 (2010), 128. doi: doi:10.2337/db09-1098. Google Scholar

[30]

D. O. Smith, J. L. Rosenheimer and R. E. Kalil, Delayed rectifier and a-type potassium channels associated with $k_v2.1$ and $k_v4.3$ expression in embryonic rat neural progenitor cells,, PLoS One, 3 (2008), 1. doi: doi:10.1371/journal.pone.0001604. Google Scholar

[31]

A. F. Strassberg and L. J. DeFelice, Limitations of the Hodgkin-Huxley formalism: Effects of single channel kinetics on transmembrane voltage dynamics,, Neural Comp., 5 (1993), 843. doi: doi:10.1162/neco.1993.5.6.843. Google Scholar

[32]

N. A. Tamarina, A. Kuznetsov, L. E. Fridlyand and L. H. Philipson, Delayed-rectifier ($k_v2.1$) regulation of pancreatic $\beta$-cell calcium responses to glucose: Inhibitor specificity and modeling,, Am. J. Physiol. Endocrinol. Metab., 289 (2005). doi: doi:10.1152/ajpendo.00054.2005. Google Scholar

[33]

T. I. Tóth and V. Crunelli, Estimation of the activation and kinetic properties of $i_{Na}$ and $i_k$ from the time course of the action potential,, J. Neurosci. Meth., 111 (2001), 111. doi: doi:10.1016/S0165-0270(01)00433-2. Google Scholar

[34]

E. Tudurí, E. Filiputti, E. M. Carneiro and I. Quesada, Inhibition of $ca^{2+}$ signaling and glucagon secretion in mouse pancreatic $\alpha$-cells by extracellular atp and purinergic receptors,, Am. J. Physiol. Endocrinol. Metab., 294 (2008). doi: doi:10.1152/ajpendo.00641.2007. Google Scholar

[35]

C. A. Vandenberg and F. Bezanilla, A sodium channel gating model based on single channel, macroscopic ionic, and gating currents in the squid giant axon,, Biophys. J., 60 (1991), 1511. doi: doi:10.1016/S0006-3495(91)82186-5. Google Scholar

[36]

S. Vignali, V. Leiss, R. Karl, F. Hofmann and A. Welling, Characterization of voltage-dependent sodium and calcium channels in mouse pancreatic a- and b-cells,, J. Physiol., 572 (2006), 691. Google Scholar

[37]

J. Villanueva, C. J. Torregrosa-Hetland, A. Gil, V. González-Vélez, J. Segura, S. Viniegra and L. M. Gutiérrez, The organization of the secretory machinery in neuroendocrine chromaffin cells as a major factor to model exocytosis,, HFSP Journal, 4 (2010), 85. doi: doi:10.2976/1.3338707. Google Scholar

[38]

S. Wang, V. E. Bondarenko, Y. Qu, M. J. Morales, R. L. Rasmusson and H. C. Strauss, Activation properties of $k_v4.3$ channels: Time, voltage and $[k^+]_o$ dependence,, J. Physiol., 557 (2004), 705. doi: doi:10.1113/jphysiol.2003.058578. Google Scholar

[39]

A. R. Willms, Neurofit: Software for fitting hodgkin-huxley models to voltage-clamp data,, J. Neurosci. Meth., 121 (2002), 139. doi: doi:10.1016/S0165-0270(02)00227-3. Google Scholar

[40]

A. R. Willms, D. J. Baro, R. M. Harris-Warrick and J. Guckenheimer, An improved parameter estimation method for hodgkin-huxley models,, J. Comp. Neurosci., 6 (1999), 145. doi: doi:10.1023/A:1008880518515. Google Scholar

[41]

G. Zamponi, "Voltage-gated Calcium Channels,", 1st ed., (2005). Google Scholar

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