American Institute of Mathematical Sciences

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

Minimal state models for ionic channels involved in glucagon secretion

Virginia González-Vélez 1, , Amparo Gil 2, and  Iván Quesada 3,

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.
Keywords: Markov models, Monte Carlo., ionic channels, pancreatic alpha cell.
Mathematics Subject Classification: Primary: 92B05; Secondary: 92C3.
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), 583-595.

[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), 1500-1510. doi: doi:10.2337/diabetes.49.9.1500.

[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), 3308-3323. doi: doi:10.1016/S0006-3495(01)75964-4.

[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), 931-941. doi: doi:10.1529/biophysj.107.128124.

[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), 1087-1097. doi: doi:10.1369/jhc.5C6684.2005.

[6]

R. C. Cannon and G. D'Alessandro, The ion channel inverse problem: Neuroinformatics meets biophysics, PLoS Computational Biology, 2 (2006), 862-869. doi: doi:10.1371/journal.pcbi.0020091.

[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), 209-229. doi: doi:10.1007/s10867-006-9013-0.

[8]

C. P. Fall, E. S. Marland, J. M. Wagner and J. J. Tyson, "Computational Cell Biology," 1st ed., Springer-Verlag, New York, 2002.

[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), 5726-5730. doi: doi:10.1074/jbc.C000791200.

[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), 65-76. doi: doi:10.1007/s10827-009-0187-x.

[11]

V. González-Vélez and H. González-Vélez, Parallel stochastic simulation of macroscopic calcium currents, J. Bioinf. Comput. Biol., 5 (2007), 755-772. doi: doi:10.1142/S0219720007002679.

[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), 509-520. doi: doi:10.1111/j.1469-7793.2000.00509.x.

[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), 217-228. doi: doi:10.1085/jgp.110.3.217.

[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), 84-116. doi: doi:10.1210/er.2006-0007.

[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), 1633-1647. doi: doi:10.1371/journal.pcbi.0030169.

[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), 161-188.

[17]

J. Klingauf and E. Neher, Modeling buffered ca$^{2+}$ diffusion near the membrane: Implications for secretion in neuroendocrine cells, Biophys. J., 72 (1997), 674-690. doi: doi:10.1016/S0006-3495(97)78704-6.

[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), 340-344. doi: doi:10.1016/j.bbrc.2006.04.066.

[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), 4766-4775. doi: doi:10.1210/en.2005-0803.

[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), 1236-1247. doi: doi:10.1371/journal.pbio.0050143.

[21]

M. E. Meyer-Hermann, The electrophysiology of the $\beta$-cell based on single transmembrane protein characteristics, Biophys. J., 93 (2007), 2952-2968. doi: doi:10.1529/biophysj.107.106096.

[22]

L. S. Milescu, G. Akk and F. Sachs, Maximum likelihood estimation of ion channel kinetics from macroscopic currents, Biophys. J., 88 (2005), 2494-2515. doi: doi:10.1529/biophysj.104.053256.

[23]

F. Qin, Principles of single-channel kinetic analysis, Meth. Mol. Biol., 288 (2007), 253-286. doi: doi:10.1007/978-1-59745-529-9_17.

[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), 5-19. doi: doi:10.1677/JOE-08-0290.

[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), 1229-1243. doi: doi:10.1016/S0006-3495(89)82770-5.

[26]

J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters, Biophys. J., 79 (2000), 1771-1786. doi: doi:10.1016/S0006-3495(00)76429-0.

[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), 185-201. doi: doi:10.1085/jgp.114.2.185.

[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), 13-22. doi: doi:10.1016/S0006-3495(87)83183-1.

[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), 128-134. doi: doi:10.2337/db09-1098.

[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), 1-9. doi: doi:10.1371/journal.pone.0001604.

[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), 843-855. doi: doi:10.1162/neco.1993.5.6.843.

[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), E578-E585. doi: doi:10.1152/ajpendo.00054.2005.

[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), 111-126. doi: doi:10.1016/S0165-0270(01)00433-2.

[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), E952-E960. doi: doi:10.1152/ajpendo.00641.2007.

[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), 1511-1533. doi: doi:10.1016/S0006-3495(91)82186-5.

[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), 691-706.

[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), 85-92. doi: doi:10.2976/1.3338707.

[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), 705-717. doi: doi:10.1113/jphysiol.2003.058578.

[39]

A. R. Willms, Neurofit: Software for fitting hodgkin-huxley models to voltage-clamp data, J. Neurosci. Meth., 121 (2002), 139-150. doi: doi:10.1016/S0165-0270(02)00227-3.

[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), 145-168. doi: doi:10.1023/A:1008880518515.

[41]

G. Zamponi, "Voltage-gated Calcium Channels," 1st ed., Kluwer Academic/Plenum Publishers, New York, 2005.

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), 583-595.

[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), 1500-1510. doi: doi:10.2337/diabetes.49.9.1500.

[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), 3308-3323. doi: doi:10.1016/S0006-3495(01)75964-4.

[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), 931-941. doi: doi:10.1529/biophysj.107.128124.

[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), 1087-1097. doi: doi:10.1369/jhc.5C6684.2005.

[6]

R. C. Cannon and G. D'Alessandro, The ion channel inverse problem: Neuroinformatics meets biophysics, PLoS Computational Biology, 2 (2006), 862-869. doi: doi:10.1371/journal.pcbi.0020091.

[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), 209-229. doi: doi:10.1007/s10867-006-9013-0.

[8]

C. P. Fall, E. S. Marland, J. M. Wagner and J. J. Tyson, "Computational Cell Biology," 1st ed., Springer-Verlag, New York, 2002.

[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), 5726-5730. doi: doi:10.1074/jbc.C000791200.

[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), 65-76. doi: doi:10.1007/s10827-009-0187-x.

[11]

V. González-Vélez and H. González-Vélez, Parallel stochastic simulation of macroscopic calcium currents, J. Bioinf. Comput. Biol., 5 (2007), 755-772. doi: doi:10.1142/S0219720007002679.

[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), 509-520. doi: doi:10.1111/j.1469-7793.2000.00509.x.

[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), 217-228. doi: doi:10.1085/jgp.110.3.217.

[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), 84-116. doi: doi:10.1210/er.2006-0007.

[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), 1633-1647. doi: doi:10.1371/journal.pcbi.0030169.

[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), 161-188.

[17]

J. Klingauf and E. Neher, Modeling buffered ca$^{2+}$ diffusion near the membrane: Implications for secretion in neuroendocrine cells, Biophys. J., 72 (1997), 674-690. doi: doi:10.1016/S0006-3495(97)78704-6.

[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), 340-344. doi: doi:10.1016/j.bbrc.2006.04.066.

[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), 4766-4775. doi: doi:10.1210/en.2005-0803.

[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), 1236-1247. doi: doi:10.1371/journal.pbio.0050143.

[21]

M. E. Meyer-Hermann, The electrophysiology of the $\beta$-cell based on single transmembrane protein characteristics, Biophys. J., 93 (2007), 2952-2968. doi: doi:10.1529/biophysj.107.106096.

[22]

L. S. Milescu, G. Akk and F. Sachs, Maximum likelihood estimation of ion channel kinetics from macroscopic currents, Biophys. J., 88 (2005), 2494-2515. doi: doi:10.1529/biophysj.104.053256.

[23]

F. Qin, Principles of single-channel kinetic analysis, Meth. Mol. Biol., 288 (2007), 253-286. doi: doi:10.1007/978-1-59745-529-9_17.

[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), 5-19. doi: doi:10.1677/JOE-08-0290.

[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), 1229-1243. doi: doi:10.1016/S0006-3495(89)82770-5.

[26]

J. Segura, A. Gil and B. Soria, Modeling study of exocytosis in neuroendocrine cells: Influence of the geometrical parameters, Biophys. J., 79 (2000), 1771-1786. doi: doi:10.1016/S0006-3495(00)76429-0.

[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), 185-201. doi: doi:10.1085/jgp.114.2.185.

[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), 13-22. doi: doi:10.1016/S0006-3495(87)83183-1.

[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), 128-134. doi: doi:10.2337/db09-1098.

[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), 1-9. doi: doi:10.1371/journal.pone.0001604.

[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), 843-855. doi: doi:10.1162/neco.1993.5.6.843.

[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), E578-E585. doi: doi:10.1152/ajpendo.00054.2005.

[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), 111-126. doi: doi:10.1016/S0165-0270(01)00433-2.

[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), E952-E960. doi: doi:10.1152/ajpendo.00641.2007.

[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), 1511-1533. doi: doi:10.1016/S0006-3495(91)82186-5.

[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), 691-706.

[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), 85-92. doi: doi:10.2976/1.3338707.

[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), 705-717. doi: doi:10.1113/jphysiol.2003.058578.

[39]

A. R. Willms, Neurofit: Software for fitting hodgkin-huxley models to voltage-clamp data, J. Neurosci. Meth., 121 (2002), 139-150. doi: doi:10.1016/S0165-0270(02)00227-3.

[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), 145-168. doi: doi:10.1023/A:1008880518515.

[41]

G. Zamponi, "Voltage-gated Calcium Channels," 1st ed., Kluwer Academic/Plenum Publishers, New York, 2005.

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