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Global existence and uniqueness of a three-dimensional model of cellular electrophysiology
1. | Graduate School of Mathematical Sciences, University of Tokyo, Komaba Tokyo, 153-8914 |
2. | School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis MN, 55414, United States |
References:
[1] |
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. |
[2] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 367-380.
doi: 10.1016/j.nonrwa.2004.09.002. |
[3] |
W. Arendt, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, ch. B-II, "Characterization of Positive Semigroups on $C_0(X)$," Springer-Verlag, 1980. |
[4] |
V. Barcilon, J. D. Cole and R. S. Eisenberg, A singular perturbation analysis of induced electric fields in nerve cells, SIAM Journal on Applied Mathematics, 21 (1971), 339-354.
doi: 10.1137/0121036. |
[5] |
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge University Press, 1990. |
[6] |
R. S. Eisenberg and E. A. Johnson, Three-dimensional electrical field problems in physiology, Prog. Biophys. Mol. Biol, 20 (1970), 1-65.
doi: 10.1016/0079-6107(70)90013-1. |
[7] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Mathematische Zeitschrift, 210 (1992), 413-439.
doi: 10.1007/BF02571805. |
[8] |
G. B. Folland, "Introduction to Partial Differential Equations," Princeton University Press, 1995. |
[9] |
P. C. Franzone and G. Savare, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, Evolution Equations, Semigroups and Functional Analysis: In memory of Brunello Terreni, 50 (2002), 49-78. |
[10] |
C. Gold, D. A. Henze, C. Koch and G. Buzsaki, On the origin of the extracellular action potential waveform: A modeling study, Journal of Neurophysiology, 95 (2006), 3113-3128.
doi: 10.1152/jn.00979.2005. |
[11] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Mathematical Society, 1988. |
[12] |
T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Royal Soc. Edinburgh A, 113 (1989), 43-60. |
[13] |
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies, Journal of Computational Neuroscience, 6 (1999), 169-184.
doi: 10.1023/A:1008832702585. |
[14] |
J. P. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998. |
[15] |
C. Koch, "Biophysics of Computation," Oxford University Press, New York, 1999. |
[16] |
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math, 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[17] |
M. Léonetti, E. Dubois-Violette and F. Homblé, Pattern formation of stationaly transcellular ionic currents in Fucus, Proc. Natl. Acad. Sci. USA, 101 (2004), 10243-10248.
doi: 10.1073/pnas.0402335101. |
[18] |
J. L. Lions and M. E., "Non-Homogeneous Boundary Value Problems and Applications," Springer-Verlag New York, 1972. |
[19] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, 1995. |
[20] |
J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, Journal of Differential Equations, 22 (1976), 331-348.
doi: 10.1016/0022-0396(76)90032-2. |
[21] |
R. Mané, "On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps," Dynamical Systems and Turbulence, Warwick 1980 (1981), 230-242.
doi: 10.1007/BFb0091916. |
[22] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 20 (1989), 816.
doi: 10.1137/0520057. |
[23] |
Y. Mori, G. I. Fishman and C. S. Peskin, Ephaptic conduction in a cardiac strand model with 3d electrodiffusion, Proceedings of the National Academy of Sciences, 105 (2008), 6463-6468.
doi: 10.1073/pnas.0801089105. |
[24] |
Y. Mori, J. W. Jerome and C. S. Peskin, "A Three-Dimensional Model of Cellular Electrical Activity," Bulletin of the Institute of Mathematics, Academia Sinica, Taiwan, 2007. |
[25] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. |
[26] |
E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs, vol. 31, Princeton University Press, 2005. |
[27] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, 1983. |
[28] |
M. Pennacchio, G. Savare and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2006), 1333.
doi: 10.1137/040615249. |
[29] |
W. Rall, Distribution of potential in cylindrical coordinates and time constants for a membrane cylinder, Biophys. J., 9 (1969), 1509-1541.
doi: 10.1016/S0006-3495(69)86468-4. |
[30] |
J. Rauch and J. Smoller, Qualitative theory of the fitzhugh-nagumo equations, Advances in Mathematics, 27 (1978), 12-44.
doi: 10.1016/0001-8708(78)90075-0. |
[31] | |
[32] |
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," Walter de Gruyter, 1996. |
[33] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer Verlag, 2002. |
[34] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Grundlehren der mathematischen Wissenschaften, vol. 258, Springer-Verlag, 1994. |
[35] |
M. E. Taylor, "Partial Differential Equations, vol. I, II, III," Springer-Verlag, 1996. |
[36] |
M. Veneroni, Reaction diffusion systems for the microscopic cellular model of the cardiac electric field, Mathematical Methods in the Applied Sciences, 29 (2006), 1631-1661.
doi: 10.1002/mma.740. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, 2003. |
[2] |
M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Existence and uniqueness for an elliptic problem with evolution arising in electrodynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 367-380.
doi: 10.1016/j.nonrwa.2004.09.002. |
[3] |
W. Arendt, One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184, ch. B-II, "Characterization of Positive Semigroups on $C_0(X)$," Springer-Verlag, 1980. |
[4] |
V. Barcilon, J. D. Cole and R. S. Eisenberg, A singular perturbation analysis of induced electric fields in nerve cells, SIAM Journal on Applied Mathematics, 21 (1971), 339-354.
doi: 10.1137/0121036. |
[5] |
E. B. Davies, "Heat Kernels and Spectral Theory," Cambridge University Press, 1990. |
[6] |
R. S. Eisenberg and E. A. Johnson, Three-dimensional electrical field problems in physiology, Prog. Biophys. Mol. Biol, 20 (1970), 1-65.
doi: 10.1016/0079-6107(70)90013-1. |
[7] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions, Mathematische Zeitschrift, 210 (1992), 413-439.
doi: 10.1007/BF02571805. |
[8] |
G. B. Folland, "Introduction to Partial Differential Equations," Princeton University Press, 1995. |
[9] |
P. C. Franzone and G. Savare, Degenerate evolution systems modeling the cardiac electric field at micro and macroscopic level, Evolution Equations, Semigroups and Functional Analysis: In memory of Brunello Terreni, 50 (2002), 49-78. |
[10] |
C. Gold, D. A. Henze, C. Koch and G. Buzsaki, On the origin of the extracellular action potential waveform: A modeling study, Journal of Neurophysiology, 95 (2006), 3113-3128.
doi: 10.1152/jn.00979.2005. |
[11] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Amer. Mathematical Society, 1988. |
[12] |
T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Royal Soc. Edinburgh A, 113 (1989), 43-60. |
[13] |
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies, Journal of Computational Neuroscience, 6 (1999), 169-184.
doi: 10.1023/A:1008832702585. |
[14] |
J. P. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998. |
[15] |
C. Koch, "Biophysics of Computation," Oxford University Press, New York, 1999. |
[16] |
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math, 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[17] |
M. Léonetti, E. Dubois-Violette and F. Homblé, Pattern formation of stationaly transcellular ionic currents in Fucus, Proc. Natl. Acad. Sci. USA, 101 (2004), 10243-10248.
doi: 10.1073/pnas.0402335101. |
[18] |
J. L. Lions and M. E., "Non-Homogeneous Boundary Value Problems and Applications," Springer-Verlag New York, 1972. |
[19] |
A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems," Birkhäuser, 1995. |
[20] |
J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, Journal of Differential Equations, 22 (1976), 331-348.
doi: 10.1016/0022-0396(76)90032-2. |
[21] |
R. Mané, "On the Dimension of the Compact Invariant Sets of Certain Non-Linear Maps," Dynamical Systems and Turbulence, Warwick 1980 (1981), 230-242.
doi: 10.1007/BFb0091916. |
[22] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 20 (1989), 816.
doi: 10.1137/0520057. |
[23] |
Y. Mori, G. I. Fishman and C. S. Peskin, Ephaptic conduction in a cardiac strand model with 3d electrodiffusion, Proceedings of the National Academy of Sciences, 105 (2008), 6463-6468.
doi: 10.1073/pnas.0801089105. |
[24] |
Y. Mori, J. W. Jerome and C. S. Peskin, "A Three-Dimensional Model of Cellular Electrical Activity," Bulletin of the Institute of Mathematics, Academia Sinica, Taiwan, 2007. |
[25] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. |
[26] |
E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Mathematical Society Monographs, vol. 31, Princeton University Press, 2005. |
[27] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, 1983. |
[28] |
M. Pennacchio, G. Savare and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2006), 1333.
doi: 10.1137/040615249. |
[29] |
W. Rall, Distribution of potential in cylindrical coordinates and time constants for a membrane cylinder, Biophys. J., 9 (1969), 1509-1541.
doi: 10.1016/S0006-3495(69)86468-4. |
[30] |
J. Rauch and J. Smoller, Qualitative theory of the fitzhugh-nagumo equations, Advances in Mathematics, 27 (1978), 12-44.
doi: 10.1016/0001-8708(78)90075-0. |
[31] | |
[32] |
T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations," Walter de Gruyter, 1996. |
[33] |
G. R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer Verlag, 2002. |
[34] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Grundlehren der mathematischen Wissenschaften, vol. 258, Springer-Verlag, 1994. |
[35] |
M. E. Taylor, "Partial Differential Equations, vol. I, II, III," Springer-Verlag, 1996. |
[36] |
M. Veneroni, Reaction diffusion systems for the microscopic cellular model of the cardiac electric field, Mathematical Methods in the Applied Sciences, 29 (2006), 1631-1661.
doi: 10.1002/mma.740. |
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