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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.
doi: 10.1016/j.nonrwa.2004.09.002. |
[3] |
W. Arendt, One-parameter semigroups of positive operators,, Lecture Notes in Mathematics, 1184 (1980). Google Scholar |
[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.
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.
doi: 10.1016/0079-6107(70)90013-1. |
[7] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Mathematische Zeitschrift, 210 (1992), 413.
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, 50 (2002), 49.
|
[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.
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.
|
[13] |
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies,, Journal of Computational Neuroscience, 6 (1999), 169.
doi: 10.1023/A:1008832702585. |
[14] |
J. P. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998).
|
[15] |
C. Koch, "Biophysics of Computation,", Oxford University Press, (1999). Google Scholar |
[16] |
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math, 42 (1989), 1097.
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.
doi: 10.1073/pnas.0402335101. |
[18] |
J. L. Lions and M. E., "Non-Homogeneous Boundary Value Problems and Applications,", Springer-Verlag New York, (1972). Google Scholar |
[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.
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, (1981), 230.
doi: 10.1007/BFb0091916. |
[22] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM Journal on Mathematical Analysis, 20 (1989).
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.
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, (2007).
|
[25] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues,, Critical Reviews in Biomedical Engineering, 21 (1993), 137. Google Scholar |
[26] |
E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs, 31 (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).
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.
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.
doi: 10.1016/0001-8708(78)90075-0. |
[31] |
W. Rudin, "Real and Complex Analysis,", McGraw-Hill, (1987).
|
[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, 258 (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.
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.
doi: 10.1016/j.nonrwa.2004.09.002. |
[3] |
W. Arendt, One-parameter semigroups of positive operators,, Lecture Notes in Mathematics, 1184 (1980). Google Scholar |
[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.
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.
doi: 10.1016/0079-6107(70)90013-1. |
[7] |
J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Mathematische Zeitschrift, 210 (1992), 413.
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, 50 (2002), 49.
|
[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.
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.
|
[13] |
G. R. Holt and C. Koch, Electrical interactions via the extracellular potential near cell bodies,, Journal of Computational Neuroscience, 6 (1999), 169.
doi: 10.1023/A:1008832702585. |
[14] |
J. P. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998).
|
[15] |
C. Koch, "Biophysics of Computation,", Oxford University Press, (1999). Google Scholar |
[16] |
J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements,, Comm. Pure Appl. Math, 42 (1989), 1097.
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.
doi: 10.1073/pnas.0402335101. |
[18] |
J. L. Lions and M. E., "Non-Homogeneous Boundary Value Problems and Applications,", Springer-Verlag New York, (1972). Google Scholar |
[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.
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, (1981), 230.
doi: 10.1007/BFb0091916. |
[22] |
M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM Journal on Mathematical Analysis, 20 (1989).
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.
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, (2007).
|
[25] |
J. C. Neu and W. Krassowska, Homogenization of syncytial tissues,, Critical Reviews in Biomedical Engineering, 21 (1993), 137. Google Scholar |
[26] |
E. M. Ouhabaz, "Analysis of Heat Equations on Domains,", London Mathematical Society Monographs, 31 (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).
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.
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.
doi: 10.1016/0001-8708(78)90075-0. |
[31] |
W. Rudin, "Real and Complex Analysis,", McGraw-Hill, (1987).
|
[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, 258 (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.
doi: 10.1002/mma.740. |
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