Global existence and uniqueness of a three-dimensional model of cellular electrophysiology

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October 2011, 29(4): 1573-1636. doi: 10.3934/dcds.2011.29.1573

Global existence and uniqueness of a three-dimensional model of cellular electrophysiology

Hiroshi Matano ^1, and Yoichiro Mori ^2,

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

Received August 2010 Revised October 2010 Published December 2010

Abstract
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We study a three-dimensional model of cellular electrical activity, which is written as a pseudodifferential equation on a closed surface $\Gamma$ in $\R^3$ coupled with a system of ordinary differential equations on $\Gamma$. Previously the existence of a global classical solution was not known, due mainly to the lack of a uniform $L^\infty$ bound. The main difficulty lies in the fact that, unlike the Laplace operator that appears in traditional models, the pseudodifferential operator in the present model does not satisfy the maximum principle. We overcome this difficulty by introducing the notion of "quasipositivity principle" and prove a uniform $L^\infty$ bound of solutions -- hence the existence of global classical solutions -- for a large class of nonlinearities including the FitzHugh-Nagumo and the Hodgkin-Huxley kinetics. We then study the asymptotic behavior of solutions to show that the system possesses a finite dimensional global attractor consisting entirely of smooth functions despite the fact that the system is only partially dissipative. We also show that ordinary differential equation models without spatial extent, often used in modeling studies, can be obtained from the present model in the small-cell-size limit.

Keywords: global attractor, 3D cable model, asymptotic smoothing, pseudodifferential operator., electrophysiology, quasipositivity, partially dissipative systems.

Mathematics Subject Classification: Primary: 35Q80, 35B41; Secondary: 35K55, 35S0.

Citation: Hiroshi Matano, Yoichiro Mori. Global existence and uniqueness of a three-dimensional model of cellular electrophysiology. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1573-1636. doi: 10.3934/dcds.2011.29.1573

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]	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, 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]	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, 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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