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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

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

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.
Citation: Hiroshi Matano, Yoichiro Mori. Global existence and uniqueness of a three-dimensional model of cellular electrophysiology. Discrete & Continuous Dynamical Systems - A, 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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1990). Google Scholar

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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. Google Scholar

[7]

J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Mathematische Zeitschrift, 210 (1992), 413. doi: 10.1007/BF02571805. Google Scholar

[8]

G. B. Folland, "Introduction to Partial Differential Equations,", Princeton University Press, (1995). Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Mathematical Society, (1988). Google Scholar

[12]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Royal Soc. Edinburgh A, 113 (1989), 43. Google Scholar

[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. Google Scholar

[14]

J. P. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM Journal on Mathematical Analysis, 20 (1989). doi: 10.1137/0520057. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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). Google Scholar

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

W. Rudin, "Real and Complex Analysis,", McGraw-Hill, (1987). Google Scholar

[32]

T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,", Walter de Gruyter, (1996). Google Scholar

[33]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer Verlag, (2002). Google Scholar

[34]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Grundlehren der mathematischen Wissenschaften, 258 (1994). Google Scholar

[35]

M. E. Taylor, "Partial Differential Equations, vol. I, II, III,", Springer-Verlag, (1996). Google Scholar

[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. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", Academic Press, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

E. B. Davies, "Heat Kernels and Spectral Theory,", Cambridge University Press, (1990). Google Scholar

[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. Google Scholar

[7]

J. Escher, Nonlinear elliptic systems with dynamic boundary conditions,, Mathematische Zeitschrift, 210 (1992), 413. doi: 10.1007/BF02571805. Google Scholar

[8]

G. B. Folland, "Introduction to Partial Differential Equations,", Princeton University Press, (1995). Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Amer. Mathematical Society, (1988). Google Scholar

[12]

T. Hintermann, Evolution equations with dynamic boundary conditions,, Proc. Royal Soc. Edinburgh A, 113 (1989), 43. Google Scholar

[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. Google Scholar

[14]

J. P. Keener and J. Sneyd, "Mathematical Physiology,", Springer-Verlag, (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

M. Marion, Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems,, SIAM Journal on Mathematical Analysis, 20 (1989). doi: 10.1137/0520057. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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). Google Scholar

[27]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Springer, (1983). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

W. Rudin, "Real and Complex Analysis,", McGraw-Hill, (1987). Google Scholar

[32]

T. Runst and W. Sickel, "Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations,", Walter de Gruyter, (1996). Google Scholar

[33]

G. R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer Verlag, (2002). Google Scholar

[34]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Grundlehren der mathematischen Wissenschaften, 258 (1994). Google Scholar

[35]

M. E. Taylor, "Partial Differential Equations, vol. I, II, III,", Springer-Verlag, (1996). Google Scholar

[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. Google Scholar

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