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March  2020, 25(3): 815-839. doi: 10.3934/dcdsb.2019191

Derivation of cable equation by multiscale analysis for a model of myelinated axons

Carlos Jerez-Hanckes 1, , Irina Pettersson 2, and  Volodymyr Rybalko 3,

1. 

Faculty of Engineering and Sciences, Universidad Adolfo Ibáñez, Diagonal Las Torres 2700, Peñalolén, Santiago, Chile
2. 

Department of Electrical Engineering, Mathematics and Science, University of Gävle, SE-801 76 Gävle, Sweden
3. 

B. Verkin Institute for Low Temperature Physics and Engineering of NASU, 47 Nauky Ave., 61103 Kharkiv, Ukraine

Received  September 2018 Revised  March 2019 Published  March 2020 Early access  September 2019

Fund Project: This research was supported by the Swedish Foundation for International Cooperation in Research and Higher Education STINT (research grant IB 2017-7370) and Chile Fondecyt Regular 1171491.

We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order $ \varepsilon $, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to $ \varepsilon $ which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model.

Keywords: Hodgkin-Huxley model, nonlinear cable equation, cellular electrophysiology, multiscale modeling, homogenization.
Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q92.
Citation: Carlos Jerez-Hanckes, Irina Pettersson, Volodymyr Rybalko. Derivation of cable equation by multiscale analysis for a model of myelinated axons. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 815-839. doi: 10.3934/dcdsb.2019191
References:
[1]

G. Allaire, Commissariat 'a L'energie Atomique, Alain Damlamian, and Ulrich Hornung, Two-scale convergence on periodic surfaces and applications, Mathematical Modelling of Flow through Porous Media, 1995.

[2]

M. Amar, D. Andreucci, P. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.  doi: 10.1002/mma.709.

[3]

M. Amar, D. Andreucci, P. Bisegna, R. Gianni and et al., A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case, Differential and Integral Equations, 26 (2013), 885-912. 

[4]

M. Amar, D. Andreucci and R. Gianni, Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 48, 24 pp. doi: 10.1007/s00030-016-0396-8.

[5]

H. Ammari, L. Giovangigli, H. Kwon, J. K. Seo and T. Wintz, Spectroscopic conductivity imaging of a cell culture, Asymptotic Analysis, 100 (2016), 87-109.  doi: 10.3233/ASY-161387.

[6]

H. Ammari, T. Widlak and W. Zhang, Towards monitoring critical microscopic parameters for electropermeabilization, Quart. Appl. Math., 75 (2017), 1-17.  doi: 10.1090/qam/1449.

[7]

T. Arbogast, J. Douglas Jr and and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM Journal on Mathematical Analysis, 21 (1990), 823-836.  doi: 10.1137/0521046.

[8]

P. J. Basser, Cable equation for a myelinated axon derived from its microstructure, Medical & Biological Engineering & Computing, 31 (1993), S87–S92. doi: 10.1007/BF02446655.

[9]

P. J. Basser and B. J. Roth, New currents in electrical stimulation of excitable tissues, Annu. Rev. Biomed. Eng., 2 (2000), 377-397.  doi: 10.1146/annurev.bioeng.2.1.377.

[10]

C. Bollini and F. Cacheiro, Peripheral nerve stimulation, Tech. Reg. Anesth. Pain Manag., 10 (2006), 79-88.  doi: 10.1053/j.trap.2006.07.007.

[11]

S. O. Choi, Y. C. Kim, J. W. Lee, J. H. Park, M. R. Prausnitz and M. G. Allen, Intracellular protein delivery and gene transfection by electroporation using a microneedle electrode array, Small, 10 (2012), 1081-1091.  doi: 10.1002/smll.201101747.

[12]

P. Colli-Franzone, L. F. Pavarino and S. Scacchi, Mathematical and numerical methods for reaction-diffusion models in electrocardiology, In Davide Ambrosi, Alfio Quarteroni, and Gianluigi Rozza, editors, Modeling of Physiological Flows, Springer Milan, Milano, 2012,107–141.

[13]

S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, volume 2., Tokyo, Japan: Springer Series, A First Course in On Silico Medicine, 2010. doi: 10.1007/978-4-431-53862-2.

[14]

F. Henríquez and C. Jerez-Hanckes, Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation, ESAIM: M2AN, 52 (2018), 659-703.  doi: 10.1051/m2an/2018019.

[15]

F. Henríquez, C. Jerez-Hanckes and F. Altermatt, Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation, Numer. Math., 136 (2016), 101-145.  doi: 10.1007/s00211-016-0835-9.

[16]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500.

[17]

J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, Springer-Verlag, New York, 1998.

[18]

E. Y. Khruslov, Homogenized models of strongly inhomogeneous media, In Proceedings of the International Congress of Mathematicians, Springer, 1995, 1270–1278.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. 

[20]

D. C. Li and Q. Li, Electrical stimulation of cortical neurons promotes oligodendrocyte development and remyelination in the injured spinal, Neural Regeneration Research, 12 (2017), 1613-1615.  doi: 10.4103/1673-5374.217330.

[21]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh"auser/Springer Basel AG, Basel, 1995.

[22]

V. A. Marchenko and E. Y. Khruslov, Boundary-value problems with fine-grained boundary, Matematicheskii Sbornik, 65 (1964), 458-472. 

[23]

H. Matano and Y. Mori, Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst, 29 (2011), 1573-1636.  doi: 10.3934/dcds.2011.29.1573.

[24]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, 1985. doi: 10.1007/978-3-662-09922-3.

[25]

C. C. McIntyre, A. G. Richardson and W. M. Grill, Modeling the excitability of mammalian nerve fibers: Influence of afterpotentials on the recovery cycle, Journal of Neurophysiology, 87 (2002), 995-1006.  doi: 10.1152/jn.00353.2001.

[26]

H. Meffin, B. Tahayori, E. N. Sergeev, I. M. Y. Mareels, D. B. Grayden and A. N. Burkitt, Modelling extracellular electrical stimulation: Ⅲ. derivation and interpretation of neural tissue equations, Journal of Neural Engineering, 11 (2014), 065004. doi: 10.1088/1741-2560/11/6/065004.

[27]

C. Meunier and B. Lamotte d'Incamps, Extending cable theory to heterogeneous dendrites, Neural Computation, 20 (2008), 1732-1775.  doi: 10.1162/neco.2008.12-06-425.

[28]

L. M. Mir, M. F. Bureau, J. Gehl, R. Rangara, D. Rouy, J. M. Caillaud, P. Delaere, D. Branellec, B. Schwartz and D. Scherman, High-efficiency gene transfer into skeletal muscle mediated by electric pulses, Proc. Nat. Acad. Sci. USA., 96 (1999), 4262-4267.  doi: 10.1073/pnas.96.8.4262.

[29]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. 

[30]

M. Neuss-Radu, Some extensions of two-scale convergence, Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 322 (1996), 899–904.

[31]

P. D. O'Brien, L. M. Hinder, B. C. Callaghan and E. L. Feldman, Neurological consequences of obesity, The Lancet Neurology, 16 (2017), 465-477.  doi: 10.1016/S1474-4422(17)30084-4.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

M. Pennacchio, G. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.  doi: 10.1137/040615249.

[34]

I. Pettersson, Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary, Differential Equations & Applications, 9 (2017), 393-412.  doi: 10.7153/dea-2017-09-28.

[35]

C. Pham-Dang, O. Kick, T. Collet, F. Gouin and M. Pinaud, Continuous peripheral nerve blocks with stimulating catheters, Reg. Anesth. Pain Med., 28 (2003), 83-88. 

[36]

R. Plonsey and R. C. Barr, Bioelectricity: A Quantitative Approach, Springer, Boston, MA, 2007.

[37]

W. Rall, Time constants and electrotonic length of membrane cylinders and neurons, Biophysical Journal, 9 (1969), 1483-1508.  doi: 10.1016/S0006-3495(69)86467-2.

[38]

F. Rattay, Electrical Nerve Stimulation. Theory, Experiments and Applications, Vienna, Austria: Springer–Verlag, 1990.

[39]

J. E. Riggs, The aging population, Neurologic Clinics, 16 (1998), 555-560.  doi: 10.1016/S0733-8619(05)70079-7.

[40]

J. Teissié, N. Eynard, B. Gabriel and M. P. Rols, Electropermeabilization of cell membranes, Adv. Drug. Del. Rev, 35 (1999), 3-19. 

[41]

V. Zhikov, On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

show all references

References:
[1]

G. Allaire, Commissariat 'a L'energie Atomique, Alain Damlamian, and Ulrich Hornung, Two-scale convergence on periodic surfaces and applications, Mathematical Modelling of Flow through Porous Media, 1995.

[2]

M. Amar, D. Andreucci, P. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787.  doi: 10.1002/mma.709.

[3]

M. Amar, D. Andreucci, P. Bisegna, R. Gianni and et al., A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: the nonlinear case, Differential and Integral Equations, 26 (2013), 885-912. 

[4]

M. Amar, D. Andreucci and R. Gianni, Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues, Nonlinear Differ. Equ. Appl., 23 (2016), Art. 48, 24 pp. doi: 10.1007/s00030-016-0396-8.

[5]

H. Ammari, L. Giovangigli, H. Kwon, J. K. Seo and T. Wintz, Spectroscopic conductivity imaging of a cell culture, Asymptotic Analysis, 100 (2016), 87-109.  doi: 10.3233/ASY-161387.

[6]

H. Ammari, T. Widlak and W. Zhang, Towards monitoring critical microscopic parameters for electropermeabilization, Quart. Appl. Math., 75 (2017), 1-17.  doi: 10.1090/qam/1449.

[7]

T. Arbogast, J. Douglas Jr and and U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM Journal on Mathematical Analysis, 21 (1990), 823-836.  doi: 10.1137/0521046.

[8]

P. J. Basser, Cable equation for a myelinated axon derived from its microstructure, Medical & Biological Engineering & Computing, 31 (1993), S87–S92. doi: 10.1007/BF02446655.

[9]

P. J. Basser and B. J. Roth, New currents in electrical stimulation of excitable tissues, Annu. Rev. Biomed. Eng., 2 (2000), 377-397.  doi: 10.1146/annurev.bioeng.2.1.377.

[10]

C. Bollini and F. Cacheiro, Peripheral nerve stimulation, Tech. Reg. Anesth. Pain Manag., 10 (2006), 79-88.  doi: 10.1053/j.trap.2006.07.007.

[11]

S. O. Choi, Y. C. Kim, J. W. Lee, J. H. Park, M. R. Prausnitz and M. G. Allen, Intracellular protein delivery and gene transfection by electroporation using a microneedle electrode array, Small, 10 (2012), 1081-1091.  doi: 10.1002/smll.201101747.

[12]

P. Colli-Franzone, L. F. Pavarino and S. Scacchi, Mathematical and numerical methods for reaction-diffusion models in electrocardiology, In Davide Ambrosi, Alfio Quarteroni, and Gianluigi Rozza, editors, Modeling of Physiological Flows, Springer Milan, Milano, 2012,107–141.

[13]

S. Doi, J. Inoue, Z. Pan and K. Tsumoto, Computational Electrophysiology, volume 2., Tokyo, Japan: Springer Series, A First Course in On Silico Medicine, 2010. doi: 10.1007/978-4-431-53862-2.

[14]

F. Henríquez and C. Jerez-Hanckes, Multiple traces formulation and semi-implicit scheme for modelling biological cells under electrical stimulation, ESAIM: M2AN, 52 (2018), 659-703.  doi: 10.1051/m2an/2018019.

[15]

F. Henríquez, C. Jerez-Hanckes and F. Altermatt, Boundary integral formulation and semi-implicit scheme coupling for modeling cells under electrical stimulation, Numer. Math., 136 (2016), 101-145.  doi: 10.1007/s00211-016-0835-9.

[16]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500.

[17]

J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, Springer-Verlag, New York, 1998.

[18]

E. Y. Khruslov, Homogenized models of strongly inhomogeneous media, In Proceedings of the International Congress of Mathematicians, Springer, 1995, 1270–1278.

[19]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Investigation of the equation of diffusion combined with increasing of the substance and its application to a biology problem, Bull. Moscow State Univ. Ser. A: Math. Mech, 1 (1937), 1-25. 

[20]

D. C. Li and Q. Li, Electrical stimulation of cortical neurons promotes oligodendrocyte development and remyelination in the injured spinal, Neural Regeneration Research, 12 (2017), 1613-1615.  doi: 10.4103/1673-5374.217330.

[21]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkh"auser/Springer Basel AG, Basel, 1995.

[22]

V. A. Marchenko and E. Y. Khruslov, Boundary-value problems with fine-grained boundary, Matematicheskii Sbornik, 65 (1964), 458-472. 

[23]

H. Matano and Y. Mori, Global existence and uniqueness of a three-dimensional model of cellular electrophysiology, Discrete Contin. Dyn. Syst, 29 (2011), 1573-1636.  doi: 10.3934/dcds.2011.29.1573.

[24]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, 1985. doi: 10.1007/978-3-662-09922-3.

[25]

C. C. McIntyre, A. G. Richardson and W. M. Grill, Modeling the excitability of mammalian nerve fibers: Influence of afterpotentials on the recovery cycle, Journal of Neurophysiology, 87 (2002), 995-1006.  doi: 10.1152/jn.00353.2001.

[26]

H. Meffin, B. Tahayori, E. N. Sergeev, I. M. Y. Mareels, D. B. Grayden and A. N. Burkitt, Modelling extracellular electrical stimulation: Ⅲ. derivation and interpretation of neural tissue equations, Journal of Neural Engineering, 11 (2014), 065004. doi: 10.1088/1741-2560/11/6/065004.

[27]

C. Meunier and B. Lamotte d'Incamps, Extending cable theory to heterogeneous dendrites, Neural Computation, 20 (2008), 1732-1775.  doi: 10.1162/neco.2008.12-06-425.

[28]

L. M. Mir, M. F. Bureau, J. Gehl, R. Rangara, D. Rouy, J. M. Caillaud, P. Delaere, D. Branellec, B. Schwartz and D. Scherman, High-efficiency gene transfer into skeletal muscle mediated by electric pulses, Proc. Nat. Acad. Sci. USA., 96 (1999), 4262-4267.  doi: 10.1073/pnas.96.8.4262.

[29]

J. C. Neu and W. Krassowska, Homogenization of syncytial tissues, Critical Reviews in Biomedical Engineering, 21 (1993), 137-199. 

[30]

M. Neuss-Radu, Some extensions of two-scale convergence, Comptes Rendus de l'Académie des Sciences. Série 1, Mathématique, 322 (1996), 899–904.

[31]

P. D. O'Brien, L. M. Hinder, B. C. Callaghan and E. L. Feldman, Neurological consequences of obesity, The Lancet Neurology, 16 (2017), 465-477.  doi: 10.1016/S1474-4422(17)30084-4.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

M. Pennacchio, G. Savaré and P. C. Franzone, Multiscale modeling for the bioelectric activity of the heart, SIAM Journal on Mathematical Analysis, 37 (2005), 1333-1370.  doi: 10.1137/040615249.

[34]

I. Pettersson, Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary, Differential Equations & Applications, 9 (2017), 393-412.  doi: 10.7153/dea-2017-09-28.

[35]

C. Pham-Dang, O. Kick, T. Collet, F. Gouin and M. Pinaud, Continuous peripheral nerve blocks with stimulating catheters, Reg. Anesth. Pain Med., 28 (2003), 83-88. 

[36]

R. Plonsey and R. C. Barr, Bioelectricity: A Quantitative Approach, Springer, Boston, MA, 2007.

[37]

W. Rall, Time constants and electrotonic length of membrane cylinders and neurons, Biophysical Journal, 9 (1969), 1483-1508.  doi: 10.1016/S0006-3495(69)86467-2.

[38]

F. Rattay, Electrical Nerve Stimulation. Theory, Experiments and Applications, Vienna, Austria: Springer–Verlag, 1990.

[39]

J. E. Riggs, The aging population, Neurologic Clinics, 16 (1998), 555-560.  doi: 10.1016/S0733-8619(05)70079-7.

[40]

J. Teissié, N. Eynard, B. Gabriel and M. P. Rols, Electropermeabilization of cell membranes, Adv. Drug. Del. Rev, 35 (1999), 3-19. 

[41]

V. Zhikov, On an extension and an application of the two-scale convergence method, Sb. Math., 191 (2000), 973-1014.  doi: 10.1070/SM2000v191n07ABEH000491.

Figure 1.  Simplified cross-section of a cylindrical myelinated axon (left) and the periodicity cell $ Y $ (right)
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Figure 2.  2D surface generating by revolution the periodicity cell in the neighborhood of a Ranvier node
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Figure 3.  Overlapping cells $ \tilde Y_k $ covering $ \Omega_ \varepsilon $
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