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Finite difference/spectral approximations for the fractional cable equation. (English) Zbl 1220.78107
Summary: The Cable equation has been one of the most fundamental equations for modeling neuronal dynamics. In this paper, we consider the numerical solution of the fractional Cable equation, which is a generalization of the classical Cable equation by taking into account the anomalous diffusion in the movement of the ions in neuronal system. A schema combining a finite difference approach in the time direction and a spectral method in the space direction is proposed and analyzed. The main contribution of this work is threefold: 1) We construct a finite difference/Legendre spectral schema for discretization of the fractional Cable equation. 2) We give a detailed analysis of the proposed schema by providing some stability and error estimates. Based on this analysis, the convergence of the method is rigourously established. We prove that the overall schema is unconditionally stable, and the numerical solution converges to the exact one with order \( O(\triangle t^{2-\max\{\alpha,\beta\}}+ \triangle t^{-1}N^{-m})\), where \( \triangle t,N\) and \( m\) are respectively the time step size, polynomial degree, and regularity in the space variable of the exact solution. \( \alpha\) and \( \beta\) are two different exponents between 0 and 1 involved in the fractional derivatives. 3) Finally, some numerical experiments are carried out to support the theoretical claims.

MSC:
78A70 Biological applications of optics and electromagnetic theory
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78M22 Spectral, collocation and related methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis, and S. Leibler. Subdiffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett., 77:4470, 1996.
[2] E. Barkai, R. Metzler, and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E (3) 61 (2000), no. 1, 132 – 138. · doi:10.1103/PhysRevE.61.132 · doi.org
[3] Christine Bernardi and Yvon Maday, Approximations spectrales de problèmes aux limites elliptiques, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 10, Springer-Verlag, Paris, 1992 (French, with French summary). · Zbl 0773.47032
[4] Christine Bernardi and Yvon Maday, Spectral methods, Handbook of numerical analysis, Vol. V, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 209 – 485. · Zbl 0991.65124 · doi:10.1016/S1570-8659(97)80003-8 · doi.org
[5] Jean-Philippe Bouchaud and Antoine Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990), no. 4-5, 127 – 293. · doi:10.1016/0370-1573(90)90099-N · doi.org
[6] E. Brown, E. Wu, W. Zipfel, and W. Webb. Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. Biophys. J., 77:2837-2849, 1999.
[7] Weihua Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 204 – 226. · Zbl 1416.65344 · doi:10.1137/080714130 · doi.org
[8] M. Dentz, A. Cortis, H. Scher, and B. Berkowitz. Time behaviour of solute in heterogeneous media: Transition from anomalous to normal transport. Adv. Water Resources, 27:155-173, 2004.
[9] Vincent J. Ervin and John Paul Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations 22 (2006), no. 3, 558 – 576. · Zbl 1095.65118 · doi:10.1002/num.20112 · doi.org
[10] T. Feder, I. Brust-Mascher, J. Slattery, B. Baird, and W. Webb. Constrained diffusion or immobile fraction on cell surfaces: A new interpretation. Biophys. J., 70:2767-2773, 1996.
[11] R. Ghosh. Mobility and clustering of individual low density lipoprotein receptor molecures on the surface of human skin fibroblasts. Ph.D. thesis. Cornell University, Ithaca, NY.
[12] R. Ghosh and W. Webb. Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecures. Biophys. J., 66:1301-1318, 1994.
[13] I. Goychuk, E. Heinsalu, M. Patriarca, G. Schmid, and Pänggi. Current and universal scaling in anomalous transport. Phys. Rev. E, 73:020101, 2006.
[14] B. I. Henry, T. A. M. Langlands, and S. L. Wearne. Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett., 100(12):128103, 2008.
[15] A. Hodgkin and A. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117:500-544, 1952.
[16] J. Jack, D. Noble, and R. Tsien. Electrical current flow in excitable cells. Oxford University Press, Oxford, 1975.
[17] D. Junge. Nerve and Muscle Excitation . Sinauer Associates, Inc., Sunderland, Massachusetts, 1981.
[18] C. Koch. Biophysics of Computation, Information Processing in Single neurons, Computational Neuroscience. Oxford University Press, New York, 1999.
[19] A. Kusumi, C. Nakada, K. Ritchie, K. Murase, K. Suzuki, H. Murakoshi, R. Kasai, J. Kondo, and T. Fujiwara. Paradigm shift of the plasma membrane concept from two-dimensional continuum fluid to the partitioned fluid: Highspeed single-molecure tracking of membrane molecures. Annu. Rev. Biophys. Biomol. Struct., 34:351-378, 2005.
[20] T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), no. 2, 719 – 736. · Zbl 1072.65123 · doi:10.1016/j.jcp.2004.11.025 · doi.org
[21] E. K. Lenzi, R. S. Mendes, Kwok Sau Fa, L. C. Malacarne, and L. R. da Silva, Anomalous diffusion: fractional Fokker-Planck equation and its solutions, J. Math. Phys. 44 (2003), no. 5, 2179 – 2185. · Zbl 1062.82043 · doi:10.1063/1.1566452 · doi.org
[22] Xianjuan Li and Chuanju Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2108 – 2131. · Zbl 1193.35243 · doi:10.1137/080718942 · doi.org
[23] Yumin Lin and Chuanju Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533 – 1552. · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 · doi.org
[24] F. Mainardi. Fractional diffusive waves in viscoelastic solids. Nonlinear Waves in Solids, pages 93-97, 1995.
[25] Ralf Metzler and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3 · doi.org
[26] R. Metzler, J. Klafter, and I. Sokolov. Anomalous transport in external fields: Continuous time random walks and fractional diffusion equations extended. Phys. Rev. E, 58:1621-1633, 1998.
[27] H. P. Müller, R. Kimmich, and J. Weis. NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius. Phys. Rev. E, 54:5278-5285, 1996.
[28] Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. · Zbl 0924.34008
[29] N. Qian and T. Sejnowski. An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons. Biol. Cybern., 62:1-15, 1989. · Zbl 0683.92004
[30] W. Rall. Branching dendritic trees and motoneuron membrane resistivity. volume 1, pages 491-527. 1959.
[31] W. Rall. Core conductor theory and cable properties of neurons. In R. Poeter, editor, Handbook of Physiology: The Nervous System, Vol. 1 (Chapter 3), pages 39-97. American Physiological Society, Bethesda, MD, 1977.
[32] K. Ritchie. Detection of non-Browian diffusion in the cell membrane in single molecure tracking. Biophys. J., 88:2266-2277, 2005.
[33] H. Scher and M. Lax, Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B (3) 7 (1973), no. 10, 4491 – 4502.
[34] H. Scher and E. Montroll. Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B, 12:2455-2477, 1975.
[35] I. Segev, J. Fleshman, and R. Burke. Compartmental models of complex neurons. In C. Koch and I. Segev, editors, Methods in Neuronal Modelling. MIT Press, Cambridge, MA.
[36] R. Simson, B. Yang, S. Moore, P. Doherty, F. Walsh, and K. Jacobson. Structural mosaicism on the submicron scale in the plasma membrane. Biophys. J., 74:297-308, 1998.
[37] P. Smith, I. Morrison, K. Wilson, N. Fernandez, and R. Cherry. Constrained diffusion or immobile fraction on cell surfaces: A new interpretation. Biophys. J., 76:3331-3344, 1999.
[38] M. Wachsmuth, T. Weidemann, G. Muller, U. Hoffmann-Rohrer, T.Knoch, W. Waldeck, and J. Langowski. Analyzing intracellular binding and diffusion with continuous fluorescence photobleaching. Biophys. J., 84:3353-3363, 2003.
[39] E. R. Weeks. Experimental studies of anomalous diffusion, blocking phenomena, and two-dimensional turbulence. Ph.D. thesis. University of Texas at Austin.
[40] Walter Wyss, The fractional diffusion equation, J. Math. Phys. 27 (1986), no. 11, 2782 – 2785. · Zbl 0632.35031 · doi:10.1063/1.527251 · doi.org
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