Fractional Cable Equation Models for Anomalous Electrodiffusion in Nerve Cells: Finite Domain Solutions

In recent work we introduced fractional Nernst–Planck equations and related fractional cable equations to model electrodiffusion of ions in nerve cells with anomalous subdiffusion along and across the nerve cells. This work was motivated by many computational and experimental studies showing that anomalous diffusion is ubiquitous in biological systems with binding, crowding, or trapping. For example, recent experiments have shown that anomalous subdiffusion occurs along the axial direction in spiny dendrites due to trapping by the spines. We modeled the subdiffusion in two ways leading to two fractional cable equations and presented fundamental solutions on infinite and semi-infinite domains. Here we present solutions on finite domains for mixed Robin boundary conditions. The finite domain solutions model passive electrotonic properties of spiny dendritic branch segments with ends that are voltage clamped, sealed, or killed. The behavior of the finite domain solutions is similar for both fractional cable ...

[1]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[2]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[3]  Daniel S. Banks,et al.  Anomalous diffusion of proteins due to molecular crowding. , 2005, Biophysical journal.

[4]  T. J. Sejnowski,et al.  An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites, spines and axons , 1989, Biological Cybernetics.

[5]  Karina Weron,et al.  Modeling of subdiffusion in space-time-dependent force fields beyond the fractional Fokker-Planck equation. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  A. Compte,et al.  The generalized Cattaneo equation for the description of anomalous transport processes , 1997 .

[7]  I M Sokolov,et al.  Reaction-subdiffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Arak M. Mathai,et al.  The H-Function with Applications in Statistics and Other Disciplines. , 1981 .

[9]  Vivien A. Casagrande,et al.  Biophysics of Computation: Information Processing in Single Neurons , 1999 .

[10]  K. Jacobson,et al.  Transient confinement of a glycosylphosphatidylinositol-anchored protein in the plasma membrane. , 1997, Biochemistry.

[11]  Rajesh Sharma,et al.  Asymptotic analysis , 1986 .

[12]  R. Cherry,et al.  Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking. , 1999, Biophysical journal.

[13]  Long-range correlation effects, generalized Brownian motion and anomalous diffusion , 1994 .

[14]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[15]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[16]  M. Saxton,et al.  Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study. , 2001, Biophysical journal.

[17]  Igor Podlubny,et al.  The Laplace Transform Method for Linear Differential Equations of the Fractional Order , 1997, funct-an/9710005.

[18]  Thermodynamics and fractional Fokker-Planck equations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[20]  Igor M. Sokolov,et al.  Field-induced dispersion in subdiffusion. , 2006 .

[21]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[22]  Denis S Grebenkov,et al.  Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  P. Hänggi,et al.  Fractional diffusion modeling of ion channel gating. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  M. Saxton.  Anomalous diffusion due to obstacles: a Monte Carlo study. , 1994, Biophysical journal.

[25]  Gernot Guigas,et al.  The degree of macromolecular crowding in the cytoplasm and nucleoplasm of mammalian cells is conserved , 2007, FEBS letters.

[26]  Damián H. Zanette,et al.  Macroscopic current in fractional anomalous diffusion , 1998 .

[27]  Gabriele Müller,et al.  Analyzing intracellular binding and diffusion with continuous fluorescence photobleaching. , 2003, Biophysical journal.

[28]  W. Rall.  Core Conductor Theory and Cable Properties of Neurons , 2011 .

[29]  B. Hille.  Ionic channels of excitable membranes , 2001 .

[30]  LONG-TIME CORRELATION-EFFECTS AND FRACTAL BROWNIAN-MOTION , 1990 .

[31]  K. Jacobson,et al.  Structural mosaicism on the submicron scale in the plasma membrane. , 1998, Biophysical journal.

[32]  W. Rall.  Branching dendritic trees and motoneuron membrane resistivity. , 1959, Experimental neurology.

[33]  Wong,et al.  Exponential Asymptotics of the Mittag—Leffler Function , 2002 .

[34]  S. Furini,et al.  Application of the Poisson-Nernst-Planck theory with space-dependent diffusion coefficients to KcsA. , 2006, Biophysical journal.

[35]  Vicenç Méndez,et al.  Non-Markovian model for transport and reactions of particles in spiny dendrites. , 2008, Physical review letters.

[36]  T. A. M. Langlands,et al.  Solution of a modified fractional diffusion equation , 2006 .

[37]  E. Lutz.  Fractional Langevin equation. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Linear response in complex systems: CTRW and the fractional Fokker-Planck equations , 2001, cond-mat/0107632.

[39]  S. Schnell,et al.  Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. , 2004, Progress in biophysics and molecular biology.

[40]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[41]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

[42]  P Hänggi,et al.  Use and abuse of a fractional Fokker-Planck dynamics for time-dependent driving. , 2007, Physical review letters.

[43]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[44]  S. Wearne,et al.  Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions , 2009, Journal of mathematical biology.

[45]  Akihiro Kusumi,et al.  Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. , 2005, Biophysical journal.

[46]  S L Wearne,et al.  Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Yuan-Chung Cheng,et al.  Aging continuous time random walks , 2003 .

[48]  Wang.  Long-time-correlation effects and biased anomalous diffusion. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[49]  E. Schutter,et al.  Anomalous Diffusion in Purkinje Cell Dendrites Caused by Spines , 2006, Neuron.

[50]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[51]  Arak M. Mathai,et al.  The H-function with applications in statistics and other disciplines , 1978 .

[52]  M. Weiss,et al.  Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. , 2004, Biophysical journal.

[53]  R. Koeller.  Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[54]  Nelson Spruston,et al.  Factors mediating powerful voltage attenuation along CA1 pyramidal neuron dendrites , 2005, The Journal of physiology.

[55]  Igor M. Sokolov,et al.  Field-induced dispersion in subdiffusion , 2008, VALUETOOLS.

[56]  S. Wearne,et al.  Fractional cable models for spiny neuronal dendrites. , 2008, Physical review letters.

[57]  I. Turner,et al.  Two New Implicit Numerical Methods for the Fractional Cable Equation , 2011 .

[58]  S. Wearne,et al.  Anomalous subdiffusion with multispecies linear reaction dynamics. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[59]  W. Gan,et al.  Dendritic spine dynamics. , 2009, Annual review of physiology.

[60]  M. Saxton.  Anomalous diffusion due to binding: a Monte Carlo study. , 1996, Biophysical journal.

[61]  N. Spruston,et al.  Determinants of Voltage Attenuation in Neocortical Pyramidal Neuron Dendrites , 1998, The Journal of Neuroscience.

[62]  Evelyn Buckwar,et al.  Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations , 1998 .

[63]  Andy M. Reynolds,et al.  On the anomalous diffusion characteristics of membrane-bound proteins , 2005 .

[64]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[65]  W. Webb,et al.  Constrained diffusion or immobile fraction on cell surfaces: a new interpretation. , 1996, Biophysical journal.

[66]  B. Eisenberg,et al.  Ion permeation and glutamate residues linked by Poisson-Nernst-Planck theory in L-type calcium channels. , 1998, Biophysical journal.

[67]  Ralf Metzler,et al.  Deriving fractional Fokker-Planck equations from a generalised master equation , 1999 .

[68]  G. V. van Heijst,et al.  Diffusion in a time-dependent external field. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Micha E. Spira,et al.  Low Mobility of the Ca2+ Buffers in Axons of Cultured Aplysia Neurons , 1997, Neuron.

[70]  W. Webb,et al.  Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. , 1999, Biophysical journal.