Medical & Biological Engineering & Computing

, Volume 50, Issue 3, pp 243–251 | Cite as

Asymptotic model of electrical stimulation of nerve fibers

  • Jonathan P. Cranford
  • Brian J. Kim
  • Wanda Krassowska NeuEmail author
Original Article


We present a novel theory and computational algorithm for modeling electrical stimulation of nerve fibers in three dimensions. Our approach uses singular perturbation to separate the full 3D boundary value problem into a set of 2D “transverse” problems coupled with a 1D “longitudinal” problem. The resulting asymptotic model contains not one but two activating functions (AF): the longitudinal AF that drives the slow development of the mean transmembrane potential and the transverse AF that drives the rapid polarization of the fiber in the transverse direction. The asymptotic model is implemented for a prototype 3D cylindrical fiber with a passive membrane in an isotropic extracellular region. The validity of this approach is tested by comparing the numerical solution of the asymptotic model to the analytical solutions. The results show that the asymptotic model predicts steady-state transmembrane potential directly under the electrodes with the root mean square error of 0.539 mV, i.e., 1.04% of the maximum transmembrane potential. Thus, this work has created a computationally efficient algorithm that facilitates studies of the complete spatiotemporal dynamics of nerve fibers in three dimensions.


Electric potential Passive fiber Cable equation Activating function Transverse field 

Supplementary material

11517_2012_870_MOESM1_ESM.pdf (57 kb)
Supplementary material 1 (PDF 57 kb)


  1. 1.
    Altman KW, Plonsey R (1988) Development of a model for point source electric fibre bundle stimulation. Med Biol Eng Comput 26:466–475PubMedCrossRefGoogle Scholar
  2. 2.
    Basser PJ (1993) Cable equation for a myelinated axon derived from its microstructure. Med Biol Eng Comput 31:S87–S92PubMedCrossRefGoogle Scholar
  3. 3.
    Brette R, Rudolph M, Carnevale T, Hines M, Beeman D, Bower JM, Diesmann M, Morrison A, Goodman PH, Harris FC Jr, Zirpe M, Natschlger T, Pecevski D, Ermentrout B, Djurfeldt M, Lansner A, Rochel O, Vieville T, Muller E, Davison AP, Boustani SE, Destexhe A (2007) Simulation of networks of spiking neurons: a review of tools and strategies. Comput Neurosci 23:349–398CrossRefGoogle Scholar
  4. 4.
    Clark J, Plonsey R (1966) A mathematical evaluation of the core conductor model. Biophys J 6:95–112PubMedCrossRefGoogle Scholar
  5. 5.
    Crank J, Nicolson P (1947) Practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proc Camb Phil Soc 43:50–67CrossRefGoogle Scholar
  6. 6.
    Dahlquist G, Björck A (1974) Numerical Methods. Prentice-Hall, Englewood CliffsGoogle Scholar
  7. 7.
    Dixon C (1971) Applied mathematics of science and engineering. Wiley, New YorkGoogle Scholar
  8. 8.
    Greenberg RJ, Velte TJ, Humayun MS, Scarlatis GN, De Juan E Jr (1999) A computational model of electrical stimulation of the retinal ganglion cell. IEEE Trans Biomed Eng 46:505–514PubMedCrossRefGoogle Scholar
  9. 9.
    Grill W, Kirsch R (2000) Neuroprosthetic applications of electrical stimulation. Assist Technol 12:6–20PubMedCrossRefGoogle Scholar
  10. 10.
    Hinch EJ (1991) Perturbation methods. Cambridge University Press, Cambridge, UKGoogle Scholar
  11. 11.
    Hodgkin AL, Rushton WAH (1946) The electrical constants of a crustacean nerve fibre. Proc Roy Soc B 133:444–479CrossRefGoogle Scholar
  12. 12.
    Holsheimer J (1998) Computer modelling of spinal cord stimulation and its contribution to therapeutic efficacy. Spinal Cord 36:531–540PubMedCrossRefGoogle Scholar
  13. 13.
    Joshi RP, Song J (2010) Model analysis of electric fields induced by high-voltage pulsing in cylindrical nerves. IEEE Trans Plasma Sci 38:2894–2900CrossRefGoogle Scholar
  14. 14.
    Joucla S, Yvert B (2009) The “mirror” estimate: an intuitive predictor of membrane polarization during extracellular stimulation. Biophys J 96:3495–3508PubMedCrossRefGoogle Scholar
  15. 15.
    Krassowska W, Neu JC (1994) Response of a single cell to an external electric field. Biophys J 66:1768–1776PubMedCrossRefGoogle Scholar
  16. 16.
    Leon LJ, Hogues H, Roberge FA (1993) A model study of extracellular stimulation of cardiac cells. IEEE Trans Biomed Eng 40:1307–1319PubMedCrossRefGoogle Scholar
  17. 17.
    Lopreore CL, Bartol TM, Coggan JS, Keller DX, Sosinsky GE, Ellisman MH, Sejnowski TJ (2008) Computational modeling of three-dimensional electrodiffusion in biological systems: application to the node of Ranvier. Biophys J 95:2624–2635PubMedCrossRefGoogle Scholar
  18. 18.
    Mahnam A, Hashemi SMR, Grill WM (2008) Computational evaluation of methods for measuring the spatial extent of neural activation. J Neurosci Methods 173:153–164PubMedCrossRefGoogle Scholar
  19. 19.
    Mclntyre CC, Miocinovic S, Butson CR (2007) Computational analysis of deep brain stimulation. Expert Rev Med Devices 4:615–622CrossRefGoogle Scholar
  20. 20.
    McNeal DR (1976) Analysis of a model for excitation of myelinated nerve. IEEE Trans Biomed Eng 23:329–337PubMedCrossRefGoogle Scholar
  21. 21.
    Miranda P, Correia L, Salvador R, Basser P (2007) Tissue heterogeneity as a mechanism for localized neural stimulation by applied electric fields. Phys Med Biol 52:5603–5617PubMedCrossRefGoogle Scholar
  22. 22.
    Neu JC, Krassowska W (1993) Homogenization of syncytial tissues. Crit Rev Biomed Eng 21:137–199PubMedGoogle Scholar
  23. 23.
    Pickard W (1968) A contribution to the electromagnetic theory of the unmyelinated axon. Math Biosc 2:111–121CrossRefGoogle Scholar
  24. 24.
    Pucihar G, Miklavčič D, Kotnik T (2009) A time-dependent numerical model of transmembrane voltage inducement and electroporation of irregularly shaped cells. IEEE Trans Biomed Eng 56:1491–1501PubMedCrossRefGoogle Scholar
  25. 25.
    Rattay F (1986) Analysis of models for external stimulation of axons. IEEE Trans Biomed Eng 33:974–977PubMedCrossRefGoogle Scholar
  26. 26.
    Rattay F, Resatz S, Lutter P, Minassian K, Jilge B, Dimitrijevic M (2003) Mechanisms of electrical stimulation with neural prostheses. Neuromodulation 6:42–56PubMedCrossRefGoogle Scholar
  27. 27.
    Roth BJ (1994) Mechanisms for electrical stimulation of excitable tissue. Crit Rev Biomed Eng 22:253–305PubMedGoogle Scholar
  28. 28.
    Ruohonen J, Panizza M, Nilsson J, Ravazzani P, Grandori F, Tognola G (1996) Transverse-field activation mechanism in magnetic stimulation of peripheral nerves. Electroencephalogr Clin Neurophysiol 101:167–174PubMedCrossRefGoogle Scholar
  29. 29.
    Rutten W (2002) Selective electrical interfaces with the nervous system. Annu Rev Biomed Eng 4:407–452PubMedCrossRefGoogle Scholar
  30. 30.
    Schnabel V, Struijk JJ (2001) Evaluation of the cable model for electrical stimulation of unmyelinated nerve fibers. IEEE Trans Biomed Eng 48:1027–1033PubMedCrossRefGoogle Scholar
  31. 31.
    Stewart DA, Gowrishankar TR, Weaver JC (2004) Transport lattice approach to describing cell electroporation: use of a local asymptotic model. IEEE Trans Plasma Sci 32:1696–1708CrossRefGoogle Scholar
  32. 32.
    Wang Y, Shen Q, Jiang D (2001) A modified cable function for represent the excitation of peripheral nerves by transverse field induced by pulsed magnetic field. In: Proceedings of the 23rd annual EMBS international conference, Istanbul, Turkey, October 25–26, pp 896–898Google Scholar
  33. 33.
    Wikswo JP Jr, Lin SF, Abbas RA (1995) Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation. Biophys J 69:2195–2210PubMedCrossRefGoogle Scholar
  34. 34.
    Ying W, Henriquez CS (2007) Hybrid finite element method for describing the electrical response of biological cells to applied fields. IEEE Trans Biomed Eng 54:611–620PubMedCrossRefGoogle Scholar

Copyright information

© International Federation for Medical and Biological Engineering 2012

Authors and Affiliations

  • Jonathan P. Cranford
    • 1
  • Brian J. Kim
    • 1
    • 2
  • Wanda Krassowska Neu
    • 1
    Email author
  1. 1.Department of Biomedical EngineeringDuke UniversityDurhamUSA
  2. 2.Department of Biomedical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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