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An Application in Neuroscience: Heterogeneous Cable Equation

  • Alexandre L. Madureira
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

We consider here a simplified version of an equation that models the voltage transmission along neurons, modeled here by a “cable.” If the thickness of the cable is much smaller than its length, it originates a singular perturbed behavior equation that does not differ substantially from what was investigated in Chap.  2 Moreover, other interesting asymptotics arise when considering a large number of synapses.

We also show here that the Multiscale Finite Element Method yields good approximations under all asymptotic regimes, even when the Galerkin Method fails.

References

  1. 13.
    Antonietti, P. F., Brezzi, F., & Marini, L. D. (2009). Bubble stabilization of discontinuous Galerkin methods. Computational Methods in Applied Mechanical Engineering, 198(21–26), 1651–1659. MR2517937.Google Scholar
  2. 21.
    Auricchio, F., Bisegna, P., & Lovadina, C. (2001). Finite element approximation of piezoelectric plates. International Journal for Numerical Methods in Engineering, 50(6), 1469–1499. doi:10.1002/10970207(20010228)50:6 1469::AID-NME823.0.CO;2-I. MR1811534.Google Scholar
  3. 27.
    Baer, S. M., Crook, S., Dur-E-Ahmad, M., & Jackiewicz, Z. (2009). Numerical solution of calcium-mediated dendritic branch model. Journal of Computational and Applied Mathematics, 229(2), 416–424. doi:10.1016/j.cam.2008.04.011.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 32.
    Bower, J. M., & Beeman, D. (2003). The book of GENESIS: Exploring realistic neural models with the general neural simulation system. Santa Clara, CA: TELOS.zbMATHGoogle Scholar
  5. 36.
    Bressloff, P. C. (2001). Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physics D, 155(1–2), 83–100. MR1837205 (2002d:92001).Google Scholar
  6. 37.
    Bressloff, P. C., Earnshaw, B. A., & Ward, M. J. (2008). Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps. SIAM Journal of Applied Mathematics, 68(5), 1223–1246. MR2407121 (2009g:92017).Google Scholar
  7. 46.
    Cai, D., Tao, L., Rangan, A. V., & McLaughlin, D. W. (2006). Kinetic theory for neuronal network dynamics. Communications in Mathematical Science, 4(1), 97–127. MR2204080 (2007a:82053).Google Scholar
  8. 47.
    Canic, S., Piccoli, B., Qiu, J.-M., & Ren, T. (2015). Runge-Kutta discontinuous Galerkin method for traffic flow model on networks. Journal of Scientific Computing, 63(1), 233–255. doi:10.1007/s10915-014-9896-z. MR3315275.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 48.
    Carnevale, N., & Hines, M. L. (2006). The NEURON book. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  10. 51.
    Chenais, D., & Paumier, J.-C. (1994). On the locking phenomenon for a class of elliptic problems. Numerical Mathematics, 67(4), 427–440. doi:10.1007/s002110050036. MR1274440.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 66.
    Dayan, P., & Abbott, L. F. (2001). Theoretical neuroscience: Computational and mathematical modeling of neural systems. Computational Neuroscience. Cambridge, MA: MIT Press. MR1985615 (2004g:92008).Google Scholar
  12. 68.
    De Schutter, E. (2001). Computational neuroscience: More math is needed to understand the human brain. In Mathematics unlimited—2001 and beyond (pp. 381–391). Berlin: Springer. MR1852166.Google Scholar
  13. 75.
    Efendiev, Y., & Hou, T. Y. (2008). Multiscale computations for flow and transport in heterogeneous media. In Quantum transport. Lecture Notes in Mathematics (Vol. 1946, pp. 169–248). Berlin: Springer. MR2497877.Google Scholar
  14. 76.
    Efendiev, Y., & Hou, T. Y. (2009). Multiscale finite element methods: Theory and applications. Surveys and Tutorials in the Applied Mathematical Sciences (Vol. 4). New York: Springer. MR2477579.Google Scholar
  15. 80.
    Efendiev, Y., & Pankov, A. (2003). Numerical homogenization of monotone elliptic operators. Multiscale Modelling and Simulation, 2(1), 62–79 (electronic). MR2044957 (2005a:65153).Google Scholar
  16. 81.
    Efendiev, Y. R., & Wu, X.-H. (2002). Multiscale finite element for problems with highly oscillatory coefficients. Numerical Mathematics, 90(3), 459–486. MR1884226 (2002m:65114).Google Scholar
  17. 84.
    Ermentrout, B. (1998). Neural networks as spatial-temporal pattern-forming systems. Reports on Progress in Physics, 61, 353–430.CrossRefGoogle Scholar
  18. 94.
    Franca, L. P., Madureira, A. L., Tobiska, L., & Valentin, F. (2005). Convergence analysis of a multiscale finite element method for singularly perturbed problems. Multiscale Modelling and Simulation, 4(3), 839–866 (electronic). MR2203943 (2006k:65316).Google Scholar
  19. 95.
    Franca, L. P., Madureira, A. L., & Valentin, F. (2005).Towards multiscale functions: Enriching finite element spaces with local but not bubble-like functions. Computational Methods in Applied Mechanical Engineering, 194(27–29), 3006–3021. MR2142535 (2006a:65159).Google Scholar
  20. 113.
    Hansel, D., Mato, G., Meunier, C., & Neltner, L. (1998). On numerical simulations of integrate-and-fire neural networks. Neural Computation, 10(2), 467–483.CrossRefGoogle Scholar
  21. 118.
    Haroske, D. D., & Triebel, H. (2008). Distributions, Sobolev spaces, elliptic equations. EMS Textbooks in Mathematics. Zürich: European Mathematical Society (EMS). MR2375667 (2009a:46003).Google Scholar
  22. 120.
    Herz, A. V. M., Gollisch, T., Machens, C. K., & Jaeger, D. (2006). Modeling single-neuron dynamics and computations: A balance of detail and abstraction. Science, 314(5796), 80–85. MR2253402 (2007d:92020).Google Scholar
  23. 121.
    Hesthaven, J. S., & Warburton, T. (2008). Nodal discontinuous Galerkin methods: Algorithms, analysis, and applications. Texts in Applied Mathematics (Vol. 54). New York: Springer. MR2372235 (2008k:65002).Google Scholar
  24. 122.
    Hines, M. (1984). Efficient computation of branched nerve equations. International Journal of Bio-Medical Computing, 15, 69–76.CrossRefGoogle Scholar
  25. 125.
    Hines, M. L., Markram, H., & Schürmann, F. (2008). Fully implicit parallel simulation of single neurons. Journal of Computational Neuroscience, 25, 439–448. doi: 10.1007/s10827-008-0087-5.MathSciNetCrossRefGoogle Scholar
  26. 127.
    Hou, T. Y. (2003). Numerical approximations to multiscale solutions in partial differential equations. In Frontiers in numerical analysis (Durham, 2002) (pp. 241–301). MR2006969 (2004m:65219).Google Scholar
  27. 129.
    Hou, T. Y., & Wu, X.-H. (1997). A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of Computational Physics, 134(1), 169–189. MR1455261 (98e:73132).Google Scholar
  28. 146.
    Laing, C. R., Frewen, T. A., & Kevrekidis, I. G. (2007). Coarse-grained dynamics of an activity bump in a neural field model. Nonlinearity, 20, 2127–2146. doi:10.1088/0951-7715/20/9/007.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 149.
    Madureira, A. L. (2009). A multiscale finite element method for partial differential equations posed in domains with rough boundaries. Mathematics of Computation, 78(265), 25–34. MR2448695.Google Scholar
  30. 151.
    Madureira, A. L., Madureira, D. Q. M., & Pinheiro, P. O. (2012). A multiscale numerical method for the heterogeneous cable equation. Neurocomputing, 77(1), 48–57.CrossRefGoogle Scholar
  31. 156.
    Mclaughlin, D., Shapley, R., & Shelley, M. (2003). Large-scale modeling of the primary visual cortex: Influence of cortical architecture upon neuronal response. Journal of Physiology-Paris, 97, 237–252.CrossRefGoogle Scholar
  32. 158.
    Meunier, C., & Lamotte d’Incamps, B. (2008). Extending cable theory to heterogeneous dendrites. Neural Computing, 20(7), 1732–1775. MR2417105 (2009e:92021).Google Scholar
  33. 159.
    Meunier, C., & Segev, I. (2002). Playing the devil’s advocate: Is the Hodgkin-Huxley model useful? Trends in Neuroscience, 25(11), 558–563.CrossRefGoogle Scholar
  34. 166.
    Omurtag, A., Knight, B. W., & Sirocich, L. (2000). On the simulation of large populations of neurons. Journal of Computational Neuroscience, 8(8), 51–63.CrossRefzbMATHGoogle Scholar
  35. 170.
    Pokornyi, Y. V., & Borovskikh, A. V. (2004). Differential equations on networks (geometric graphs). Journal of Mathematical Science (N. Y.), 119(6), 691–718. doi:10.1023/B:JOTH.0000012752.77290.fa. MR2070600.Google Scholar
  36. 172.
    Rangan, A. V., & Cai, D. (2007). Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks. Journal of Computational Neuroscience, 22, 81–100. doi:10.1007/s10827-006-8526-7.MathSciNetCrossRefGoogle Scholar
  37. 175.
    Rempe, M. J., & Chopp, D. L. (2006). A predictor-corrector algorithm for reaction-diffusion equations associated with neural activity on branched structures. SIAM Journal of Scientific Computing, 28(6), 2139–2161 (electronic). MR2272255 (2008f:65148).Google Scholar
  38. 176.
    Rempe, M. J., Spruston, N., Kath, W. L., & Chopp, D. L. (2008). Compartmental neural simulations with spatial adaptivity. Journal of Computational Neuroscience, 25, 465–480. doi:10.1007/s10827-008-0089-3.MathSciNetCrossRefGoogle Scholar
  39. 178.
    Roggensack, A. (2013). A kinetic scheme for the one-dimensional open channel flow equations with applications on networks. Calcolo, 50(4), 255–282. doi:10.1007/s10092-012-0066-0. MR3118265.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 179.
    Roos, H.-G., Stynes, M., & Tobiska, L. (2008). Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Springer Series in Computational Mathematics (2nd ed., Vol. 24). Berlin: Springer. MR2454024 (2009f:65002).Google Scholar
  41. 180.
    Rubin, J., & Wechselberger, M. (2008). The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales. Chaos, 18(1), 015105, 12. MR2404661 (2009a:37194).Google Scholar
  42. 183.
    Shelley, M. J., & Mclaughlin, D. W. (2002). Coarse-grained reduction and analysis of a network model of cortical response, I: Drifting grating stimuli. Journal of Computational Neuroscience, 12, 97–122.CrossRefzbMATHGoogle Scholar
  43. 184.
    Shelley, M. J., & Tao, L. (2001). Efficient and accurate time-stepping schemes for integrate-and-fire neuronal networks. Journal of Computational Neuroscience, 11, 111–119.CrossRefGoogle Scholar
  44. 186.
    Tuckwell, H. C. (1988). Introduction to theoretical neurobiology. Vol. 1: Linear cable theory and dendritic structure. Cambridge Studies in Mathematical Biology (Vol. 8). Cambridge: Cambridge University Press. MR947344 (90a:92003a).Google Scholar
  45. 187.
    Tuckwell, H. C. (1988). Introduction to theoretical neurobiology. Vol. 2: Nonlinear and stochastic theories. Cambridge Studies in Mathematical Biology (Vol. 8). Cambridge: Cambridge University Press. MR947345 (90a:92003b).Google Scholar
  46. 193.
    Weinan E, & Engquist, B. (2003). Multiscale modeling and computation. Notices of the American Mathematical Society, 50(9), 1062–1070. MR2002752 (2004m:65163).Google Scholar
  47. 200.
    Zhuang, Y. (2006). A parallel and efficient algorithm for multicompartment neuronal modelling. Neurocomputing, 69(10–12), 1035–1038. doi:10.1016/j.neucom.2005.12.040.CrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Alexandre L. Madureira
    • 1
    • 2
  1. 1.Laboratório Nacional de Computação Científica-LNCCPetrópolisBrazil
  2. 2.Fundação Getúlio Vargas-FGVRio de JaneiroBrazil

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