Advertisement

Mathematics of Nerve Signals

  • Tanel PeetsEmail author
  • Kert Tamm
Chapter
Part of the Mathematics of Planet Earth book series (MPE, volume 6)

Abstract

Since the classical works of Hodgkin and Huxley (J. Physiol. 117(4), 500–544 (1952)), it has become evident that the nerve function is a richer phenomenon than a set of electrical action potentials (AP ) alone. The propagation of an AP is accompanied by mechanical and thermal effects. These include the pressure wave (PW) in axoplasm , the longitudinal wave (LW) in a biomembrane , the transverse displacement (TW) of a biomembrane and temperature changes (Θ). The whole nerve signal is, therefore, an ensemble of waves. The primary components (AP , LW, PW) are characterised by corresponding velocities. The secondary components (TW, Θ) are derived from the primary components and have no independent velocities of their own. In this chapter, the emphasis is on mathematical models rather than the physiological aspects. Based on models of single waves, a coupled model for the nerve signal is presented in the form of a nonlinear system of partial differential equations. The model equations are solved numerically by making use of the Fourier transform based pseudospectral method .

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are indebted to prof. Jüri Engelbrecht for his guidance and support over the years and for introducing the authors to the exciting field of mathematical physics.

This research was supported by the European Union through the European Regional Development fund (Estonian Programme TK 124) and by the Estonian Research Council (projects IUT 33-24, PUT 434).

Section 10.2.4 is derived in part from the article (Tamm et al., 2019) published in J. Non-Equilib. Thermodyn. (2019) aop: Ⓒ De Gruyter, available online:  https://doi.org/10.1515/jnet-2019-0012.

References

  1. Abbott, B.C., Hill, A.V., Howarth, J.V.: The positive and negative heat production associated with a nerve impulse. Proc. R. Soc. B Biol. Sci. 148(931), 149–187 (1958).  https://doi.org/10.1098/rspb.1958.0012 CrossRefGoogle Scholar
  2. Andersen, S.S.L., Jackson, A.D., Heimburg, T.: Towards a thermodynamic theory of nerve pulse propagation. Prog. Neurobiol. 88(2), 104–13 (2009). https://doi.org/10.1016/j.pneurobio.2009.03.002 CrossRefGoogle Scholar
  3. Andronov, A., Witt, A., Khaikin, S.: Theory of Oscillations. Phys. Math. Publ. Moscow (In Russian) (1959)Google Scholar
  4. Appali, R., Petersen, S., Van Rienen, U.: A comparison of Hodgkin–Huxley and soliton neural theories. Adv. Radio Sci. 8, 75–79 (2010).  https://doi.org/10.5194/ars-8-75-2010 CrossRefGoogle Scholar
  5. Bennett, M.V.L.: Electrical synapses, a personal perspective (or history). Brain Res. Rev. 32(1), 16–28 (2000). https://doi.org/10.1016/S0165-0173(99)00065-X CrossRefGoogle Scholar
  6. Billingham, J., King, A.C.: Wave Motion. Cambridge Texts in Applied Mathematics. Cambridge University Press (2001).  https://doi.org/10.1017/CBO9780511841033
  7. Bonhoeffer, K.F.: Activation of passive iron as a model for the excitation of nerve. J. Gen. Physiol. 32(1), 69–91 (1948).  https://doi.org/10.1085/jgp.32.1.69 CrossRefGoogle Scholar
  8. Bountis, T., Starmer, C.F., Bezerianos, A.: Stationary pulses and wave front formation in an excitable medium. Prog. Theor. Phys. Suppl. 139, 12–33 (2000).  https://doi.org/10.1143/PTPS.139.12 CrossRefGoogle Scholar
  9. Bresslof, P.C.: Waves in Neural Media. Springer, New York (2014)CrossRefGoogle Scholar
  10. Carslaw, H., Jaeger, J.: Conduction of Heat in Solids, 2nd edn. Oxford Science Publications, Oxford (1959)zbMATHGoogle Scholar
  11. Christov, C.I., Maugin, G.A., Porubov, A.V.: On Boussinesq’s paradigm in nonlinear wave propagation. C. R. Mec. 335(9-10), 521–535 (2007). https://doi.org/10.1016/j.crme.2007.08.006 zbMATHCrossRefGoogle Scholar
  12. Clay, J.R.: Axonal excitability revisited. Prog. Biophys. Mol. Biol. 88(1), 59–90 (2005). https://doi.org/10.1016/j.pbiomolbio.2003.12.004 CrossRefGoogle Scholar
  13. Courtemanche, M., Ramirez, R.J., Nattel, S.: Ionic mechanisms underlying human atrial action potential properties: insights from a mathematical model. Am. J. Physiol. 275(1), 301–321 (1998).  https://doi.org/10.1152/ajpheart.1998.275.1.H301 Google Scholar
  14. Coveney, P.V., Fowler, P.W.: Modelling biological complexity: a physical scientist’s perspective. J. R. Soc. Interface 2(4), 267–280 (2005).  https://doi.org/10.1098/rsif.2005.0045 CrossRefGoogle Scholar
  15. Debanne, D., Campanac, E., Bialowas, A., Carlier, E., Alcaraz, G.: Axon physiology. Physiol. Rev. 91(2), 555–602 (2011). doi10.1152/physrev.00048.2009CrossRefGoogle Scholar
  16. Downing, A.C., Gerard, R.W., Hill, A.V.: The heat production of nerve. Proc. R. Soc. B Biol. Sci. 100(702), 223–251 (1926).  https://doi.org/10.1098/rspb.1926.0044 CrossRefGoogle Scholar
  17. Drukarch, B., Holland, H.A., Velichkov, M., Geurts, J.J., Voorn, P., Glas, G., de Regt, H.W.: Thinking about the nerve impulse: A critical analysis of the electricity-centered conception of nerve excitability. Prog. Neurobiol. 169, 172–185 (2018). https://doi.org/10.1016/j.pneurobio.2018.06.009 CrossRefGoogle Scholar
  18. El Hady, A., Machta, B.B.: Mechanical surface waves accompany action potential propagation. Nat. Commun. 6, 6697 (2015).  https://doi.org/10.1038/ncomms7697 CrossRefGoogle Scholar
  19. Engelbrecht, J.: On theory of pulse transmission in a nerve fibre. Proc. R. Soc. Lond. 375(1761), 195–209 (1981).  https://doi.org/10.1098/rspa.1981.0047 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Engelbrecht, J.: Nonlinear Wave Processes of Deformation in Solids. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 16. Pitman Advanced Publishing Program (1983)Google Scholar
  21. Engelbrecht, J.: An Introduction to Asymmetric Solitary Waves. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol 56. Longman Scientific & Technical, Harlow (1996)Google Scholar
  22. Engelbrecht, J.: Complexity in engineering and natural sciences. Proc. Estonian Acad. Sci. 64(3), 249–255 (2015a).  https://doi.org/10.3176/proc.2015.3.07 CrossRefGoogle Scholar
  23. Engelbrecht, J.: Questions About Elastic Waves. Springer International Publishing, Cham (2015b). https://doi.org/10.1007/978-3-319-14791-8 zbMATHCrossRefGoogle Scholar
  24. Engelbrecht, J., Tamm, K., Peets, T.: On mathematical modelling of solitary pulses in cylindrical biomembranes. Biomech. Model. Mechanobiol. 14(1), 159–167 (2015). https://doi.org/10.1007/s10237-014-0596-2 CrossRefGoogle Scholar
  25. Engelbrecht, J., Tamm, K., Peets, T.: On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes. Phil. Mag. 97(12), 967–987 (2017). https://doi.org/10.1080/14786435.2017.1283070 CrossRefGoogle Scholar
  26. Engelbrecht, J., Peets, T., Tamm, K.: Electromechanical coupling of waves in nerve fibres. Biomech. Model. Mechanobiol. 17(6), 1771–1783 (2018). https://doi.org/10.1007/s10237-018-1055-2 CrossRefGoogle Scholar
  27. Engelbrecht, J., Peets, T., Tamm, K., Laasmaa, M., Vendelin, M.: On the complexity of signal propagation in nerve fibres. Proc. Estonian Acad. Sci. 67(1), 28–38 (2018).  https://doi.org/10.3176/proc.2017.4.28 MathSciNetzbMATHCrossRefGoogle Scholar
  28. Engelbrecht, J., Tamm, K., Peets, T.: Modeling of complex signals in nerve fibers. Med. Hypotheses 120, 90–95 (2018). https://doi.org/10.1016/j.mehy.2018.08.021 CrossRefGoogle Scholar
  29. Engelbrecht, J., Tamm, K., Peets, T.: Primary and secondary components of nerve signals. arXiv:1812.05335 [physics.bio-ph] (2018)Google Scholar
  30. Engelbrecht, J., Tamm, K., Peets, T.: Mathematics of nerve signals. arXiv:1902.00011 [physics.bio-ph] (2019)Google Scholar
  31. Engelbrecht, J., Tobias, T.: On a model stationary nonlinear wave in an active medium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. A411(1840), 139–154 (1987).  https://doi.org/10.1098/rspa.1987.0058 zbMATHCrossRefGoogle Scholar
  32. Fillafer, C., Mussel, M., Muchowski, J., Schneider, M.F.: On cell surface deformation during an action potential. arXiv:1703.04608 [physics.bio-ph] (2017)Google Scholar
  33. FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445–466 (1961). https://doi.org/10.1016/S0006-3495(61)86902-6 CrossRefGoogle Scholar
  34. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1998).  https://doi.org/10.1017/CBO9780511626357
  35. Frigo, M., Johnson, S.: The design and implementation of FFTW 3. Proc. IEEE 93(2), 216–231 (2005).  https://doi.org/10.1109/jproc.2004.840301 CrossRefGoogle Scholar
  36. Gilbert, D.S.: Axoplasm architecture and physical properties as seen in the Myxicola giant axon. J. Physiol. 253, 257–301 (1975).  https://doi.org/10.1113/jphysiol.1975.sp011190 CrossRefGoogle Scholar
  37. Gonzalez-Perez, A., Mosgaard, L., Budvytyte, R., Villagran-Vargas, E., Jackson, A., Heimburg, T.: Solitary electromechanical pulses in lobster neurons. Biophys. Chem. 216, 51–59 (2016). https://doi.org/10.1016/j.bpc.2016.06.005 CrossRefGoogle Scholar
  38. Heimburg, T., Jackson, A.D.: On soliton propagation in biomembranes and nerves. Proc. Natl. Acad. Sci. USA 102(28), 9790–5 (2005).  https://doi.org/10.1073/pnas.0503823102 CrossRefGoogle Scholar
  39. Heimburg, T., Jackson, A.: On the action potential as a propagating density pulse and the role of anesthetics. Biophys. Rev. Lett. 2(1), 57–78 (2007). https://doi.org/10.1142/s179304800700043x CrossRefGoogle Scholar
  40. Heimburg, T., Jackson, A.D.: Thermodynamics of the nervous impulse. In: Nag, K. (ed.) Structure and Dynamics of Membranous Interfaces, pp. 317–339. John Wiley & Sons, Hoboken, NJ, USA (2014). https://doi.org/10.1002/9780470388495.ch12 Google Scholar
  41. Hindmarsh, A.: ODEPACK, A Systematized Collection of ODE Solvers, vol. 1. North-Holland, Amsterdam (1983)Google Scholar
  42. Hodgkin, A.L.: The Conduction of the Nervous Impulse. Liverpool University Press (1964)Google Scholar
  43. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–544 (1952).  https://doi.org/10.1113/jphysiol.1952.sp004764 CrossRefGoogle Scholar
  44. Howarth, J.V., Keynes, R.D., Ritchie, J.M.: The origin of the initial heat associated with a single impulse in mammalian non-myelinated nerve fibres. J. Physiol. 194(3), 745–93 (1968).  https://doi.org/10.1113/jphysiol.1968.sp008434 CrossRefGoogle Scholar
  45. Ilison, L., Salupere, A.: Propagation of sech2-type solitary waves in hierarchical KdV-type systems. Math. Comput. Simul. 79(11), 3314–3327 (2009).zbMATHCrossRefGoogle Scholar
  46. Ilison, L., Salupere, A., Peterson, P.: On the propagation of localized perturbations in media with microstructure. Proc. Estonian Acad. Sci. Phys. Math. 56(2), 84–92 (2007)MathSciNetzbMATHGoogle Scholar
  47. Iwasa, K., Tasaki, I., Gibbons, R.: Swelling of nerve fibers associated with action potentials. Science 210(4467), 338–339 (1980).  https://doi.org/10.1126/science.7423196 CrossRefGoogle Scholar
  48. Jérusalem, A., García-Grajales, J.A., Merchán-Pérez, A., Peña, J.M.: A computational model coupling mechanics and electrophysiology in spinal cord injury. Biomech. Model. Mechanobiol. 13(4), 883–896 (2014). https://doi.org/10.1007/s10237-013-0543-7 CrossRefGoogle Scholar
  49. Johnson, A.S., Winlow, W.: The soliton and the action potential – primary elements underlying sentience. Front. Physiol. 9 (2018).  https://doi.org/10.3389/fphys.2018.00779
  50. Jones, E., Oliphant, T., Peterson, P.: SciPy: open source scientific tools for Python (2007). http://www.scipy.org
  51. Kaufmann, K.: Action Potentials and Electromechanical Coupling in the Macroscopic Chiral Phospholipid Bilayer. Caruaru, Brazil (1989)Google Scholar
  52. Kitano, H.: Systems biology: a brief overview. Science 295(5560), 1662–1664 (2002).  https://doi.org/10.1126/science.1069492 CrossRefGoogle Scholar
  53. Lieberstein, H.H.: On the Hodgkin–Huxley partial differential equation. Math. Biosci. 1(1), 45–69 (1967). https://doi.org/10.1016/0025-5564(67)90026-0 MathSciNetCrossRefGoogle Scholar
  54. Lin, T., Morgan, G.: Wave propagation through fluid contained in a cylindrical elastic shell. J. Acoust. Soc. Am. 28(6), 1165–1176 (1956)CrossRefGoogle Scholar
  55. Malmivuo, J., Plonsey, R.: Bioelectromagnetism. Principles and Applications of Bioelectric and Biomagnetic Fields. Oxford University Press (1995).  https://doi.org/10.1093/acprof:oso/9780195058239.001.0001
  56. Maugin, G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  57. Maurin, F., Spadoni, A.: Wave propagation in periodic buckled beams. Part I: Analytical models and numerical simulations. Wave Motion 66, 190–209 (2016). https://doi.org/10.1016/j.wavemoti.2016.05.008 Google Scholar
  58. Maurin, F., Spadoni, A.: Wave propagation in periodic buckled beams. Part II: Experiments. Wave Motion 66, 210–219 (2016). https://doi.org/10.1016/j.wavemoti.2016.05.009 Google Scholar
  59. Mueller, J.K., Tyler, W.J.: A quantitative overview of biophysical forces impinging on neural function. Phys. Biol. 11(5), 051001 (2014). https://doi.org/10.1088/1478-3975/11/5/051001 CrossRefGoogle Scholar
  60. Mussel, M., Schneider, M.F.: It sounds like an action potential: unification of electrical, chemical and mechanical aspects of acoustic pulses in lipids. J. R. Soc. Interface 16(151), 20180743 (2019).  https://doi.org/10.1098/rsif.2018.0743 CrossRefGoogle Scholar
  61. Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50(10), 2061–2070 (1962).  https://doi.org/10.1109/JRPROC.1962.288235 CrossRefGoogle Scholar
  62. Neu, J.C., Preissig, R., Krassowska, W.: Initiation of propagation in a one-dimensional excitable medium. Physica D 102(3–4), 285–299 (1997). https://doi.org/10.1016/S0167-2789(96)00203-5 MathSciNetzbMATHCrossRefGoogle Scholar
  63. Noble, D.: Chair’s introduction. In: Bock, G., Goode, J.A. (eds.) ‘In Silico’ Simulation of Biological Processes, pp. 1–3. John Wiley & Sons, Chichester (2002). https://doi.org/10.1002/0470857897.ch1 Google Scholar
  64. Nolte, D.D.: Introduction to Modern Dynamics: Chaos, Networks, Space and Time. Oxford University Press, Oxford (2015)zbMATHGoogle Scholar
  65. Peets, T., Tamm, K.: On mechanical aspects of nerve pulse propagation and the Boussinesq paradigm. Proc. Estonian Acad. Sci. 64(3S), 331–337 (2015).  https://doi.org/10.3176/proc.2015.3S.02 zbMATHCrossRefGoogle Scholar
  66. Peets, T., Tamm, K., Engelbrecht, J.: On the role of nonlinearities in the Boussinesq-type wave equations. Wave Motion 71, 113–119 (2017). https://doi.org/10.1016/j.wavemoti.2016.04.003 MathSciNetCrossRefGoogle Scholar
  67. Peets, T., Tamm, K., Simson, P., Engelbrecht, J.: On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: A soliton doublet. Wave Motion 85, 10–17 (2019). https://doi.org/10.1016/j.wavemoti.2018.11.001 MathSciNetCrossRefGoogle Scholar
  68. Perez-Camacho, M.I., Ruiz-Suarez, J.: Propagation of a thermo-mechanical perturbation on a lipid membrane. Soft Matter 13, 6555–6561 (2017). https://doi.org/10.1039/C7SM00978J CrossRefGoogle Scholar
  69. Peterson, P.: F2PY: Fortran to Python interface generator. http://cens.ioc.ee/projects/f2py2e/ (2005)
  70. van der Pol, B.: On “relaxation-oscillations”. London, Edinburgh, Dublin Philos. Mag. J. Sci. 2(11), 978–992 (1926). https://doi.org/10.1080/14786442608564127 CrossRefGoogle Scholar
  71. Porubov, A.V.: Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore (2003). https://doi.org/10.1142/5238
  72. Reissig, R., Sansone, G., Conti, R.: Qualitative Theorie Nichtlinearer Differentialgleichungen. Edizioni Cremonese, Roma (1963)Google Scholar
  73. Ritchie, J.M., Keynes, R.D.: The production and absorption of heat associated with electrical activity in nerve and electric organ. Q. Rev. Biophys. 18(4), 451 (1985). https://doi.org/10.1017/S0033583500005382 CrossRefGoogle Scholar
  74. Salupere, A.: The pseudospectral method and discrete spectral analysis. In: Quak, E., Soomere, T. (eds.) Applied Wave Mathematics. Selected Topics in Solids, Fluids and Mathematical Methods, pp. 301–334. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00585-5_16 zbMATHGoogle Scholar
  75. Scott, A.: Nonlinear Science. Emergence & Dynamics of Coherent Structures. Oxford University Press (1999)Google Scholar
  76. Tamm, K., Peets, T.: On solitary waves in case of amplitude-dependent nonlinearity. Chaos Solitons Fractals 73, 108–114 (2015). https://doi.org/10.1016/j.chaos.2015.01.013 MathSciNetzbMATHCrossRefGoogle Scholar
  77. Tamm, K., Engelbrecht, J., Peets, T.: Temperature changes accompanying signal propagation in axons. J. Non-Equilibr. Thermodyn. 44(3), 277–284 (2019).  https://doi.org/10.1515/jnet-2019-0012 CrossRefGoogle Scholar
  78. Taniuti, T., Nishihara, K.: Nonlinear Waves. Pitman, Boston (1983)zbMATHGoogle Scholar
  79. Tasaki, I.: A macromolecular approach to excitation phenomena: mechanical and thermal changes in nerve during excitation. Physiol. Chem. Phys. Med. NMR 20(4), 251–268 (1988)Google Scholar
  80. Tasaki, I., Byrne, P.M.: Heat production associated with a propagated impulse in bullfrog myelinated nerve fibers. Jpn. J. Physiol. 42(5), 805–813 (1992).  https://doi.org/10.2170/jjphysiol.42.805 CrossRefGoogle Scholar
  81. Tasaki, I., Kusano, K., Byrne, P.M.: Rapid mechanical and thermal changes in the garfish olfactory nerve associated with a propagated impulse. Biophys. J. 55(6), 1033–1040 (1989). https://doi.org/10.1016/s0006-3495(89)82902-9 CrossRefGoogle Scholar
  82. Terakawa, S.: Potential-dependent variations of the intracellular pressure in the intracellularly perfused squid giant axon. J. Physiol. 369(1), 229–248 (1985).  https://doi.org/10.1113/jphysiol.1985.sp015898 CrossRefGoogle Scholar
  83. Thompson, J.: Instabilities and Catastrophes in Science and Engineering. Wiley, New York (1982)zbMATHCrossRefGoogle Scholar
  84. Tritton, J.: Physical Fluid Dynamics. Oxford Sci. Publ. (1988)zbMATHGoogle Scholar
  85. Vendelin, M., Saks, V., Engelbrecht, J.: Principles of mathematical modeling and in silico studies of integrated cellular energetics. In: Saks, V. (ed.) Molecular System Bioenergetics: Energy for Life, pp. 407–433. Wiley, Weinheim (2007). https://doi.org/10.1002/9783527621095.ch12 CrossRefGoogle Scholar
  86. Wilke, E.: On the problem of nerve excitation in the light of the theory of waves. Pflügers Arch. 144, 35–38 (1912)CrossRefGoogle Scholar
  87. Yang, Y., Liu, X.W., Wang, H., Yu, H., Guan, Y., Wang, S., Tao, N.: Imaging action potential in single mammalian neurons by tracking the accompanying sub-nanometer mechanical motion. ACS Nano p. acsnano.8b00867 (2018).  https://doi.org/10.1021/acsnano.8b00867 CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Cybernetics, School of ScienceTallinn University of TechnologyTallinnEstonia

Personalised recommendations