Continuum mechanics and signals in nerves; pp. 3–18

Full article in PDF format | 10.3176/proc.2021.1.02

Jüri Engelbrecht, Kert Tamm, Tanel Peets


The review describes how ideas from the fields of physics and mathematics have influenced the studies on signal propagation in nerves, which has classically been related to electrophysiology and chemistry.


1. Allen, D. H. How Mechanics Shaped the Modern World. Springer, Cham, 2014.

2. Altenbach, H. and Öchsner, A. (eds). Encyclopedia of Continuum Mechanics. Springer, Cham, 2020.

3. Engelbrecht, J., Tamm, K., and Peets, T. On mechanisms of electromechanophysiological interactions between the components of nerve signals in axons. Proc. Estonian Acad. Sci., 2020, 69(2), 81–96.

4. Kaufmann, K. Action Potentials and Electromechanical Coupling in the Macroscopic Chiral Phospholipid Bilayer. Caruaru, 1989.

5. Keener, J. and Sneyd, J. Mathematical Physiology. Springer, New York, NY, 1998.

6. Cohen, J. E. Mathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better. PLoS Biol., 2004, 2(12), e439.

7. Hobbie, R. K. and Roth, B. J. Intermediate Physics for Medicine and Biology. Springer, Cham, 2015.

8. Kitano, H. Systems biology: a brief overview. Science, 2002, 295(5560), 1662–1664.

9. Noble, D. The rise of computational biology. Nat. Rev. Mol. Cell Biol., 2002, 3(6), 459–463.

10. Gavaghan, D., Garny, A., Maini, P. K., and Kohl, P.  Mathematical models in physiology. Philos. Trans. Royal Soc. A364(1842), 2006, 1099–1106.

11. Bialek, W. Perspectives on theory at the interface of physics and biology. Reports Prog. Phys., 2018, 81(1), 012601.

12. Clay, J. R. Axonal excitability revisited. Prog. Biophys. Mol. Biol., 2005, 88(1), 59–90.

13. Debanne, D., Campanac, E., Bialowas, A., Carlier, E., and Alcaraz, G. Axon physiology. Physiol. Rev., 2011, 91(2), 555–602.

14. de Lichtervelde, A. C. L., de Souza, J. P., and Bazant, M. Z. Heat of nervous conduction: A thermodynamic framework. Phys. Rev. E., 2020, 101(2), 022406.

15. Drukarch, B., Holland, H. A., Velichkov, M., Geurts, J. J., Voorn, P., Glas, G., et al. Thinking about the nerve impulse: a critical analysis of the electricity-centered conception of nerve excitability. Prog. Neurobiol., 2018, 169, 172–185.

16. Hodgkin, A. L. The Conduction of the Nervous Impulse. Liverpool University Press, 1964.

17. Nagumo, J., Arimoto, S., and Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proc. IRE, 1962, 50(10), 2061–2070.

18. Askar, A. Lattice Dynamical Foundations of Continuum Theories. World Scientific, Singapore, 1986.

19. Maugin, G. A. Nonlinear Waves in Elastic Crystals. Oxford University Press, 1999.

20. Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics. Springer, Berlin, 1965.

21. Berezovski, A., Engelbrecht, J., Salupere, A., Tamm, K., Peets, T., and Berezovski, M. Dispersive waves in microstructured solids. Int. J. Solids Struct., 2013, 50, 1981–1990.

22. Eringen, A. C. Nonlinear Theory of Continuous Media. McGraw-Hill Book Company, New York, NY, 1962.

23. Eringen, A. C. and Maugin, G. A. Electrodynamics of Continua I. Springer, New York, NY, 1990.

24. Engelbrecht, J. Questions About Elastic Waves. Springer , Cham, 2015.

25. Johnston, D. and Wu, S. M.-S. Foundations of Cellular Neurophysiology. The MIT Press, Cambridge, MA, 1995.

26. Lieberstein, H. On the Hodgkin-Huxley partial differential equation. Math. Biosci., 1967, 1(1), 45–69.

27. Chen, H., Garcia-Gonzalez, D., and Jérusalem, A. Computational model of the mechanoelectrophysiological coupling in axons with application to neuromodulation. Phys. Rev. E, 2019, 99(3), 032406.

28. FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J., 1961, 1(6), 445–466.

29. Engelbrecht, J., Peets, T., and Tamm, K. Electromechanical coupling of waves in nerve fibres. Biomech. Model. Mechanobiol., 2018, 17(6), 1771–1783.

30. Engelbrecht, J., Peets, T., Tamm, K., Laasmaa, M., and Vendelin, M. On the complexity of signal propagation in nerve fibres. Proc. Estonian Acad. Sci., 2018, 67(1), 28–38.

31. Heimburg, T. and Jackson, A. D. On soliton propagation in biomembranes and nerves. Proc. Natl. Acad. Sci. USA, 2005, 102(28), 9790–9795.

32. Christov, C. I., Maugin, G. A., and Porubov, A. V. On Boussinesq’s paradigm in nonlinear wave propagation. Comptes Rendus Mécanique, 2007, 335(9–10), 521–535.

33. Engelbrecht, J., Tamm, K., and Peets, T. On mathematical modelling of solitary pulses in cylindrical biomembranes. Biomech. Model. Mechanobiol., 2015, 14(1), 159–167.

34. Abbott, B. C., Hill, A. V., and Howarth, J. V. The positive and negative heat production associated with a nerve impulse. Proc. R. Soc. B Biol. Sci.148(931), 1958, 149–187.

35. Tasaki, I. A macromolecular approach to excitation phenomena: mechanical and thermal changes in nerve during excitation. Physiol. Chem. Phys. Med. NMR, 1988, 20(4), 251–268.

36. Kolsky, H. Stress Waves in Solids. Dover Publications, New York, NY, 1963.

37. Graff, K. F. Wave Motion in Elastic Solids. Dover Publications, New York, NY, 1975.

38. Porubov, A. V. Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore, 2003.

39. Engelbrecht, J., Tamm, K., and Peets, T. Internal variables used for describing the signal propagation in axons. Contin. Mech. Thermodyn., 2020, 32(6), 1619–1627.

40. Maugin, G. A. The saga of internal variables of state in continuum thermo-mechanics (1893-2013). Mech. Res. Commun., 2015, 69, 79–86.

41. Maugin, G. A. Internal variables and dissipative structures. J. Non-Equilib. Thermodyn., 1990, 15(2), 173–192.

42. Berezovski, A. and Ván, P. Internal Variables in Thermoelasticity. Springer, Cham, 2017.

43. Hodgkin, A. L. The ionic basis of nervous conduction. Science, 1964, 145(3637), 1148–1154.

44. Maugin, G. A. and Engelbrecht, J. A thermodynamical viewpoint on nerve pulse dynamics. J. Non-Equilib. Thermodyn., 1994, 19(1), 9–23.

45. Morris, C. and Lecar, H. Voltage oscillations in the barnacle giant muscle fiber. Biophys. J., 1981, 35(1), 193–213.

46. Hodgkin, A. L. and Huxley, A. F. Action potentials recorded from inside a nerve fibre. Nature, 1939, 144(3651), 710–711.

47. Appali, R., Petersen, S., and van Rienen, U.  A comparision of Hodgkin-Huxley and soliton neural theories. Adv. Radio Sci., 2010, 8, 75–79.

48. Petrov, A. G. Electricity and mechanics of biomembrane systems: flexoelectricity in living membranes. Anal. Chim. Acta, 2006, 568(1-2), 70–83.

49. Gross, D., Williams, W. S., and Connor, J. A. Theory of electromechanical effects in nerve. Cell. Mol. Neurobiol., 1983, 3(2), 89–111.

50. Terakawa, S. Potential-dependent variations of the intracellular pressure in the intracellularly perfused squid giant axon. J. Physiol., 1985, 369(1), 229–248.

51. Howarth, J. V., Keynes, R. D., and Ritchie, J. M. The origin of the initial heat associated with a single impulse in mammalian non-myelinated nerve fibres. J. Physiol., 1968, 194(3), 745–93.

52. Tamm, K., Engelbrecht, J., and Peets, T. Temperature changes accompanying signal propagation in axons. J. NonEquilib. Thermodyn., 2019, 44(3), 277–284.

53. Heimburg, T. and Jackson, A. D. Thermodynamics of the nervous impulse. In Structure and Dynamics of Membranous Interfaces (Kaushik, N., ed.). John Wiley & Sons, 2008, 318–337.

54. Andersen, S. S., Jackson, A. D., and Heimburg, T. Towards a thermodynamic theory of nerve pulse propagation. Prog. Neurobiol., 2009, 88(2), 104–113.

55. Bressloff, P. C. Waves in Neural Media. Springer, New York, NY, 2014.

56. Ermentrout, G. B. and Terman, D. H. Mathematical Foundations of Neuroscience. Springer, New York, NY, 2010.

57. Garfinkel, A., Shevtsov, J., and Guo, Y. Modeling Life. Springer, Cham, 2017.

58. Nelson, P. C., Radosavljevic, M., and Bromberg, S. Biological Physics: Energy, Information, Life. W.H. Freeman and Company, New York, NY, 2003.

59. McCulloch, A. D. and Huber, G. Integrative biological modelling in silico. In ‘In Silico’ Simul. Biol. Process. (Bock, G. and Goode, J. A., eds). John Wiley & Sons, Chichester, 2002, 4–25.

60. Engelbrecht, J., Tamm, K., and Peets, T. On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes. Philos. Mag., 2017, 97(12), 967–987.

61. Peets, T., Tamm, K., Simson, P., and Engelbrecht, J. On solutions of a Boussinesq-type equation with displacementdependent nonlinearity: a soliton doublet. Wave Motion, 2019, 85, 10–17.

62. Peets, T. and Tamm, K. Mathematics of nerve signals. In Applied Wave Mathematics II (Berezovski, A. and Soomere, T., eds), Vol. 6, Springer, Cham, 2019, 207–238. 

63. Engelbrecht, J., Tamm, K., and Peets, T. Modelling of processes in nerve fibres at the interface of physiology and mathematics. Biomech. Model. Mechanobiol., 2020, 19, 2491–2498. 

64. Bennett, M. V. Electrical synapses, a personal perspective (or history). Brain Res. Rev., 2000, 32(1), 16–28.

65. Hormuzdi, S. G., Filippov, M. A., Mitropoulou, G., Monyer, H., and Bruzzone, R. Electrical synapses: a dynamic signaling system that shapes the activity of neuronal networks. – BBA Biomembr., 2004, 1662(1–2), 113–137.

66. Mueller, J. K. and Tyler, W. J. A quantitative overview of biophysical forces impinging on neural function. Phys. Biol., 2014, 11(5), 051001.

67. Engelbrecht, J., Tamm, K., and Peets, T. Modeling of complex signals in nerve fibers. Med. Hypotheses, 2018, 120, 90–95.

68. Bini, D., Cherubini, C., and Filippi, S. Heat transfer in Fitzhugh-Nagumo models. Phys. Rev. E, 2006, 74(4), 041905.

69. Scott, A. Nonlinear Science. Emergence and Dynamics of Coherent Structures. Oxford University Press, 1999.

70. Pietruszka, M., Stolarek, J., and Pazurkiewicz-Kocot, K. Time evolution of the action potential in plant cells. J. Biol. Phys, 1997, 23(4), 219–232.

71. El Hady, A. and Machta, B. B. Mechanical surface waves accompany action potential propagation. Nat. Commun., 2015, 6, 6697.

72. Jérusalem, A., García-Grajales, J. A., Merchán-Pérez, A., and Peña, J. M. A computational model coupling mechanics and electrophysiology in spinal cord injury. Biomech. Model. Mechanobiol., 2014, 13(4), 883–896.

73. Rvachev, M. M. On axoplasmic pressure waves and their possible role in nerve impulse propagation. Biophys. Rev. Lett., 2010, 5(2), 73–88.

74. Kappler, J., Shrivastava, S., Schneider, M. F., and Netz, R. R. Nonlinear fractional waves at elastic interfaces. Phys. Rev. Fluids, 2017, 2(11), 1–18.

75. Mussel, M. and Schneider, M. F. Similarities between action potentials and acoustic pulses in a van der Waals fluid. Sci. Rep., 2019, 9(1), 1–10.

76. Mussel, M. and 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, 2019, 16(151), 20180743.

77. Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.

78. Kotthaus, J. P. A mechatronics view at nerve conduction. arXiv:1909.06313 [physics.bio-ph], 2019.

79. Holland, L., de Regt, H. W., and Drukarch, B. Thinking about the nerve impulse: the prospects for the development of a comprehensive account of nerve impulse propagation. Front. Cell. Neurosci., 2019, 13(208), 1–12.

80. Hall, C. W. Laws and Models: Science, Engineering, and Technology. CRC Press, Boca Raton, 1999.

81. Heimburg, T. The important consequences of the reversible heat production in nerves and the adiabaticity of the action potential. arXiv:2002.06031v1 [physics.bio-ph], 2020.

82. Wooley, J. C. and Lin, H. S. (eds). Catalyzing Inquiry at the Interface of Computing and Biology. National Research Council, The National Academies Press, Washington, D.C., 2005.

83. Goriely, A., Geers, M. G., Holzapfel, G. A., Jayamohan, J., Jérusalem, A., Sivaloganathan, S., et al. Mechanics of the brain: perspectives, challenges, and opportunities. Biomech. Model. Mechanobiol., 2015, 14(5), 931–965.

84. Dyson, F. A meeting with Enrico Fermi. How one intuitive physicist rescued a team from fruitless research. Nature, 2004, 427(6972), 297.



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