Abstract
The propagation of an action potential in nerves is accompanied by mechanical and thermal effects. Several mathematical models explain the deformation of the unmyelinated axon wall (a mechanical wave). In this paper, the deformation of the myelinated axon wall is studied. The mathematical model is inspired by the mechanics of microstructured materials with multiple scales. The model involves a Boussinesq-type equation together with a modification that describes the process in the myelin sheath. The dispersion analysis of such a model explains the behaviour of group and phase velocities. In addition, it is shown how dissipative effects may influence the process. Numerical calculations demonstrate the changes in velocities and wave profiles in the myelinated axon wall.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersen SS, Jackson AD, Heimburg T (2009) Towards a thermodynamic theory of nerve pulse propagation. Prog Neurobiol 88(2):104–113. https://doi.org/10.1016/j.pneurobio.2009.03.002
Berezovski A, Engelbrecht J, Peets T (2010) Multiscale modeling of microstructured solids. Mech Res Commun 37(6):531–534. https://doi.org/10.1016/j.mechrescom.2010.07.020
Berezovski A, Engelbrecht J, Maugin GA (2011) Thermoelasticity with dual internal variables. J Therm Stress 34(5–6):413–430. https://doi.org/10.1080/01495739.2011.564000
Berezovski A, Engelbrecht J, Salupere A, Tamm K, Peets T, Berezovski M (2013) Dispersive waves in microstructured solids. Int J Solids Struct 50:1981–1990. https://doi.org/10.1016/j.ijsolstr.2013.02.018
Bishop GH (1956) Natural history of the nerve impulse. Physiol Rev 36(3):376–399. https://doi.org/10.1152/physrev.1956.36.3.376
Bressloff PC (2014) Waves in neural media. Lecture notes on mathematical modelling in the life sciences. Springer, New York. https://doi.org/10.1007/978-1-4614-8866-8
Chen H, Garcia-Gonzalez D, Jérusalem A (2019) Computational model of the mechanoelectrophysiological coupling in axons with application to neuromodulation. Phys Rev E 99(3):032406. https://doi.org/10.1103/PhysRevE.99.032406
Christov CI, Maugin GA, Porubov AV (2007) On Boussinesq’s paradigm in nonlinear wave propagation. Comptes Rendus Mécanique 335(9–10):521–535. https://doi.org/10.1016/j.crme.2007.08.006
Coveney PV, Fowler PW (2005) Modelling biological complexity: a physical scientist’s perspective. J R Soc Interface 2(4):267–280. https://doi.org/10.1098/rsif.2005.0045
Debanne D, Campanac E, Bialowas A, Carlier E, Alcaraz G (2011) Axon physiology. Physiol Rev 91(2):555–602. https://doi.org/10.1152/physrev.00048.2009
Doyle DA, Cabral JM, Pfuetzner RA, Kuo A, Gulbis JM, Cohen SL, Chait BT, MacKinnon R (1998) The structure of the potassium channel: molecular basis of K\(^{+}\) conduction and selectivity. Science 280(5360):69–77. https://doi.org/10.1126/science.280.5360.69
Engelbrecht J, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philos Mag 85(33–35):4127–4141. https://doi.org/10.1080/14786430500362769
Engelbrecht J, Pastrone F, Braun M, Berezovski A (2006) Hierarchies of waves in nonclassical materials. In: Delsanto P (ed) Universality of nonclassical nonlinearity. Springer, New York, pp 29–47
Engelbrecht J, Tamm K, Peets T (2015) On mathematical modelling of solitary pulses in cylindrical biomembranes. Biomech Model Mechanobiol 14(1):159–167. https://doi.org/10.1007/s10237-014-0596-2
Engelbrecht J, Tamm K, Peets T (2017) On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes. Philos Mag 97(12):967–987. https://doi.org/10.1080/14786435.2017.1283070
Engelbrecht J, Peets T, Tamm K, Laasmaa M, Vendelin M (2018) On the complexity of signal propagation in nerve fibres. Proc Estonian Acad Sci 67(1):28–38. https://doi.org/10.3176/proc.2017.4.28
Engelbrecht J, Tamm K, Peets T (2020a) On mechanisms of electromechanophysiological interactions between the components of nerve signals in axons. Proc Estonian Acad Sci 69(2):81–96. https://doi.org/10.3176/proc.2020.2.03
Engelbrecht J, Tamm K, Peets T (2020b) Signals in nerves from a philosophical viewpoint. PhilSci Arch, p 18135
Engelbrecht J, Tamm K, Peets T (2021) Modelling of complex signals in nerves. Springer, Cham. https://doi.org/10.1007/978-3-030-75039-8
Fields RD (2014) Myelin—more than insulation. Science 344(6181):264–266. https://doi.org/10.1126/science.1253851
Fillafer C, Mussel M, Muchowski J, Schneider MF (2018) Cell surface deformation during an action potential. Biophys J 114(2):410–418. https://doi.org/10.1016/j.bpj.2017.11.3776
FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466. https://doi.org/10.1016/S0006-3495(61)86902-6
Fornberg B (1998) A practical guide to pseudospectral methods. Cambridge University Press, Cambridge
Gonzalez-Perez A, Mosgaard L, Budvytyte R, Villagran-Vargas E, Jackson A, Heimburg T (2016) Solitary electromechanical pulses in lobster neurons. Biophys Chem 216:51–59. https://doi.org/10.1016/j.bpc.2016.06.005
Hanig J, Negi G (2018) Myelin: structure, function, pathology, and targeted therapeutics, 2nd edn. Elsevier Inc., Amsterdam. https://doi.org/10.1016/B978-0-12-809405-1.00004-3
Heimburg T, Jackson AD (2005) On soliton propagation in biomembranes and nerves. Proc Natl Acad Sci USA 102(28):9790–9795. https://doi.org/10.1073/pnas.0503823102
Heimburg T, Jackson A (2007) On the action potential as a propagating density pulse and the role of anesthetics. Biophys Rev Lett 02(01):57–78. https://doi.org/10.1142/S179304800700043X
Hill WD, Marioni RE, Maghzian O, Ritchie SJ, Hagenaars SP, McIntosh AM, Gale CR, Davies G, Deary IJ (2019) A combined analysis of genetically correlated traits identifies 187 loci and a role for neurogenesis and myelination in intelligence. Mol Psychiatry 24(2):169–181. https://doi.org/10.1038/s41380-017-0001-5
Hodgkin AL (1964) The conduction of the nervous impulse. Liverpool University Press, Liverpool
Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544. https://doi.org/10.1113/jphysiol.1952.sp004764
Hodgkin AL, Katz B (1949) The effect of temperature on the electrical activity of the giant axon of the squid. J Physiol 109(1–2):240–249. https://doi.org/10.1113/jphysiol.1949.sp004388
Huang CYM, Rasband MN (2018) Axon initial segments: structure, function, and disease. Ann N Y Acad Sci 1420(1):46–61. https://doi.org/10.1111/nyas.13718
Ilison L, Salupere A (2009) Propagation of sech\(^2\)-type solitary waves in hierarchical KdV-type systems. Math Comput Simul 79(11):3314–3327. https://doi.org/10.1016/j.matcom.2009.05.003
Ilison L, Salupere A, Peterson P (2007) On the propagation of localized perturbations in media with microstructure. Proc Eststonian Acad Sci Phys Math 56(2):84–92
Iwasa K, Tasaki I, Gibbons R (1980) Swelling of nerve fibers associated with action potentials. Science 210(4467):338–339. https://doi.org/10.1126/science.7423196
Jerusalem A, Al-Rekabi Z, Chen H, Ercole A, Malboubi M, Tamayo-Elizalde M, Verhagen L, Contera S (2019) Electrophysiological-mechanical coupling in the neuronal membrane and its role in ultrasound neuromodulation and general anaesthesia. Acta Biomater 97:116–140. https://doi.org/10.1016/j.actbio.2019.07.041
Lodish H, Berk A, Matsudaira P, Kaiser CA, Krieger M, Scott MP (2004) Transport of ions and small molecules across cell membranes. Mol Cell Biol 7:245–300
Michailov GV, Sereda M, Brinkmann B, Fischer T, Haug B, Birchmeier C, Role L, Lai C, Schwab M, Nave KA (2004) Axonal neuregulin-1 regulates myelin sheath thickness. Science 304(5671):700–703. https://doi.org/10.1126/science.1095862
Mindlin R (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78
Mueller JK, Tyler WJ (2014) A quantitative overview of biophysical forces impinging on neural function. Phys Biol 11(5):051001. https://doi.org/10.1088/1478-3975/11/5/051001
Mussel M, Schneider MF (2019) 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. https://doi.org/10.1098/rsif.2018.0743
Nelson PC, Radosavljevic M, Bromberg S (2003) Biological physics: energy, information, life. W.H. Freeman and Company, New York
Peets T (2016) Internal scales and dispersive properties of microstructured materials. Math Comput Simul 127:220–228. https://doi.org/10.1016/j.matcom.2014.03.006
Peets T, Tamm K (2015) On mechanical aspects of nerve pulse propagation and the Boussinesq paradigm. Proc Estonian Acad Sci 64(3):331. https://doi.org/10.3176/proc.2015.3S.02
Peets T, Tamm K (2019) Mathematics of nerve signals. In: Berezovski A, Soomere T (eds) Appl. wave math. II. Mathematics of planet earth, vol 6. Springer, Cham, pp 207–238
Peets T, Kartofelev D, Tamm K, Engelbrecht J (2013) Waves in microstructured solids and negative group velocity. EPL (Europhys Lett) 103(1):16001. https://doi.org/10.1209/0295-5075/103/16001
Peets T, Tamm K, Simson P, Engelbrecht J (2019) On solutions of a Boussinesq-type equation with displacement-dependent nonlinearity: a soliton doublet. Wave Motion 85:10–17. https://doi.org/10.1016/j.wavemoti.2018.11.001
Perez-Camacho MI, Ruiz-Suarez J (2017) Propagation of a thermo-mechanical perturbation on a lipid membrane. Soft Matter 13:6555–6561. https://doi.org/10.1039/C7SM00978J
Porubov AV (2003) Amplification of nonlinear strain waves in solids. World Scientific, Singapore
Raasakka A, Ruskamo S, Kowal J, Han H, Baumann A, Myllykoski M, Fasano A, Rossano R, Riccio P, Bürck J, Ulrich AS, Stahlberg H, Kursula P (2019) Molecular structure and function of myelin protein P0 in membrane stacking. Sci Rep 9(1):1–15. https://doi.org/10.1038/s41598-018-37009-4
Ravera S, Morelli AM, Panfoli I (2020) Myelination increases chemical energy support to the axon without modifying the basic physicochemical mechanism of nerve conduction. Neurochem Int 141(104):883. https://doi.org/10.1016/j.neuint.2020.104883
Salupere A (2009) The pseudospectral method and discrete spectral analysis. In: Quak E, Soomere T (eds) Appl. wave math. Springer, Berlin, pp 301–334. https://doi.org/10.1007/978-3-642-00585-5
Sanders FK (1948) The thickness of the myelin sheaths of normal and regenerating peripheral nerve fibres. Proc R Soc Lond Ser B Biol Sci 135(880):323–357. https://doi.org/10.1098/rspb.1948.0015
Schmidt H, Knösche TR (2019) Action potential propagation and synchronisation in myelinated axons. PLOS Comput Biol 15(10):10070e04. https://doi.org/10.1371/journal.pcbi.1007004
Schwiening CJ (2012) A brief historical perspective: Hodgkin and Huxley. J Physiol 590(11):2571–2575. https://doi.org/10.1113/jphysiol.2012.230458
Sula A, Booker J, Ng LC, Naylor CE, Decaen PG, Wallace BA (2017) The complete structure of an activated open sodium channel. Nat Commun 8:2–10. https://doi.org/10.1038/ncomms14205
Suminaite D, Lyons DA, Livesey MR (2019) Myelinated axon physiology and regulation of neural circuit function. Glia 67(11):2050–2062. https://doi.org/10.1002/glia.23665
Tamm K, Peets T, Engelbrecht J, Kartofelev D (2017) Negative group velocity in solids. Wave Motion 71:127–138. https://doi.org/10.1016/j.wavemoti.2016.04.010
Tasaki I (1988) A macromolecular approach to excitation phenomena: mechanical and thermal changes in nerve during excitation. Physiol Chem Phys Med NMR 20(4):251–268
Tasaki I, Byrne PM (1992) Heat production associated with a propagated impulse in Bullfrog myelinated nerve fibers. Jpn J Physiol 42(5):805–813. https://doi.org/10.2170/jjphysiol.42.805
Tasaki I, Kusano K, Byrne PM (1989) Rapid mechanical and thermal changes in the garfish olfactory nerve associated with a propagated impulse. Biophys J 55(6):1033–1040. https://doi.org/10.1016/S0006-3495(89)82902-9
Terakawa S (1985) Potential-dependent variations of the intracellular pressure in the intracellularly perfused squid giant axon. J Physiol 369(1):229–248. https://doi.org/10.1113/jphysiol.1985.sp015898
Tomassy GS, Berger DR, Chen HH, Kasthuri N, Hayworth KJ, Vercelli A, Seung HS, Lichtman JW, Arlotta P (2014) Distinct profiles of myelin distribution along single axons of pyramidal neurons in the neocortex. Science 344(6181):319–324. https://doi.org/10.1126/science.1249766
Ván P, Berezovski A, Engelbrecht J (2008) Internal variables and dynamic degrees of freedom. J Non-Equilib Thermodyn. https://doi.org/10.1515/JNETDY.2008.010
Ván P, Kovács R, Vázquez F (2022) Spectral properties of dissipation. J Non-Equilib Thermodyn 47(1):95–102. https://doi.org/10.1515/jnet-2021-0050
Watanabe A (1986) Mechanical, thermal, and optical changes of the nerve membrane associated with excitation. Jpn J Physiol 36:625–643. https://doi.org/10.2170/jjphysiol.36.625
Winlow W, Johnson AS (2021) Nerve impulses have three interdependent functions: communication, modulation, and computation. Bioelectricity 3(3):161–170. https://doi.org/10.1089/bioe.2021.0001
Yang Y, Liu XW, Wang H, Yu H, Guan Y, Wang S, Tao N (2018) Imaging action potential in single mammalian neurons by tracking the accompanying sub-nanometer mechanical motion. ACS Nano 12(5):4186–4193. https://doi.org/10.1021/acsnano.8b00867
Acknowledgements
This research was supported by the Estonian Research Council (IUT 33-24, PRG 1227). Jüri Engelbrecht acknowledges the support from the Estonian Academy of Sciences. Authors would like to thank all reviewers for thought-provoking questions and suggesting improvements that made the paper easier to follow.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tamm, K., Peets, T. & Engelbrecht, J. Mechanical waves in myelinated axons. Biomech Model Mechanobiol 21, 1285–1297 (2022). https://doi.org/10.1007/s10237-022-01591-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-022-01591-4