Mathematics of Nerve Signals
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 .
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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).
- Andronov, A., Witt, A., Khaikin, S.: Theory of Oscillations. Phys. Math. Publ. Moscow (In Russian) (1959)Google Scholar
- Billingham, J., King, A.C.: Wave Motion. Cambridge Texts in Applied Mathematics. Cambridge University Press (2001). https://doi.org/10.1017/CBO9780511841033
- 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
- 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
- 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
- Engelbrecht, J., Tamm, K., Peets, T.: Primary and secondary components of nerve signals. arXiv:1812.05335 [physics.bio-ph] (2018)Google Scholar
- Engelbrecht, J., Tamm, K., Peets, T.: Mathematics of nerve signals. arXiv:1902.00011 [physics.bio-ph] (2019)Google Scholar
- 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
- Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1998). https://doi.org/10.1017/CBO9780511626357
- Hindmarsh, A.: ODEPACK, A Systematized Collection of ODE Solvers, vol. 1. North-Holland, Amsterdam (1983)Google Scholar
- Hodgkin, A.L.: The Conduction of the Nervous Impulse. Liverpool University Press (1964)Google Scholar
- 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
- Jones, E., Oliphant, T., Peterson, P.: SciPy: open source scientific tools for Python (2007). http://www.scipy.org
- Kaufmann, K.: Action Potentials and Electromechanical Coupling in the Macroscopic Chiral Phospholipid Bilayer. Caruaru, Brazil (1989)Google Scholar
- 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
- Peterson, P.: F2PY: Fortran to Python interface generator. http://cens.ioc.ee/projects/f2py2e/ (2005)
- Porubov, A.V.: Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore (2003). https://doi.org/10.1142/5238
- Reissig, R., Sansone, G., Conti, R.: Qualitative Theorie Nichtlinearer Differentialgleichungen. Edizioni Cremonese, Roma (1963)Google Scholar
- Scott, A.: Nonlinear Science. Emergence & Dynamics of Coherent Structures. Oxford University Press (1999)Google Scholar
- 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
- 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