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 .
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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.
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Peets, T., Tamm, K. (2019). Mathematics of Nerve Signals. In: Berezovski, A., Soomere, T. (eds) Applied Wave Mathematics II. Mathematics of Planet Earth, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-29951-4_10
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