Alexander Mengnjo¹, Alain M. Dikandé², Gideon A. Ngwa¹

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¹ Department of Mathematics, Faculty of Science, University of Buea, Buea, Cameroon.
² Department of Physics, Faculty of Science, University of Buea, Buea, Cameroon.
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Abstract: Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.

Keywords: Nerve Impulse, Mechanosensory Response, Hodgkin-Huxley Equation, Korteweg-de Vries Equation, Associated Legendre Polynomials

Cite this paper: Mengnjo, A., Dikandé, A.M. and Ngwa, G.A. (2020) Model of the Nerve Impulse with Account of Mechanosensory Processes: Stationary Solutions.. Journal of Applied Mathematics and Physics, 8, 2091-2102. doi: 10.4236/jamp.2020.810156.

References

[1]   Hodgkin, A.L. and Huxley, A.F. (1952) A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. The Journal of Physiology, 117, 500-544.
https://doi.org/10.1113/jphysiol.1952.sp004764

[2]   Engelbretcht, J. (1981) On Theory of Pulse Transmission in a Nerve Fibre. Proceedings of the Royal Society of London, 375, 195.
https://doi.org/10.1098/rspa.1981.0047

[3]   Tasaki, I. and Matsumoto. G. (2002) On the Cable Theory of Nerve Conduction. Bulletin of Mathematical Biology, 64, Article ID: 1069.
https://doi.org/10.1006/bulm.2002.0310

[4]   Heimburg, T. and Jackson, A.D. (2005) On Soliton Propagation in Biomembranes and Nerves. Proceedings of the National Academy of Sciences of the United States of America, 102, 9790-9795.
https://doi.org/10.1073/pnas.0503823102

[5]   Dikandé, A.M. and Ga-Akeku, B. (2008) Localized Short Impulses in a Nerve Model with Self-Excitable Membrane. Physical Review E, 80, Article ID: 041904.
https://doi.org/10.1103/PhysRevE.80.041904

[6]   Keynes, R.D. and Aidley, D.J. (1982) Nerve and Muscle. Cambridge University Press, Cambridge, MA.

[7]   Fongang Achu, G., Moukam-Kakmeni, F.M. and Dikandé, A.M. (2018) Breathing Pulses in the Damped-Soliton Model for Nerves. Physical Review E, 97, Article ID: 012211.
https://doi.org/10.1103/PhysRevE.97.012211

[8]   Engelbrecht, J., Peets, T., Tamm, J., Laasmaa, M. and Vendelin, M. (2018) On the Complexity of Signal Propagation in Nerve Fibres. Proceedings of Estonian Academy of Science, 67, 28-38.
https://doi.org/10.3176/proc.2017.4.28

[9]   Warman, E.N., Grill, W.M. and Duran, D. (1992) Modeling the Effects of Electric Fields on Nerve Fibers: Determination of Excitation Thresholds. IEEE Transactions on Biomedical Engineering, 39, 1244-1254.
https://doi.org/10.1109/10.184700

[10]   Grill, W.M. (1999) Modeling the Effects of Electric Fields on Nerve Fibers: Influence of Tissue Electrical Properties. IEEE Transactions on Biomedical Engineering, 46, 918-928.
https://doi.org/10.1109/10.775401

[11]   Mueller, J.K. and Tyler, W.J. (2014) A Quantitative Overview of Biophysical Forces Impinging on Neural Function. Physical Biology, 11, Article ID: 051001.
https://doi.org/10.1088/1478-3975/11/5/051001

[12]   Heimburg, T. (2014) Mechanical Aspects of Membrane Thermodynamics. Estimation of the Mechanical Properties of Lipid Membranes Close to the Chain Melting Transition from Calorimetry. Biochimica et Biophysica Acta (BBA)-Biomembranes, 1415, 147-162.
https://doi.org/10.1016/S0005-2736(98)00189-8

[13]   Tasaki, I., Kusano, K. and Byrne, M. (1989) Rapid Mechanical and Thermal Changes in the Garfish Olfactory Nerve Associated with a Propagated Impulse. Biophysical Journal, 55, 1033-1040.
https://doi.org/10.1016/S0006-3495(89)82902-9

[14]   Tasaki, I. and Byrne, M. (1990) Volume Expansion of Nonmyelinated Nerve Fibers during Impulse Conduction. Biophysical Journal, 57, 633-635.
https://doi.org/10.1016/S0006-3495(90)82580-7

[15]   Blunck, R. and Batulan, Z. (2012) Mechanism of Electromechanical Coupling in Voltage-Gated Potassium Channels. Frontiers in Pharmacology, 3, 166.
https://doi.org/10.3389/fphar.2012.00166

[16]   Covarrubias, M., Barber, A.F., Carnevale, V., Treptow, W. and Eckenhof, R.G. (2015) Mechanistic Insights into the Modulation of Voltage-Gated Ion Channels by Inhalational Anesthetics. Biophysical Journal, 109, 2003-2011.
https://doi.org/10.1016/j.bpj.2015.09.032

[17]   Heimburg, T. (2004) Mechanistic Insights into the Modulation of Voltage-Gated Ion Channels by Inhalational Anesthetics. Biochimica et Biophysica Acta-Biomembranes, 1662, 113.

[18]   Wallace, D.C. (1965) Thermal Expansion and Other Anharmonic Properties of Crystals. Physical Review, 139, A877.
https://doi.org/10.1103/PhysRev.139.A877

[19]   Petrov, A.G. (1965) Flexoelectricity of Model and Living Membranes. Biochimica et Biophysica Acta (BBA)-Biomembranes, 1561, 1-25.
https://doi.org/10.1016/S0304-4157(01)00007-7

[20]   Gross, D., Williams, W.S. and Connor, J.A. (1983) Theory of Electromechanical Effects in Nerve. Cellular and Molecular Neurobiology, 3, 89-111.
https://doi.org/10.1007/BF00735275

[21]   Heimburg, T., Blicher, A., Mosgaard, L.D. and Zecchi, K. (2004) Electromechanical Properties of Biomembranes and Nerves. Journal of Physics: Conference Series, 558, Article ID: 012018.
https://doi.org/10.1088/1742-6596/558/1/012018

[22]   Plaksin, M., Shoham, S. and Kimmel, E. (2014) Intramembrane Cavitation as a Predictive Bio-Piezoelectric Mechanism for Ultrasonic Brain Stimulation. Physical Review X, 4, Article ID: 011004.
https://doi.org/10.1103/PhysRevX.4.011004

[23]   Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M. (1967) Method for Solving the Korteweg-de Vries Equation. Physical Review Letters, 19, 1095-1097.
https://doi.org/10.1103/PhysRevLett.19.1095

[24]   Lax, P.D. (1968) Integrals of Nonlinear Equations of Evolution and Solitary Waves. Communications on Pure and Applied Mathematics, 31, 467-490.
https://doi.org/10.1002/cpa.3160210503

[25]   Gelfand, I.M. and Levitan, B.M. (1951) On the Determination of a Differential Equation from Its Spectral Function. Izv. Akad. Nauk SSSR Ser. Mat., 15, 309-360.

[26]   Marchenko, V.A. (1955) Reconstruction of the Potential Energy from the Phases of the Scattered Waves. Doklady Akademii Nauk SSSR, 104, 695-698.

[27]   Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with Formula, Graphs and Mathematical Tables. National Bureau of Standards, Dover, USA.