Classic and Non-Classic Soliton Like Structures for Traveling Nerve Pulses


After some reduction procedure made on the nonlinear evolution equation for nerve pulses, based on thermodynamic principles, new classic and non-classic traveling solutions have been obtained. We have studied this model for particular values in the parameter space, and obtained both the bell and compacton like solutions. These nonlinear traveling waves could be responsible for transmitting efficiently the necessary information along the axons. The non-classic structures named as compactons, due to their robust configuration, could be considered in some sense a more realistic type of nonlinear chargers of information. The last solutions do not have tails and as adiabatic waves could propagate along the nerve with constant velocity that could be equal, smaller or higher than the sound velocity.

Share and Cite:

F. Contreras, H. Cervantes, M. Aguero and M. Najera, "Classic and Non-Classic Soliton Like Structures for Traveling Nerve Pulses," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 1, 2013, pp. 7-13. doi: 10.4236/ijmnta.2013.21002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. L. Hodgkin and A. F. Huxley, “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve,” The Journal of Physiology, Vol. 117, No. 4, 1952, pp. 500-544.
[2] B. Katz, “Nerve, Muscle, and Synapse,” Mc Graw-Hill, New York, 1966.
[3] R. Fitzhugh, “Impulses and Physiological States in Theoretical Models,” Biophysical Journal, Vol. 1, No. 6, 1961, pp. 445-466. doi:10.1016/S0006-3495(61)86902-6
[4] J. Nagumo, S. Arimoto and S. Yoshizawa, “An Active Pulse Transmission Line Simulating Nerve Axon,” Proceedings of the IRE, Vol. 50, No. 10, 1962, pp. 2061-2070.
[5] F. Ongay and M. Agüero, “Bifurcaciones del Sistema Fitzhugh Nagumo,” Ciencia Ergo Sum, Vol. 17, No. 3, 2011, pp. 295-306.
[6] C. B. Muratov, “A Quantitative Approximation Scheme for the Traveling Wave Solutions in the Hodgkin-Huxley Model,” Biophysical Journal, Vol. 79, No. 6, 2000, pp. 2893-2901. doi:10.1016/S0006-3495(00)76526-X
[7] T. Heinburg and A. Jackson, “On Soliton Propagation in Biomembranes and Nerves,” PNAS, Vol. 102, No. 28, 2005, pp. 9790-9795. doi:10.1073/pnas.0503823102
[8] B. Lautrup, R. Appali, A. D. Jackson and T. Heimburg, “The Stability of Solitons in Biomembranes and Nerves,” European Physical Journal E, Vol. 34, No. 6, 2011, pp. 1-9. doi:10.1140/epje/i2011-11057-0
[9] R. Appali, U. Van Rienenand and T. Heimburg, “A Comparison of the Hodgkin-Huxley Model and the Soliton Theory for the Action Potential in Nerves,” Advances in Planar Lipid Bilayers and Liposomas, Vol. 16, 2012, pp. 271-279. doi:10.1016/B978-0-12-396534-9.00009-X
[10] E. V. Vargas, et al., “Periodic Solutions and Refractory Periods in the Soliton Theory for Nerves and the Locust Femoral Nerve,” Biophysical Chemistry, Vol. 153 No. 2-3, 2011, pp. 159-167. doi:10.1016/B978-0-12-396534-9.00009-X
[11] G. B. Whitham, “Linear and Nonlinear Wave,” John Wiley and Son Inc., Hoboken, 1999. doi:10.1002/9781118032954
[12] M. Aguero and R. Alvarado, “Bright and Singular Solitons in the Boussinesq Like Equation,” Physica Scripta, Vol. 62, No. 4, 2000, pp. 232-237. doi:10.1238/Physica.Regular.062a00232
[13] V. G. Makhankov, “Soliton Phenomenology,” Kluwer Academic Publishers, Dordrecht, 1990. doi:10.1007/978-94-009-2217-4
[14] P. Rosenau and J. M. Hyman, “Compactons: Solitons with Finite Wavelength,” Physical Review Letters, Vol. 70, No. 5, 1993, pp. 564-567. doi:10.1103/PhysRevLett.70.564
[15] A. L. Bertozzi and M. Pugh, “The Lubrication Approximation for Thin Viscous Films: Regularity and Long-Time Behavior of Weak Solutions,” Communications on Pure and Applied Mathematics, Vol. 49, No. 2, 1996, pp. 85-123. doi:10.1002/(SICI)1097-0312(199602)49:2<85::AID-CPA1>3.0.CO;2-2
[16] M. Aguero, V. Serkin and T. Belyaeva, “Compacton Anti-Compacton Pair for Hydrogen Bonds and Rotational Waves in Dna Dynamics,” Communications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 8, 2011 pp. 3071-3080. doi:10.1016/j.cnsns.2010.10.025
[17] R. H. J. Grimshaw, L. A. Ostrovsky, V. I. Shrira and Y. A. Stepanyants, “Long Nonlinear Surface and Internal Gravity Waves in a Rotating Ocean,” Surveys in Geophysics, Vol. 19, No, 4, 1998, pp. 289-338. doi:10.1023/A:1006587919935
[18] S. Dusuel, P. Michaux and M. Remoissenet, “From Kinks to Compactonlike Kinks,” Physical Review E, Vol. 57, No. 2, 1998, pp. 2320-2326. doi:10.1103/PhysRevE.57.2320
[19] B. Mihaila, et al., ¨Stability and Dynamical Properties of Rosenau—Hyman Compactons Using Pade Approximants,” Physical Review E, Vol. 81, No. 5, 2010, p. 056708. doi:10.1103/PhysRevE.81.056708

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.