IJMNTA  Vol.2 No.4 , December 2013
Non-Topological Solitons as Traveling Pulses along the Nerve
ABSTRACT

Several new soliton-like structures have been obtained under the consideration of non trivial boundary condition for the difference value of density in the thermodynamic model of nerve pulses. The model is based on thermodynamic principles of zero transfer of energy to the media. We have studied these solutions for particular values in the parameter space, and obtained both bell soliton on the condensate and bubble like solutions as typical non-topological representative solutions. The solutions will propagate along the nerve with constant velocity. The analysis of the properties of the solutions provides us with available permitted velocities and the prediction of the constant density value of the background at long distances far from the excited zone in the nerve.


Cite this paper
F. Contreras, F. Ongay, O. Pavón and M. Aguero, "Non-Topological Solitons as Traveling Pulses along the Nerve," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 4, 2013, pp. 195-200. doi: 10.4236/ijmnta.2013.24027.
References
[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,” McGraw-Hill, New York, 1966.

[3]   R. FitzHugh, “Impulses and Physiological States in Theoretical Models,” Biophysical Journal, Vol. 1, No. 6, 1961 pp. 445-466. http://dx.
http://dx.doi.org/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. http://dx.doi.org/10.1109/JRPROC.1962.288235

[5]   S. Abbasbandy, “Soliton Solutions for the Fitzhugh-Nagumo Equation with the Homotopy Analysis Method,” Applied Mathematical Modelling, Vo. 32, No. 12, 2008, pp. 2706-2714.
http://dx.doi.org/10.1016/j.apm.2007.09.019

[6]   F. Ongay and M. Aguero, “Bifurcations in the Fitzhugh— Nagumo System,” Ciencia Ergo Sum, Vol. 17, No. 3, 2011, pp. 295-306.

[7]   Y. Kobatake, I. Tasaki and A. Watanabe, “Phase Transition in Membrane with Reference to Nerve Excitation,” Advances in Biophysics, Vol. 2, 1971, pp. 1-31

[8]   T. Heinburg and A. Jackson, “On Soliton Propagation in Biomembranes and Nerves,” Proceedings of the National Academy of Sciences, Vol. 102, No. 28, 2005, pp. 9790-9795. http://dx.doi.org/10.1073/pnas.0503823102

[9]   B. Lautrup, R. Appali, A. D. Jackson and T. Heimburg, “The Stability of Solitons in Biomembranes and Nerves,” The European Physical Journal E, Vol. 34, 201, p. 57.
http://dx.doi.org/10.1140/epje/i2011-11057-0

[10]   E. Villagran, A. Ludu, R. Hustert, P. Gumrich, A. D. Jackson and T. Heimburg, “Periodic Solutions and Refractory Periods in the Soliton Theory for Nerves He Locust Femoral Nerve,” Biophysical Chemistry, Vol. 153, No. 2-3, 2011, pp. 159-167.
http://dx.doi.org/10.1016/B978-0-12-396534-9.00009-X

[11]   R. Appali, S. Petersen and U. van Rienen, “A Comparison of Hodgkin-Huxley and Soliton Neural Theories,” Advances in Radio Science, Vol. 8, 2010, pp. 75-79.
http://dx.doi.org/10.5194/ars-8-75-2010

[12]   V. G. Makhankov, “Soliton Phenomenology,” Kluwer Academic Publishers, Norwell, 1990.
dhttp://dx.doi.org/10.1007/978-94-009-2217-4

[13]   N. N. Akhmediev and A. Ankevich, “Solitons, Nonlinear Pulses and Beams,” Chapman and Hall, London, 1997.

[14]   D. E. Pelinovsky and P. G. Kevrekidis, “Dark Solitons in External Potentials,” Zeitschrift für angewandte Mathematik und Physik, Vol. 59, No. 4, 2008, pp. 559-599.
http://dx.doi.org/10.1007/s00033-007-6120-0

[15]   V. N. Serkin, A. Hasegawa, “Novel Soliton Solutions of the Nonlinear Schrodinger Equation Model,” Physical Review Letters, Vol. 85, No. 21, 2000, pp. 4502-4505.
http://dx.doi.org/10.1103/PhysRevLett.85.4502

[16]   S. Rajendran, P. Muruganandam and M. Lakshmanan, “Bright and Dark Solitons in a Quasi 1D Bose-Einstein Condensates Modelled by 1D Gross-Pitaevskii Equation with Time-Dependent Parameters,” Physica D, Vol. 239, No. 7, 2010, pp. 366-386.
http://dx.doi.org/10.1016/j.physd.2009.12.005

[17]   Yu. Kivshar and B. Luther-Davies, “Dark Optical Solitons: Physics and Applications,” Physics Reports, Vol. 298, No. 2-3, 1998, p. 81.
http://dx.doi.org/10.1016/S0370-1573(97)00073-2

[18]   D. J. Frantzeskakis, “Dark Solitons in Atomic Bose-Einstein Condensates: From Theory to Experiments,” Journal of Physics A: Mathematical and Theoretical, Vol. 43, No. 21, 2010, Article ID: 21300.
http://dx.doi.org/10.1088/1751-8113/43/21/213001

 
 
Top