Research Article
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On the new travelling wave solution of a neural communication model

Year 2019, Volume: 21 Issue: 2, 666 - 678, 28.06.2019

Gülnur Yel

https://doi.org/10.25092/baunfbed.636782
Cited By: 2

Abstract

The aim of this study is to present some new travelling wave solutions of conformable time-fractional Fitzhugh–Nagumo equation that model the transmission of nerve impulses.  For this purpose, the improved Bernoulli sub-equation function method has been used.  The obtained results are shown by way of the the 3D-2D graphs and contour surfaces for the suitable values.

Keywords

Time-Fractional Fitzhugh–Nagumo equation conformable fractional derivative travelling wave solutions

References

  • Keener, J. P. and Sneyd, J., Mathematical Physiology, Springer, New York, (1998).
  • Murray, J. D., Mathematical Biology I and II, Springer, New York, (2002).
  • Fisher, R. A., The wave of advantageous genes, Annals of Eugenics. 7, 355-369, (1937).
  • Zeldovich, Y. B. and Frank-Kamenetskii, D. A. Zhurnal Fis. Khimii, 12, 1938, 100; Acts Physico-them. URSS, 9, 341, (1938).
  • Wilhelmsson, H. and Lazzaro, E., Reaction–diffusion problems in the physics of hot plasmas, Bristol and Philadelphia, Bristol and Philadelphia: Institute of Physics Publishing, (2001).
  • Hundsdorfer, W. and Verwer, J. G., Numerical solution of time dependent advection-diffusion-reaction equations, Berlin: Springer, (2003).
  • Hodgkin, A. L. and Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117, 500–544, (1952).
  • Fitzhugh, R., Impulse and physiological states in models of nerve membrane, Biophysical Journal, 1, 445–466, (1961).
  • Nagumo, J. S., Arimoto, S. and Yoshizawa, S., An active pulse transmission line simulating nurve axon, Proceedings of the Institute of Radio Engineers, 50, 2061–2070, (1962).
  • Jost, J., Mathematical Methods in Biology and Neurobiology, Springer, (2014).
  • Wang, J., Zhang, T. and Deng, B., Synchronization of FitzHugh Nagumo neurons in external electrical stimulation via nonlinear control, Chaos, Solitons and Fractals, 31, 30–38, (2007).
  • Murray, J. D., Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, Springer, (2003).
  • Quininao, C. and Touboul, J. D., Clamping and Synchronization in the Strongly Coupled FitzHugh–Nagumo Model, submitted, (2018).
  • Tabi, C. B., Dynamical analysis of the FitzHugh–Nagumo oscillations through a modified Van der Pol equation with fractional-order derivative term, International Journal of Non-Linear Mechanics 105,173–178, (2018).
  • Momani, S., Freihat, A., and AL-Smadi, M., Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method, Abstract and Applied Analysis, Article ID 276279, 10p, (2014).
  • Markov, N., Ushenin, K., and Hendy A., Performance Evaluation of Space Fractional FitzHugh-Nagumo Model: an Implementation with PETSc Library, CEUR Workshop Proceedings, 1729, 12, (2016).
  • Armanyos, M. and Radwan, A. G., Fractional-Order Fitzhugh-Nagumo and Izhikevich Neuron Models, 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), (IEEE 2016), 1-5, Thailand, (2016) .
  • Sungu, I. C. and Demir, H., A new approach and solution technique to solve time fractional nonlinear reaction-diffusion equations, Mathematical Problems in Engineering, 2015, Article ID 457013, p.13, (2015).
  • Brandibur, O. and Kaslik, E., Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model, Mathematical Methods in the Applied Sciences, 41(17), 7182-7194, (2018).
  • Ascione, G., and Pirozzi, E., On Fractional Stochastic Modeling of Neuronal Activity Including Memory Effects, Computer Aided Systems Theory – EUROCAST 2017, 3-11, Spain, (2018).
  • Khanday, F. A., Kant, N. A., Dar, R. M. and Zulkifli, T. Z. A., Low-Voltage Low-Power Integrable CMOS Circuit Implementation of Integer- and Fractional-Order FitzHugh-Nagumo Neuron Model, IEEE Transactions on Neural Networks and Learning Systems, 99, 1-15, (2018).
  • Kumar, D., Singh, J. and Baleanu, D., A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses, Nonlinear Dynamics, 91, 307–317, (2018).
  • Veeresha, P., Prakasha, D. G. and Baskonus, H. M., New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29, 013119, (2019).
  • Gencoglu, M. T., Baskonus, H. M. and Bulut, H., Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, AIP Conference Proceedings, 020103 (1798), 1-9, (2017).
  • De Pillis, L. G., Gua, W. and Radunskaya, A. E., Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, Journal of Theoretical Biology, 238, 841–862, (2006).
  • Buzsaki, G. and Draguhn, A., Neuronal oscillations in cortical networks, Science, 304, 1926-1929, (2004).
  • Wang, X. J., Neurophysiological and computational principles of cortical rhythms in cognition, Physiological reviews, 90, 1195-1268, (2010).
  • Yokus, A., Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32(29), 1850365, (2018).
  • Yokus, A., Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 57(3), 2085-2091, (2018).
  • Yokus, A. and Bulut, H., On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications, 9(1), 18, (2018).
  • Yokus, A., Baskonus, H. M., Sulaiman, T. A. and Bulut, H., Numerical simulation and solutions of the two‐component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34(1), 211-227, (2018).
  • Yavuz, M. and Ozdemir, N., On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative, ITM Web of Conferences, 22, 01045, (2018).
  • Yokus, A. and Tuz, M., An application of a new version of (G′/G)-expansion method, AIP Conference Proceedings, 1798(1), 020165, (2017).
  • Yel, G., Baskonus, H. M. and Bulut, H., Novel Archetypes of New Coupled Konno-Oono Equation by Using sine-Gordon Expansion Method, Optical and Quantum Electronics, 49, 285, (2017).
  • Kumar, D., Hosseini, K. and Samadani, F., The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics, Optik - International Journal for Light and Electron Optics, 149, 439-446, (2017).
  • Khan, K., Akbar, M. A., Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 15(1), (2013).
  • Baskonus, H. M., Sulaiman, T. A. and Bulut, H., On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics, Indian Journal of Physics, 1-7, (2018).
  • Sulaiman, T. A., Bulut, H., Yel, G. and Atas, S. S., Optical solitons to the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model, Optical and Quantum Electronics, 50, 372, (2018).
  • Sulaiman, T. A., Yel, G. and Bulut, H., M-fractional solitons and periodic wave solutions to the Hirota Maccari system, Modern Physics Letters B, 33, No. 0, 1950052, (2019).
  • Baskonus, H. M., Yel, G. and Bulut, H., Novel wave surfaces to the fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, American Institute of Physics, 1863(560084) (2017).
  • Kocak, Z. F. and Yel, G., Trigonometric Function Solutions of Fractional Drinfeld's Sokolov -Wilson System, ITM Web of Conferences 13, 01006, (2017).
  • Veeresha, P., Prakasha D. G. and Baskonus H. M., Novel simulations to the time-fractional Fisher’s equation, Mathematical Science,13, (2019).
  • Merdan, M., Solutions of time-fractional reaction–diffusion equation with modified Riemann–Liouville derivative, International. Journal of. Physical. Sciences. 7(15), 2317–2326 (2012).
  • Uçar, S., Uçar, E., Özdemir, N. and Hammouch, Z., Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons& Fractals, 118, 300-306, (2019).
  • Atangana, A. and Alkahtani, B., Analysis of the Keller–Segel model with a fractional derivative without singular kernel, Entropy, 17, 4439-4453, (2015).
  • Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications, 6, 75-83, (2016).
  • Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress in. Fractional Differentiation and Applications, 1, 73-85, (2015).
  • Atangana, A. and Baleanu, D., New fractional derivatives with non-local and non-singular kernel theory and applications to heat transfer model, Thermal Science, 20, 763-769, (2016).
  • Khalila, R., Al Horania, M., Yousefa, A. and Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
  • Atangana, A., Baleanu, D. and Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13, 889–898, (2015).
  • Khan, N. A., Khan, N. U., Ara, A. and Jamil, M., Approximate analytical solutions of fractional reaction-diffusion equations, Journal of King Saud University - Science, 24, 111-118,(2012).
  • Rida, S. Z., El-Sayed, A. M. A. and Arafa, A. A. M., On the Solutions of Time-Fractional Reaction-Diffusion Equations, Communication in Nonlinear Science & Numerical Simulation, 15, 3847-3854, (2010) .
  • Pandir, Y. and Tandogan, Y. A., Exact solutions of the time-fractional Fitzhugh-Nagumo equation, AIP Conference Proceedings 1558, 1919, (2013).

Bir sinirsel iletişim modelinin yeni salınımlı dalga çözümleri üzerinde

Year 2019, Volume: 21 Issue: 2, 666 - 678, 28.06.2019

Gülnur Yel

https://doi.org/10.25092/baunfbed.636782
Cited By: 2

Abstract

Bu çalışmanın amacı, sinir uyarılarının iletişimini modelleyen uyumlu zaman-kesirli türevli Fitzhugh–Nagumo denkleminin bazı yeni salınımlı dalga çözümlerini sunmaktır.  Bu amaçla, geliştirilmiş Bernoulli alt denklem fonksiyon metodu kullanılmıştır.  Elde edilen çözümler uygun değerler için 2-3 boyutlu grafikler ve kontur yüzeyleri ile gösterilmiştir. 

Keywords

Zaman-kesirli türevli Fitzhugh–Nagumo denklemi uyumlu kesirli türev salınımlı dalga çözümleri

References

  • Keener, J. P. and Sneyd, J., Mathematical Physiology, Springer, New York, (1998).
  • Murray, J. D., Mathematical Biology I and II, Springer, New York, (2002).
  • Fisher, R. A., The wave of advantageous genes, Annals of Eugenics. 7, 355-369, (1937).
  • Zeldovich, Y. B. and Frank-Kamenetskii, D. A. Zhurnal Fis. Khimii, 12, 1938, 100; Acts Physico-them. URSS, 9, 341, (1938).
  • Wilhelmsson, H. and Lazzaro, E., Reaction–diffusion problems in the physics of hot plasmas, Bristol and Philadelphia, Bristol and Philadelphia: Institute of Physics Publishing, (2001).
  • Hundsdorfer, W. and Verwer, J. G., Numerical solution of time dependent advection-diffusion-reaction equations, Berlin: Springer, (2003).
  • Hodgkin, A. L. and Huxley, A. F., A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117, 500–544, (1952).
  • Fitzhugh, R., Impulse and physiological states in models of nerve membrane, Biophysical Journal, 1, 445–466, (1961).
  • Nagumo, J. S., Arimoto, S. and Yoshizawa, S., An active pulse transmission line simulating nurve axon, Proceedings of the Institute of Radio Engineers, 50, 2061–2070, (1962).
  • Jost, J., Mathematical Methods in Biology and Neurobiology, Springer, (2014).
  • Wang, J., Zhang, T. and Deng, B., Synchronization of FitzHugh Nagumo neurons in external electrical stimulation via nonlinear control, Chaos, Solitons and Fractals, 31, 30–38, (2007).
  • Murray, J. D., Mathematical Biology: I. An Introduction, Interdisciplinary Applied Mathematics, Springer, (2003).
  • Quininao, C. and Touboul, J. D., Clamping and Synchronization in the Strongly Coupled FitzHugh–Nagumo Model, submitted, (2018).
  • Tabi, C. B., Dynamical analysis of the FitzHugh–Nagumo oscillations through a modified Van der Pol equation with fractional-order derivative term, International Journal of Non-Linear Mechanics 105,173–178, (2018).
  • Momani, S., Freihat, A., and AL-Smadi, M., Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method, Abstract and Applied Analysis, Article ID 276279, 10p, (2014).
  • Markov, N., Ushenin, K., and Hendy A., Performance Evaluation of Space Fractional FitzHugh-Nagumo Model: an Implementation with PETSc Library, CEUR Workshop Proceedings, 1729, 12, (2016).
  • Armanyos, M. and Radwan, A. G., Fractional-Order Fitzhugh-Nagumo and Izhikevich Neuron Models, 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), (IEEE 2016), 1-5, Thailand, (2016) .
  • Sungu, I. C. and Demir, H., A new approach and solution technique to solve time fractional nonlinear reaction-diffusion equations, Mathematical Problems in Engineering, 2015, Article ID 457013, p.13, (2015).
  • Brandibur, O. and Kaslik, E., Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model, Mathematical Methods in the Applied Sciences, 41(17), 7182-7194, (2018).
  • Ascione, G., and Pirozzi, E., On Fractional Stochastic Modeling of Neuronal Activity Including Memory Effects, Computer Aided Systems Theory – EUROCAST 2017, 3-11, Spain, (2018).
  • Khanday, F. A., Kant, N. A., Dar, R. M. and Zulkifli, T. Z. A., Low-Voltage Low-Power Integrable CMOS Circuit Implementation of Integer- and Fractional-Order FitzHugh-Nagumo Neuron Model, IEEE Transactions on Neural Networks and Learning Systems, 99, 1-15, (2018).
  • Kumar, D., Singh, J. and Baleanu, D., A new numerical algorithm for fractional Fitzhugh–Nagumo equation arising in transmission of nerve impulses, Nonlinear Dynamics, 91, 307–317, (2018).
  • Veeresha, P., Prakasha, D. G. and Baskonus, H. M., New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos, 29, 013119, (2019).
  • Gencoglu, M. T., Baskonus, H. M. and Bulut, H., Numerical simulations to the nonlinear model of interpersonal relationships with time fractional derivative, AIP Conference Proceedings, 020103 (1798), 1-9, (2017).
  • De Pillis, L. G., Gua, W. and Radunskaya, A. E., Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, Journal of Theoretical Biology, 238, 841–862, (2006).
  • Buzsaki, G. and Draguhn, A., Neuronal oscillations in cortical networks, Science, 304, 1926-1929, (2004).
  • Wang, X. J., Neurophysiological and computational principles of cortical rhythms in cognition, Physiological reviews, 90, 1195-1268, (2010).
  • Yokus, A., Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32(29), 1850365, (2018).
  • Yokus, A., Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 57(3), 2085-2091, (2018).
  • Yokus, A. and Bulut, H., On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications, 9(1), 18, (2018).
  • Yokus, A., Baskonus, H. M., Sulaiman, T. A. and Bulut, H., Numerical simulation and solutions of the two‐component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34(1), 211-227, (2018).
  • Yavuz, M. and Ozdemir, N., On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative, ITM Web of Conferences, 22, 01045, (2018).
  • Yokus, A. and Tuz, M., An application of a new version of (G′/G)-expansion method, AIP Conference Proceedings, 1798(1), 020165, (2017).
  • Yel, G., Baskonus, H. M. and Bulut, H., Novel Archetypes of New Coupled Konno-Oono Equation by Using sine-Gordon Expansion Method, Optical and Quantum Electronics, 49, 285, (2017).
  • Kumar, D., Hosseini, K. and Samadani, F., The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics, Optik - International Journal for Light and Electron Optics, 149, 439-446, (2017).
  • Khan, K., Akbar, M. A., Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 15(1), (2013).
  • Baskonus, H. M., Sulaiman, T. A. and Bulut, H., On the new wave behavior to the Klein–Gordon–Zakharov equations in plasma physics, Indian Journal of Physics, 1-7, (2018).
  • Sulaiman, T. A., Bulut, H., Yel, G. and Atas, S. S., Optical solitons to the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model, Optical and Quantum Electronics, 50, 372, (2018).
  • Sulaiman, T. A., Yel, G. and Bulut, H., M-fractional solitons and periodic wave solutions to the Hirota Maccari system, Modern Physics Letters B, 33, No. 0, 1950052, (2019).
  • Baskonus, H. M., Yel, G. and Bulut, H., Novel wave surfaces to the fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation, American Institute of Physics, 1863(560084) (2017).
  • Kocak, Z. F. and Yel, G., Trigonometric Function Solutions of Fractional Drinfeld's Sokolov -Wilson System, ITM Web of Conferences 13, 01006, (2017).
  • Veeresha, P., Prakasha D. G. and Baskonus H. M., Novel simulations to the time-fractional Fisher’s equation, Mathematical Science,13, (2019).
  • Merdan, M., Solutions of time-fractional reaction–diffusion equation with modified Riemann–Liouville derivative, International. Journal of. Physical. Sciences. 7(15), 2317–2326 (2012).
  • Uçar, S., Uçar, E., Özdemir, N. and Hammouch, Z., Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons& Fractals, 118, 300-306, (2019).
  • Atangana, A. and Alkahtani, B., Analysis of the Keller–Segel model with a fractional derivative without singular kernel, Entropy, 17, 4439-4453, (2015).
  • Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications, 6, 75-83, (2016).
  • Caputo, M. and Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress in. Fractional Differentiation and Applications, 1, 73-85, (2015).
  • Atangana, A. and Baleanu, D., New fractional derivatives with non-local and non-singular kernel theory and applications to heat transfer model, Thermal Science, 20, 763-769, (2016).
  • Khalila, R., Al Horania, M., Yousefa, A. and Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
  • Atangana, A., Baleanu, D. and Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13, 889–898, (2015).
  • Khan, N. A., Khan, N. U., Ara, A. and Jamil, M., Approximate analytical solutions of fractional reaction-diffusion equations, Journal of King Saud University - Science, 24, 111-118,(2012).
  • Rida, S. Z., El-Sayed, A. M. A. and Arafa, A. A. M., On the Solutions of Time-Fractional Reaction-Diffusion Equations, Communication in Nonlinear Science & Numerical Simulation, 15, 3847-3854, (2010) .
  • Pandir, Y. and Tandogan, Y. A., Exact solutions of the time-fractional Fitzhugh-Nagumo equation, AIP Conference Proceedings 1558, 1919, (2013).
There are 53 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Gülnur Yel

Publication Date June 28, 2019
Submission Date March 4, 2019
Published in Issue Year 2019 Volume: 21 Issue: 2

Cite

APA Yel, G. (2019). On the new travelling wave solution of a neural communication model. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21(2), 666-678. https://doi.org/10.25092/baunfbed.636782
AMA Yel G. On the new travelling wave solution of a neural communication model. BAUN Fen. Bil. Enst. Dergisi. June 2019;21(2):666-678. doi:10.25092/baunfbed.636782
Chicago Yel, Gülnur. “On the New Travelling Wave Solution of a Neural Communication Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21, no. 2 (June 2019): 666-78. https://doi.org/10.25092/baunfbed.636782.
EndNote Yel G (June 1, 2019) On the new travelling wave solution of a neural communication model. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21 2 666–678.
IEEE G. Yel, “On the new travelling wave solution of a neural communication model”, BAUN Fen. Bil. Enst. Dergisi, vol. 21, no. 2, pp. 666–678, 2019, doi: 10.25092/baunfbed.636782.
ISNAD Yel, Gülnur. “On the New Travelling Wave Solution of a Neural Communication Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 21/2 (June 2019), 666-678. https://doi.org/10.25092/baunfbed.636782.
JAMA Yel G. On the new travelling wave solution of a neural communication model. BAUN Fen. Bil. Enst. Dergisi. 2019;21:666–678.
MLA Yel, Gülnur. “On the New Travelling Wave Solution of a Neural Communication Model”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 21, no. 2, 2019, pp. 666-78, doi:10.25092/baunfbed.636782.
Vancouver Yel G. On the new travelling wave solution of a neural communication model. BAUN Fen. Bil. Enst. Dergisi. 2019;21(2):666-78.

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