Araştırma Makalesi
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On the new travelling wave solution of a neural communication model

Yıl 2019, Cilt: 21 Sayı: 2, 666 - 678, 28.06.2019

Gülnur Yel

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • 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

Yıl 2019, Cilt: 21 Sayı: 2, 666 - 678, 28.06.2019

Gülnur Yel

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

Öz

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. 

Anahtar Kelimeler

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

Kaynakça

  • 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).
Toplam 53 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Gülnur Yel

Yayımlanma Tarihi 28 Haziran 2019
Gönderilme Tarihi 4 Mart 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 21 Sayı: 2

Kaynak Göster

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. Haziran 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, sy. 2 (Haziran 2019): 666-78. https://doi.org/10.25092/baunfbed.636782.
EndNote Yel G (01 Haziran 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, c. 21, sy. 2, ss. 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 (Haziran 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, c. 21, sy. 2, 2019, ss. 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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