# American Institute of Mathematical Sciences

March  2020, 13(3): 781-795. doi: 10.3934/dcdss.2020044

## Analysis of the Fitzhugh Nagumo model with a new numerical scheme

 Department of Mathematics, Gyan Ganga Institute of Technology and Sciences, Near Tilwara Ghat, Jabalpur, Pin-482003, M.P., India

Received  May 2018 Revised  May 2018 Published  March 2019

The model describing a prototype of an excitable system was extended using the newly established concept of fractional differential operators with non-local and non-singular kernel in this paper. We presented a detailed discussion underpinning the well-poseness of the extended model. Due to the non-linearity of the modified model, we solved it using a newly established numerical scheme for partial differential equations that combines the fundamental theorem of fractional calculus, the Laplace transform and the Lagrange interpolation approximation. We presented some numerical simulations that, of course reflect asymptotically the real world observed behaviors.

Citation: Jyoti Mishra. Analysis of the Fitzhugh Nagumo model with a new numerical scheme. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 781-795. doi: 10.3934/dcdss.2020044
##### References:
 [1] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [2] A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, (2016). [3] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. [4] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010. [5] K. Ervin Lenzi, A. Tateishi Angel and V. Haroldo Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics, 5 (2017), 1-9.  doi: 10.3389/fphy.2017.00052. [6] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278. [7] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J, 1 (1961), 445-466. [8] R. FitzHugh, Mathematical models of excitation and propagation in nerve, Chapter 1in H.P. Schwan, ed. Biological Engineering, McGrawHill Book Co., N.Y., 1 (1969), 1-85. [9] J. F. Gomez-Aguilar, Analytical and numerical solutions of the telegraph equation using the Atangana Caputo fractional order derivative, Journal of Electromagnetic Waves and Applications, 32 (2017), 695-712. [10] 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, 117 (1952), 500-544. [11] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070. [12] H. Ypez-Martnez and J. F. Gmez-Aguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002.

show all references

##### References:
 [1] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [2] A. Atangana and B. Dumitru, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, (2016). [3] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. [4] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp. doi: 10.1051/mmnp/2018010. [5] K. Ervin Lenzi, A. Tateishi Angel and V. Haroldo Ribeiro, The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics, 5 (2017), 1-9.  doi: 10.3389/fphy.2017.00052. [6] R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278. [7] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J, 1 (1961), 445-466. [8] R. FitzHugh, Mathematical models of excitation and propagation in nerve, Chapter 1in H.P. Schwan, ed. Biological Engineering, McGrawHill Book Co., N.Y., 1 (1969), 1-85. [9] J. F. Gomez-Aguilar, Analytical and numerical solutions of the telegraph equation using the Atangana Caputo fractional order derivative, Journal of Electromagnetic Waves and Applications, 32 (2017), 695-712. [10] 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, 117 (1952), 500-544. [11] J. Nagumo, S. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070. [12] H. Ypez-Martnez and J. F. Gmez-Aguilar, Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 13, 17 pp. doi: 10.1051/mmnp/2018002.
Numerical solution for $\alpha = 0.15$
Numerical solution for the value $\alpha = 0.35$
Numerical solution for the value $\alpha = 0.70$
Numerical solution for the value $\alpha = 0.75$
Numerical solution value $\alpha = 0.95$
Numerical solution value $\alpha = 1$
Contour plot value $\alpha = 0.15$
Contour plot value $\alpha = 0.35$
Contour plot for $\alpha = 0.75$
Contourplot for $\alpha = 1$
 [1] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055 [2] G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027 [3] Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021 [4] Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031 [5] Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173 [6] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435 [7] Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3719-3746. doi: 10.3934/dcdss.2020430 [8] Francesco Cordoni, Luca Di Persio. Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. Evolution Equations and Control Theory, 2018, 7 (4) : 571-585. doi: 10.3934/eect.2018027 [9] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [10] Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022063 [11] Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134 [12] Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216 [13] Anna Cattani. FitzHugh-Nagumo equations with generalized diffusive coupling. Mathematical Biosciences & Engineering, 2014, 11 (2) : 203-215. doi: 10.3934/mbe.2014.11.203 [14] Yiqiu Mao. Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3935-3947. doi: 10.3934/dcdsb.2018118 [15] Vyacheslav Maksimov. Some problems of guaranteed control of the Schlögl and FitzHugh-Nagumo systems. Evolution Equations and Control Theory, 2017, 6 (4) : 559-586. doi: 10.3934/eect.2017028 [16] John Guckenheimer, Christian Kuehn. Homoclinic orbits of the FitzHugh-Nagumo equation: The singular-limit. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 851-872. doi: 10.3934/dcdss.2009.2.851 [17] Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172 [18] Anhui Gu, Bixiang Wang. Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1689-1720. doi: 10.3934/dcdsb.2018072 [19] Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150 [20] Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou, Timoléon Crépin Kofané. Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2229-2243. doi: 10.3934/dcdss.2020397

2020 Impact Factor: 2.425