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 & Continuous Dynamical Systems - S, 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.  Google Scholar

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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). Google Scholar

[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. Google Scholar

[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.  Google Scholar

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K. Ervin LenziA. 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.  Google Scholar

[6]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.   Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J, 1 (1961), 445-466.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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). Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

K. Ervin LenziA. 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.  Google Scholar

[6]

R. FitzHugh, Mathematical models of threshold phenomena in the nerve membrane, Bull. Math. Biophysics, 17 (1955), 257-278.   Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J, 1 (1961), 445-466.   Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[11]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.   Google Scholar

[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.  Google Scholar

Figure 1.  Numerical solution for $ \alpha = 0.15 $
Figure 2.  Numerical solution for the value $ \alpha = 0.35 $
Figure 3.  Numerical solution for the value $ \alpha = 0.70 $
Figure 4.  Numerical solution for the value $\alpha = 0.75 $
Figure 5.  Numerical solution value $\alpha = 0.95$
Figure 6.  Numerical solution value $\alpha = 1$
Figure 7.  Contour plot value $\alpha = 0.15$
Figure 8.  Contour plot value $\alpha = 0.35$
Figure 9.  Contour plot for $\alpha = 0.75$
Figure 10.  Contourplot for $\alpha = 1$
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