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Mathematical Biosciences and Engineering

2008, Volume 5, Issue 2: 239-260. doi: 10.3934/mbe.2008.5.239
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"Traveling wave'' solutions of Fitzhugh model with cross-diffusion

  • 1.
    Department of Mathematics, Howard University, Washington D.C., 20059
  • 2.
    Department of Mathematical Sciences and Applied Computing, Arizona State University, Glendale, AZ 85306
  • 3.
    National Centre for Biotechnological Information, National Institutes of Health, Bethesda, MD 20894
  • Received: 01 May 2007 Accepted: 29 June 2018 Published: 01 March 2008

  • MSC : Primary: 34C20, 34C23; Secondary: 92C99.

  • The FitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron FIring and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, be- sides giving rise to the typical fast traveling wave solution exhibited in the original ''diffusion'' FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where ''normal'' impulse propagation is possible.

    Citation: F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. 'Traveling wave'' solutions of Fitzhugh model with cross-diffusion[J]. Mathematical Biosciences and Engineering, 2008, 5(2): 239-260. doi: 10.3934/mbe.2008.5.239

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  • The FitzHugh-Nagumo equations have been used as a caricature of the Hodgkin-Huxley equations of neuron FIring and to capture, qualitatively, the general properties of an excitable membrane. In this paper, we utilize a modified version of the FitzHugh-Nagumo equations to model the spatial propagation of neuron firing; we assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, be- sides giving rise to the typical fast traveling wave solution exhibited in the original ''diffusion'' FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where ''normal'' impulse propagation is possible.


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    2. Argha Mondal, Ranjit Kumar Upadhyay, Arnab Mondal, Sanjeev Kumar Sharma, Dynamics of a modified excitable neuron model: Diffusive instabilities and traveling wave solutions, 2018, 28, 1054-1500, 113104, 10.1063/1.5048119
    3. Connie L. McNeely, Erika Camacho, Conceptualizing STEM Workforce Migration in the Modern World Polity, 2010, 1556-5068, 10.2139/ssrn.1593393
    4. Tian Xiang, On a class of Keller–Segel chemotaxis systems with cross-diffusion, 2015, 259, 00220396, 4273, 10.1016/j.jde.2015.05.021
    5. Evgeny P. Zemskov, Mikhail A. Tsyganov, Werner Horsthemke, Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion, 2019, 99, 2470-0045, 10.1103/PhysRevE.99.062214
    6. Evgeny P. Zemskov, Mikhail A. Tsyganov, Werner Horsthemke, Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion, 2018, 97, 2470-0045, 10.1103/PhysRevE.97.062206
    7. B Z Essimbi, A Stöhr, I Jäger, D Jäger, Neuronlike impulses in a travelling wave structure loaded with resonant tunneling diodes and air bridges, 2019, 3, 2399-6528, 085010, 10.1088/2399-6528/ab2661
    8. Faina Berezovskaya, 2016, Chapter 1, 978-3-319-31321-4, 1, 10.1007/978-3-319-31323-8_1
    9. Evgeny P. Zemskov, Mikhail A. Tsyganov, Klaus Kassner, Werner Horsthemke, Nonlinear waves in a quintic FitzHugh–Nagumo model with cross diffusion: Fronts, pulses, and wave trains, 2021, 31, 1054-1500, 033141, 10.1063/5.0043919
    10. Evgeny P. Zemskov, Mikhail A. Tsyganov, Genrich R. Ivanitsky, Werner Horsthemke, Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection, 2022, 105, 2470-0045, 10.1103/PhysRevE.105.014207
    11. G. Gambino, M. C. Lombardo, R. Rizzo, M. Sammartino, Excitable FitzHugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023, 0035-5038, 10.1007/s11587-023-00814-9
    12. Daniel Cebrián-Lacasa, Pedro Parra-Rivas, Daniel Ruiz-Reynés, Lendert Gelens, Six decades of the FitzHugh–Nagumo model: A guide through its spatio-temporal dynamics and influence across disciplines, 2024, 1096, 03701573, 1, 10.1016/j.physrep.2024.09.014
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