IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v313y2017icp245-258.html
   My bibliography  Save this article

Pattern formation by super-diffusion in FitzHugh–Nagumo model

Author

Listed:
  • Iqbal, Naveed
  • Wu, Ranchao
  • Liu, Biao

Abstract

The aim of this paper is to examine the Turing instability and pattern formation in the FitzHugh–Nagumo model with super-diffusion in two dimensional numerical simulation. We also studied the effects of the super-diffusive exponent on pattern formation concluding that with the presence of super-diffusion the stable homogenous steady state becomes unstable. By using the stability analysis of local equilibrium point, we procure the conditions which ensure that the Turing and Hopf bifurcations occur. For pattern selection, the weak nonlinear multi-scale analysis is used to derive the amplitude equations of the stationary patterns. We then apply amplitude equations and observe that this model has very rich dynamical behaviors, such as stripes, spots and hexagon patterns. The complexity of the dynamics in this system is theoretically discussed and graphically displayed in numerical simulation. The simulation helps us to show the effectiveness of theoretical analysis and patterns which appear numerically.

Suggested Citation

  • Iqbal, Naveed & Wu, Ranchao & Liu, Biao, 2017. "Pattern formation by super-diffusion in FitzHugh–Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 245-258.
    • Handle: RePEc:eee:apmaco:v:313:y:2017:i:c:p:245-258
    DOI: 10.1016/j.amc.2017.05.072
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300317303843
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2017.05.072?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Perc, Matjaž, 2007. "Effects of small-world connectivity on noise-induced temporal and spatial order in neural media," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 280-291.
    2. Perc, Matjaž, 2007. "Fluctuating excitability: A mechanism for self-sustained information flow in excitable arrays," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1118-1124.
    3. Buceta, J. & Lindenberg, Katja, 2003. "Patterns in reaction–diffusion systems generated by global alternation of dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 325(1), pages 230-242.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zheng, Qianqian & Shen, Jianwei, 2020. "Turing instability induced by random network in FitzHugh-Nagumo model," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    2. Mondal, Argha & Hens, Chittaranjan & Mondal, Arnab & Antonopoulos, Chris G., 2021. "Spatiotemporal instabilities and pattern formation in systems of diffusively coupled Izhikevich neurons," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    3. Tah, Forwah Amstrong & Tabi, Conrad Bertrand & Kofane, Timoléon Crépin, 2021. "Pattern formation in the Fitzhugh–Nagumo neuron with diffusion relaxation," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    4. Fateev, I. & Polezhaev, A., 2024. "Chimera states in a lattice of superdiffusively coupled neurons," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    5. Mondal, Arnab & Upadhyay, Ranjit Kumar & Mondal, Argha & Sharma, Sanjeev Kumar, 2022. "Emergence of Turing patterns and dynamic visualization in excitable neuron model," Applied Mathematics and Computation, Elsevier, vol. 423(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sun, Xiaojuan & Lu, Qishao & Kurths, Jürgen, 2008. "Correlated noise induced spatiotemporal coherence resonance in a square lattice network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(26), pages 6679-6685.
    2. Wang, Qingyun & Zheng, Yanhong & Ma, Jun, 2013. "Cooperative dynamics in neuronal networks," Chaos, Solitons & Fractals, Elsevier, vol. 56(C), pages 19-27.
    3. Yuan, Guoyong & Xu, Lin & Xu, Aiguo & Wang, Guangrui & Yang, Shiping, 2011. "Spiral waves in excitable media due to noise and periodic forcing," Chaos, Solitons & Fractals, Elsevier, vol. 44(9), pages 728-738.
    4. Upadhyay, Ranjit Kumar & Paul, Chinmoy & Mondal, Argha & Vishwakarma, Gajendra K., 2018. "Estimation of biophysical parameters in a neuron model under random fluctuations," Applied Mathematics and Computation, Elsevier, vol. 329(C), pages 364-373.
    5. Yu, Haitao & Guo, Xinmeng & Wang, Jiang & Deng, Bin & Wei, Xile, 2015. "Vibrational resonance in adaptive small-world neuronal networks with spike-timing-dependent plasticity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 170-179.
    6. Li, Fan, 2020. "Effect of field coupling on the wave propagation in the neuronal network," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    7. Yu, Haitao & Guo, Xinmeng & Wang, Jiang & Deng, Bin & Wei, Xile, 2015. "Spike coherence and synchronization on Newman–Watts small-world neuronal networks modulated by spike-timing-dependent plasticity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 307-317.
    8. Rostami, Zahra & Rajagopal, Karthikeyan & Khalaf, Abdul Jalil M. & Jafari, Sajad & Perc, Matjaž & Slavinec, Mitja, 2018. "Wavefront-obstacle interactions and the initiation of reentry in excitable media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 1162-1173.
    9. Guo, Xinmeng & Wang, Jiang & Liu, Jing & Yu, Haitao & Galán, Roberto F. & Cao, Yibin & Deng, Bin, 2017. "Optimal time scales of input fluctuations for spiking coherence and reliability in stochastic Hodgkin–Huxley neurons," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 381-390.
    10. Xu, Ying & Ren, Guodong & Ma, Jun, 2023. "Patterns stability in cardiac tissue under spatial electromagnetic radiation," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    11. Ma, Jun & Wu, Fuqiang & Hayat, Tasawar & Zhou, Ping & Tang, Jun, 2017. "Electromagnetic induction and radiation-induced abnormality of wave propagation in excitable media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 508-516.
    12. Gafiychuk, V.V. & Datsko, B.Yo., 2006. "Pattern formation in a fractional reaction–diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 300-306.
    13. Hao, Yinghang & Gong, Yubing & Wang, Li & Ma, Xiaoguang & Yang, Chuanlu, 2011. "Single or multiple synchronization transitions in scale-free neuronal networks with electrical or chemical coupling," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 260-268.
    14. Li, Fan & Liu, Shuai & Li, Xiaola, 2022. "Pattern selection in thermosensitive neuron network induced by noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 589(C).
    15. Song, Aiguo & Duan, Jianghai & Wu, Juan & Li, Huijun, 2009. "Design 2D nonlinear system for information storage," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 157-163.
    16. Levinsohn, Erik A. & Mendoza, Steve A. & Peacock-López, Enrique, 2012. "Switching induced complex dynamics in an extended logistic map," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 426-432.
    17. Ford, Neville J. & Lima, Pedro M. & Lumb, Patricia M., 2017. "Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh–Nagumo equation," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 448-460.
    18. Fedaravičius, Augustinas Povilas & Cao, Maosen & Ragulskis, Minvydas, 2016. "Control of a dendritic neuron driven by a phase-independent stimulation," Chaos, Solitons & Fractals, Elsevier, vol. 85(C), pages 77-83.
    19. Xie, Huijuan & Gong, Yubing & Wang, Baoying, 2018. "Spike-timing-dependent plasticity optimized coherence resonance and synchronization transitions by autaptic delay in adaptive scale-free neuronal networks," Chaos, Solitons & Fractals, Elsevier, vol. 108(C), pages 1-7.
    20. Wang, Jiang & Guo, Xinmeng & Yu, Haitao & Liu, Chen & Deng, Bin & Wei, Xile & Chen, Yingyuan, 2014. "Stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses," Chaos, Solitons & Fractals, Elsevier, vol. 60(C), pages 40-48.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:313:y:2017:i:c:p:245-258. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.