IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i12p2041-d837263.html
   My bibliography  Save this article

Transport Phenomena in Excitable Systems: Existence of Bounded Solutions and Absorbing Sets

Author

Listed:
  • Monica De Angelis

    (Department of Mathematics and Applications “R. Caccioppoli”, University of Naples “Federico II”, Via Cinthia 26, 80126 Naples, Italy)

Abstract

In this paper, the transport phenomena of synaptic electric impulses are considered. The FitzHugh–Nagumo and FitzHugh–Rinzel models appear mathematically appropriate for evaluating these scientific issues. Moreover, applications of such models arise in several biophysical phenomena in different fields such as, for instance, biology, medicine and electronics, where, by means of nanoscale memristor networks, scientists seek to reproduce the behavior of biological synapses. The present article deals with the properties of the solutions of the FitzHugh–Rinzel system in an attempt to achieve, by means of a suitable “energy function”, conditions ensuring the boundedness and existence of absorbing sets in the phase space. The results obtained depend on several parameters characterizing the system, and, as an example, a concrete case is considered.

Suggested Citation

  • Monica De Angelis, 2022. "Transport Phenomena in Excitable Systems: Existence of Bounded Solutions and Absorbing Sets," Mathematics, MDPI, vol. 10(12), pages 1-11, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2041-:d:837263
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/12/2041/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/12/2041/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Lin, Y. & Liu, W.B. & Bao, H. & Shen, Q., 2020. "Bifurcation mechanism of periodic bursting in a simple three-element-based memristive circuit with fast-slow effect," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    2. Capone, F. & Carfora, M.F. & De Luca, R. & Torcicollo, I., 2019. "Turing patterns in a reaction–diffusion system modeling hunting cooperation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 172-180.
    Full references (including those not matched with items on IDEAS)

    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. Maria Francesca Carfora & Isabella Torcicollo, 2022. "Traveling Band Solutions in a System Modeling Hunting Cooperation," Mathematics, MDPI, vol. 10(13), pages 1-11, July.
    2. Deng, Yue & Li, Yuxia, 2021. "Bifurcation and bursting oscillations in 2D non-autonomous discrete memristor-based hyperchaotic map," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    3. Zhang, Shaohua & Wang, Cong & Zhang, Hongli & Ma, Ping & Li, Xinkai, 2022. "Dynamic analysis and bursting oscillation control of fractional-order permanent magnet synchronous motor system," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    4. Maria Francesca Carfora & Isabella Torcicollo, 2020. "Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense," Mathematics, MDPI, vol. 8(8), pages 1-20, July.
    5. Pandey, Soumik & Ghosh, Uttam & Das, Debashis & Chakraborty, Sarbani & Sarkar, Abhijit, 2024. "Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 49-76.
    6. Yangyang Shao & Yan Meng & Xinyue Xu, 2022. "Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect," Mathematics, MDPI, vol. 10(9), pages 1-15, May.
    7. Danjin Zhang & Youhua Qian, 2023. "Bursting Oscillations in General Coupled Systems: A Review," Mathematics, MDPI, vol. 11(7), pages 1-16, April.
    8. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2024. "Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 128-147.
    9. Djilali, Salih & Cattani, Carlo, 2021. "Patterns of a superdiffusive consumer-resource model with hunting cooperation functional response," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    10. Shang, Zuchong & Qiao, Yuanhua & Duan, Lijuan & Miao, Jun, 2021. "Bifurcation analysis in a predator–prey system with an increasing functional response and constant-yield prey harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 976-1002.
    11. Lin, Yi & Liu, Wenbo & Hang, Cheng, 2023. "Revelation and experimental verification of quasi-periodic bursting, periodic bursting, periodic oscillation in third-order non-autonomous memristive FitzHugh-Nagumo neuron circuit," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    12. Zhang, Xiufang & Yao, Zhao & Guo, Yeye & Wang, Chunni, 2021. "Target wave in the network coupled by thermistors," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    13. Bao, Bocheng & Chen, Liuhui & Bao, Han & Chen, Mo & Xu, Quan, 2024. "Bifurcations to bursting oscillations in memristor-based FitzHugh-Nagumo circuit," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).

    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:gam:jmathe:v:10:y:2022:i:12:p:2041-:d:837263. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.