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Emergence of Canard induced mixed mode oscillations in a slow–fast dynamics of a biophysical excitable model

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  • Sharma, Sanjeev Kumar
  • Mondal, Arnab
  • Mondal, Argha
  • Aziz-Alaoui, M.A.
  • Upadhyay, Ranjit Kumar
  • Ma, Jun

Abstract

We study the dynamics of a biophysically motivated slow–fast FitzHugh–Rinzel (FHR) model neurons in understanding the complex dynamical behavior of neural computation. We discuss the mathematical frameworks of diverse excitabilities and repetitive firing responses due to the applied stimulus using the slow–fast system. The results focus on the multiple time scale dynamics that include canard phenomenon induced mixed mode oscillations (MMOs) and mixed mode bursting oscillations (MMBOs). The bifurcation structure of the system is examined with injected current stimulus as the relevant parameter. We use the folded node theory to study the canards near the fold points. Further, we demonstrate the homoclinic bifurcation and the transition route to chaos through MMOs. It helps us in understanding the fundamentals of such complex rich neuronal responses. To show the chaotic nature in certain parameter regime, we compute the Lyapunov spectrum as a function of time and predominant parameter, I, that establishes our findings. Finally, we conclude that our observed results may have major significance and discuss the potential applications of MMOs in neural dynamics.

Suggested Citation

  • Sharma, Sanjeev Kumar & Mondal, Arnab & Mondal, Argha & Aziz-Alaoui, M.A. & Upadhyay, Ranjit Kumar & Ma, Jun, 2022. "Emergence of Canard induced mixed mode oscillations in a slow–fast dynamics of a biophysical excitable model," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s0960077922008487
    DOI: 10.1016/j.chaos.2022.112669
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    References listed on IDEAS

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    1. An, Xinlei & Qiao, Shuai, 2021. "The hidden, period-adding, mixed-mode oscillations and control in a HR neuron under electromagnetic induction," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    2. Zhang, Chun & Ma, Xindong & Bi, Qinsheng, 2022. "Complex mixed-mode oscillations based on a modified Rayleigh-Duffing oscillator driven by low-frequency excitations," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
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    Cited by:

    1. Zhao, Heqi & Ma, Xindong & Yang, Weijie & Zhang, Zhao & Bi, Qinsheng, 2023. "The mechanism of periodic and chaotic bursting patterns in an externally excited memcapacitive system," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    2. Zhou, Ping & Ma, Jun & Xu, Ying, 2023. "Phase synchronization between neurons under nonlinear coupling via hybrid synapse," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).

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