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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Non-Trivial Dynamics in the FizHugh-Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions

Benjamin Ambrosio1,2, M A Aziz-Alaoui1, Argha Mondal3,4

  • 1UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Normandie University, 76600 Le Havre, France.

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|July 29, 2023
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Summary

This study analyzes complex dynamics in neuroscience models like the FitzHugh-Rinzel (FHR) and non-homogeneous FitzHugh-Nagumo (Nh-FHN) systems. It introduces novel methods to understand phenomena such as canards and mixed-mode oscillations in these mathematical models.

Keywords:
FitzHugh–NagumoFitzHugh–Rinzel modelbifurcationbursting oscillationscanardfast-slow dynamicsmixed mode oscillationsneurosciencewaves

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Area of Science:

  • Computational Neuroscience
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Neuroscience models are crucial for understanding brain function.
  • Complex dynamics, including oscillations and bifurcations, are observed in neural systems.
  • Existing models may not fully capture the nuances of neural signal propagation.

Purpose of the Study:

  • To qualitatively analyze complex dynamics in the FitzHugh-Rinzel (FHR) model.
  • To investigate dynamics in non-homogeneous FitzHugh-Nagumo (Nh-FHN) reaction-diffusion systems.
  • To present original methods for characterizing these complex dynamics and their emergence.

Main Methods:

  • Qualitative analysis of ordinary differential equations (ODEs).
  • Analysis of spatially extended reaction-diffusion systems.
  • Characterization of bifurcations, canards, and mixed-mode oscillations (MMOs).

Main Results:

  • Demonstrated complex dynamics in the 3D FHR model.
  • Illustrated dynamics in Nh-FHN reaction-diffusion systems.
  • Highlighted emergence of canards, MMOs, and Hopf-bifurcations in both model types.

Conclusions:

  • The FHR and Nh-FHN models effectively generate complex dynamics relevant to neuroscience.
  • The proposed methods offer novel insights into the emergence of these phenomena.
  • This work contributes to a deeper understanding of neural signal processing and wave propagation.