Non-Trivial Dynamics in the FizHugh-Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions Systems

Benjamin Ambrosio; M A Aziz-Alaoui; Argha Mondal; Arnab Mondal; Sanjeev K Sharma; Ranjit Kumar Upadhyay

doi:10.3390/biology12070918

Non-Trivial Dynamics in the FizHugh-Rinzel Model and Non-Homogeneous Oscillatory-Excitable Reaction-Diffusions Systems

Biology (Basel). 2023 Jun 27;12(7):918. doi: 10.3390/biology12070918.

Authors

Benjamin Ambrosio^{1

2}, M A Aziz-Alaoui¹, Argha Mondal^{3

4}, Arnab Mondal⁵, Sanjeev K Sharma⁵, Ranjit Kumar Upadhyay⁵

Affiliations

¹ UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Normandie University, 76600 Le Havre, France.
² The Hudson School of Mathematics, New York, NY 10001, USA.
³ Department of Mathematics, Sidho-Kanho-Birsha University, Purulia 723104, India.
⁴ Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK.
⁵ Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India.

Abstract

This article focuses on the qualitative analysis of complex dynamics arising in a few mathematical models in neuroscience context. We first discuss the dynamics arising in the three-dimensional FitzHugh-Rinzel (FHR) model and then illustrate those arising in a class of non-homogeneous FitzHugh-Nagumo (Nh-FHN) reaction-diffusion systems. FHR and Nh-FHN models can be used to generate relevant complex dynamics and wave-propagation phenomena in neuroscience context. Such complex dynamics include canards, mixed-mode oscillations (MMOs), Hopf-bifurcations and their spatially extended counterpart. Our article highlights original methods to characterize these complex dynamics and how they emerge in ordinary differential equations and spatially extended models.

Keywords: FitzHugh–Nagumo; FitzHugh–Rinzel model; bifurcation; bursting oscillations; canard; fast-slow dynamics; mixed mode oscillations; neuroscience; waves.

Grants and funding

This research was funded by the LMAH, Région Normandie, CNRS project IEA00134, and the HSM.