Investigating the dynamics of bursting by combining two fast–slow analyses with codimension-2 bifurcations in the embryonic pre-BötC neuron model

The network activity of the pre-Bötzinger complex (pre-BötC) in the mammalian brainstem controls the inspiratory phase of the respiratory rhythm. Because fluctuations in the action potentials of neurons can generate a magnetic field in the medium, we consider a flux-controlled memristor in the model of the isolated embryonic pre-BötC neuron featuring two distinct bursting mechanisms: depolarization block (DB) bursting and mixed bursting in this work. As the slow variables h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document} and Cai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{Ca}}_{i}$$\end{document} are at almost the same time scale, we combine two fast–slow analyses with the codimension-2 bifurcation analysis of these variables to identify the dynamical mechanisms of DB bursting and mixed bursting. We precisely define the bursting discharges according to topology by determining the bifurcations ranging from the resting state to the spiking state and vice versa, as well as bifurcations of the hysteresis loop for the upper and lower states with bistability. Our analysis provides a new perspective on the fast–slow analysis of the bursting mechanism. The results not only provide a reference for research on the effects of magnetic current on the firing patterns of neurons, but also offer insights for further exploring bursting from a dynamic perspective.