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As we observe the electrical activities of the secretory cells (GnRH neurons, $\beta$-cells of the pancreas,...) we observe an alternation between an \textit{active} phase and a \textit{quiescent} one. A lot of mathematical models have been developed to represent this behavior. They are \textit{dynamical systems} which take the form~:
$$
\dot{x}=f(x,\alpha),
$$
where $x \in \mathbb{R}^n\;(n\geq3)$ are the variables and $\alpha \in \mathbb{R}^m$ are the parameters. For such a system to have bursting solutions, at least two fast variables and one slow variable are needed; the Hindmarsh-Rose (HR) model is one of the most famous bursters. Even if it is overall well known, many things remain to be understood in the spike adding mechanism. In this poster we present a work about two mathematical phenomena which occur as a spike is added to the burst in the HR model. The first one is the \textit{Dal\'{i} canard} which is based on the form of the center manifold near the upper fold of the fast nullcline. The second one is a \textit{singular Hopf bifurcation} that we treat with a new method based on the study of the integrable normal form of this bifurcation. |
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