We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

中文翻译：

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神经兴奋性和奇异分叉。

我们从几何奇异摄动理论的角度讨论了二维慢/快神经模型中的兴奋性概念。我们关注慢/快速神经模型的固有奇异性质，并通过奇异分叉定义兴奋性。特别是，我们表明I型兴奋性与新型奇异的Bogdanov-Takens / SNIC分叉有关，而II型兴奋性与奇异的Andronov-Hopf分叉有关。在这两种情况下，在了解这些奇异的分叉结构的展开过程中，鸭ard起着重要作用。我们还解释了两种兴奋性类型之间的过渡，并突出显示了所涉及的所有分叉，从而基于几何奇异摄动理论提供了对兴奋性的完整分析。