In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current $I_{app}$ , the maximal conductance of the M-current $g_{M}$ and the conductance of the leak current $g_{L}$ . We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying $I_{app}$ and $g_{M}$ . We discuss the case when the BT point becomes a Bogdanov–Takens–cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

中文翻译：

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M电流诱导的Bogdanov–Takens分叉和神经元兴奋性类别的转换

在这项工作中，我们考虑了一个基于电导的一般神经元模型，其中包括对乙酰胆碱敏感的M电流。我们研究了参数空间中的分叉，该分叉包括施加的电流$ I_ {app} $，M电流$ g_ {M} $的最大电导和泄漏电流$ g_ {L} $的电导。我们为模型提供了精确的条件，以确保存在Bogdanov–Takens（BT）点，并表明可以通过改变$ I_ {app} $和$ g_ {M} $来发生此点。我们讨论了当BT点成为Bogdanov–Takens–cusp（BTC）点时的情况，并说明了这种点可以出现在三维参数空间中。分叉分析的结果应用于不同的神经元模型，并通过使用MATCONT软件包生成的数字分叉图进行验证和补充。