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In this work, we consider a general conductance-based neuron model with theinclusion of the acetycholine sensitive, M-current. We study bifurcations inthe parameter space consisting of the applied current, $I_{app}$ the maximalconductance 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 aBogdanov-Takens (BT) point and show such a point can occur by varying $I_{app}$and $g_{M}$. We discuss the case when the BT point becomes aBogdanov-Takens-Cusp (BTC) point and show that such a point can occur in thethree dimensional parameter space. The results of the bifurcation analysis areapplied to different neuronal models and are verified and supplemented bynumerical bifurcation diagrams generated using the package MATCONT. We concludethat there is a transition in the neuronal excitability type organized by theBT point and the neuron switches from Class-I to Class-II as conductance of theM-current increases.

M-current Induced Bogdanov-Takens Bifurcation and Switching of Neuron Excitability Class