  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 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 organized by the
BT point and the neuron switches from Class-I to Class-II as conductance of the
M-current increases.
