In this paper we define a class of formal neuron models being computationally efficient and biologically plausible, i.e., able to reproduce a wide range of behaviors observed in in vivo or in vitro recordings of cortical neurons. This class includes, for instance, two models widely used in computational neuroscience, the Izhikevich and the Brette–Gerstner models. These models consist of a 4-parameter dynamical system. We provide the full local bifurcation diagram of the members of this class and show that they all present the same bifurcations: an Andronov–Hopf bifurcation manifold, a saddle-node bifurcation manifold, a Bogdanov–Takens bifurcation, and possibly a Bautin bifurcation, i.e., all codimension two local bifurcations in a two-dimensional phase space except the cusp. Among other global bifurcations, this system shows a saddle homoclinic bifurcation curve. We show how this bifurcation diagram generates the most prominent cortical neuron behaviors. This study leads us to introduce a new neuron model, the quartic model, able to reproduce among all the behaviors of the Izhikevich and Brette–Gerstner models self-sustained subthreshold oscillations, which are of great interest in neuroscience.