The intrinsic electrophysiological properties of single neurons can be described by a broad spectrum of models, from realistic Hodgkin-Huxley-type models with numerous detailed mechanisms to the phenomenological models. The adaptive exponential integrate-and-fire (AdEx) model has emerged as a convenient middle-ground model. With a low computational cost but keeping biophysical interpretation of the parameters, it has been extensively used for simulations of large neural networks. However, because of its current-based adaptation, it can generate unrealistic behaviors. We show the limitations of the AdEx model, and to avoid them, we introduce the conductance-based adaptive exponential integrate-and-fire model (CAdEx). We give an analysis of the dynamics of the CAdEx model and show the variety of firing patterns it can produce. We propose the CAdEx model as a richer alternative to perform network simulations with simplified models reproducing neuronal intrinsic properties.