We present a process to automatically generate three-dimensional mesh representations of the complex, arborized cell membrane surface of cortical neurons (the principal information processing cells of the brain) from nonuniform morphological measurements. Starting from manually sampled morphological points (3D points and diameters) from neurons in a brain slice preparation, we construct a polygonal mesh representation that realistically represents the continuous membrane surface, closely matching the original experimental data. A mapping between the original morphological points and the newly generated mesh enables simulations of electrophysiolgical activity to be visualized on this new membrane representation. We compare the new mesh representation with the state of the art and present a series of use cases and applications of this technique to visualize simulations of single neurons and networks of multiple neurons.