Digital reconstruction of neuronal structures from 3D microscopy images is critical for the quantitative investigation of brain circuits and functions. It is a challenging task that would greatly benefit from automatic neuron reconstruction methods. In this paper, we propose a novel method called SPE-DNR that combines spherical-patches extraction (SPE) and deep-learning for neuron reconstruction (DNR). Based on 2D Convolutional Neural Networks (CNNs) and the intensity distribution features extracted by SPE, it determines the tracing directions and classifies voxels into foreground or background. This way, starting from a set of seed points, it automatically traces the neurite centerlines and determines when to stop tracing. To avoid errors caused by imperfect manual reconstructions, we develop an image synthesizing scheme to generate synthetic training images with exact reconstructions. This scheme simulates 3D microscopy imaging conditions as well as structural defects, such as gaps and abrupt radii changes, to improve the visual realism of the synthetic images. To demonstrate the applicability and generalizability of SPE-DNR, we test it on 67 real 3D neuron microscopy images from three datasets. The experimental results show that the proposed SPE-DNR method is robust and competitive compared with other state-of-the-art neuron reconstruction methods.