Glottal Closure Instants (GCI) detection consists in automatically detecting temporal locations of most significant excitation of the vocal tract from the speech signal. It is used in many speech analysis and processing applications, and various algorithms have been proposed for this purpose. Recently, new approaches using convolutional neural networks have emerged, with encouraging results. Following this trend, we propose a simple approach that performs a mapping from the speech waveform to a target signal from which the GCIs are obtained by peak-picking. However, the ground truth GCIs used for training and evaluation are usually extracted from EGG signals, which are not perfectly reliable and often not available. To overcome this problem, we propose to train our network on high-quality synthetic speech with perfect ground truth. The performances of the proposed algorithm are compared with three other state-of-the-art approaches using publicly available datasets, and the impact of using controlled synthetic or real speech signals in the training stage is investigated. The experimental results demonstrate that the proposed method obtains similar or better results than other state-of-the-art algorithms and that using large synthetic datasets with many speakers offers a better generalization ability than using a smaller database of real speech and EGG signals.