This paper proposes a novel neuronal current source localization method based on Deep Prior that represents a more complicated prior distribution of current source using convolutional networks. Deep Prior has been suggested as a means of an unsupervised learning approach that does not require learning using training data, and randomly-initialized neural networks are used to update a source location using a single observation. In our previous work, a Deep-Prior-based current source localization method in the brain has been proposed but the performance was not almost the same as those of conventional approaches, such as sLORETA. In order to improve the Deep-Prior-based approach, in this paper, a depth weight of the current source is introduced for Deep Prior, where depth weighting amounts to assigning more penalty to the superficial currents. Its effectiveness is confirmed by experiments of current source estimation on simulated MEG data.