The size of publicly available data in cognitive neuro-imaging has increased a lot in recent years, thanks to strong research and community efforts. Exploiting this wealth of data demands new methods to turn the heterogeneous cognitive information held in different task-fMRI studies into common-universal-cognitive models. In this paper, we pool data from large fMRI repositories to predict psychological conditions from statistical brain maps across different studies and subjects. We leverage advances in deep learning, intermediate representations and multi-task learning to learn universal interpretable low-dimensional representations of brain images, usable for predicting psychological stimuli in all input studies. The method improves decoding performance for 80% of studies, by permitting cognitive information to flow from every study to the others: it notably gives a strong performance boost when decoding studies of small size. The trained low-dimensional representation-task-optimized networks-is interpretable as a set of basis cognitive dimensions relevant to meaningful categories of cognitive stimuli. Our approach opens new ways of extracting information from brain maps, overcoming the low power of typical fMRI studies.