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Quantifying the amount of information communicated between neural population is crucial to understand brain dynamics. To address this question, many tools for the analysis of time series of neural activity, such as Granger causality, Transfer Entropy, Directed Information have been proposed. However, none of these popular model-free measures can reveal what information has been exchanged. Yet, understanding what information is exchanged is key to be able to infer, from brain recordings, the nature and the mechanisms of brain computation. To provide the mathematical tools needed to address this issue, we developed a new measure, exploiting benefits of novel Partial Information Decomposition framework, that determines how much information about each specific stimulus or task feature has been transferred between two neuronal populations. We tested this methodology on simulated neural data and showed that it captures the specific information being transmitted very well, and it is also highly robust to several of the confounds that have proven to be problematic for previous methods. Moreover, the measure was significantly better in detection of the temporal evolution of the information transfer and the directionality of it than the previous measures. We also applied the measure to an EEG dataset acquired during a face detection task that revealed interesting patterns of interhemispheric phase-specific information transfer. We finally analyzed high gamma activity in an MEG dataset of a visuomotor associations. Our measure allowed for tracing of the stimulus information flow and it confirmed the notion that dorsal fronto-parietal network is crucial for the visuomotor computations transforming visual information into motor plans. Altogether our work suggests that our new measure has potential to uncover previously hidden specific information transfer dynamics in neural communication.