We present an approach to assess redundant and synergistic interactions in network systems via the information-theoretic analysis of multivariate physiological processes. The approach sets up a strategy to decompose the information shared between the present states of a group of random processes and their own past states into unique contributions arising from the past of subgroups of processes and redundant and synergistic contributions arising from the dynamic interaction among the subgroups. The method is illustrated in a theoretical example of linearly interacting Gaussian processes, showing that redundancy and synergy are related mostly to unidirectional coupling and to bidirectional coupling with internal dynamics. It is then applied to the network of short-term heart period, arterial pressure and respiratory variability probed in healthy subjects, showing that redundancy and synergy prevail respectively in cardiorespiratory interactions and in cardiovascular interactions in the resting state, and that postural stress increases the predictive information and the redundancy of physiological interactions.