The spatio-temporal structure of brain dynamics at the macroscopic level has been hypothesized to reflect neuronal communication for cognitive processes. In particular, recent studies have focused on time-irreversibility and its relationship with consciousness [1,2]. Beyond this example of high-level cognitive function, this raises the question of the role of different ingredients that shape the structure-function relationship in whole-brain dynamics, which can be formalized using the Ornstein-Uhlenbeck (OU) process that combines input “noise” and effective connectivity as key ingredients. Here we extend previous studies [3,4] by considering an OU system with non-stationary inputs, starting with its estimation from fMRI data and interpreting its resulting dynamics. We show how the connectivity interplays with the input properties to shape a manifold of brain/neuronal activity that can be in a non-equilibrium regime (as characterized by entropy production). This approach provides a model-based analysis of network activity that defines a dynamic complexity. Reviewing to previous interpretations that relates this entropy to various (bio)physical mechanisms, we here consider the link with a controlled sampling of a distribution of activity patterns [5] that can be regulated by the intrinsic or extrinsic inputs to the network.