A core feature of complex systems is that the interactions between elements in the present causally constrain each-other as the system evolves through time. To fully model all of these interactions (between elements, as well as ensembles of elements), we can decompose the total information flowing from past to future into a set of non-overlapping temporal interactions that describe all the different modes by which information can flow. To achieve this, we propose a novel information-theoretic measure of temporal dependency ($I_{τsx}$) based on informative and misinformative local probability mass exclusions. To demonstrate the utility of this framework, we apply the decomposition to spontaneous spiking activity recorded from dissociated neural cultures of rat cerebral cortex to show how different modes of information processing are distributed over the system. Furthermore, being a localizable analysis, we show that $I_{τsx}$ can provide insight into the computational structure of single moments. We explore the time-resolved computational structure of neuronal avalanches and find that different types of information atoms have distinct profiles over the course of an avalanche, with the majority of non-trivial information dynamics happening before the first half of the cascade is completed. These analyses allow us to move beyond the historical focus on single measures of dependency such as information transfer or information integration, and explore a panoply of different relationships between elements (and groups of elements) in complex systems.