  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
($I_{\tau sx}$) 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 $I_{\tau sx}$ 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.
