One in a series of talks from the 2019 Models of Consciousness conference. Pedro Mediano Department of Computing, Imperial College London In a seminal series of papers, Tononi, Sporns, and Edelman (TSE) introduced the idea that the neural dynamics underlying conscious states are characterised by a balance of integration and differentiation between system components. This idea remains prevalent in consciousness research today, influencing theoretical and experimental work. Such work has faced a number of challenges. For example, distinct measures designed to measure such a balance behave very differently in practice, making it hard to choose which is the "right one", and dynamics of conscious and unconscious brains defy some of the predictions of this framework. We argue that these problems arise, at least in part, from the non-specific nature of the concepts of integration and differentiation. Here, we present a revised mathematical theory of neural complexity: we introduce a new measure, called O-information, that quantifies the balance between redundancy and synergy within a system, and is more effective than TSE’s original measure at describing phenomena where large-scale correlation and short-scale independence coexist; and develop a formalism to decompose different "modes" of information dynamics, providing an exhaustive taxonomy of redundant and synergistic effects. These developments allow us to place previous measures within a common framework and explain their their similarities and differences. Filmed at the Models of Consciousness conference, University of Oxford, September 2019. Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/