This paper addresses the question of how the brain of an animal achieves cognitive integration—that is to say how it manages to bring its fullest resources to bear on an ongoing situation. To fully exploit its cognitive resources, whether inherited or acquired through experience, it must be possible for unanticipated coalitions of brain processes to form. This facilitates the novel recombination of the elements of an existing behavioural repertoire, and thereby enables innovation. But in a system comprising massively many anatomically distributed assemblies of neurons, it is far from clear how such open-ended coalition formation is possible. The present paper draws on contemporary findings in brain connectivity and neurodynamics, as well as the literature of artificial intelligence, to outline a possible answer in terms of the brain’smost richly connected and topologically central structures, its so-called connective core.