Integrated Information Theory (IIT) has become nowadays the most sensible general theory of consciousness. In addition to very important statements, it opens the door for an abstract (mathematical) formulation of the theory. Given a mechanism in a particular state, IIT identifies a conscious experience with a conceptual structure, an informational object which exists, is composed of identified parts, is informative, integrated and maximally irreducible. This paper introduces a space-time continuous version of the concept of integrated information. To this aim, a graph and a dynamical systems treatment is used to define, for a given mechanism in a state for which a dynamics is settled, an Informational Structure, which is associated to the global attractor at each time of the system. By definition, the informational structure determines all the past and future behavior of the system, possesses an informational nature and, moreover, enriches all the points of the phase space with cause-effect power by means of its associated Informational Field. A detailed description of its inner structure by invariants and connections between them allows to associate a transition probability matrix to each informational structure and to develop a measure for the level of integrated information of the system.

Electroencephalography (EEG) monitors -by either intrusive or noninvasive electrodes-time and frequency variations and spectral content of voltage fluctuations or waves, known as brain rhythms, which in some way uncover activity during both rest periods and specific events in which the subject is under stimulus. This is a useful tool to explore brain behavior, as it complements imaging techniques that have a poorer temporal resolution. We here approach the understanding of EEG data from first principles by numerical simulating and studying a networked model of excitatory and inhibitory neurons which generates a variety of comparable waves. In fact, we thus numerically reproduce oscillatory behavior similar to α, β, γ and other rhythms as observed by EEG recordings, and identify the details of the respectively involved complex phenomena, including a precise relationship between an input and the collective response to it. It ensues the potentiality of our model to better understand actual brain oscillatory activity in normal and pathological situations, and we also describe kind of stochastic resonance phenomena which could be useful to locate main qualitative changes of brain activity in (e.g.) humans.

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