Abstract
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 spacetime 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 causeeffect 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.
Author summary
In this paper we introduce a spacetime continuous version for the level of integrated information of a network on which a dynamics is defined. The concept of integrated information comes from the IIT of consciousness. By a strict mathematical formulation, we complement the existing IIT theoretical framework from a dynamical systems perspective. In other words, we develop the bases for a continuous mathematical approach to IIT introducing a dynamical system as the driving rule of a given mechanism. We also introduce and define the concepts of Informational Structure and Informational Field as the complex network with the power to ascertain the dynamics (past and future scenarios) of the studied phenomena. The detailed description of an informational structure is showing the causeeffect power of a mechanism in a state and thus, a characterization of the quantity and quality of information, and the way this is integrated. We firstly introduce how network patterns arise from dynamic phenomena on networks, leading to the concept of informational structure. Then, we formally introduce the mathematical objects supporting the theory, from graphs to informational structures, throughout the integration of dynamics on graphs with a global model of differential equations. After this, we formally present some of the IIT’s postulates associated to a given mechanism. Finally, we provide the quantitative and qualitative characterization of the integrated information, and how it depends on the geometry of the mechanism.
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