The ability to integrate information into a unified coherent whole is a fundamental capacity of many biological and cognitive systems. Integrated Information Theory (IIT) provides a mathematical approach to quantify the level of integration in a system . Nevertheless , it is not yet well understood how integration scales up with the size of a system or with different temporal scales of activity, nor how a system maintains its integration as its interacts with its environment. We argue that modelling and understanding how measures of information integration operate in these scenarios is fundamental to adequately capture integration in systems such as the brain. We propose a simplified and modified version of integrated information \phi%DIFDELCMD < } %%% in order to explore these questions. Using mean field approximations, we measure \phi%DIFDELCMD < } %%% in a kinetic Ising models of infinite size. We find that information integration diverges in some cases when the system is near critical pointsin continuous phase transitions . Moreover, we find that we can delimit the boundary of a system with respect to its environment by comparing the divergent integrative tendencies of system and system-environment processes respectively . Finally, we model a system that maintains integrated information when interacting with a range of different environments by generating a critical surface spanning a parametric region of the environment. We conclude by discussing how thinking about integrated information in the thermodynamic limit opens fruitful research avenues for studying URLanization of biological  and cognitive systems.
The capacity to integrate information is a prominent feature of biological and cognitive systems. Integrated Information Theory (IIT) provides a mathematical approach to quantify the level of integration in a system , yet its computational cost generally precludes its applications beyond relatively small models. In consequence , it is not yet well understood how integration scales up with the size of a system or with different temporal scales of activity, nor how a system maintains its integration as its interacts with its environment. Here, we show for the first time how measures of information integration %DIFDELCMD < } %%% %DIFDELCMD < } %%% scale when systems become very large. Using kinetic Ising models and mean-field approximations from statistical mechanics, we show that information integration diverges in the thermodynamic limit at certain critical points . Moreover, by comparing different divergent tendencies of blocks of a system at these critical points, we delimit the boundary between an integrated unit and its environment . Finally, we present a model that adaptively maintains its integration despite changes in its environment by generating a critical surface where its integrity is preserved. We argue that the exploration of integrated information for these limit cases helps in addressing a variety of poorly understood questions about URLanization of biological , neural, and cognitive systems.
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