  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 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 the organization of biological, neural, and
cognitive systems.
