  Motivated by the possible applications that a better understanding of
consciousness might bring, we follow Tononi's idea and calculate analytically a
complexity index for two systems of Ising spins with parallel update dynamics,
the homogeneous and a modular infinite range models. Using the information
geometry formulation of integrated information theory, we calculate the
geometric integrated information index, $\phi_G(\Pi)$ for a fixed partition
$\Pi$ with $K$ components and $\Phi=$max$_\Pi\phi_G(\Pi)$ for $K=2$ or $3$. For
systems in the deep ferromagnetic phase, the optimal partition undergoes a
transition such that the smallest (largest) component is above (resp. below)
its critical temperature. The effects of partitioning are taken into account by
introducing site dilution.
