Evaluating Approximations and Heuristic Measures of Integrated Information

Integrated information theory (IIT) proposes a measure of integrated information, termed Phi (phi), to capture the level of consciousness of a physical system in a given state. Unfortunately, calculating phi itself is currently possible only for very small model systems and far from computable for the kinds of system typically associated with consciousness (brains). Here, we considered several proposed heuristic measures and computational approximations, some of which can be applied to larger systems, and tested if they correlate well with phi. While these measures and approximations capture intuitions underlying IIT and some have had success in practical applications, it has not been shown that they actually quantify the type of integrated information specified by the latest version of IIT and, thus, whether they can be used to test the theory. In this study, we evaluated these approximations and heuristic measures considering how well they estimated the phi values of model systems and not on the basis of practical or clinical considerations. To do this, we simulated networks consisting of 3-6 binary linear threshold nodes randomly connected with excitatory and inhibitory connections. For each system, we then constructed the system's state transition probability matrix (TPM) and generated observed data over time from all possible initial conditions. We then calculated phi, approximations to phi, and measures based on state differentiation, coalition entropy, state uniqueness, and integrated information. Our findings suggest that phi can be approximated closely in small binary systems by using one or more of the readily available approximations (r > 0.95) but without major reductions in computational demands. Furthermore, the maximum value of phi across states (a state-independent quantity) correlated strongly with measures of signal complexity (LZ, r(s) = 0.722), decoder-based integrated information (phi*, r(s) = 0.816), and state differentiation (D1, r(s) = 0.827). These measures could allow for the efficient estimation of a system's capacity for high phi or function as accurate predictors of low- (but not high-)phi systems. While it is uncertain whether the results extend to larger systems or systems with other dynamics, we stress the importance that measures aimed at being practical alternatives to phi be, at a minimum, rigorously tested in an environment where the ground truth can be established.