System Integrated Information

Abstract

Integrated information theory (IIT) starts from consciousness itself and identifies a set of properties (axioms) that are true of every conceivable experience. The axioms are translated into a set of postulates about the substrate of consciousness (called a complex), which are then used to formulate a mathematical framework for assessing both the quality and quantity of experience. The explanatory identity proposed by IIT is that an experience is identical to the cause-effect structure unfolded from a maximally irreducible substrate (a Φ-structure). In this work we introduce a definition for the integrated information of a system (φs) that is based on the existence, intrinsicality, information, and integration postulates of IIT. We explore how notions of determinism, degeneracy, and fault lines in the connectivity impact system-integrated information. We then demonstrate how the proposed measure identifies complexes as systems, the φs of which is greater than the φs of any overlapping candidate systems.

Keywords: causation; consciousness; integrated information; intrinsic.

Figure 1

A set of example systems used to explore the impact of indeterminism and degeneracy on system intrinsic information. Dark colored bars and grey bars to the right of the transition probabilities in (B–D) represent the quantities for informativeness (constrained and unconstrained) and light colored bars for selectivity. (A) A base system constituted of four units. The units were all-to-all connected, and each unit had a unique input–output function, described in the state-by-node transition probability matrix (each column defines the probability a unit is ON, given the previous state defined by the row). Black circles and capital letters indicate the current state is ON, while white circles with lower case letters indicate the current state is OFF. (B) The state-by-state transition probability matrix (${\mathcal{T}}_S$) of the system in (A), and the corresponding cause and effect repertoires used to identify $s_{c/e}^{\prime }$. The system was deterministic (one non-zero value in each row), non-degenerate (one non-zero value in each column), and had high values of intrinsic cause and effect information $({\mathrm{ii}}_c={\mathrm{ii}}_e=4)$. (C) The same system as in (B), but with unit D noisy, so that it went into the state specified by system (B) with probability 0.6, and the opposite state with probability 0.4. The system was now non-deterministic (more than one non-zero entry in each row) and degenerate (more than one non-zero entry in each column), and the cause and effect intrinsic information decreased $({\mathrm{ii}}_c=1.95,{\mathrm{ii}}_e=1.95)$. (D) The same system as in (B), but unit D was changed so that it’s input–output function was identical to that of unit A. The system was deterministic (one non-zero entry in each row) but degenerate (more than one non-zero entry in each column), and had reduced cause and effect intrinsic information (${\mathrm{ii}}_c=1.5,{\mathrm{ii}}_e=3$).

Figure 2

A set of example systems used to explore the impact of connectivity on system integrated information. In this example, the systems were constituted of four units in the OFF state, each with a sigmoidal activation function (Equation (2); $k=3,l=1$) and varying weights. The minimum partition for each system is indicated by the dashed orange line(s). (A) The first system had a symmetric connectivity structure with no fault lines. Two equivalent minimum partitions were identified, cutting $\left\{AB\right\}$ away from $\left\{CD\right\}$ or $\left\{AD\right\}$ away from $\left\{BC\right\}$. The integrated information was 48.1% of the intrinsic information. (B) A system with 3 strongly interconnected units $\{A,B,C\}$, and a fourth unit {D} that was weakly connected with the rest. A directional partition of $\left\{ABC\right\}$ from $\left\{D\right\}$ was the minimum partition of the system, and, as a result, the integrated information was only 10.0% of the intrinsic information. (C) A system with 2 strongly interconnected sets of units $\left\{AB\right\}$ and $\left\{CD\right\}$ with weak connections between them. Cutting $\left\{AB\right\}$ away from $\left\{CD\right\}$ (bidirectionally) was the minimum partition of the system, and the resulting integrated information was only 21.2% of the intrinsic information.

Figure 3

A universal substrate used to explore how degeneracy, determinism, and fault lines impacted how it condensed into non-overlapping complexes. (A) A universal substrate of 8 units, each having a sigmoidal activation function (Equation (2) ($l=1$). Units $\{A,B,C,D,E,F\}$ has a moderate level of determinsism ($k=2$) and units $\{G,H\}$ had a low level of determinism ($k=0.2$). A cluster of 5 units $\{A,B,C,D,E\}$ had stronger (but varying) connections within the cluster than without, a unit $\left\{F\right\}$ had a strong self-connection and weak connections with everything else, and two units $\{G,H\}$ were strongly connected between themselves with weak self-connections and weak connections to the rest of the units. (B) The system integrated information values for the potential systems $S\subseteq U$. (C) The universal substrate condensed into three non-overlapping complexes, $\{A,B,C,D,E\}$, $\left\{F\right\}$, and $\{G,H\}$, according to the algorithm in Appendix C. A solid blue line around a set of units indicates that it constituted a complex. (D) A sample of potential systems (dashed grey outline) that were excluded because they were a subset of a complex ($\{C,D,E\}$), a superset of a complex ($\{A,B,C,D,E,G\}$), or a “paraset”, partially overlapping a complex ($\{A,B,H\}$). Each of these candidate systems had lower ${\phi }_s$ than $\{A,B,C,D,E\}$. (E) The intrinsic cause and effect information of a nested sequence of candidate systems. (F) The integrated information of the systems in (E). The intrinsic information increased with each new unit added to the candidate system, but only when all five units were considered were there no fault lines in the system, leading to a maximum of integrated information.