# Integrated information theory

(Redirected from Integrated Information Theory)
Phi, the symbol for integrated information.

Integrated information theory (IIT) attempts to explain what consciousness is and why it might be associated with certain physical systems. Given any such system, the theory predicts whether that system is conscious, to what degree it is conscious, and what particular experience it is having (see Central identity). According to IIT, a system's consciousness is determined by its causal properties and is therefore an intrinsic, fundamental property of any physical system.[1]

IIT was proposed by neuroscientist Giulio Tononi in 2004, and has been continuously developed over the past decade. The latest version of the theory, labeled IIT 3.0, was published in 2014.[2][3]

## Overview

### Relationship to the "hard problem of consciousness"

David Chalmers has argued that any attempt to explain consciousness in purely physical terms (i.e. to start with the laws of physics as they are currently formulated and derive the necessary and inevitable existence of consciousness) eventually runs into the so-called "hard problem". Rather than try to start from physical principles and arrive at consciousness, IIT "starts with consciousness" (accepts the existence of consciousness as certain) and reasons about the properties that a postulated physical substrate would have to have in order to account for it. The ability to perform this jump from phenomenology to mechanism rests on IIT's assumption that if a conscious experience can be fully accounted for by an underlying physical system, then the properties of the physical system must be constrained by the properties of the experience.

Specifically, IIT moves from phenomenology to mechanism by attempting to identify the essential properties of conscious experience (dubbed "axioms") and, from there, the essential properties of conscious physical systems (dubbed "postulates").

### Axioms: essential properties of experience

Axioms and postulates of integrated information theory.

The axioms are intended to capture the essential aspects of every conscious experience. Every axiom should apply to every possible experience.

The wording of the axioms has changed slightly as the theory has developed, and the most recent and complete statement of the axioms is as follows:

### Postulates: properties required of the physical substrate

The axioms describe regularities in conscious experience, and IIT seeks to explain these regularities. What could account for the fact that every experience exists, is structured, is differentiated, is unified, and is definite? IIT argues that the existence of an underlying causal system with these same properties offers the most parsimonious explanation. Thus a physical system, if conscious, is so by virtue of its causal properties.

The properties required of a conscious physical substrate are called the "postulates," since the existence of the physical substrate is itself only postulated (remember, IIT maintains that the only thing one can be sure of is the existence of one's own consciousness). In what follows, a "physical system" is taken to be a set of elements, each with two or more internal states, inputs that influence that state, and outputs that are influenced by that state (neurons or logic gates are the natural examples). Given this definition of "physical system", the postulates are:

### Mathematics: formalization of the postulates

For a complete and thorough account of the mathematical formalization of IIT, see reference.[2] What follows is intended as a brief summary, adapted from,[4] of the most important quantities involved. Pseudocode for the algorithms used to calculate these quantities can be found at reference.[5]

A system refers to a set of elements, each with two or more internal states, inputs that influence that state, and outputs that are influenced by that state. A mechanism refers to a subset of system elements. The mechanism-level quantities below are used to assess the integration of any given mechanism, and the system-level quantities are used to assess the integration of sets of mechanisms ("sets of sets").

In order to apply the IIT formalism to a system, its full transition probability matrix (TPM) must be known. The TPM specifies the probability with which any state of a system transitions to any other system state. Each of the following quantities is calculated in a bottom-up manner from the system's TPM.

Mechanism-level quantities
A cause-effect repertoire ${\textstyle {\textrm {CER}}(m_{t},\,Z_{t\pm 1})=\{p_{\textrm {cause}}(z_{t-1}|m_{t}),\,p_{\textrm {effect}}(z_{t+1}|m_{t})\}}$ is a set of two probability distributions, describing how the mechanism ${\textstyle M_{t}}$ in its current state ${\textstyle m_{t}}$ constrains the past and future states of the sets of system elements ${\textstyle Z_{t-1}}$ and ${\textstyle Z_{t+1}}$, respectively.

Note that ${\textstyle Z_{t-1}}$ may be different from ${\textstyle Z_{t+1}}$, since the elements that a mechanism affects may be different from the elements that affect it.

A partition ${\textstyle P=\{M_{1},Z_{1};M_{2},Z_{2}\}}$ is a grouping of system elements, where the connections between the parts ${\textstyle \{M_{1},Z_{1}\}}$ and ${\textstyle \{M_{2},Z_{2}\}}$ are injected with independent noise. For a simple binary element ${\textstyle A}$ which outputs to a simple binary element ${\textstyle B}$, injecting the connection ${\textstyle A\to B}$ with independent noise means that the input value which ${\textstyle A}$ receives, ${\textstyle 0}$ or ${\textstyle 1}$, is entirely independent of the actual state of ${\textstyle B}$, thus rendering ${\textstyle B}$ causally ineffective.

${\textstyle P_{t\pm 1}}$ denotes a pair of partitions, one of which is considered when looking at a mechanism's causes, and the other of which is considered when looking at its effects.

The earth mover's distance ${\textstyle {\textrm {EMD}}(p_{1},\,p_{2})}$ is used to measure distances between probability distributions ${\textstyle p_{1}}$ and ${\textstyle p_{2}}$. The EMD depends on the user's choice of ground distance between points in the metric space over which the probability distributions are measured, which in IIT is the system's state space. When computing the EMD with a system of simple binary elements, the ground distance between system states is chosen to be their Hamming distance.
Integrated information ${\textstyle \varphi }$ measures the irreducibility of a cause-effect repertoire with respect to partition ${\textstyle P_{t\pm 1}}$, obtained by combining the irreducibility of its constituent cause and effect repertoires with respect to the same partitioning.

The irreducibility of the cause repertoire with respect to ${\textstyle P_{t-1}}$ is given by ${\textstyle \varphi _{\textrm {cause}}(m_{t},\,Z_{t-1},\,P_{t-1})={\textrm {EMD}}(p_{\textrm {cause}}(z_{t-1}|m_{t}),\,p_{\textrm {cause}}(z_{1,t-1}|m_{1,t})\times p_{\textrm {cause}}(z_{2,t-1}|m_{2,t}))}$, and similarly for the effect repertoire.

Combined, ${\textstyle \varphi _{\textrm {cause}}}$ and ${\textstyle \varphi _{\textrm {effect}}}$ yield the irreducibility of the ${\textstyle {\textrm {CER}}}$ as a whole: ${\textstyle \varphi (m_{t},\,Z_{t\pm 1},\,P_{t\pm 1})=\min(\varphi _{\textrm {cause}}(m_{t},\,Z_{t-1},\,P_{t-1}),\varphi _{\textrm {effect}}(m_{t},\,Z_{t+1},\,P_{t+1})).}$.

The minimum-information partition of a mechanism and its purview is given by ${\textstyle {\textrm {MIP}}(m_{t},\,Z_{t\pm 1})=\operatorname {*} {\arg \,\min }_{P_{t\pm 1}}\,(\varphi (m_{t},\,Z_{t\pm 1},\,P_{t\pm 1}))}$. The minimum-information partition is the partitioning that least affects a cause-effect repertoire. For this reason, it is sometimes called the minimum-difference partition.

Note that the minimum-information "partition", despite its name, is really a pair of partitions. We call these partitions ${\textstyle {\textrm {MIP}}_{\textrm {cause}}}$ and ${\textstyle {\textrm {MIP}}_{\textrm {effect}}}$.

There is at least one choice of elements over which a mechanism's cause-effect repertoire is maximally irreducible (in other words, over which its ${\textstyle \varphi }$ is highest). We call this choice of elements ${\textstyle Z_{t\pm 1}^{*}=\{Z_{t-1}^{*},\,Z_{t+1}^{*}\}}$, and say that this choice specifies a maximally irreducible cause-effect repertoire.

Formally, ${\textstyle Z_{t-1}^{*}=\{\operatorname {*} {\arg \,\max }_{Z_{t-1}}\,(\varphi _{\textrm {cause}}(m_{t},\,Z_{t-1},\,{\textrm {MIP}}_{\textrm {cause}}))\}}$ and ${\textstyle Z_{t+1}^{*}=\{\operatorname {*} {\arg \,\max }_{Z_{t+1}}\,(\varphi _{\textrm {effect}}(m_{t},\,Z_{t+1},\,{\textrm {MIP}}_{\textrm {effect}}))\}}$.

The concept ${\textstyle {\textrm {CER}}(m_{t},\,Z_{t\pm 1}^{*})=\{p_{\textrm {cause}}(z_{t-1}^{*}|m_{t}),\,p_{\textrm {effect}}(z_{t+1}^{*}|m_{t})\}}$ is the maximally irreducible cause-effect repertoire of mechanism${\textstyle M_{t}}$ in its current state ${\textstyle m_{t}}$ over ${\textstyle Z_{t\pm 1}^{*}}$, and describes the causal role of ${\textstyle M_{t}}$ within the system. Informally, ${\textstyle Z_{t\pm 1}^{*}}$ is the concept's purview, and specifies what the concept "is about".

The intrinsic cause-effect power of ${\textstyle m_{t}}$ is the concept's strength, and is given by: ${\textstyle \varphi ^{\textrm {Max}}(m_{t})=\varphi (m_{t},\,Z_{t\pm 1}^{*},\,{\textrm {MIP}})=\min(\varphi _{\textrm {cause}}(m_{t},\,Z_{t-1}^{*},\,{\textrm {MIP}}_{\textrm {cause}}),\,\varphi _{\textrm {effect}}(m_{t},\,Z_{t+1}^{*},\,{\textrm {MIP}}_{\textrm {effect}}))}$

System-level quantities
A cause-effect structure ${\textstyle C(s_{t})}$ is the set of concepts specified by all mechanisms with ${\textstyle \varphi ^{\textrm {Max}}(m_{t})>0}$ within the system ${\textstyle S_{t}}$ in its current state ${\textstyle s_{t}}$. If a system turns out to be conscious, its cause-effect structure is often referred to as a conceptual structure.
A unidirectional partition ${\textstyle P_{\to }=\{S_{1},S_{2}\}}$ is a grouping of system elements where the connections from the set of elements ${\textstyle S_{1}}$ to ${\textstyle S_{2}}$ are injected with independent noise.
The extended earth mover's distance ${\textstyle {\textrm {XEMD}}(C_{1},\,C_{2})}$ is used to measure the minimal cost of transforming cause-effect structure ${\textstyle C_{1}}$ into structure ${\textstyle C_{2}}$. Informally, one can say that–whereas the EMD transports the probability of a system state over the distance between two system states–the XEMD transports the strength of a concept over the distance between two concepts.

In the XEMD, the "earth" to be transported is intrinsic cause-effect power (${\textstyle \varphi ^{\textrm {Max}}}$), and the ground distance between concepts ${\textstyle A}$ and ${\textstyle B}$ with cause repertoires ${\textstyle A_{\textrm {cause}}}$ and ${\textstyle B_{\textrm {cause}}}$ and effect repertoires ${\textstyle A_{\textrm {effect}}}$ and ${\textstyle B_{\textrm {effect}}}$ is given by ${\textstyle {\textrm {EMD}}(A_{\textrm {cause}},\,B_{\textrm {cause}})+{\textrm {EMD}}(A_{\textrm {effect}},\,B_{\textrm {effect}})}$.

Integrated (conceptual) information ${\textstyle \Phi (s_{t},\,P_{\to })={\textrm {XEMD}}(C(s_{t})|C(s_{t},\,P_{\to }))}$ measures the irreducibility of a cause-effect structure with respect to a unidirectional partition. ${\textstyle \Phi }$ captures how much the cause-effect repertoires of the system's mechanisms are altered and how much intrinsic cause effect power (${\textstyle \varphi ^{\textrm {Max}}}$) is lost due to partition ${\textstyle P_{\to }}$.
The minimum-information partition of a set of elements in a state is given by ${\textstyle {\textrm {MIP}}(s_{t})=\operatorname {*} {\arg \,\min }_{P_{\to }}\,(\Phi (s_{t},\,\,P_{\to }))}$. The minimum-information partition is the unidirectional partition that least affects a cause-effect structure ${\textstyle C(s_{t})}$.
The intrinsic cause-effect power of a set of elements in a state is given by ${\textstyle \Phi ^{\textrm {Max}}(s_{t}^{*})=\Phi (s_{t}^{*},\,{\textrm {MIP}}(s_{t}^{*}))}$, such that for any other ${\textstyle S_{t}}$ with ${\textstyle (S_{t}\cap S_{t}^{*})\neq \emptyset }$, ${\textstyle \Phi (s_{t})\leq \Phi (s_{t}^{*})}$. According to IIT, a system's ${\textstyle \Phi ^{\textrm {Max}}}$ is the degree to which it can be said to exist.
A complex is a set of elements ${\textstyle S_{t}^{*}}$ with ${\textstyle \Phi ^{\textrm {Max}}=\Phi (s_{t}^{*})>0}$, and thus specifies a maximally irreducible cause-effect structure, also called a conceptual structure. According to IIT, complexes are conscious entities.

#### Cause-effect space

For a system of ${\displaystyle N}$ simple binary elements, cause-effect space is formed by ${\displaystyle 2\times 2^{N}}$ axes, one for each possible past and future state of the system. Any cause-effect repertoire ${\displaystyle R}$, which specifies the probability of each possible past and future state of the system, can be easily plotted as a point in this high-dimensional space: The position of this point along each axis is given by the probability of that state as specified by ${\displaystyle R}$. If a point is also taken to have a scalar magnitude (which can be informally thought of as the point's "size", for example), then it can easily represent a concept: The concept's cause-effect repertoire specifies the location of the point in cause-effect space, and the concept's ${\displaystyle \varphi ^{\textrm {Max}}}$ value specifies that point's magnitude.

In this way, a conceptual structure ${\displaystyle C}$ can be plotted as a constellation of points in cause-effect space. Each point is called a star, and each star's magnitude (${\displaystyle \varphi ^{\textrm {Max}}}$) is its size.

### Central identity

IIT addresses the mind-body problem by proposing an identity between phenomenological properties of experience and causal properties of physical systems: The conceptual structure specified by a complex of elements in a state is identical to its experience.

Specifically, the form of the conceptual structure in cause-effect space completely specifies the quality of the experience, while the irreducibility ${\displaystyle \Phi ^{\textrm {Max}}}$ of the conceptual structure specifies the level to which it exists (i.e., the complex's level of consciousness). The maximally irreducible cause-effect repertoire of each concept within a conceptual structure specifies what the concept contributes to the quality of the experience, while its irreducibility ${\displaystyle \varphi ^{\textrm {Max}}}$ specifies how much the concept is present in the experience.

According to IIT, an experience is thus an intrinsic property of a complex of mechanisms in a state.

## Extensions

The calculation of even a modestly-sized system's ${\displaystyle \Phi ^{\textrm {Max}}}$ is often computationally intractable, so efforts have been made to develop heuristic or proxy measures of integrated information. For example, Masafumi Oizumi has developed ${\displaystyle \Phi ^{*}}$, a practical approximation for integrated information that solves the theoretical shortcomings of previously proposed proxy measures,[6] such as the one proposed by Adam Barrett.[7]

A significant computational challenge in calculating integrated information is finding the Minimum Information Partition of a neural system, which requires iterating through all possible network partitions. To solve this problem, Daniel Toker has suggested using the most modular decomposition of a network as an extremely quick proxy for the Minimum Information Partition.[8]

## Related experimental work

While the algorithm[5] for assessing a system's ${\displaystyle \Phi ^{\textrm {Max}}}$ and conceptual structure is relatively straightforward, its high time complexity makes it computationally intractable for many systems of interest. Heuristics and approximations can sometimes be used to provide ballpark estimates of a complex system's integrated information, but precise calculations are often impossible. These computational challenges, combined with the already difficult task of reliably and accurately assessing consciousness under experimental conditions, make testing many of the theory's predictions difficult.

Despite these challenges, researchers have attempted to use measures of information integration and differentiation to assess levels of consciousness in a variety of subjects.[9][10] For instance, a recent study using a less computationally-intensive proxy for ${\displaystyle \Phi ^{\textrm {Max}}}$ was able to reliably discriminate between varying levels of consciousness in wakeful, sleeping (dreaming vs. non-dreaming), anesthetized, and comatose (vegetative vs. minimally-conscious vs. locked-in) individuals.[11]

IIT also makes several predictions which fit well with existing experimental evidence, and can be used to explain some counterintuitive findings in consciousness research.[12] For example, IIT can be used to explain why some brain regions, such as the cerebellum do not appear to contribute to consciousness, despite their size and/or functional importance.

## Reception

### Support

Neuroscientist Christof Koch, who has helped to develop the theory, has called IIT "the only really promising fundamental theory of consciousness".[13] Technologist Virgil Griffith says "IIT is currently the leading theory of consciousness."[14]

### Criticism

Challenges to IIT:

• IIT proposes conditions which are necessary for consciousness, but are not entirely sufficient.[15]
• IIT claims that all of its axioms are self-evident.[16]
• Since IIT is not a functionalist theory of consciousness, criticisms of non-functionalism have been levied against it.[16]
• The limits of IIT's definition of consciousness have led to criticism.[15][16]

## References

1. ^ Tononi, Giulio; Boly, Melanie; Massimini, Marcello; Koch, Christof. "Integrated information theory: from consciousness to its physical substrate". Nature Reviews Neuroscience. 17 (7): 450–461. doi:10.1038/nrn.2016.44. PMID 27225071.
2. ^ a b Oizumi, Masafumi; Albantakis, Larissa; Tononi, Giulio (2014-05-08). "From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0". PLoS Comput Biol. 10 (5): e1003588. Bibcode:2014PLSCB..10E3588O. doi:10.1371/journal.pcbi.1003588. PMC 4014402. PMID 24811198.
3. ^ a b c "Integrated information theory - Scholarpedia". www.scholarpedia.org. Retrieved 2015-11-23.
4. ^ Albantakis, Larissa; Tononi, Giulio (2015-07-31). "The Intrinsic Cause-Effect Power of Discrete Dynamical Systems—From Elementary Cellular Automata to Adapting Animats". Entropy. 17 (8): 5472–5502. Bibcode:2015Entrp..17.5472A. doi:10.3390/e17085472.
5. ^ a b "CSC-UW/iit-pseudocode". GitHub. Retrieved 2016-01-29.
6. ^ Oizumi, Masafumi; Amari, Shun-ichi; Yanagawa, Toru; Fujii, Naotaka; Tsuchiya, Naotsugu (2015-05-17). "Measuring integrated information from the decoding perspective". PLOS Computational Biology. 12 (1): e1004654. arXiv:1505.04368. Bibcode:2016PLSCB..12E4654O. doi:10.1371/journal.pcbi.1004654.
7. ^ Barrett, A.B.; Seth, A.K. (2011). "Practical measures of integrated information for time-series data". PLoS Comput. Biol. 7 (1): e1001052. Bibcode:2011PLSCB...7E1052B. doi:10.1371/journal.pcbi.1001052.
8. ^ Toker, Daniel; Sommer, Friedrich (2016-05-03). "Moving Past the Minimum Information Partition: How to Quickly and Accurately Calculate Integrated Information". arXiv:1605.01096.
9. ^ Massimini, M.; Ferrarelli, F.; Murphy, Mj; Huber, R.; Riedner, Ba; Casarotto, S.; Tononi, G. (2010-09-01). "Cortical reactivity and effective connectivity during REM sleep in humans". Cognitive Neuroscience. 1 (3): 176–183. doi:10.1080/17588921003731578. ISSN 1758-8936. PMC 2930263. PMID 20823938.
10. ^ Ferrarelli, Fabio; Massimini, Marcello; Sarasso, Simone; Casali, Adenauer; Riedner, Brady A.; Angelini, Giuditta; Tononi, Giulio; Pearce, Robert A. (2010-02-09). "Breakdown in cortical effective connectivity during midazolam-induced loss of consciousness". Proceedings of the National Academy of Sciences of the United States of America. 107 (6): 2681–2686. Bibcode:2010PNAS..107.2681F. doi:10.1073/pnas.0913008107. ISSN 1091-6490. PMC 2823915. PMID 20133802.
11. ^ Casali, Adenauer G.; Gosseries, Olivia; Rosanova, Mario; Boly, Mélanie; Sarasso, Simone; Casali, Karina R.; Casarotto, Silvia; Bruno, Marie-Aurélie; Laureys, Steven; Massimini, Marcello (2013-08-14). "A Theoretically Based Index of Consciousness Independent of Sensory Processing and Behavior". Science Translational Medicine. 5 (198): 198ra105–198ra105. doi:10.1126/scitranslmed.3006294. ISSN 1946-6234. PMID 23946194.
12. ^ "Integrated information theory - Scholarpedia". www.scholarpedia.org. Retrieved 2016-01-28.
13. ^ Zimmer, Carl (2010-09-20). "Sizing Up Consciousness by Its Bits". The New York Times. ISSN 0362-4331. Retrieved 2015-11-23.
14. ^
15. ^ a b "Shtetl-Optimized » Blog Archive » Why I Am Not An Integrated Information Theorist (or, The Unconscious Expander)". www.ScottAaronson.com. Retrieved 23 November 2015.
16. ^ a b c Cerullo, Michael A.; Kording, Konrad P. (17 September 2015). "The Problem with Phi: A Critique of Integrated Information Theory". PLOS Computational Biology. 11 (9): e1004286. Bibcode:2015PLSCB..11E4286C. doi:10.1371/journal.pcbi.1004286.