Integrated information theory provides a mathematical framework to fully characterize the cause-effect structure of a physical system. Here, we introduce PyPhi, a Python software package that implements this framework for causal analysis and unfolds the full cause-effect structure of discrete dynamical systems of binary elements. The software allows users to easily study these structures, serves as an up-to-date reference implementation of the formalisms of integrated information theory, and has been applied in research on complexity, emergence, and certain biological questions. We first provide an overview of the main algorithm and demonstrate PyPhi’s functionality in the course of analyzing an example system, and then describe details of the algorithm’s design and implementation. PyPhi can be installed with Python’s package manager via the command ‘pip install pyphi’ on Linux and macOS systems equipped with Python 3.4 or higher. PyPhi is open-source and licensed under the GPLv3; the source code is hosted on GitHub at https://github.com/wmayner/pyphi. Comprehensive and continually-updated documentation is available at https://pyphi.readthedocs.io. The pyphi-users mailing list can be joined at https://groups.google.com/forum/#!forum/pyphi-users. A web-based graphical interface to the software is available at http://integratedinformationtheory.org/calculate.html.

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 $\mathsf{\Phi }$-structure). In this work we introduce a definition for the integrated information of a system (${\phi }_s$) 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 ${\phi }_s$ of which is greater than the ${\phi }_s$ of any overlapping candidate systems.

Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information ($\mathsf{\Phi }$). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.

The target article misrepresents the foundations of integrated information theory (IIT) and ignores many essential publications. It, thus, falls to this lead commentary to outline the axioms and postulates of IIT and correct major misconceptions. The commentary also explains why IIT starts from phenomenology and why it predicts that only select physical substrates can support consciousness. Finally, it highlights that IIT's account of experience – a cause–effect structure quantified by integrated information – has nothing to do with “information transfer.”

Despite significant progress in understanding neural coding, it remains unclear how the coordinated activity of large populations of neurons relates to what an observer actually perceives. Since neurophysiological differences must underlie differences among percepts,differentiation analysis—quantifying distinct patterns of neurophysiological activity—has been proposed as an “inside-out” approach that addresses this question. This methodology contrasts with “outside-in” approaches such as feature tuning and decoding analyses, which are defined in terms of extrinsic experimental variables. Here, we used two-photon calcium imaging in mice of both sexes to systematically survey stimulus-evoked neurophysiological differentiation (ND) in excitatory neuronal populations in layers (L)2/3, L4, and L5 across five visual cortical areas (primary, lateromedial, anterolateral, posteromedial, and anteromedial) in response to naturalistic and phase-scrambled movie stimuli. We find that unscrambled stimuli evoke greater ND than scrambled stimuli specifically in L2/3 of the anterolateral and anteromedial areas, and that this effect is modulated by arousal state and locomotion. By contrast, decoding performance was far above chance and did not vary substantially across areas and layers. Differentiation also differed within the unscrambled stimulus set, suggesting that differentiation analysis may be used to probe the ethological relevance of individual stimuli.

Integrated information theory (IIT) starts from the existence of consciousness and characterizes its essential properties: every experience is intrinsic, specific, unitary, definite, and structured. IIT then formulates existence and its essential properties operationally in terms of cause–effect power of a substrate of units. Here, we address IIT’s operational requirements for existence by considering that, to have cause–effect power, to have it intrinsically, and to have it specifically, substrate units in their actual state must both (i) ensure the intrinsic availability of a repertoire of cause–effect states, and (ii) increase the probability of a specific cause–effect state. We showed previously that requirement (ii) can be assessed by the intrinsic difference of a state’s probability from maximal differentiation. Here, we show that requirement (i) can be assessed by the intrinsic difference from maximal specification. These points and their consequences for integrated information are illustrated using simple systems of micro units. When applied to macro units and systems of macro units such as neural systems, a tradeoff between differentiation and specification is a necessary condition for intrinsic existence—and therefore, according to IIT, for consciousness.