Originally developed as a theory of consciousness, integrated information theory provides a mathematical framework to quantify the causal irreducibility of systems and subsets of units in the system. Specifically, mechanism integrated information quantifies how much of the causal powers of a subset of units in a state, also referred to as a mechanism, cannot be accounted for by its parts. If the causal powers of the mechanism can be fully explained by its parts, it is reducible and its integrated information is zero. Here, we study the upper bound of this measure and how it is achieved. We study mechanisms in isolation, groups of mechanisms, and groups of causal relations among mechanisms. We put forward new theoretical results that show mechanisms that share parts with each other cannot all achieve their maximum. We also introduce techniques to design systems that can maximize the integrated information of a subset of their mechanisms or relations. Our results can potentially be used to exploit the symmetries and constraints to reduce the computations significantly and to compare different connectivity profiles in terms of their maximal achievable integrated information.

This paper presents Integrated Information Theory (IIT) 4.0. IIT aims to account for the properties of experience in physical (operational) terms. It identifies the essential properties of experience (axioms), infers the necessary and sufficient properties that its substrate must satisfy (postulates), and expresses them in mathematical terms. In principle, the postulates can be applied to any system of units in a state to determine whether it is conscious, to what degree, and in what way. IIT offers a parsimonious explanation of empirical evidence, makes testable predictions concerning both the presence and the quality of experience, and permits inferences and extrapolations. IIT 4.0 incorporates several developments of the past ten years, including a more accurate formulation of the axioms as postulates and mathematical expressions, the introduction of a unique measure of intrinsic information that is consistent with the postulates, and an explicit assessment of causal relations. By fully unfolding a system's irreducible cause-effect power, the distinctions and relations specified by a substrate can account for the quality of experience.

The understanding of human consciousness based on brain connectivity is considered important for brain- machine interfacing. In this study, we investigated changes in causal connectivity in electroencephalography data related to conscious and unconscious experiences during non-rapid eye movement sleep after parietal transcranial magnetic stimulation (TMS). A serial awakening paradigm was used to determine whether subjects had had a conscious experience or not. We calculated direct transfer function (DTF) as a measure of effective connectivity in five frequency bands focusing on frontal and parietal-occipital regions. The DTF showed significant differences in frontal-to-parietal flow between reported unconsciousness and consciousness. During the first 100 ms after TMS, the outward links of the parietal region at low frequencies were higher in no conscious experience than in conscious experience. During the next 100 ms, however, the outward links of the frontal region were higher in the conscious experience than the no conscious experience at low frequencies. Changes with causal connectivity over time after TMS indicate that the spatial roles in brain regions associated with consciousness are different. These findings may help clarify the cortical mechanisms related to conscious experience.

Reductionism assumes that causation in the physical world occurs at the micro level, excluding the emergence of macro-level causation. We challenge this reductionist assumption by employing a principled, well-defined measure of intrinsic cause-effect power-integrated information (Φ), and showing that, according to this measure, it is possible for a macro level to "beat" the micro level. Simple systems were evaluated for Φ across different spatial and temporal scales by systematically considering all possible black boxes. These are macro elements that consist of one or more micro elements over one or more micro updates. Cause-effect power was evaluated based on the inputs and outputs of the black boxes, ignoring the internal micro elements that support their input-output function. We show how black-box elements can have more common inputs and outputs than the corresponding micro elements, revealing the emergence of high-order mechanisms and joint constraints that are not apparent at the micro level. As a consequence, a macro, black-box system can have higher Φ than its micro constituents by having more mechanisms (higher composition) that are more interconnected (higher integration). We also show that, for a given micro system, one can identify local maxima of Φ across several spatiotemporal scales. The framework is demonstrated on a simple biological system, the Boolean network model of the fission-yeast cell-cycle, for which we identify stable local maxima during the course of its simulated biological function. These local maxima correspond to macro levels of organization at which emergent cause-effect properties of physical systems come into focus, and provide a natural vantage point for scientific inquiries.

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.

Natural selection favors the evolution of brains that can capture fitness-relevant features of the environment's causal structure. We investigated the evolution of small, adaptive logic-gate networks ("animats") in task environments where falling blocks of different sizes have to be caught or avoided in a 'Tetris-like' game. Solving these tasks requires the integration of sensor inputs and memory. Evolved networks were evaluated using measures of information integration, including the number of evolved concepts and the total amount of integrated conceptual information. The results show that, over the course of the animats' adaptation, i) the number of concepts grows; ii) integrated conceptual information increases; iii) this increase depends on the complexity of the environment, especially on the requirement for sequential memory. These results suggest that the need to capture the causal structure of a rich environment, given limited sensors and internal mechanisms, is an important driving force for organisms to develop highly integrated networks ("brains") with many concepts, leading to an increase in their internal complexity.

This paper presents Integrated Information Theory (IIT) of consciousness 3.0, which incorporates several advances over previous formulations. IIT starts from phenomenological axioms: information says that each experience is specific--it is what it is by how it differs from alternative experiences; integration says that it is unified--irreducible to non-interdependent components; exclusion says that it has unique borders and a particular spatio-temporal grain. These axioms are formalized into postulates that prescribe how physical mechanisms, such as neurons or logic gates, must be configured to generate experience (phenomenology). The postulates are used to define intrinsic information as "differences that make a difference" within a system, and integrated information as information specified by a whole that cannot be reduced to that specified by its parts. By applying the postulates both at the level of individual mechanisms and at the level of systems of mechanisms, IIT arrives at an identity: an experience is a maximally irreducible conceptual structure (MICS, a constellation of concepts in qualia space), and the set of elements that generates it constitutes a complex. According to IIT, a MICS specifies the quality of an experience and integrated information ΦMax its quantity. From the theory follow several results, including: a system of mechanisms may condense into a major complex and non-overlapping minor complexes; the concepts that specify the quality of an experience are always about the complex itself and relate only indirectly to the external environment; anatomical connectivity influences complexes and associated MICS; a complex can generate a MICS even if its elements are inactive; simple systems can be minimally conscious; complicated systems can be unconscious; there can be true "zombies"--unconscious feed-forward systems that are functionally equivalent to conscious complexes.

The advent of non von Neumann computational models, specifically neuromorphic architectures, has engendered a new class of challenges for computer architects. On the one hand, each neuron-like computational element must consume minimal power and area to enable scaling up to biological scales of billions of neurons; this rules out direct support for complex and expensive features like floating point and transcendental functions. On the other hand, to fully benefit from cortical properties and operations, neuromorphic architectures must support complex non-linear neuronal behaviors. This semantic gap between the simple and power-efficient processing elements and complex neuronal behaviors has rekindled a RISC vs. CISC-like debate within the neuromorphic hardware design community. In this paper, we address the aforementioned semantic gap for a recently-described digital neuromorphic architecture that constitutes simple Linear-Leak Integrate-and-Fire (LLIF) spiking neurons as processing primitives. We show that despite the simplicity of LLIF primitives, a broad class of complex neuronal behaviors can be emulated by composing assemblies of such primitives with low area and power overheads. Furthermore, we demonstrate that for the LLIF primitives without built-in mechanisms for synaptic plasticity, two well-known neural learning rules-spike timing dependent plasticity and Hebbian learning-can be emulated via assemblies of LLIF primitives. By bridging the semantic gap for one such system we enable neuromorphic system developers, in general, to keep their hardware design simple and power-efficient and at the same time enjoy the benefits of complex neuronal behaviors essential for robust and accurate cortical simulation.

What is the relationship between the complexity and the fitness of evolved organisms, whether natural or artificial? It has been asserted, primarily based on empirical data, that the complexity of plants and animals increases as their fitness within a particular environment increases via evolution by natural selection. We simulate the evolution of the brains of simple organisms living in a planar maze that they have to traverse as rapidly as possible. Their connectome evolves over 10,000s of generations. We evaluate their circuit complexity, using four information-theoretical measures, including one that emphasizes the extent to which any network is an irreducible entity. We find that their minimal complexity increases with their fitness.

What is the relationship between the complexity and the fitness of evolved organisms, whether natural or artificial? It has been asserted, primarily based on empirical evidence, that the complexity of plants and animals increases as their fitness within a particular environment increases via evolution by natural selection (Bonner, 1988; McShea, 1996; Adami et al., 2000). We here derive an analytical relationship between these two quantities within an information-theoretical framework, showing that under certain conditions, complexity is a monotonically increasing function of fitness. We also simulate the adaptation of brains of digital organisms living in mazes and whose connectome evolves over 10,000s of generations in a stationary environment. We compute their circuit complexity, using an entropy-based measure (Balduzzi and Tononi, 2008). We find that their minimal complexity increases with their fitness, in line with our analytical derivation.

One of the hallmarks of biological organisms is their ability to integrate disparate information sources to optimize their behavior in complex environments. How this capability can be quantified and related to the functional complexity of an organism remains a challenging problem, in particular since organismal functional complexity is not well-defined. We present here several candidate measures that quantify information and integration, and study their dependence on fitness as an artificial agent ("animat") evolves over thousands of generations to solve a navigation task in a simple, simulated environment. We compare the ability of these measures to predict high fitness with more conventional information-theoretic processing measures. As the animat adapts by increasing its "fit" to the world, information integration and processing increase commensurately along the evolutionary line of descent. We suggest that the correlation of fitness with information integration and with processing measures implies that high fitness requires both information processing as well as integration, but that information integration may be a better measure when the task requires memory. A correlation of measures of information integration (but also information processing) and fitness strongly suggests that these measures reflect the functional complexity of the animat, and that such measures can be used to quantify functional complexity even in the absence of fitness data.

According to the integrated information theory, the quantity of consciousness is the amount of integrated information generated by a complex of elements, and the quality of experience is specified by the informational relationships it generates. This paper outlines a framework for characterizing the informational relationships generated by such systems. Qualia space (Q) is a space having an axis for each possible state (activity pattern) of a complex. Within Q, each submechanism specifies a point corresponding to a repertoire of system states. Arrows between repertoires in Q define informational relationships. Together, these arrows specify a quale -- a shape that completely and univocally characterizes the quality of a conscious experience. Phi -- the height of this shape -- is the quantity of consciousness associated with the experience. Entanglement measures how irreducible informational relationships are to their component relationships, specifying concepts and modes. Several corollaries follow from these premises. The quale is determined by both the mechanism and state of the system. Thus, two different systems having identical activity patterns may generate different qualia. Conversely, the same quale may be generated by two systems that differ in both activity and connectivity. Both active and inactive elements specify a quale, but elements that are inactivated do not. Also, the activation of an element affects experience by changing the shape of the quale. The subdivision of experience into modalities and submodalities corresponds to subshapes in Q. In principle, different aspects of experience may be classified as different shapes in Q, and the similarity between experiences reduces to similarities between shapes. Finally, specific qualities, such as the "redness" of red, while generated by a local mechanism, cannot be reduced to it, but require considering the entire quale. Ultimately, the present framework may offer a principled way for translating qualitative properties of experience into mathematics.

This paper introduces a time- and state-dependent measure of integrated information, phi, which captures the repertoire of causal states available to a system as a whole. Specifically, phi quantifies how much information is generated (uncertainty is reduced) when a system enters a particular state through causal interactions among its elements, above and beyond the information generated independently by its parts. Such mathematical characterization is motivated by the observation that integrated information captures two key phenomenological properties of consciousness: (i) there is a large repertoire of conscious experiences so that, when one particular experience occurs, it generates a large amount of information by ruling out all the others; and (ii) this information is integrated, in that each experience appears as a whole that cannot be decomposed into independent parts. This paper extends previous work on stationary systems and applies integrated information to discrete networks as a function of their dynamics and causal architecture. An analysis of basic examples indicates the following: (i) phi varies depending on the state entered by a network, being higher if active and inactive elements are balanced and lower if the network is inactive or hyperactive. (ii) phi varies for systems with identical or similar surface dynamics depending on the underlying causal architecture, being low for systems that merely copy or replay activity states. (iii) phi varies as a function of network architecture. High phi values can be obtained by architectures that conjoin functional specialization with functional integration. Strictly modular and homogeneous systems cannot generate high phi because the former lack integration, whereas the latter lack information. Feedforward and lattice architectures are capable of generating high phi but are inefficient. (iv) In Hopfield networks, phi is low for attractor states and neutral states, but increases if the networks are optimized to achieve tension between local and global interactions. These basic examples appear to match well against neurobiological evidence concerning the neural substrates of consciousness. More generally, phi appears to be a useful metric to characterize the capacity of any physical system to integrate information.

The connection matrix of the human brain (the human "connectome") represents an indispensable foundation for basic and applied neurobiological research. However, the network of anatomical connections linking the neuronal elements of the human brain is still largely unknown. While some databases or collations of large-scale anatomical connection patterns exist for other mammalian species, there is currently no connection matrix of the human brain, nor is there a coordinated research effort to collect, archive, and disseminate this important information. We propose a research strategy to achieve this goal, and discuss its potential impact.

Autonomous robotic systems need to adjust their sensorimotor coordinations so as to maintain good performance in the presence of changes in their sensory and motor characteristics. Biological systems are able to adapt to large variations in their physical and functional properties. The adjustment of orienting behavior has been carefully investigated in the barn owl, a nocturnal predator with highly developed auditory capabilities. In the optic tectum of the barn owl, an area well-known to be involved in the production of orienting behavior, neural maps of space in the visual, auditory, and motor modalities are found in close alignment with each other. As a neurophysiological correlate of the adjustment of motor responses, neural maps in the tectum tend to realign if the sensory inputs are manipulated. We have recently proposed that the development and maintenance of such map alignment can be explained through a process of learning, in which plasticity is mediated by the activation of diffuse-projecting neuromodulatory systems which respond to innate or acquired salient cues. This proposal was tested using a detailed computer model of the principal neural structures involved in the process of spatial localisation in the barn owl. Here we consider the application of this model to the control of the orienting behavior of a robotic system in the presence of auditory and visual stimulation. The system we consider is composed of a robotic head equipped with two lateral microphones and a camera. We show that the model produces accurate orienting behavior toward both auditory and visual stimuli during normal visual experience, after alteration of the visual inputs, and after the reestablishment of normal visual conditions. The results illustrate that an architecture specifically designed to account for biological phenomena can produce flexible and robust control of a robotic system.