Integrated Informational States (IISs)

• IISs are unitary, integrated informational states that capture the full set of intrinsic causal relationships within a system.
• They are quantified using measures such as φ and Φ, ensuring that the information is irreducible and distinct from the sum of its parts.
• IISs have broad applications in neuroscience, AI, and dynamical systems, providing insights into the architecture of consciousness and adaptive behavior.

Integrated Informational States (IISs) are a central construct in contemporary consciousness science and theoretical neuroscience, and have also emerged in robust models of complex information processing in both natural and artificial systems. Multiple lines of research—ranging from Integrated Information Theory (IIT), complexity measures in dynamical systems, and computational architectures such as Modular Consciousness Theory (MCT)—have contributed precise definitions, quantification frameworks, and functional interpretations of IISs. Across these approaches, IISs are conceived as unitary, integrated, and causally irreducible informational configurations, often realized as discrete packets or structures, that underlie subjective experience, adaptive memory, and action.

1. Foundational Definitions and Formalism

IISs are information-rich states that capture not only the current system configuration but also its history of informational integration. In IIT and its mathematical generalizations, an IIS is not a mere message or code in the Shannon sense; rather, it is a maximally irreducible cause–effect structure that specifies the full set of intrinsic causal relationships within a complex in a given state (Zaeemzadeh et al., 14 Dec 2024, Kleiner et al., 2020, Mayner et al., 30 Dec 2024). The defining property is that the informational content of the IIS cannot be reduced to the union of information provided by separate subsystems or mechanisms—its integrated information, quantified by measures such as φ ("small phi") or Φ ("big Phi"), strictly exceeds the sum of its parts.

The formal structure of an IIS in modern IIT is encapsulated by the "Φ-structure" (also referred to as a Q-shape in the literature), which is a constellation of all maximally irreducible distinctions (mechanism–purview pairs, each defined in a particular state) and the binding relations among them. The mathematical grounding involves:

• Specification of a candidate system S and its state s.
• Construction of transition probability matrices governing the evolution of S and identification of all possible partitions (cuts).
• For each mechanism (subset of S), computation of its cause–effect repertoires (marginalized by appropriate partitions), and quantification of its irreducibility φ by measuring the minimal informational distance between the intact and partitioned causal structures.
• Aggregation of all such maximally irreducible distinctions (the Φ-structure), with their strengths and overlaps, which jointly account for both the "meaning" (qualia) and the intensity (Φ) of experience.

In dynamical systems and the MCT framework (Gillon, 2 Oct 2025), IISs are produced as discrete events or packets—integrated informational episodes—each tagged by a multidimensional density vector that quantifies their informational richness along axes such as salience, narrative coherence, emotional impact, and autobiographical anchoring.

2. Theoretical Underpinnings: Axioms and Postulates

The theoretical basis for IISs in IIT derives from a set of phenomenological axioms—intrinsicality, information, integration, exclusion, and composition—that are mapped to physical postulates about the substrate of consciousness (Zaeemzadeh et al., 14 Dec 2024, Mayner et al., 30 Dec 2024, Marshall et al., 2022). These require that:

• The relevant substrate (complex) must exhibit intrinsic cause–effect power: it "selects" a well-specified cause–effect repertoire in its current state.
• This structure must be maximally integrated, i.e., irreducible to any partition into independent components (integration).
• Only one candidate set (the complex) at an appropriate spatiotemporal grain will maximize integrated information (exclusion).
• The quality of experience is given by the specific structure (composition) formed by all distinctions (mechanism–purview pairs) and their mutual relations.
• The intrinsic meaning of an experience is equated to the irreducible cause–effect structure, which is unique to each IIS in its specific context.

In computational theories such as MCT, these principles are rendered as system design constraints: IISs are generated in a modular architecture where information is funneled through abstraction, narrative, evaluation, and self-evaluation modules, then integrated and densely tagged, forming the discrete units of subjective experience and behavioral adaptation (Gillon, 2 Oct 2025).

3. Quantification and Decomposition of Integrated Information

The quantification of IISs capitalizes on rigorous information-theoretic and mathematical frameworks:

• Classical IIT (3.0, 4.0): The integrated information φ for a subsystem is assessed by the minimal informational loss (as measured by metrics such as the Earth Mover’s Distance or intrinsic difference) incurred by partitioning the system at its "minimum information partition" (MIP) (Kleiner et al., 2020, Marshall et al., 2022). The Φ-structure is assembled by summing the irreducible φ-values for all distinctions and relations.
• Partial Information Decomposition and ΦID: Recent generalizations (e.g., ΦID) decompose information flow not only into integrated components but also into redundant, unique, and synergistic atoms. This yields a lattice of information dynamics, classifying the modes of integration (copy, transfer, storage, synergy, erasure, etc.) within the IIS (Mediano et al., 2019).
• Dynamical Systems: In statistical physics and network theory, integrated information is formulated as the Kullback–Leibler divergence between the full joint distribution of network states and the factorized distribution over its parts, often with explicit analytic expressions (e.g., for kinetic Ising models or Gaussian networks) (Arsiwalla et al., 2017, Aguilera et al., 2018, Citton et al., 2023). In such formulations, IISs typically emerge near dynamical critical points, where integration (as measured by Φ or φ) diverges or peaks (Mediano et al., 2016, Mediano et al., 2021).
• Quantum Extensions: Mechanism integrated information (ϕ) has been extended to quantum systems, utilizing density matrix formalism and a quantum intrinsic difference (QID) measure. In quantum IIT, conditional independence is replaced with maximal entanglement partitions; integrated information quantifies the irreducibility of a mechanism’s causal impact within composite or entangled systems (Albantakis et al., 2023, McQueen et al., 2023).

4. Functional and Computational Significance

IISs, as conceived in both IIT and computational models like MCT, have well-defined functional implications:

• Memory and Behavior: In MCT, the amplitude of the information-density vector tagging each IIS predicts the likelihood of robust encoding into long-term memory and the degree of behavioral influence. Conditions such as stress that heighten density produce IISs with greater lasting impact and adaptive priority (Gillon, 2 Oct 2025).
• Subjective Experience: The structure and magnitude of IISs encode the "richness" and "intensity" of conscious episodes. Variability in internal modular processing (abstraction, narrative, evaluation) contributes directly to the quality of each IIS. Systems failing to integrate or tag information (e.g., under certain pathologies) may have diminished or altered experiences.
• Perception and Meaning: IISs formalize perception as structured interpretation. External stimuli act as triggers, selecting, but not determining, which intrinsic meanings (distinctions in the Φ-structure) are activated. The matching between different environments and the system’s differentiation structure (the union of all IISs engaged by typical stimulus sequences) quantifies environmental meaningfulness (Mayner et al., 30 Dec 2024).
• Hierarchical and Modular Architectures: The explicit modular division and labeling of ISSs in MCT enables mapping to distinct neural substrates (e.g., narrative and integration modules) and supports the design of artificial agents with quantifiable consciousness-like states.

5. Comparative and General Theoretical Perspectives

IISs distinguish themselves from representations in other consciousness theories:

Theory	Unit of Analysis / Marker	Integration Mechanism	Subjectivity Encoding
Global Workspace Theory	Global activation ("ignition")	Broadcasting information to distributed modules	Not explicitly tagged within packets
IIT	Φ-structure (maximally irreducible structure)	Causal irreducibility, structural integration across all scales	Encoded intrinsically in structure
Modular CT (MCT)	IIS + information-density vector	Modular integration followed by explicit tagging	Density vector signals subjectivity
Higher-Order Thought	Higher-order representation	Meta-representation about lower-order states	Explicit reflective representation

A core distinction of IISs in MCT is the explicit multidimensional tagging of integrated packets—a mechanism not present in standard IIT or workspace models, which tend to use scalar measures (Φ) or global access, respectively. This enables fine-grained predictions regarding memory, emotion, and behavioral readiness based on the makeup of individual IISs (Gillon, 2 Oct 2025).

6. Implications for Physical and Biological Systems

IISs provide a principled basis for assessing not only the presence but also the scope and structure of meaningful, subjective processing in physical and biological substrates:

• Criticality and Scaling: High integrated information (Φ or related measures) and thus IISs, emerge in networks poised near critical points—an insight applicable to neural systems exhibiting critical dynamics, as well as to large-scale kinetic or modular Ising models (Arsiwalla et al., 2017, Aguilera et al., 2018, Citton et al., 2023).
• Agent–Environment Boundary: By tracking the divergence of integration scales across agent and agent–environment composites, one can delimit the boundary of an IIS and, by extension, of an autonomous individual in a coupled system (Aguilera et al., 2018).
• Matching and Adaptive Interpretation: In complex adaptive systems, the degree of match between the system's intrinsic repertoire of IISs and the structure of environmental inputs predicts both perceptual sophistication and evolutionary viability (Mayner et al., 30 Dec 2024).

7. Applications and Future Directions

The IIS formalism is extensible to multiple domains:

• Empirical and Clinical Neuroscience: Analyses of brain networks with IIT or MCT-inspired approaches may identify the loci and structure of conscious complexes, provide biomarkers for states of consciousness, and elucidate the neural correlates of memory prioritization or behavioral anomalies.
• Artificial Intelligence and Robotics: MCT provides a blueprint for constructing artificial agents capable of synthesizing IISs, equipping them with modulated subjective experience-like signatures and adaptable memory/action frameworks.
• Quantum and Hybrid Systems: Advanced models support the definition and quantification of IISs in quantum circuits, exploring the implications of superposed or entangled conscious-like states (Albantakis et al., 2023, McQueen et al., 2023).
• Communication of Meaning: The transfer of meaning between systems is not guaranteed by symbol transmission alone; effective communication of "meaning" requires structural alignment of the cause–effect (Φ) structures between sender and receiver (Zaeemzadeh et al., 14 Dec 2024).
• Testable Predictions: MCT specifically predicts that experiences of stress or high salience yield IISs with higher density, and thus enhanced memory encoding—a claim amenable to empirical validation. Disorders disrupting IIS formation or density tagging are hypothesized to underlie clinical conditions such as dissociative or affective disturbances.

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Integrated Informational States unify the structural, quantitative, and computational underpinnings of subjective experience across theoretical frameworks. They provide precise targets for both empirical paper and system design, forming the groundwork for a science of consciousness and adaptive information processing tightly coupled to formal theories of causation, integration, and meaning.