Computational studies of consciousness

Paris Mathematical Models of Consciousness

For a study group centered around the topic of Mathematical Models of Consciousness, named the "Paris Mathematical Models of Consciousness." see here PMMC.

Computational Phenomenology: Context and State-of-the-Art

The pursuit of a comprehensive theory of consciousness is a central focus in cognitive science, holding the potential to address essential questions in psychology, neuroscience, artificial intelligence, and robotics. Numerous theories have emerged, but no consensus exists on the most promising contender. While much research has centered on identifying the neural correlates of consciousness, it is essential not to limit the study of consciousness to the study of its neural underpinning. Investigating its underlying principles and mechanisms independently of its neural implementation can provide a foundation for more quantitative and model-based research, potentially overcome methodological challenges associated with neural models and neuroscience research more generally, and offer a valuable starting point for a mathematical theory of consciousness.

Such investigations can begin from properties typically ascribed to consciousness on the basis of introspection: it integrates diverse cognitive and sensory functions and processes to enable coherent cognition, facilitate learning, and generate resilient behaviors, essentially playing a cybernetic role for adaptive systems.

Two prominent models currently dominate consciousness studies. Integrated Information Theory (IIT) defines consciousness as the dynamic interactions between a system's parts that maximize an information quantity (measured as "Phi)" that cannot be reduced to the information contained in such parts. IIT faces important challenges regarding construct validity, the specificity of its predictions, and its operationalization for empirical research. The Global Workspace Theory (GWT) defines consciousness as an integrative workspace with limited capacity for decision-making, information processing, and simulation. GWT lacks mathematical models that comprehensively capture the functions and interactions it ascribes to consciousness, hindering its operationalization for simulations and empirical research.

The Projective Consciousness Model (PCM)

In a way that is more deliberate and systematic than conventional approaches, we propose to first focus on modeling the phenomenological aspects of consciousness. Consciousness involves a subjective perspective, characterized by viewpoint-structured organization, synchronic and diachronic unities, embodiment, and an internal representation of the world in perspective from a specific standpoint. To enable decision-making based on noisy sensory inputs, agents need to construct an internal state-space comprising extracted features. Our innovative concept involves imbuing this space with a geometric structure that facilitates the integration of features into a coherent whole by encoding various perspectives the agent can adopt directly into the state space. Each perspective corresponds to a 'frame’ used by the agent to orient itself in its environment. This idea can be encapsulated mathematically using the concept of a G-space, in which a group G acts continuously on a topological space X. In this framework, the agent's state-space encompasses all potential frames it can adopt, with each frame change being associated with an element of the group.

Such a G-space serves as a global workspace within which features are organized; perspectives can be those of other potential agents or those induced by the agent's own actions, such as its movements.

Various models of agency exist in the literature, one of the most common being stochastic optimal control, including Markov Decision Processes and Partially Observable Markov Decision Processes. Active inference is another notable example. All these previous formalisms rely on an internal state-space. We propose to adapt such formalisms to cases in which the state space is a G-space. This adaptation opens new theoretical and algorithmic possibilities, which we have been actively researching in recent years. For a comprehensive overview of the framework and our results, please refer to Section 2 of 'The Projective Consciousness Model: projective geometry at the core of consciousness and the integration of perception, imagination, motivation, emotion, social cognition, and action' ( link ).

Specifically, our research yields experimental and numerical findings related to perceptual illusions, as well as insights into areas such as social cognition, adaptive and maladaptive behaviors (including imagination, emotion regulation, drives, empathy, and emotional expression).

Selected papers

  1. The Projective Consciousness Model: projective geometry at the core of consciousness and the integration of perception, imagination, motivation, emotion, social cognition and action.
    D. Rudrauf, G. Sergeant-Perthuis, Y. Tisserand, G. Poloudenny, K. Williford and M-A. Amorim, In Press, 2023.

  2. Action of the Euclidean versus Projective group on an agent's internal space in curiosity-driven exploration: a formal analysis.
    G. Sergeant-Perthuis, D. Rudrauf, D. Ognibene, and Y. Tisserand, Preprint, arXiv:2304.00188, 2023

  3. Pre-Reflective Self-Consciousness & Projective Geometry.
    Kenneth Williford, Daniel Bennequin and David Rudrauf, Review of Philosophy and Psychology, 2022

  4. The integrated information theory of consciousness: a case of mistaken identity.
    Bjorn Merker, Kenneth Williford, and David Rudrauf, Behavioral and Brain Sciences, 2022.

  5. Modeling the subjective perspective of consciousness and its role in the control of behaviors.
    D. Rudrauf, G. Sergeant-Perthuis, O. Belli, Y. Tisserand, G. Di Marzo Serugendo, Journal of Theoretical Biology, 2022.

  6. The moon illusion explained by the projective consciousness model.
    David Rudrauf, Daniel Bennequin, and Kenneth Williford, Journal of Theoretical Biology, 2020.

  7. The Projective Consciousness Model and Phenomenal Selfhood.
    Kenneth Williford, Daniel Bennequin, Karl Friston, David Rudrauf, 2018.

  8. A mathematical model of embodied consciousness.
    David Rudrauf, Daniel Bennequin, Isabela Granic, Gregory Landini, Karl Friston, and Kenneth Williford, Journal of Theoretical Biology, 2017.


  1. Agency with structured latent state-spaces: a model for computational phenomenology.
    ActInf GuestStream (talk), Active Inference Institue, 16 September, 2023, Gregoire Sergeant-Perthuis Video
  2. The Projective Consciousness Model: Formal developments from projective geometry to epistemic drives.
    MoC4, Oxford (talk), 6 September, 2023, Kenneth Williford, Gregoire Sergeant-Perthuis, David Rudrauf, Video
  3. The Projective Consciousness Model: Phenomenological Prolegomena.
    MoC4, Oxford (talk), 6 September, 2023, Kenneth Williford, Gregoire Sergeant-Perthuis, David Rudrauf Video