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The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity...
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The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Entropy and Biology".

Deadline for manuscript submissions: 20 August 2025 | Viewed by 4362

Special Issue Editors

Dr. Giulio Ruffini

E-Mail Website
Guest Editor
Brain Modeling Department, Neuroelectrics, 08035 Barcelona, Spain
Interests: biophysics; computational neuroscience; algorithmic information theory; AI; neuroscience of consciousness; brain stimulation
Dr. Johannes Kleiner

E-Mail Website
Guest Editor
1. Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich, 80539 Munich, Germany
2. Graduate School of Systemic Neurosciences, Ludwig Maximilian University of Munich, 82152 Planegg-Martinsried, Germany
3. Institute for Psychology, University of Bamberg, 96047 Bamberg, Germany
Interests: mathematical consciousness science; mathematical physics; calculus of variations
Special Issues, Collections and Topics in MDPI journals
Dr. Ryota Kanai

E-Mail Website
Guest Editor
Araya Inc., Tokyo 107-6024, Japan
Interests: information; consciousness; neuroscience; AI; agency
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is motivated by the Kolmogorov theory of consciousness (KT) and related mathematical frameworks in cognition and consciousness, including active inference and predictive coding. It invites contributions that shed light on the intricate link between brain dynamics and the experiential phenomena they induce. Central to this discourse is the proposition that agents (computational entities) construct compressive models (algorithms) of the world to track world data and guide action planning through objective function evaluation. This perspective underscores the profound impact of model mathematical structure on the brain's dynamic trajectories, or the "dynamical landscape," and on the resulting qualia (structured experience). It opens exciting avenues for empirical investigation and methodological innovation, leveraging advanced concepts from dynamical systems theory, geometry, topology, algorithmic information theory, and critical phenomena theory. Success in this endeavor promises to significantly enrich fields like fundamental neuroscience, computational neuropsychiatry, and artificial intelligence, offering novel insights and approaches.

Titled “The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain”, this Special Issue aims to explore several key areas in this program:

  1. Characteristics of compressive world models: We aim to delve into the nature of the world models created and run by natural and artificial agents. What do we mean, precisely, by a world model? What is the connection between program structure and the resulting dynamics? What is the role of symmetry and criticality in shaping world models and programs, and how do they enable agents to encapsulate the world's complexity in a comprehensible form?
  2. Mapping models to dynamical systems: A crucial exploration will be how compressive world model characteristics translate into the workings of agents as dynamical systems and their features, especially recurrent neural networks. Topics include the study of geometry and topology of invariant manifolds, dimensionality reduction (manifold hypothesis), dynamical latent spaces, and their connection with algorithmic concepts such as compression and symmetry. We invite contributions that use tools from dynamical systems theory, geometry, topology, and criticality to characterize and understand the underlying dynamics of biological or artificial systems and their relation to the data they generate (neuroimaging/neurophysiology or other data).
  3. Empirical paradigms for validation: This Special Issue also seeks to address the design of experimental paradigms aimed at validating these concepts. Specifically, we are interested in establishing connections between features derived from structured experience reports or other behavior and the observed structure in the brain (or, more generally, complex systems) dynamics (e.g., as measured by neuroimaging techniques) using the tools mentioned in the previous points. Application areas include the study of states of consciousness and disorders of consciousness, as well as non-human consciousness, using currently available datasets or through the design of specific experiments.
  4. Implications for the design of artificial intelligence and computational models of the brain. How does this perspective influence research on artificial systems or computational models of the brain? What are the design principles inspired by the mathematics of algorithmic agenthood that can be used in artificial intelligence or computational neuroscience? 

Through these explorations, this Special Issue aims to stimulate a multidisciplinary dialogue that bridges abstract mathematical concepts in algorithmic information theory, dynamics, geometry, and topology with neural phenomena and, ultimately, first-person experience. Our objective is to enhance our understanding of how the brain encodes, processes, and manifests structured experience and how this understanding can inform new computational models and therapeutic approaches for neuroscience and clinical neuropsychiatric as well as artificial intelligence.

We welcome submissions of original research, reviews, and perspective pieces that contribute to this field. Emphasis should be on the theoretical underpinnings, methodological approaches, and potential implications of integrating mathematics with the study of the brain and the consciousness of complex systems.

References:

Dr. Giulio Ruffini
Dr. Johannes Kleiner
Dr. Ryota Kanai
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical theories of consciousness
  • Kolmogorov complexity
  • AIT
  • manifold hypothesis
  • symmetry
  • world models
  • dynamics
  • topology
  • AI

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Published Papers (3 papers)

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Research

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24 pages, 7866 KiB  
Article
The Epistemic Uncertainty Gradient in Spaces of Random Projections
by Jeffrey F. Queißer, Jun Tani and Jochen J. Steil
Entropy 2025, 27(2), 144; https://doi.org/10.3390/e27020144 - 1 Feb 2025
Viewed by 661
Abstract
This work presents a novel approach to handling epistemic uncertainty estimates with motivation from Bayesian linear regression. We propose treating the model-dependent variance in the predictive distribution—commonly associated with epistemic uncertainty—as a model for the underlying data distribution. Using high-dimensional random feature transformations, [...] Read more.
This work presents a novel approach to handling epistemic uncertainty estimates with motivation from Bayesian linear regression. We propose treating the model-dependent variance in the predictive distribution—commonly associated with epistemic uncertainty—as a model for the underlying data distribution. Using high-dimensional random feature transformations, this approach allows for a computationally efficient, parameter-free representation of arbitrary data distributions. This allows assessing whether a query point lies within the distribution, which can also provide insights into outlier detection and generalization tasks. Furthermore, given an initial input, minimizing the uncertainty using gradient descent offers a new method of querying data points that are close to the initial input and belong to the distribution resembling the training data, much like auto-completion in associative networks. We extend the proposed method to applications such as local Gaussian approximations, input–output regression, and even a mechanism for unlearning of data. This reinterpretation of uncertainty, alongside the geometric insights it provides, offers an innovative and novel framework for addressing classical machine learning challenges. Full article
(This article belongs to the Special Issue The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain)
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55 pages, 18955 KiB  
Article
Structured Dynamics in the Algorithmic Agent
by Giulio Ruffini, Francesca Castaldo and Jakub Vohryzek
Entropy 2025, 27(1), 90; https://doi.org/10.3390/e27010090 - 19 Jan 2025
Cited by 1 | Viewed by 920
Abstract
In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how [...] Read more.
In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether’s theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent’s constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain. Full article
(This article belongs to the Special Issue The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain)
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Review

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20 pages, 2127 KiB  
Review
Entropy and Complexity Tools Across Scales in Neuroscience: A Review
by Rodrigo Cofré and Alain Destexhe
Entropy 2025, 27(2), 115; https://doi.org/10.3390/e27020115 - 24 Jan 2025
Viewed by 1527
Abstract
Understanding the brain’s intricate dynamics across multiple scales—from cellular interactions to large-scale brain behavior—remains one of the most significant challenges in modern neuroscience. Two key concepts, entropy and complexity, have been increasingly employed by neuroscientists as powerful tools for characterizing the interplay between [...] Read more.
Understanding the brain’s intricate dynamics across multiple scales—from cellular interactions to large-scale brain behavior—remains one of the most significant challenges in modern neuroscience. Two key concepts, entropy and complexity, have been increasingly employed by neuroscientists as powerful tools for characterizing the interplay between structure and function in the brain across scales. The flexibility of these two concepts enables researchers to explore quantitatively how the brain processes information, adapts to changing environments, and maintains a delicate balance between order and disorder. This review illustrates the main tools and ideas to study neural phenomena using these concepts. This review does not delve into the specific methods or analyses of each study. Instead, it aims to offer a broad overview of how these tools are applied within the neuroscientific community and how they are transforming our understanding of the brain. We focus on their applications across scales, discuss the strengths and limitations of different metrics, and examine their practical applications and theoretical significance. Full article
(This article belongs to the Special Issue The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain)
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