The Poincaré-Shannon Machine: Statistical Physics and Machine Learning Aspects of Information Cohomology

A hands-on tutorial on network and topological neuroscience

The brain is an extraordinarily complex system that facilitates the optimal integration of information from different regions to execute its functions. With the recent advances in technology, researchers can now collect enormous amounts of data from the brain using neuroimaging at different scales and from numerous modalities. With that comes the need for sophisticated tools for analysis. The field of network neuroscience has been trying to tackle these challenges, and graph theory has been one of its essential branches through the investigation of brain networks. Recently, topological data analysis has gained more attention as an alternative framework by providing a set of metrics that go beyond pairwise connections and offer improved robustness against noise. In this hands-on tutorial, our goal is to provide the computational tools to explore neuroimaging data using these frameworks and to facilitate their accessibility, data visualisation, and comprehension for newcomers to the field. We will start by giving a concise (and by no means complete) overview of the field to introduce the two frameworks and then explain how to compute both well-established and newer metrics on resting-state functional magnetic resonance imaging. We use an open-source language (Python) and provide an accompanying publicly available Jupyter Notebook that uses the 1000 Functional Connectomes Project dataset. Moreover, we would like to highlight one part of our notebook dedicated to the realistic visualisation of high order interactions in brain networks. This pipeline provides three-dimensional (3-D) plots of pairwise and higher-order interactions projected in a brain atlas, a new feature tailor-made for network neuroscience.

The Design of Global Correlation Quantifiers and Continuous Notions of Statistical Sufficiency

Using first principles from inference, we design a set of functionals for the purposes of ranking joint probability distributions with respect to their correlations. Starting with a general functional, we impose its desired behavior through the Principle of Constant Correlations (PCC), which constrains the correlation functional to behave in a consistent way under statistically independent inferential transformations. The PCC guides us in choosing the appropriate design criteria for constructing the desired functionals. Since the derivations depend on a choice of partitioning the variable space into n disjoint subspaces, the general functional we design is the n-partite information (NPI), of which the total correlation and mutual information are special cases. Thus, these functionals are found to be uniquely capable of determining whether a certain class of inferential transformations, ρ → ∗ ρ ' , preserve, destroy or create correlations. This provides conceptual clarity by ruling out other possible global correlation quantifiers. Finally, the derivation and results allow us to quantify non-binary notions of statistical sufficiency. Our results express what percentage of the correlations are preserved under a given inferential transformation or variable mapping.

Elements of qualitative cognition: An information topology perspective

Elementary quantitative and qualitative aspects of consciousness are investigated conjointly from the biology, neuroscience, physic and mathematic point of view, by the mean of a theory written with Bennequin that derives and extends information theory within algebraic topology. Information structures, that accounts for statistical dependencies within n-body interacting systems are interpreted a la Leibniz as a monadic-panpsychic framework where consciousness is information and physical, and arise from collective interactions. The electrodynamic intrinsic nature of consciousness, sustained by an analogical code, is illustrated by standard neuroscience and psychophysic results. It accounts for the diversity of the learning mechanisms, including adaptive and homeostatic processes on multiple scales, and details their expression within information theory. The axiomatization and logic of cognition are rooted in measure theory expressed within a topos intrinsic probabilistic constructive logic. Information topology provides a synthesis of the main models of consciousness (Neural Assemblies, Integrated Information, Global Neuronal Workspace, Free Energy Principle) within a formal Gestalt theory, an expression of information structures and patterns in correspondence with Galois cohomology and discrete symmetries. The methods provide new formalization of deep neural network with homologicaly imposed architecture applied to challenges in AI-machine learning.