We present a theory on the neural basis of aesthetic experience, and judgment of beauty. It is based on both empirical facts concerning brain mechanisms and theoretical mathematical theories. We first recall previous evidence that the brain uses several non-Euclidian geometries for perception and action at different scales of space (personal, peri-personal, near locomotor, environmental, imaginative). This is supported by neuroscience data (brain imaging, neuropsychology, movement control, etc.). For example, the movement of drawing obeys specific kinematic rules, that reflect the control by Euclidian and affine geometries. We already formulated the corresponding geometries in brain's networks by using Topos and Stacks theory of the mathematician Alexander Grothendieck. The present article extends the previous proposals by suggesting that a meta-geometry provides the binding between these specialized geometries, by using known higher structures and dynamics (like n-Topos and n-Stacks) for joint perceptions and movements, and other modalities, as concepts, memories or emotions, at different spatial scales domains. We suggest that a form, an object, a movement, an environment, an event, an idea, is perceived as beautiful if the data provided by the senses and programs are embedded in these higher geometries, providing a sort of dynamic recognition, through relations of generalized proportions.

Accurate protein synthesis requires ribosomes to integrate signals from distant functional sites and execute complex dynamics. Despite advances in understanding ribosome structure and function, two key questions remain: how information is transmitted between these distant sites, and how ribosomal movements are synchronized? We recently highlighted the existence of ribosomal protein networks, likely evolved to participate in ribosome signaling. Here, we investigate the relationship between ribosomal protein networks and ribosome dynamics. Our findings show that major motion centers in the bacterial ribosome interact specifically with r-proteins, and that ribosomal RNA exhibits high mobility around each r-protein. This suggests that periodic electrostatic changes in the context of negatively charged residues (Glu and Asp) induce RNA-protein 'distance-approach' cycles, controlling key ribosomal movements during translocation. These charged residues play a critical role in modulating electrostatic repulsion between RNA and proteins, thus coordinating ribosomal dynamics. We propose that r-protein networks synchronize ribosomal dynamics through an 'electrostatic domino' effect, extending the concept of allostery to the regulation of movements within supramolecular assemblies.

The perceptual and motor systems appear to have a set of movement primitives that exhibit certain geometric and kinematic invariances. Complex patterns and mental representations can be produced by (re)combining some simple motor elements in various ways using basic operations, transformations, and respecting a set of laws referred to as kinematic laws of motion. For example, point-to-point hand movements are characterized by straight hand paths with single-peaked-bell-shaped velocity profiles, whereas hand speed profiles for curved trajectories are often irregular and more variable, with speed valleys and inflections extrema occurring at the peak curvature. Curvature and speed are generically related by the 2/3 power law. Mathematically, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partially detected in various brain areas including the cerebellum and the basal ganglia. The cerebellum has been found to play an important role in the control of coordination, balance, posture, and timing over the past years. It is also assumed that the cerebellum computes forward internal models in relationship with specific cortical and subcortical brain regions but its motor relationship with the perceptual space is unclear. A renewed interest in the geometrical and spatial role of the cerebellum may enable a better understanding of its specific contribution to the action-perception loop and behavior’s adaptation. In this sense, we complete this overview with an innovative theoretical framework that describes a possible implementation and selection by the cerebellum of geometries adhering to different mathematical laws.

We argue that the projective geometrical component of the Projective Consciousness Model (PCM) can account for key aspects of pre-reflective self-consciousness (PRSC) and can relate PRSC intelligibly to another signal feature of subjectivity: perspectival character or point of view. We illustrate how the projective geometrical versions of the concepts of duality, reciprocity, polarity, closedness, closure, and unboundedness answer to salient aspects of the phenomenology of PRSC. We thus show that the same mathematics that accounts for the statics and dynamics of perspectival character also accounts for PRSC. More generally, we argue that introducing higher-level geometrical concepts into the theory of PRSC, and into the theory of consciousness broadly, as the PCM does, promises to break longstanding theoretical impasses and dialectical stalemates.

Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy of P.Baudot and D.Bennequin in 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators.

Using the functor of points, we prove that the Wess-Zumino equations for massive chiral superfields in dimension 4|4 can be represented by supersymmetric equations in terms of superfunctions in the Berezin-Kostant-Leites sense (involving ordinary fields, with real and complex valued components). Then, after introducing an appropriate supersymmetric extension of the Fourier transform, we prove explicitly that these supersymmetric equations provide a realization of the irreducible unitary representations with positive mass and zero superspin of the super Poincaré group in dimension 4|4.

In this study we explore the spontaneous apparition of visible intelligible reasoning in simple artificial networks, and we connect this experimental observation with a notion of semantic information. We start with the reproduction of a DNN model of natural neurons in monkeys, studied by Neromyliotis and Moschovakis in 2017 and 2018, to explain how "motor equivalent neurons", coding only for the action of pointing, are supplemented by other neurons for specifying the actor of the action, the eye E, the hand H, or the eye and the hand together EH. There appear inner neurons performing a logical work, making intermediary proposition, for instance E V EH. Then, we remarked that adding a second hidden layer and choosing a symmetric metric for learning, the activities of the neurons become almost quantized and more informative. Using the work of Carnap and Bar-Hillel 1952, we define a measure of the logical value for collections of such cells. The logical score growths with the depth of the layer, i.e. the information on the output decision increases, which confirms a kind of bottleneck principle. Then we study a bit more complex tasks, a priori involving predicate logic. We compare the logic and the measured weights. This shows, for groups of neurons, a neat correlation between the logical score and the size of the weights. It exhibits a form of sparsity between the layers. The most spectacular result concerns the triples which can conclude for all conditions: when applying their weight matrices to their logical matrix, we recover the classification. This shows that weights precisely perform the proofs.

In this text we present a collection of various inventions in Geometry, from Euclid to Grothendieck and Thom, passing by Kepler, Euler, Riemann and Poincaré, and we indicate for all of them a deep link with the organization and functions of the brains of animals, especially humans. In particular, we present works conducted with Tamar Flash and Alain Berthoz, and many other collaborators, using a variety of geometries, all included in the affine geometry, which are combined to prepare, plan and control the execution of human movements, for drawing or for walking. We also present a recent model, elaborated with David Rudrauf, Kenneth Williford, and other collaborators, showing that the three-dimensional projective geometry underlines a space for consciousness. Additionally, we introduce a new kind of geometry developed with Alain Berthoz, and many collaborators, which is able to organize different action spaces (at least five); this geometry corresponds to a 2-category of fibered categories of geometrical spaces over several networks in the brain, seen as different sites for different topos. The unifying theme of these works is adaptation.

To perform an accurate protein synthesis, ribosomes accomplish complex tasks involving the long-range communication between its functional centres such as the peptidyl transfer centre, the tRNA bindings sites and the peptide exit tunnel. How information is transmitted between these sites remains one of the major challenges in current ribosome research. Many experimental studies have revealed that some r-proteins play essential roles in remote communication and the possible involvement of r-protein networks in these processes have been recently proposed. Our phylogenetic, structural and mathematical study reveals that of the three kingdom’s r-protein networks converged towards non-random graphs where r-proteins collectively coevolved to optimize interconnection between functional centres. The massive acquisition of conserved aromatic residues at the interfaces and along the extensions of the newly connected eukaryotic r-proteins also highlights that a strong selective pressure acts on their sequences probably for the formation of new allosteric pathways in the network.

We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies. These transformations generate the modular group, and this fact plays a crucial role in solving the system. The method suggests a more general relation between conditional probabilities and arithmetic.

The Moon often appears larger near the perceptual horizon and smaller high in the sky though the visual angle subtended is invariant. We show how this illusion results from the optimization of a projective geometrical frame for conscious perception through free energy minimization, as articulated in the Projective Consciousness Model. The model accounts for all documented modulations of the illusion without anomalies (e.g., the size-distance paradox), surpasses other theories in explanatory power, makes sense of inter- and intra-subjective variability vis-a-vis the illusion, and yields new quantitative and qualitative predictions.

We introduce an original notion of extra-fine sheaf on a topological space, for which Cech cohomology in strictly positive dimension vanishes. We provide a characterization of such sheaves when the topological space is a partially ordered set (poset) equipped with the Alexandrov topology. Then we further specialize our results to some sheaves of vector spaces and injective maps, where extra-fineness is (essentially) equivalent to the decomposition of the sheaf into a direct sum of subfunctors, known as interaction decomposition, and can be expressed by a sum-intersection condition. We use these results to compute the dimension of the space of global sections when the presheaves are freely generated over a functor of sets, generalizing classical counting formulae for the number of solutions of the linearized marginal problem (Kellerer and Matus). We finish with a comparison theorem between the Cech cohomology associated to a covering and the topos cohomology of the poset with coefficients in the presheaf, which is also the cohomology of a cosimplicial local system over the nerve of the poset. For that, we give a detailed treatment of cosimplicial local systems on simplicial sets. The appendixes present presheaves, sheaves and Cech cohomology, and their application to the marginal problem.

During evolution, living systems, actively interacting with their environment, developed the ability, through sensorimotor contingencies, to construct functional spaces shaping their perception and their movements. These geometries were modularly embedded in specific functional neuro-architectures. In particular, human movements were shown to obey several empirical laws, such as the 2/3 power law, isochrony, or jerk minimization principles, which constrain and adapt motor planning and execution. Outstandingly, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partly detected in several brain areas including the cerebellum and the basal ganglia. Reviving Pellionisz and Llinas general hypothesis regarding the cerebrum and the cerebellum as geometric machines, we speculate that the cerebellum should be involved in implementing and/or selecting task-specific geometries for motor and cognitive skills. More precisely, the cerebellum is assumed to compute forward internal models to help specific cortical and subcortical regions to select appropriate geometries among, at least, Euclidean and affine geometries. We emphasize that the geometrical role of the cerebellum deserves a renewal of interest, which may provide a better understanding of its specific contributions to motor and associative (cognitive) functions.

This paper presents methods that quantify the structure of statistical interactions within a given data set, and were applied in a previous article. It establishes new results on the k-multivariate mutual-information ( I k ) inspired by the topological formulation of Information introduced in a serie of studies. In particular, we show that the vanishing of all I k for 2 ≤ k ≤ n of n random variables is equivalent to their statistical independence. Pursuing the work of Hu Kuo Ting and Te Sun Han, we show that information functions provide co-ordinates for binary variables, and that they are analytically independent from the probability simplex for any set of finite variables. The maximal positive I k identifies the variables that co-vary the most in the population, whereas the minimal negative I k identifies synergistic clusters and the variables that differentiate–segregate the most in the population. Finite data size effects and estimation biases severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and the k-dependences. We give an example of application of these methods to genetic expression and unsupervised cell-type classification. The methods unravel biologically relevant subtypes, with a sample size of 41 genes and with few errors. It establishes generic basic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism. We propose that higher-order statistical interactions and non-identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity. The topological information data analysis presented here allows for precisely estimating this higher-order structure characteristic of biological systems.

In the past few decades, studies on translation have converged towards the metaphor of a “ribosome nanomachine”; they also revealed intriguing ribosome properties challenging this view. Many studies have shown that to perform an accurate protein synthesis in a fluctuating cellular environment, ribosomes sense, transfer information and even make decisions. This complex “behaviour” that goes far beyond the skills of a simple mechanical machine has suggested that the ribosomal protein networks could play a role equivalent to nervous circuits at a molecular scale to enable information transfer and processing during translation. We analyse here the significance of this analogy and establish a preliminary link between two fields: ribosome structure-function studies and the analysis of information processing systems. This cross-disciplinary analysis opens new perspectives about the mechanisms of information transfer and processing in ribosomes and may provide new conceptual frameworks for the understanding of the behaviours of unicellular organisms.

Human movements are characterized by their invariant spatiotemporal features. The kinematic features and internal movement timing were accounted for by the mixture of geometries model using a combination of Euclidean, affine and equi-affine geometries. Each geometry defines a unique parametrization along a given curve and the net tangential velocity arises from a weighted summation of the logarithms of the geometric velocities. The model was also extended to deal with geometrical singularities forcing unique constraints on the allowed geometric mixture. Human movements were shown to optimize different costs. Specifically, hand trajectories were found to maximize motion smoothness by minimizing jerk. The minimum jerk model successfully accounted for a range of human end-effector motions including unconstrained and path-constrained trajectories. The two modeling approaches involving motion optimality and the geometries’ mixture model are here further combined to form a joint model whereby specific compositions of geometries can be selected to generate an optimal behavior. The optimization serves to define the timing along a path. Additionally, new notions regarding the nature of movement primitives used for the construction of complex movements naturally arise from the consideration of the two modelling approaches. In particular, we suggest that motion primitives may consist of affine orbits; trajectories arising from the group of full-affine transformations. Affine orbits define the movement’s shape. Particular mixtures of geometries achieve the smoothest possible motions, defining timing along each orbit. Finally, affine orbits can be extracted from measured human paths, enabling movement segmentation and an affine-invariant representation of hand trajectories.

We summarize our recently introduced Projective Consciousness Model (PCM) (Rudrauf et al. 2017) and relate it to outstanding conceptual issues in the theory of consciousness. The PCM combines a projective geometrical model of the perspectival phenomenological structure of the field of consciousness with a variational Free Energy minimization model of active inference, yielding an account of the cybernetic function of consciousness, viz., the modulation of the field's cognitive and affective dynamics for the effective control of embodied agents. The geometrical and active inference components are linked via the concept of projective transformation, which is crucial to understanding how conscious organisms integrate perception, emotion, memory, reasoning, and perspectival imagination in order to control behavior, enhance resilience, and optimize preference satisfaction. The PCM makes substantive empirical predictions and fits well into a (neuro)computationalist framework. It also helps us to account for aspects of subjective character that are sometimes ignored or conflated: pre-reflective self-consciousness, the first-person point of view, the sense of minenness or ownership, and social self-consciousness. We argue that the PCM, though still in development, offers us the most complete theory to date of what Thomas Metzinger has called "phenomenal selfhood".

We introduce a mathematical model of embodied consciousness, the Projective Consciousness Model (PCM), which is based on the hypothesis that the spatial field of consciousness (FoC) is structured by a projective geometry and under the control of a process of active inference. The FoC in the PCM combines multisensory evidence with prior beliefs in memory and frames them by selecting points of view and perspectives according to preferences. The choice of projective frames governs how expectations are transformed by consciousness. Violations of expectation are encoded as free energy. Free energy minimization drives perspective taking, and controls the switch between perception, imagination and action. In the PCM, consciousness functions as an algorithm for the maximization of resilience, using projective perspective taking and imagination in order to escape local minima of free energy. The PCM can account for a variety of psychological phenomena: the characteristic spatial phenomenology of subjective experience, the distinctions and integral relationships between perception, imagination and action, the role of affective processes in intentionality, but also perceptual phenomena such as the dynamics of bistable figures and body swap illusions in virtual reality. It relates phenomenology to function, showing the computational advantages of consciousness. It suggests that changes of brain states from unconscious to conscious reflect the action of projective transformations and suggests specific neurophenomenological hypotheses about the brain, guidelines for designing artificial systems, and formal principles for psychology.

In previous works we reanalyzed the kinematics of hand movements and locomotion, and suggested that several geometries are used conjointly by the brain for according the shape and the duration along trajectories; this was done in collaboration with Tamar Flash and her collaborators [10, 64, 67], and with Quang-Cuong Pham [79]. The variety of geometries which were implied in this process, were associated to sub-groups of the affine group of a plane: full affine, equi-affine and Euclidean. Other studies have shown how the above geometries constrain the production of the movements [92], or began to use the affine geometry in Robotics [80]. In this article, we propose to use a new variety of geometries which extends the preceding series in another direction, to cover wider contexts and more complex movements, like prehension, initiation of walking, locomotion, navigation, imagined motion. The new spaces adapted to those geometries have no points; they come from topos theory, which is an extension of set theory replacing sets by fields and graphs of dynamics. Any given topos generates a variety of different geometries, which can be mixed as in the preceding studies. Such geometries take into account efforts, forces and dynamics; they do not neglect them aside as does traditional geometry. In this preliminary report we indicate the simplest characteristics of spaces which underly the above examples. The hypothesis is also that these spaces are implemented in different, although overlapping, central nervous system networks in the brain, corresponding to the different action spaces mentioned above. Here, as for the known classical geometries, the most concrete suggestion concerns the timing of movement: we predict that different components of the controlled system are using different intrinsic time courses, and that the mapping between these different internal durations is an important part of the dynamic under geometrical control. This reminds us of a well known psychological observation, for instance that time in imagination does not flow as ordinary clocks time, but this also suggests that reaching an object with the hand has its own time, or that equilibrium control in walking works within a specific time, which is different from the walking trajectory displacement time.