arXiv subjects

Giovanna Citti

Publications and source records attributed to Giovanna Citti.

At least 19 recordsLinked to original sources

arXiv2026

A sub-Riemannian model of neural states in the primary motor cortex

We develop a neurogeometric model for the arm area of motor cortex, which encodes complex motor primitives, ranging from simple movement features like movement direction, to short hand trajectories, termed fragments, and ultimately to more complex patterns known as neural states (Georgopoulos, Hatsopoulos, Kadmon-Harpaz et al). Based on the sub-riemannian framework introduced in 2023, we model the space of fragments as a set of short curves defined by kinematic parameters. We then introduce a geometric kernel that serves as a model for cortical connectivity and use it in a differential equation to describe cortical activity. By applying a grouping algorithm to this cortical activity model, we successfully recover the neural states observed in Kadmon-Harpaz et al, which were based on measured cortical activity. This confirms that the choice of kinematic variables and the distance metric used here are sufficient to explain the phenomena of neural state formation. The modularity of our model reflects the brain's hierarchical structure, where initial groupings in the kinematic space $\mathcal{M}$ lead to more abstract representations. This approach mimics how the brain processes stimuli at different scales, extracting both local and global properties.

q-bio.NC↗

arXiv2026

A sub-Riemannian model of the motor cortex with Wasserstein distance

This study aims to better understand the functional geometry of the motor cortex, starting from different sources of experimental evidence. Recent studies have proved that cells of the primary motor cortex (M1) are sensitive to short hand trajectories called fragments. Here, we propose a sub-Riemannian higher-dimensional geometry accounting for geometric and kinematic properties. Due to the constraints of the geometry, horizontal curves naturally satisfy a relation between geometric and kinematic properties experimentally observed. In the space of trajectories, we also apply a clustering algorithm based on the Wasserstein distance: we obtain a grouping which nicely fits the observed experimental data much more efficiently than the Sobolev distance.

q-bio.NC↗

arXiv2026

Curvature Sensitive Cells in the Modular Structures of The Visual Cortex

We propose a model of the functional architecture of curvature-sensitive cells in the primary visual cortex. The model accounts for the modular and hierarchical organization of the cortex, the horizontal connectivity, and the shape of receptive profiles of these cells as Gabor-type filters. We construct a canonical affine subbundle of the cotangent bundle of the manifold of oriented contact elements of the retina as a geometric model for these cells, and show that this subbundle carries an Engel structure related to that of the Cartan prolongation. On an open submanifold of the Cartan prolongation, we identify generators of the Engel distribution whose iterated Lie brackets span the Lie algebra of SIM(2). The identification of sim(2) as the Lie algebra of these generators determines SIM(2) as the natural symmetry group for curvature-sensitive cells. Finally, we characterize the receptive profiles of curvature-sensitive cells as minima of a SIM(2)-adapted uncertainty principle applied to the generators of the Engel structure.

q-bio.NC↗

arXiv2026

Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group

We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the Hölder regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.

math.AP↗

arXiv2025

Transport densities and congested optimal transport problem in the Heisenberg group

We adapt the problem of continuous congested optimal transport to the Heisenberg group, equipped with a sub-Riemannian metric. Originally introduced in the Euclidean setting by Carlier, Jimenez, and Santambrogio as a path-dependent variant of the Monge-Kantorovich problem, we significantly restrict the set of admissible curves to horizontal ones. We establish the existence of equilibrium configurations as solutions to a convex minimization problem over a suitable set of measures on horizontal curves. This result is achieved through the notions of horizontal transport density and horizontal traffic intensity.

math.OC↗

arXiv2025

Benamou-Brenier and Kantorovich on sub-Riemannian manifolds with no abnormal geodesics

We prove that the Benamou-Brenier formulation of the Optimal Transport problem and the Kantorovich formulation are equivalent on a sub-Riemannian connected and complete manifold $M$ without boundary and with no non-trivial abnormal geodesics, when the problems are considered between two measures with finite $2$-momentum. Furthermore, we prove the existence of a minimizer for the Benamou-Brenier formulation and link it to the optimal transport plan.

math.OC↗

arXiv2024

Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multiscale approach. The nonlocal equation, which is very similar to the Ermentrout-Cowan equation used in neurobiology, can be derived from an interacting particle model. As sub-Riemannian geometries play an important role in the models of the visual cortex proposed by Petitot and Citti-Sarti, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field equation to curvature flows which are used in image processing. From a pure mathematical point of view, it provides a new approximation and regularization of Heisenberg mean curvature flow. Using the local structure of the rototranslational group, we extend the result to cover the model by Citti and Sarti. Numerically, the parameters in our algorithm interpolate between solving an Ementrout-Cowan type of equation and a Bence-Merriman-Osher algorithm type algorithm for sub-Riemannian mean curvature. We also reproduce some known exact solutions in the Heisenberg case.

math.AP↗

arXiv2024

A representation formula for regular functions on the characteristic plane of the second Heisenberg group

The aim of this paper is to study a Laplace-type operator and its fundamental solution on the characteristic plane in the Heisenberg group $\mathbb{H}^2$. We introduce a conformal version of the Laplacian and we prove that the distance induced by the immersion in the ambient space is a good approximation of its fundamental solution. We provide in particular a representation formula for smooth functions in terms of the gradient of the function and the gradient of the approximated fundamental solution. This representation formula in the plane is stable up to its characteristic point.

math.AP↗

arXiv2024

Individuation of 3D perceptual units from neurogeometry of binocular cells

We model the functional architecture of the early stages of three-dimensional vision by extending the neurogeometric sub-Riemannian model for stereo-vision introduced in \cite{BCSZ23}. A new framework for correspondence is introduced that integrates a neural-based algorithm to achieve stereo correspondence locally while, simultaneously, organizing the corresponding points into global perceptual units. The result is an effective scene segmentation. We achieve this using harmonic analysis on the sub-Riemannian structure and show, in a comparison against Riemannian distance, that the sub-Riemannian metric is central to the solution.

q-bio.NC↗

arXiv2023

Cortical Functional architectures as contact and sub-Riemannian geometry

In a joint paper, Jean Petitot together with the authors of the present paper described the functional geometry of the visual cortex as the symplectization of a contact form to describe the family of cells sensitive to position, orientation and scale. In the present paper, as a "homage" to the enormous contribution of Jean Petitot to neurogeometry, we will extend this approach to more complex functional architectures built as a sequence of contactization or a symplectization process, able to extend the dimension of the space. We will also outline a few examples where a sub-Riemannian lifting is needed.

q-bio.NC↗

arXiv2023

The cortical V1 transform as a heterogeneous Poisson problem

Receptive profiles of V1 cortical cells are very heterogeneous and act by differentiating the stimulus image as operators changing from point to point. A lightness and color constancy image can be reconstructed as the solution of the associated inverse problem, that is a Poisson equation with heterogeneous differential operators. At the neural level the weights of short range connectivity constitute the fundamental solution of the Poisson problem adapted point by point. A first demonstration of convergence of the result towards homogeneous reconstructions is proposed by means of homogenisation techniques.

math.AP↗

arXiv2023

A sub-Riemannian model of the functional architecture of M1 for arm movement direction

In this paper we propose a neurogeometrical model of the behaviour of cells of the arm area of the primary motor cortex (M1). We mathematically express the hypercolumnar organization of M1 discovered by Georgopoulos, as a fiber bundle, as in classical sub-riemannian models of the visual cortex (Hoffmann, Petitot, Citti-Sarti). On this structure, we consider the selective tuning of M1 neurons of kinematic variables of positions and directions of movement. We then extend this model to encode the notion of fragments of movements introduced by Hatsopoulos. In our approach fragments are modelled as integral curves of vector fields in a suitable sub-Riemannian space. These fragments are in good agreements with movement decomposition from neural activity data. Here, we recover these patterns through a spectral clustering algorithm in the subriemannian structure we introduced, and compare our results with the neurophysiological ones of Kadmon-Harpaz et al.

q-bio.NC↗

arXiv2023

Internal Schauder estimates for Hörmander type equations with Dini continuous source

We study the regularity properties of a general second order Hörmander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the Hörmander condition, we consider the linear operator in $\mathbb{R}^{N}$, with $N>m+1$: \begin{equation*} \mathcal{L} u := \sum_{i, j= 1}^{m} a_{ij} X_{i}X_{j} u - X_0 u. \end{equation*} The vector field $X_0$ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if $f$ is a Dini continuous function, then the second order derivatives of the solution $u$ to the equation $\mathcal{L} u = f$ are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.

math.AP↗

arXiv2023

Regularity theory of quasilinear elliptic and parabolic equations in the Heisenberg group

This note provides a succinct survey of the existing literature concerning the Hölder regularity for the gradient of weak solutions of PDEs of the form $$\sum_{i=1}^{2n} X_i A_i(\nabla_0 u)=0 \text{ and } \partial_t u= \sum_{i=1}^{2n} X_i A_i(\nabla_0 u)$$ modeled on the $p$-Laplacian in a domain $Ω$ in the Heisenberg group $\mathbb H^n$, with $1\le p <\infty$, and of its parabolic counterpart. We present some open problems and outline some of the difficulties they present.

math.AP↗

arXiv2023

Functional architecture of M1 cells encoding movement direction

In this paper we propose a neurogeometrical model of the behaviour of cells of the arm area of the primary motor cortex (M1). We will mathematically express as a fiber bundle the hypercolumnar organization of this cortical area, first modelled by Georgopoulos in \cite{georgopoulos1982relations, georgopoulos2015columnar}. On this structure, we will consider the selective tuning of M1 neurons of kinematic variables of positions and directions of movement. We will then extend this model to encode the notion of fragments introduced by Hatsopoulos \cite{Encoding} which describes the selectivity of neurons to movement direction varying in time. This leads to consider a higher dimensional geometrical structure where fragments are represented as integral curves. A comparison with the curves obtained through numerical simulations and experimental data will be presented. Moreover, neural activity shows coherent behaviours represented in terms of movement trajectories pointing to a specific pattern of movement decomposition \cite{kadmon2019movement}. Here, we will recover this pattern through a spectral clustering algorithm in the subriemannian structure we introduced, and compare our results with the neurophysiological one of \cite{kadmon2019movement}.

q-bio.NC↗

arXiv2023

Schauder estimates up to the boundary on H-type groups: an approach via the double layer potential

We establish the Schauder estimates at the boundary away from the characteristic points for the Dirichlet problem by means of the double layer potential in a Heisenberg-type group $\mathbb{G}$. Despite its singularity we manage to invert the double layer potential restricted to the boundary thanks to a reflection technique for an approximate operator in $\mathbb{G}$. This is the first instance where a reflection-type argument appears to be useful in the sub-Riemannian setting.

math.AP↗

arXiv2023

A model of reaching via subriemannian geodesics in Engel-type group

In this paper, we propose a model of arm reaching movements expressed in terms of geodesics in a sub-Riemannian space. We will choose a set of kinematic variables to which motor cortical cells are selective with the purpose of modelling the specific task of reaching. Minimizing trajectories will be recovered as suitable geodesics of the geometric spaces arising from the selective behaviour of M1 neurons. We will then extend this model by including the direction of arm movement. On this set, we will define a suitable sub-Riemannian metric able to provide a geometric interpretation of two-dimensional task-dependent arm reaching movements.

math.OC↗

arXiv2023

Good continuation in 3D: the neurogeometry of stereo vision

Classical good continuation for image curves is based on $2D$ position and orientation. It is supported by the columnar organization of cortex, by psychophysical experiments, and by rich models of (differential) geometry. Here we extend good continuation to stereo. We introduce a neurogeometric model, in which the parametrizations involve both spatial and orientation disparities. Our model provides insight into the neurobiology, suggesting an implicit organization for neural interactions and a well-defined $3D$ association field. Our model sheds light on the computations underlying the correspondence problem, and illustrates how good continuation in the world generalizes good continuation in the plane.

q-bio.NC↗