Neurogeometry of reaching

The purpose of the thesis is to develop a model for the functional behaviour of neurons in the primary motor cortex (M1) responsible for arm reaching movements. From Georgopoulos neurophysiological data, we provide a first bundle structure compatible with the hypercolumnar organization and with the position-direction selectivity of motor cortical cells. We then extend this model to encode the direction of arm movement which varies in time, as experimentally measured by Hatsopoulos by introducing the notion of movement fragments. We provide a sub-Riemannian model which describes the time-dependent directional selectivity of cells though integral curves of the geometric structure we set up. The sub-Riemannian distance we define allows to implement a grouping algorithm able to detect a set of hand motor trajectories. These paths, identified by using a kernel defined in terms of kinematic variables, are compatible with the motor primitives obtained from neurophysiological results by spectral analysis applied directly on cortical variables. In a second part of the work, we propose geodesics in this space as an alternative model of models for arm movement trajectories. We define a special class of curves, called admissible, on which to study the geodesics problem: we provide a connectivity property in terms of admissible paths and the existence of normal length minimizers. Admissible geodesics are used as a model of reaching paths, finding a first validation through Flash and Hogan minimizing trajectories.

Exploiting molecular simulations as an atomic-level microscope to elucidate clinically relevant cation chloride cotransporters

The cation chloride cotransporters (CCCs) represent a vital family of ion transporters, with several members implicated in significant neurological disorders. Specifically, conditions such as cerebrospinal fluid accumulation, epilepsy, Down’s syndrome, Asperger’s syndrome, and certain cancers have been attributed to various CCCs. This thesis delves into these pharmacological targets using advanced computational methodologies. I primarily employed GPU-accelerated all-atom molecular dynamics simulations, deep learning-based collective variables, enhanced sampling methods, and custom Python scripts for comprehensive simulation analyses. Our research predominantly centered on KCC1 and NKCC1 transporters. For KCC1, I examined its equilibrium dynamics in the presence/absence of an inhibitor and assessed the functional implications of different ion loading states. In contrast, our work on NKCC1 revealed its unique alternating access mechanism, termed the rocking-bundle mechanism. I identified a previously unobserved occluded state and demonstrated the transporter's potential for water permeability under specific conditions. Furthermore, I confirmed the actual water flow through its permeable states. In essence, this thesis leverages cutting-edge computational techniques to deepen our understanding of the CCCs, a family of ion transporters with profound clinical significance.

A mathematical model of the early visual system and border completion via subriemannian Hamiltonian geodesics

In this thesis, we aim to discuss a simple mathematical model for the edge detection mechanism and the boundary completion problem in the human brain in a differential geometry framework. We describe the columnar structure of the primary visual cortex as the fiber bundle R2 × S1, the orientation bundle, and by introducing a first vector field on it, explain the edge detection process. Edges are detected through a lift from the domain in R2 into the manifold R2 × S1 and are horizontal to a completely non-integrable distribution. Therefore, we can construct a subriemannian structure on the manifold R2 × S1, through which we retrieve perceived smooth contours as subriemannian geodesics, solutions to Hamilton’s equations. To do so, in the first chapter, we illustrate the functioning of the most fundamental structures of the early visual system in the brain, from the retina to the primary visual cortex. We proceed with introducing the necessary concepts of differential and subriemannian geometry in chapters two and three. We finally implement our model in chapter four, where we conclude, comparing our results with the experimental findings of Heyes, Fields, and Hess on the existence of an association field.