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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.

In questa tesi studiamo le proprietà fondamentali delle funzioni armoniche. Ricaviamo le formule di media mostrando alcune proprietà importanti, quali la disuguaglianza di Harnack, il teorema di Liouville, il principio del massimo debole e forte. Infine, illustriamo un criterio di risolubilità per il problema di Dirichlet per il Laplaciano in un arbitrario dominio limitato di R^n tramite un metodo noto come metodo di Perron per le funzioni subarmoniche.

In this work we present a geometric model of motor cortex that generalizes an already existing model of visual cortex. The thesis opens by recalling the notions of fiber bundles, principal bundles, Lie groups and sub-Riemannian geometry. In particular, we enunciate Chow’s theorem which ensures that if the Hörmander condition holds, the space connectivity property is satisfied. Then we recall the visual cortex model proposed by Citti-Sarti, which describes the set of simple cells as a Lie group with sub-Riemannian metric. The original part of the thesis is the extension to the motor cortex. Based on neural data, collected by Georgopoulos, we study the set of motor cortical cells and we describe them as a principal bundle. The fiber contains the movement direction and shapes the hypercolumnar structure measured. Finally we determine the intrinsic coordinates of the motor cortex, studying the cellular response to the motor impulse.

In this work we present a geometric model of motor cortex that generalizes an already existing model of visual cortex. The thesis opens by recalling the notions of fiber bundles, principal bundles, Lie groups and sub-Riemannian geometry. In particular, we enunciate Chow’s theorem which ensures that if the Hörmander condition holds, the space connectivity property is satisfied. Then we recall the visual cortex model proposed by Citti-Sarti, which describes the set of simple cells as a Lie group with sub-Riemannian metric. The original part of the thesis is the extension to the motor cortex. Based on neural data, collected by Georgopoulos, we study the set of motor cortical cells and we describe them as a principal bundle. The fiber contains the movement direction and shapes the hypercolumnar structure measured. Finally we determine the intrinsic coordinates of the motor cortex, studying the cellular response to the motor impulse.

In questa tesi studiamo le proprietà fondamentali delle funzioni armoniche. Ricaviamo le formule di media mostrando alcune proprietà importanti, quali la disuguaglianza di Harnack, il teorema di Liouville, il principio del massimo debole e forte. Infine, illustriamo un criterio di risolubilità per il problema di Dirichlet per il Laplaciano in un arbitrario dominio limitato di R^n tramite un metodo noto come metodo di Perron per le funzioni subarmoniche.