Systematic investigations through several experimental techniques have revealed that hippocampal pyramidal neurons express voltage gated ion channels (VGICs) with well-defined gradients along their dendritic arbor. These actively maintained gradients in various dendritic VGICs effectuate several stereotypic, topographically continuous functional gradients along the topograph of the dendritic arbor, and have been referred to as intraneuronal functional maps. The prime goal of my thesis was to understand the emergence and homeostasis of the several coexistent functional maps that express within hippocampal pyramidal neurons. In the first part of the thesis, we focus only on spatial interactions between ion channels and analyzed the role of such interactions in the emergence of functional maps. We developed a generalized quantitative framework, the influence field, to analyze the extent of influence of a spatially localized VGIC cluster. Employing this framework, we showed that a localized VGIC cluster could have spatially widespread influence, and was heavily reliant on the specific physiological property and background conductances. Using the influence field model, we reconstructed functional gradients from VGIC conductance gradients, and demonstrated that the cumulative contribution of VGIC conductances in adjacent compartments plays a critical role in determining physiological properties at a given location. These results suggested that spatial interactions among spatially segregated VGIC clusters are necessary for the emergence of the functional maps. In the second part of the thesis, we assessed the specific roles of only kinetic interactions between ion channels in determining physiological properties by employing a single-compartmental model. In doing this, we analyzed the roles of interactions among several VGICs in regulating intrinsic response dynamics. Using global sensitivity analysis, we showed that functionally similar models could be achieved even when underlying parameters displayed tremendous variability and exhibited weak pair-wise correlations. These results suggested that that response homeostasis could be achieved through several non-unique channel combinations, as an emergent consequence of kinetic interactions among these channel conductances. In the final part of the thesis, we analyzed the combined impact of both spatial and kinetic interactions among ion channel conductances on the emergence and homeostasis of functional maps in a neuronal model endowed with extensive dendritic arborization. To do this, we performed global sensitivity analysis on morphologically realistic conductance-based models of hippocampal pyramidal neurons that coexpressed six functional maps. We found topographically continuous functional maps to emerge from disparate model parameters with weak pair-wise correlations between parameters. These results implied that individual channel properties need not be set at constant values in achieving overall homeostasis of several coexistent functional maps. We suggest collective channelostasis, where several channels regulate their properties and expression profiles in an uncorrelated manner, as an alternative for accomplishing functional map homeostasis. Finally, we developed a methodology to assess the contribution of individual channel conductances to the various functional measurements employing virtual knockout simulations. We found that the deletion of individual channels resulted in variable, measurement-and location-specific impacts across the model population.

En ce moment même, d'énormes quantités de données sont générées, collectées et analysées. Dans de nombreux cas, ces données sont échantillonnées sur des objets à la structure géométrique particulière. De tels objets apparaissent fréquemment dans notre vie quotidienne. Utiliser ce genre de données pour inférer la structure géométrique de tels objets est souvent ardue. Cette tâche est rendue plus difficile encore si les objets sous-jacents sont abstraits ou encore de grande dimension. Dans cette thèse, nous nous intéressons à deux problèmes concernant l'analyse géométrique de données. Dans un premier temps, nous nous penchons sur l'inférence de la métrique de structures filamentaires. En supposant que ces structures sont des espaces métriques proches d'un graphe métrique nous proposons une méthode, combinant les graphes de Reeb et l'algorithme Mapper, pour approximer la structure filamentaire via un graphe de Reeb. Notre méthode peut de plus être facilement implémentée et permet de visualiser simplement le résultat. Nous nous concentrons ensuite sur le problème de la comparaison de formes. Nous étudions un ensemble de méthodes récentes et prometteuses pour la comparaison de formes qui utilisent la notion de carte fonctionnelles. Nos résultats théoriques montrent que ces approches sont stables et peuvent être utilisées dans un contexte plus général que la comparaison de formes comme la comparaison de variétés Riemanniennes de grande dimension. Enfin, en nous basant sur notre analyse théorique, nous proposons une généralisation des cartes fonctionnelles aux nuages de points. Bien que cette généralisation ne bénéficie par des garanties théoriques, elle permet d'étendre le champ d'application des méthodes basées sur les cartes fonctionnelles. / Massive amounts of data are being generated, collected and processed all the time. A considerable portion of them are sampled from objects with geometric structures. Such objects can be tangible and ubiquitous in our daily life. Inferring the geometric information from such data, however, is not always an obvious task. Moreover, it’s not a rare case that the underlying objects are abstract and of high dimension, where the data inference is more challenging. This thesis studies two problems on geometric data analysis. The first one concerns metric reconstruction for filamentary structures. We in general consider a filamentary structure as a metric space being close to an underlying metric graph, which is not necessarily embedded in some Euclidean spaces. Particularly, by combining the Reeb graph and the Mapper algorithm, we propose a variant of the Reeb graph, which not only faithfully approximates the metric of the filamentary structure but also allows for efficient implementation and convenient visualization of the result. Then we focus on the problem of shape comparison. In this part, we study the stability properties of some recent and promising approaches for shape comparison, which are based on the notion of functional maps. Our results show that these approaches are stable in theory and potential for being used in more general setting such as comparing high-dimensional Riemannian manifolds. Lastly, we propose a pipeline for implementing the functional-maps-based frameworks under our stability analysis on unorganised point cloud data. Though our pipeline is experimental, it undoubtedly extends the range of applications of these frameworks.