Currently technologies such as EEG, EMG and EOG recordings are the established methods used in the analysis of sleep. But if these methods are to be employed to study sleep in rodents, extensive surgery and recovery is involved which can be both time consuming and costly. This thesis presents and analyzes a cost effective, non-invasive, high throughput system for detecting the sleep and wake patterns in mice using a piezoelectric sensor. This sensor was placed at the bottom of the mice cages to monitor the movements of the mice. The thesis work included the development of the instrumentation and signal acquisition system for recording the signals critical to sleep and wake classification. Classification of the mouse sleep and wake states were studied for a linear classifier and a Neural Network classifier based on 23 features extracted from the Power Spectrum (PS), Generalized Spectrum (GS), and Autocorrelation (AC) functions of short data intervals. The testing of the classifiers was done on two data sets collected from two mice, with each data set having around 5 hours of data. A scoring of the sleep and wake states was also done via human observation to aid in the training of the classifiers. The performances of these two classifiers were analyzed by looking at the misclassification error of a set of test features when run through a classifier trained by a set of training features. The best performing features were selected by first testing each of the 23 features individually in a linear classifier and ranking them according to their misclassification rate. A test was then done on the 10 best individually performing features where they were grouped in all possible combinations of 5 features to determine the feature combinations leading to the lowest error rates in a multi feature classifier. From this test 5 features were eventually chosen to do the classification. It was found that the features related to the signal energy and the spectral peaks in the 3Hz range gave the lowest errors. Error rates as low as 4% and 9% were achieved from a 5-feature linear classifier for the two data sets. The error rates from a 5-feature Neural Network classifier were found to be 6% and 12% respectively for these two data sets.

Le point de départ de cette thèse est la conjecture de Bieberbach : sa démonstration par De Branges utilise deux ingrédients, à savoir la théorie de Loewner des domaines plans croissants et une inégalité de Milin qui concerne les coefficients logarithmiques. Nous commençons par étudier les coefficients logarithmiques du whole-plane SLE en utilisant une méthode combinatoire, assistée par ordinateur. Nous retrouvons les résultats en utilisant une équation aux dérivées partielles analogue à celle obtenue par Beliaev et Smirnov. Nous généralisons ces résultats en définissant le spectre généralisé du whole-plane SLE, que nous calculons par la même méthode, à savoir en dérivant, par le calcul d’Itô, une EDP parabolique satisfaite par les quantités que nous moyennons. Cette famille à deux paramètres d’EDP admet une riche structure algébrique que nous étudions en détail. La dernière partie de la thèse concerne l’opérateur de Grunsky et ses généralisations. Plus expérimentale, nous y mettons à jour, grâce à un logiciel de calcul formel, une structure assez complexe dont nous avons commencé l’exploration. / The starting point of this thesis is Bieberbach’s conjecture: its proof, given by De Branges, uses two ingredients, namely Loewner’s theory of increasing plane domains and an inequality from Milin about the logarithmic coefficients. We start with a study of the logarithmic coefficients of the whole-plane SLE by using a combinatorial method, assisted by computer. We find the results by using a partial differential equation similar to that obtained by Beliaev and Smirnov. We generalize these results by defining the generalized spectrum of the whole-plane SLE, that we calculate by the same method, namely by deriving, thanks to Itô calculus, a parabolic PDE satisfied by the quantities of which we take the average. This two-parameter family of PDEs admits a rich algebraic structure that we study in detail. The last part of this thesis is about the Grunsky operator and its generalizations. In this part that is more experimental we update, thanks to a computer algebra system, a rather complex structure of which we began the exploration.