Les neurones interagissent à travers leur potentiel de membrane qui a en général une évolution temporelle complexe due aux nombreuses entrées synaptiques irrégulières reçues. Cette évolution est mieux décrite en termes probabilistes, en raison de ces entrées irrégulières ou «bruit synaptique». L'évolution temporelle du potentiel de membrane est stochastique mais aussi déterministe: stochastique, car conduite par des entrées synaptiques qui arrivent de façon aléatoire dans le temps, et déterministe, car un neurone biologique a une évolution temporelle très similaire quand soumis à une même séquence d'entrées synaptiques. Nous étudions les propriétés statistiques d'un modèle simplifié de neurone soumis à des entrées à taux variable d'où en résulte l'évolution non-stationnaire du potentiel de membrane. Nous considérons un modèle passif de membrane neuronale, sans mécanisme de décharge neuronale, soumis à des entrées à courant ou à conductance sous la forme d'un processus de «shot noise». Les fluctuations du potentiel de membrane sont aussi modélisées par un processus stochastique similaire, de «shot noise» filtré. Nous avons analysé les propriétés statistiques de ces processus dans le cadre des transformations de processus ponctuels de Poisson. Des propriétés de ces transformations sont dérivées les statistiques non-stationnaires du processus. Nous obtenons ainsi des expressions analytiques exactes pour les moments et cumulants du processus filtré dans le cas général des taux d'entrée variables. Ce travail ouvre de nombreuses perspectives pour l'analyse de neurones dans les conditions in vivo, en présence d'entrées synaptiques intenses et bruitées. / Neurons interact through their membrane potential that generally has a complex time evolution due to numerous irregular synaptic inputs received. This complex time evolution is best described in probabilistic terms due to this irregular or "noisy" activity. The time evolution of the membrane potential is therefore both stochastic and deterministic: it is stochastic since it is driven by random input arrival times, but also deterministic, since subjecting a biological neuron to the same sequence of input arrival times often results in very similar membrane potential traces. In this thesis, we investigated key statistical properties of a simplified neuron model under nonstationary input from other neurons that results in nonstationary evolution of membrane potential statistics. We considered a passive neuron model without spiking mechanism that is driven by input currents or conductances in the form of shot noise processes. Under such input, membrane potential fluctuations can be modeled as filtered shot noise currents or conductances. We analyzed the statistical properties of these filtered processes in the framework of Poisson Point Processes transformations. The key idea is to express filtered shot noise as a transformation of random input arrival times and to apply the properties of these transformations to derive its nonstationary statistics. Using this formalism we derive exact analytical expressions, and useful approximations, for the mean and joint cumulants of the filtered process in the general case of variable input rate. This work opens many perspectives for analyzing neurons under in vivo conditions, in the presence of intense and noisy synaptic inputs.

A equação de Poisson-Boltzmann tem uma vasta gama de aplicações, desde a ciência coloidal e microfluídica até bioquímica e biofísica. O potencial elétrico na dupla camada elétrica leva a um potencial de força, em termos das equações de Navier-Stokes que é então usado para simular o fluxo resultante. Em escoamentos bifásicos uma simplificação desta equação é usada para se obter o campo de pressão. O presente trabalho tem como principal objetivo estudar o problema de Poisson-Boltzmann com coeficiente constante e propor uma solução através da implementação do método das interfaces imersas utilizando diferenças finitas de altas ordens de precisão numérica. / The Poisson-Boltzmann equation has a wide range of applications, from colloidal and microfluidic science to biochemistry and biophysics. The electrical potential in electric double layer leads to a force potential in terms of the Navier-Stokes equations that is then used to simulate the resulting flow. In biphasic flows a simplification of this equation is used to obtain the pressure field. The present study has as main objective to study the problem of Poisson-Boltzmann with constant coefficient and propose a solution through implementation of the immersed interfaces method using high order finite difference scheme sand thus get high order numerical accuracy.

Dans cette thèse j’étudie l’impact de la géométrie de micro et nano-domaines biologiques sur les propriétés d'électrodiffusion, ceci à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck. Je considère des domaines non-triviaux ayant une forme cuspide ou elliptique. Mon objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles. Dans la première partie j’étudie le voltage à l'équilibre pour un électrolyte dans un domaine borné, et ayant un fort excès de charges positives. Je montre que le premier temps de sortie dans une boule chargée dépend de la surface et non du volume. J’étudie ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Je construis ensuite une solution asymptotique pour le voltage dans les cas 2D et 3D et je montre qu’ils sont donnés au premier ordre par la même expression. Enfin, j’obtiens la même conclusion en considérant une géométrie formée d'une ellipse, dont je construis une solution asymptotique du voltage en 2D et 3D. La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. A partir de simulations numériques, je mets en évidence le lien entre la polarisation de concentration dans l'épine et sa géométrie. Je compare ensuite mon modèle à des données de microscopie. Je développe une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir des données de microscopie. Enfin j’estime la résistance du cou et montre que celle-ci ne suit pas la loi d'Ohm. / In this PhD I study how electro-diffusion within biological micro and nano-domains is affected by their shapes using the Poisson-Nernst-Planck (PNP) partial differential equations. I consider non-trivial shapes such as domains with cusp and ellipses. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales. In the first part I estimate the steady-state voltage inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. I show the mean first passage time in a charged ball depends on the surface and not on the volume. I further study a geometry composed of a ball with an attached cusp-shaped domain. I construct an asymptotic solution for the voltage in 2D and 3D and I show that to leading order expressions for the voltage in 2D and 3D are identical. Finally, I obtain similar conclusion considering an elliptical-shaped domain for which I construct an asymptotic solution for the voltage in 2D and 3D. In the second part, I model the electrical compartmentalization in dendritic spines. Based on numerical simulations, I show how spines non-cylindrical geometry leads to concentration polarization effects. I then compare my model to experimental data of microscopy imaging. I develop a deconvolution method to recover the fast voltage dynamic from the data. I estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius.

The objective of this project is to fit a sequence of increasingly complex zero-inflated censored regression models to a known data set. It is quite common to find censored count data in statistical analyses of health-related data. Modeling such data while ignoring the censoring, zero-inflation, and overdispersion often results in biased parameter estimates. This project develops various regression models that can be used to predict a count response variable that is affected by various predictor variables. The regression parameters are estimated with Bayesian analysis using a Markov chain Monte Carlo (MCMC) algorithm. The tests for model adequacy are discussed and the models are applied to an observed data set.

[pt] Desde fevereiro de 2020 a pandemia gerada pelo novo coronavírus SarsCoV-2, vírus gerador da doença COVID-19, tem causado muitos óbitos, principalmente nos grandes centros urbanos. No Brasil, um dos estados mais afetados foi o Rio de Janeiro que, apesar de todas as ações feitas para mitigar o avanço da COVID-19, chegou em 01 de março de 2021 a uma taxa de mortalidade de 206,9 por cento, que corresponde a aproximadamente 207 óbitos a cada mil habitantes. No entanto, os municípios do RJ foram atingidos de maneira distinta, onde a cidade menos afetada alcançou 9,7 por cento e a mais afetada 331,3 por cento. Estudos prévios da literatura especializada indicam que a principal razão desta discrepância pode ser associada à fatores relacionados a população, renda, educação, saúde, economia, território e ambiente. Portanto, esse trabalho tem como principal objetivo identificar os principais fatores socioeconômicos, sociodemográficos e de acesso a recursos hospitalares que estão associadas a taxa de mortalidade oriunda do Sars-CoV-2 nos noventa e dois municípios do estado do Rio de Janeiro com base no modelo de Regressão de Poisson, no período de 01 de março de 2020 a 01 de março de 2021, contabilizando 12 meses. A partir do modelo escolhido foi possível detectar que dez dos onze fatores analisados influenciam na taxa de mortalidade. Sendo os fatores, Índice de desenvolvimento humano municipal (IDHM), Renda per capita (RDPC), Percentual de pobres (PMPOB), Produto interno bruto (PIB), Taxa de frequência bruta ao superior (T_FBSUPER), percentual de aglomerados subnormais (PER_AGSN), Densidade demográfica, Número de leitos hospitalares do SUS por habitante, Número de leitos hospitalares totais por habitante e Número de respiradores por habitante. Assim, os resultados obtidos com base nesses fatores analisados podem auxiliar na criação de ações mitigadoras mais direcionadas e eficientes, de acordo com as características dos municípios do estado do Rio de Janeiro. / [en] Since February 2020 the pandemic generated by the new coronavirus SarsCoV-2, the virus generating the disease COVID-19, has caused many deaths, mainly in large urban centers. In Brazil, one of the most affected states was Rio de Janeiro, which, despite all the actions taken to mitigate the progress of COVID19, reached on March 1, 2021 a mortality rate of 206.9 percent, which corresponds to approximately 207 deaths per thousand inhabitants. However, the Rio de Janeiro municipalities were affected differently, where the least affected city reached 9.7 percent and the most affected 331.3 percent. Previous studies in the specialized literature indicate that the main reason for this discrepancy may be associated with factors related to population, income, education, health, economy, territory, and environment. Therefore, this work has as main objective to identify the main socioeconomic, socio-demographic factors and access to hospital resources that are associated with the mortality rate from Sars-CoV-2 in the ninety-two municipalities in the state of Rio de Janeiro based on the Poisson Regression model, in the period from March 01, 2020 to March 01, 2021, accounting for 12 months. From the model chosen it was possible to detect those ten of the eleven factors analyzed influence the mortality rate. The factors being, municipal human development index (IDHM), per capita income (RDPC), percentage of poor (PMPOB), gross domestic product (GDP), gross attendance rate to higher (T_FBSUPER), percentage of subnormal settlements (PER_AGSN), demographic density, number of SUS hospital beds per inhabitant, number of total hospital beds per inhabitant and number of respirators per inhabitant. Thus, the results obtained based on these analyzed factors can help in the creation of more targeted and efficient mitigating actions, according to the characteristics of the municipalities in the state of Rio de Janeiro.

[pt] Espaços de moduli de polígonos em R(3) com comprimento dos lados fixados é um exemplo amplamente estudado de variedade simplética. Esses espaços podem ser descritos como quociente simplético de um número finito de órbitas coadjuntas pelo grupo SU(2). Nesta tese esses espaços de moduli são identificados como folhas simpléticas de uma variedade de Poisson que pode ser construída como quociente. Essa construção é a seguir generalizada ao caso de um produto de um número finito de órbitas coadjuntas pelo grupo SU(n), e o resultado principal desse trabalho de tese descreve como esses espaços de moduli de polígonos generalizados formam uma folheação em folhas simpléticas de uma variedade de Poisson. / [en] Moduli spaces of polygons in R(3)with fixed sides length are a widely studied example of symplectic manifold that can be described as the symplectic quotient of a finite number of SU(2)−coadjoint orbits by the diagonal action of the group SU(2). In this thesis these spaces are identified as the symplectic leaves of a Poisson manifold, that can itself be obtained by a quotient procedure. The construction is then generalized to the case of the quotient of a product of finitely many SU(n)−coadjoint orbits by the diagonal action of SU(n), and the main result of this thesis describes how these moduli spaces of generalized polygons fit together as the symplectic leaves of a quotient Poisson manifold.

[pt] Existe consenso entre os pesquisadores de que o modelo de séries temporais não é adequado para previsão de peças de reposição. Entretanto, a maioria das ferramentas de previsão existentes no mercado emprega o modelo de séries temporais. Este trabalho apresenta a distribuição de Poisson como alternativa para a previsão de estoque de peças eletrônicas de reposição. A partir de noções básicas de gestão de estoques utilizando séries temporais e dos conceitos de confiabilidade, disponibilidade e do Processo de Poisson é proposto um modelo alternativo. Com o uso de exemplos reais são apresentados os resultados da aplicação do modelo proposto e a comparação com o modelo SAGA, que utiliza séries temporais. A principal característica do modelo proposto é o uso da distribuição de Poisson e a Taxa de Falhas real como principais parâmetros de cálculo. A análise dos resultados mostrou que é possível reduzir os erros de previsão, o custo de estoque e o número de pedidos não atendidos, com conseqüente aumento da Disponibilidade Operacional. / [en] There is a consensus that time series model is not appropriate in forecasting replacement parts. However most of market used forecasting tools are time series models. This work presents Poisson distribution as an alternative to forecast replacement parts on electronic equipments. From basic stock management notions, using time series and trust concepts of reliability, availability, and Poisson Process, an alternative model is proposed. Using real examples, the result from proposed model and its comparison to SAGA model, which is based on time series, is presented. The major characteristic of the proposed model is the application of Poisson distribution, and the real faults rate as the main calculus parameters. The analyses results have shown that is possible to reduce forecasting errors, therefore the stock cost, and the reduction of back orders amount, increasing the Operational availability.

<p>In this thesis, we explore several stochastic models associated withthe two-parameter Poisson-Dirichlet distribution and population genetics.The impacts of mutation, selection and time onthe population evolutionary process will be studied by focusing on two aspects of the model:equilibrium and non-equilibrium. In the first chapter, we introduce relevant background on stochastic genetic models, andsummarize our main results and their motivations. In the second chapter, the two-parameter GEM distribution is constructedfrom a linear birth process with immigration. The derivationrelies on the limiting behavior of the age-ordered family frequencies. In the third chapter, to show the robustness of the sampling formula we derive the Laplace transform of the two-parameterPoisson-Dirichlet distribution from Pitman sampling formula. The correlationmeasure of the two-parameter point process is obtained in our proof. We also reverse this derivationby getting the sampling formula from the Laplace transform. Then,we establish a central limit theorem for the infinitely-many-neutral-alleles modelat a fixed time as the mutation rate goes to infinity.Lastly, we get the Laplace transform for the selectionmodel from its sampling formula. In the fourth chapter, we establisha central limit theorem for the homozygosity functions under overdominant selectionwith mutation approaching infinity. The selection intensity is given by a multiple of certain powerof the mutation rate. This result shows an asymptotic normality for the properly scaled homozygosities,resembling the neutral model without selection.This implies that the influence of selection can hardly be observed with large mutation. In the fifth chapter, the stochastic dynamics of the two-parameter extension of theinfinitely-many-neutral-alleles model is characterized by the derivation of its transition function,which is absolutely continuous with respect to the stationary distribution being the two-parameter Poisson-Dirichlet distribution.The transition density is obtained by the expansion of eigenfunctions.Combining this result with the correlation measure in Chapter 3, we obtain the probability generatingfunction of a random sampling from the two-parameter model at a fixed time. Finally, we obtain two results based on the quasi-invariance of the Gamma processwith respect to the multiplication transformation group.One is the quasi-invariance property of the two-parameter Poisson-Dirichletdistribution with respect to Markovian transformation group.The other one is the equivalence between the quasi-invarianceof the stationary distributions of aclass of branching processes and their reversibility.</p> / Doctor of Philosophy (PhD)

Multiple regression models play an important role in analyzing and making predictions about data. Prediction accuracy becomes lower when two or more explanatory variables in the model are highly correlated. One solution is to use ridge regression. The purpose of this thesis is to study the performance of available ridge regression estimators for Poisson regression models in the presence of moderately to highly correlated variables. As performance criteria, we use mean square error (MSE), mean absolute percentage error (MAPE), and percentage of times the maximum likelihood (ML) estimator produces a higher MSE than the ridge regression estimator. A Monte Carlo simulation study was conducted to compare performance of the estimators under three experimental conditions: correlation, sample size, and intercept. It is evident from simulation results that all ridge estimators performed better than the ML estimator. We proposed new estimators based on the results, which performed very well compared to the original estimators. Finally, the estimators are illustrated using data on recreational habits.

Made available in DSpace on 2016-06-02T20:06:09Z (GMT). No. of bitstreams: 1 5822.pdf: 1106242 bytes, checksum: 613a82d7af4c6f40b60637e4c7122121 (MD5) Previous issue date: 2014-03-07 / Financiadora de Estudos e Projetos / In this dissertation, two models of probability distributions for the lifetimes until the occurrence of the event produced by a specific cause for elements in a population are reviewed. The first revised model is called the Weibull-Poisson (WP) which has been proposed by Louzada et al. (2011a). This model generalizes the exponential-Poisson distributions proposed by Kus (2007) and Weibull. The second, called long-term model, has been proposed by several authors and it considers that the population is not homogeneous in relation to the risk of event occurence by the cause studied. The population has a sub-population that consists of elements who are not liable do die by the specific cause in study. These elements are considered as immune or cured. In relation to the elements who are at risk the minimum value of time of the event accurance is observed. In the review of WP the expressions of the survival function, quantile function, probability density function, and of the hazard function, as well the expression of the non-central moments of order k and the distribution of order statistics are detailed. From this review we propose, in an original way, studies of the simulation to analyze the paramenters of frequentist properties of maximum likelihood estimators for this distribution. And also we also present results related to the inference about the parameters of this distribution, both in the case in which the data set consists of complete observations of lifetimes, and also in the case in which it may contain censored observations. Furthermore, we present in this paper, in an original way a regression model in a form of location and scale when T has WP distribution. Another original contribution of this dissertation is to propose the distribution of long-term Weibull-Poisson (LWP). Besides studying the LWP in the situation in which the covariates are included in the analysis. We also described the functions that characterize this distribution (distribution function, quantile function, probability density function and the hazard function). Moreover we describe the expression of the moment of order k, and the density function of a statistical order. A study by simulation viii of this distribution is made through maximum likelihood estimators. Applications to real data set illustrate the applicability of the two considered models. / Nesta dissertação são revistos dois modelos de distribuições de probabilidade para os tempos de vida até a ocorrência do evento provocado por uma causa específica para elementos em uma população. O primeiro modelo revisto é o denominado Weibull-Poisson (WP) que foi proposto por Louzada et al. (2011a), esse modelo generaliza as distribuições exponencial Poisson proposta por Kus (2007) e Weibull. O segundo, denominado modelo de longa duração, foi proposto por vários autores e considera que a população não é homogênea em relação ao risco de ocorrência do evento pela causa em estudo. A população possui uma sub-população constituída de elementos que não estão sujeitos ao evento pela causa especifica em estudo, sendo considerados como imunes ou curados. Em relação à parcela dos elementos que estão em risco observa-se o valor mínimo dos tempos da ocorrência do evento. Na revisão sobre a WP são detalhadas as expressões da função de sobrevivência, da função quantil, da função densidade de probabilidade e da função de risco, bem como a expressão dos momentos não centrais de ordem k e a distribuição de estatísticas de ordem. A partir desta revisão, é proposta de forma original, estudos de simulação com o objetivo de analisar as propriedades frequentistas dos estimadores de máxima verossimilhança dos parâmetros desta distribuição. E apresenta-se resultados relativos à inferência sobre os parâmetros desta distribuição, tanto no caso em que o conjunto de dados consta de observações completas de tempos de vida, como no caso em que ele possa conter observações censuradas. Alem disso, apresentamos de forma original neste trabalho um modelo de regressão na forma de locação e escala quando T tem distribuição WP. Outra contribuição original dessa dissertação é propor a distribuição de longa duração Weibull-Poisson (LWP), alem de estudar a LWP na situação em que as covariáveis são incluídas na análise. Realizou-se também a descrição das funções que caracterizam essa distribuição (função distribuição, função quantil, função densidade de probabilidade e função de risco). Assim como a descrição da expressão do momento de ordem k e da função densidade da estatística de ordem. É feito um estudo por simulação desta distribuição via máxima verossimilhança. Aplicações à conjuntos de dados reais ilustram a utilidade dos dois modelos considerados.