Resumen de Clustering based on Bayesian networks with Gaussian and angular predictors: applications in neuroscience
Sergio Luengo Sánchez
One of the greatest challenges facing science today is to disentangle the functioning of the brain, with the simulation of the neuronal circuits of the human brain at different scales being an area of study that has awakened many expectations and interests. Given the incredible complexity of this goal, computer-assisted mathematical models are a fundamental tool for reasoning, making predictions and suggesting new hypotheses about the functioning and organization of neurons. In this thesis we focus on the study of the morphology of dendritic spines and somas of human pyramidal neurons from the point of view of the computational neuroanatomy.
Dendritic spines are small membranous protrusions located on the surface of the dendrites, which are in charge of receiving excitatory synapses. Their morphology has been associated with cognitive functions such as learning or memory, and it is not surprising that a wide variety of mental illnesses have been related with alterations in their morphology or density. It is therefore interesting to identify the types of dendritic spines. Kaiserman-Abramof's categorisation, which proposes four groups of dendritic spines, is the most accepted although it is discussed whether the diversity of morphologies reflects more a continuum than the existence of particular groups. For their part, somas contain the nucleus of the neuron and are responsible for generating neurotransmitters, the basic elements of synapses and therefore of brain activity. Their morphology has been identified as one of the fundamental properties for distinguishing between types of neurons.
For the development of this dissertation we used individual 3D reconstructions of dendritic spines and somas. The application of a novel feature extraction technique allowed us to univocally characterise the geometry of these 3D bodies according to several magnitudes and directions. Through cluster analysis, we automatically and objectively separated the observations into homogeneous categories. Specifically, we applied model-based clustering, a probabilistic approach that assumes that the data were generated by a statistical mixture model and whose goal is to fit it from the observed data. According to this framework, each cluster is represented by a multidimensional probability distribution. In this case, learning these models according to classical statistics presents serious problems due to their inability to handle the periodicity of directional data. Some distributions focused on modeling directional-linear data has been proposed on the directional statistics literature, but all of them exhibit important limitations for performing model-based clustering. Most of the directional-linear distributions are based on copulas to construct bivariate distributions, which present complicated theoretical results, making them difficult to extend to higher dimensions. Additionally, they require from optimisation algorithms for the estimation of the parameters, that can be prohibitive during the clustering process from a computational perspective. Multivariate directional-linear data clustering is even more challenging and it is almost limited to models that assume independence among directional and linear variables, severely reducing the expressiveness of the model and introducing an unnecessary number of clusters.
Probabilistic graphical models, and more specifically Bayesian networks, are diagrammatic representations of probability distributions that can be used to design generative models or understand the underlying relationships between random variables. In addition, they are a very useful tool for probabilistic reasoning in the presence of incomplete information. Interactions among several variables may be a consequence of a hidden variable, i.e., a variable that could not be measure or observed. Therefore, BNs provides a framework for discovering hidden variables and performing model-based clustering. In this thesis we exploit the properties of Bayesian networks to introduce for the first time the Extended Mardia-Sutton mixture model. To achieve this, we derive a new multivariate density function that captures directional-linear correlations and whose parameters can be calculated according to closed-form expressions, relaxing the limitations of previous probability distributions.
In order to understand and interpret the groups resulting from applying model-based clustering, we identify the most representative features of each cluster using hypothesis tests and rules generated by a rule induction algorithm. Finally, from the combination of the generative models implemented in this study and the univocal definition of the morphology of the neuronal components, we create a methodology for the simulation of 3D virtual somas and dendritic spines. To the best of our knowledge, this is the first attempt to fully characterise, model and simulate 3D dendritic spines and somas.
