This Ph.D. thesis deals with mathematical and statistical modeling of synaptic vesicle distribution, shape, orientation and interactions. The first major part of this thesis treats the problem of determining the effect of stress on synaptic vesicle distribution and interactions. Serial section transmission electron microscopy is used to acquire images from two experimental groups of rats: 1) rats subjected to a behavioral model of stress and 2) rats subjected to sham stress as the control group. The synaptic vesicle distribution and interactions are modeled by employing a point process approach. The model is able to correctly separate the two experimental groups.

Two different approaches to estimate the thickness of each section of specimen being imaged are introduced. The first approach uses Darboux frame and Cartan matrix to measure the isophote curvature and the second approach is based on differences of statistical measures in section and the same measures in between sections.

Three-dimensional (3D) datasets are reconstructed by using image registration techniques and estimated thicknesses. We distinguish the effect of stress by estimating the synaptic vesicle densities and modeling their distribution in 3D. A comparison to the commonly used twodimensional (2D) estimations is presented. The results indicate the differences of the two experimental groups both in 2D and 3D methods. The estimations in three dimensions include the slant of the active zone and estimated thickness of the sections, which leads to more accurate results.

Finally, we present a thorough statistical investigation of the shape, orientation and interactions of the synaptic vesicles during active time of the synapse. Focused ion beam-scanning electron microscopy images of a male mammalian brain are used for this study. As a start of the assessment, univariate analysis of shape and orientation of the vesicles with regard to the active zone are studied. The configurations of the synaptic vesicles are regarded as a marked point process, where the centre of the vesicles are the points and the marks are given by size, shape and orientation characteristics. The study shows that the synaptic vesicles are ellipsoidal and not spherical. The analysis also shows that there is a systematic pattern in vesicle's orientation in relation to the active zone and also between the neighboring vesicles.