Modelling Voxel Dependent Hemodynamic Response Function

Ph. D. Thesis.An important challenge of contemporary neuroscience is the detection and understanding of significant brain activity using functional magnetic resonance imaging (fMRI). One of the many motivations of this research, related to the data set used in this thesis, is to investigate brain activation and connectivity patterns aimed at identifying associations between these patterns and regaining motor functionality following a stroke. Much statistical modelling has attempted to interpret noisy fMRI data and detect changes in response to activity. However due to the large data sets usually involved in fMRI modelling, here as many as 150, 000 measurements in localised spatial volumes known as voxels at each time point, many simplifying assumptions are usually made to make computation feasible. This is known to have a negative impact on detecting voxel activation. In this work we fit a space-time model to a fMRI data set using a sequential approach to allow for scalability. However, the main contribution of this work is an alternative method to detect activation in the brain. Here we take the novel approach of using topological data analysis to investigate the model residuals to detect changes in the fMRI data. In particular we analyse the spatial distribution of topological features of the residuals to provide a test for normality, and also by providing a method to analyse how the spatial distribution of such features change over time, we are able to detect changes in the data in response to activity where conventional methods cannot. A recommendation for future work is to also investigate how topological features change for different filtration levels of the field, as this may provide new insights on brain activation