We present a method for detecting patterns in time-varying functional magnetic resonance imaging (fMRI) data based on topological analysis. The oxygenated blood flow measured by fMRI is widely used as an indicator of brain activity. The signal is, however, prone to noise from various sources. Random brain activity, physiological noise, and noise from the scanner can reach a strength comparable to the signal itself. Thus, extracting the underlying signal is a challenging process typically approached by applying statistical methods. The goal of this work is to investigate the possibilities of recovering information from the signal using topological feature vectors directly based on the raw signal without medical domain priors. We utilize merge trees to define a robust feature vector capturing key features within a time step of fMRI data. We demonstrate how such a concise feature vector representation can be utilized for exploring the temporal development of brain activations, connectivity between these activations, and their relation to cognitive tasks.