  Functional magnetic resonance imaging (fMRI) is a crucial technology for
gaining insights into cognitive processes in humans. Data amassed from fMRI
measurements result in volumetric data sets that vary over time. However,
analysing such data presents a challenge due to the large degree of noise and
person-to-person variation in how information is represented in the brain. To
address this challenge, we present a novel topological approach that encodes
each time point in an fMRI data set as a persistence diagram of topological
features, i.e. high-dimensional voids present in the data. This representation
naturally does not rely on voxel-by-voxel correspondence and is robust to
noise. We show that these time-varying persistence diagrams can be clustered to
find meaningful groupings between participants, and that they are also useful
in studying within-subject brain state trajectories of subjects performing a
particular task. Here, we apply both clustering and trajectory analysis
techniques to a group of participants watching the movie 'Partly Cloudy'. We
observe significant differences in both brain state trajectories and overall
topological activity between adults and children watching the same movie.
