Quaternionic views of rs-fMRI hierarchical brain activation regions. Discovery of multilevel brain activation region intensities in rs-fMRI video frames

•Quaterionic views of multi-level brain activation region intensities in resting state fMRI videos.•Discovery of varying brain activation brightness levels in spiraling activation cycles in rs-fMRI video frames.•Presentation of the natural formation of multi-level cycles that form pyramidal vortex shapes with varying diameters on multiple brain activation regions in each rs-fMRI video frame.•Topology of data approach in tracking persistent, recurring ring clusters of spiraling cycles resulting from blood oxygen level dependent (BOLD) signals in triangulated rs-fMRI video frames.•Measurement of the persistence of spiraling vortex shapes in terms of Betti numbers (counts of distinguished cycle vertexes are a source of cyclic group representations of vortex cycles) that rise and fall over time during spontaneous activity of the brain. This paper introduces quaternionic views of multi-level brain activation region intensities in resting-state functional magnetic resonance imaging (rs-fMRI) videos. Quaternions make it possible to explore rs-fMRI brain activation regions in a 4D space in which there are varying brain activation intensities in spiralling activation cycles (each with its own intensity). As a result, there is a natural formation of multi-level cycles that form pyramidal vortex shapes with varying diameters. These pyramidal vortexes reflect the fractality (self-similarity) of clusters of similar multilevel brain activation region cycles. Using a computational topology of data approach, we have found that persistent, recurring clusters of spiraling cycles resulting from blood oxygen level dependent (BOLD) signals in triangulated rs-fMRI video frames. Each brain activation region cycle is a cell complex, which is a collection path-connected vertexes that has no end vertex. Measurement of persistence of spiraling vortex shapes in BOLD signal propagation regions is carried out in terms of Betti numbers (counts of distinguished cycle vertexes called generators) that rise and fall over time during spontaneous activity of the brain. A main result given here is that every quaternionic brain activation region vortex has a free group presentation. In addition, we introduce 3D barcodes of brain activation videos that help visualize and quantify the fractality of clusters of multilevel vortexes arising naturally from triangulated brain activation regions in rs-fMRI video frames. We have made freely available downloadable archives of videos that exhibit the resu