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 resulting clusters of spiraling brain activation cycles.

本文介绍了静息态功能磁共振成像 (rs-fMRI) 视频中多级大脑激活区域强度的四元数视图。四元数使得在 4D 空间中探索 rs-fMRI 大脑激活区域成为可能，其中在螺旋激活周期（每个都有自己的强度）中有不同的大脑激活强度。结果，自然形成了多级循环，形成具有不同直径的金字塔形涡旋形状。这些金字塔形涡旋反映了类似多级大脑激活区域循环簇的分形（自相似性）。使用数据方法的计算拓扑，我们发现持续的、重复的螺旋循环集群由三角化 rs-fMRI 视频帧中的血氧水平依赖 (BOLD) 信号产生。每个大脑激活区域周期都是一个细胞复合体，这是一个集合路径连接的顶点，没有结束顶点。BOLD 信号传播区域中螺旋涡旋形状的持久性的测量是根据 Betti 数（称为发生器的不同循环顶点的计数）进行的，这些数在大脑自发活动期间随时间上升和下降。这里给出的一个主要结果是每个四元大脑激活区域漩涡都有一个自由群呈现。此外，我们引入了大脑激活视频的 3D 条形码，有助于可视化和量化由 rs-fMRI 视频帧中的三角化大脑激活区域自然产生的多级涡旋簇的分形。我们制作了可免费下载的视频档案，这些档案展示了由此产生的螺旋式大脑激活周期集群。