  Developing reliable methods to discriminate different transient brain states
that change over time is a key neuroscientific challenge in brain imaging
studies. Topological data analysis (TDA), a novel framework based on algebraic
topology, can handle such a challenge. However, existing TDA has been somewhat
limited to capturing the static summary of dynamically changing brain networks.
We propose a novel dynamic-TDA framework that builds persistent homology over a
time series of brain networks. We construct a Wasserstein distance based
inference procedure to discriminate between time series of networks. The method
is applied to the resting-state functional magnetic resonance images of human
brain. We demonstrate that our proposed dynamic-TDA approach can distinctly
discriminate between the topological patterns of male and female brain
networks. MATLAB code for implementing this method is available at
https://github.com/laplcebeltrami/PH-STAT.
