Understanding biological and social interactions in a dynamical system framework amounts to identifying stable patterns of activity and tracking transitions between them. When the system’s dimension is low (few
interacting elements), one can often visually inspect the time series and write down the relevant variables and differential equations. Once we venture into higher dimensions, this traditional modeling approach becomes intractable: what constitutes a “stable pattern” or a “transition” in the empirical data depends on the spatiotemporal scales at which we look at the system. We show how persistent homology serves as a natural tool for characterizing such multiscale dynamics in rhythmic social interaction. Furthermore, a key goal of studying transitions is to understand the relations between distinct stable dynamic patterns – they
provide access to the global organization of the dynamical system. We show how existing topological data analysis (TDA) tools can be adapted to understand the network of transitions in brain dynamics. The talk focuses on the empirical and dynamical-system context in which TDA is applied in the hope of eliciting new
conversations across the boundary of empirical science, dynamical systems modeling, and TDA.