  Living systems exhibit complex yet organized behavior on multiple
spatiotemporal scales. To investigate the nature of multiscale coordination in
living systems, one needs a meaningful and systematic way to quantify the
complex dynamics, a challenge in both theoretical and empirical realms. The
present work shows how integrating approaches from computational algebraic
topology and dynamical systems may help us meet this challenge. In particular,
we focus on the application of multiscale topological analysis to coordinated
rhythmic processes. First, theoretical arguments are introduced as to why
certain topological features and their scale-dependency are highly relevant to
understanding complex collective dynamics. Second, we propose a method to
capture such dynamically relevant topological information using persistent
homology, which allows us to effectively construct a multiscale topological
portrait of rhythmic coordination. Finally, the method is put to test in
detecting transitions in real data from an experiment of rhythmic coordination
in ensembles of interacting humans. The recurrence plots of topological
portraits highlight collective transitions in coordination patterns that were
elusive to more traditional methods. This sensitivity to collective transitions
would be lost if the behavioral dynamics of individuals were treated as
separate degrees of freedom instead of constituents of the topology that they
collectively forge. Such multiscale topological portraits highlight collective
aspects of coordination patterns that are irreducible to properties of
individual parts. The present work demonstrates how the analysis of multiscale
coordination dynamics can benefit from topological methods, thereby paving the
way for further systematic quantification of complex, high-dimensional dynamics
in living systems.
