Single cells, population dynamics, and Euler characteristic profiles

The Euler characteristic profile (ECP) of a multifiltered simplicial complex records the Euler characteristic at each point in the filtration poset. While cruder than multiparameter persistent homology, ECPs are computationally much more tractable and often still sensitive enough to detect changes in the topology of the underlying data. For example, ECPs based on vector field data are able to differentiate between dynamical systems in 2 and 3 dimensions. In this talk, I present a recent project for similarly extracting dynamical information from high-dimensional point cloud data equipped with a vector field. The motivating application is single-cell RNA sequence data, where RNA velocity provides a proxy for the direction and rate of cellular state transitions. We construct multifiltered flag complexes where edge weights are derived from distances and velocities. On synthetic data generated by a stochastic model of gene expression dynamics with known ground-truth transition graphs, the resulting ECPs distinguish between competing state transition networks. Ultimately we want to use these ideas to investigate neural stem cell differentiation -- both in homeostasis and when it goes wrong, as in glioblastoma. This is joint work with Marta Marszewska, Justyna Signerska-Rynkowska, Paweł Dłotko, Anna Marciniak-Czochra, and Ana Martín-Vilalba.