I will describe how tools from applied topology may be used to identify components of time series most responsible for cyclic dynamics observed in orbits of an underlying dynamical system. In this setting, I will show that derivatives of the persistent homology may be computed explicitly and describe a simple algorithm for gradient descent. As an example, we will consider neuronal data from the model organism C. elegans and identify subsets of neurons driving global cyclic brain dynamics in the spirit of dimensionality reduction.

This is based on joint work with Peter Bubenik.