  One of the paramount challenges in neuroscience is to understand the dynamics
of individual neurons and how they give rise to network dynamics when
interconnected. Historically, researchers have resorted to graph theory,
statistics, and statistical mechanics to describe the spatiotemporal structure
of such network dynamics. Our novel approach employs tools from algebraic
topology to characterize the global properties of network structure and
dynamics.
  We propose a method based on persistent homology to automatically classify
network dynamics using topological features of spaces built from various
spike-train distances. We investigate the efficacy of our method by simulating
activity in three small artificial neural networks with different sets of
parameters, giving rise to dynamics that can be classified into four regimes.
We then compute three measures of spike train similarity and use persistent
homology to extract topological features that are fundamentally different from
those used in traditional methods. Our results show that a machine learning
classifier trained on these features can accurately predict the regime of the
network it was trained on and also generalize to other networks that were not
presented during training. Moreover, we demonstrate that using features
extracted from multiple spike-train distances systematically improves the
performance of our method.
