Computational Topology in Neuroscience

Computational topology is a set of algorithmic methods developed to understand topological invariants such as loops and holes in high-dimensional data sets. In particular, a method know as persistent homology has been used to understand such shapes and their persistence in point clouds and networks. It has only been applied to neuronal networks in recent years. While most tools from network science focus solely on local properties based on pairwise connections, the topological tools reveal more global features. We apply persistent homology to neuronal networks to see which properties these tools can uncover, which might be invisible to existing methods. We give an introduction to relevant concepts from algebraic topology such as topological spaces, simplicial complexes and filtrations. Filtrations are the main ingredients for methods from persistent homology and can be imagined as an embedded sequence of networks with some form of geometrical object built from the edges and nodes in each sequence step. We use three different filtrations: a filtration by weights, a weight rank clique filtration and a modified version of the Vietoris-Rips complex to analyse networks. Our example networks consist of data from neuroscientific experiments and the output of a non-linear oscillator model, the Kuramoto model. Our results reveal that all three methods can be used to investigate different aspects of such networks, but that the methods and their interpretation still need to be developed further. In particular, computational scaling needs to be improved on the more sophisticated methods so that they can be used on networks of a reasonably large size and density.