Topological Neuroscience: Linking Circuits to Function

We review recent developments of the use of topology in neuroscience. From grid cells and head direction cells to the geometry of olfactory space, modern applied topology methods such as persistent homology are increasingly being used to study neural circuits and perception. In addition to outlining the big picture and reviewing various applications of topological data analysis (TDA) to neuroscience, we take a deep dive into the basic homology computation to make the underlying mathematics more accessible to neuroscientists. A discussion of practical considerations and pointers to TDA software are also included.