  Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.
