The effective number of shared dimensions between neural populations

A number of recent studies have explored the analysis of multi-area neural recording data and, in particular, the shared communication, computation and representational subspaces between brain areas. These have largely relied upon model-based methods to quantify shared variability between areas, but a standard, model-free measure is lacking. In this work we 1) present a candidate for this measure that we call the effective number of shared dimensions (ENSD), 2) demonstrate the utility and computational efficiency of our measure by visiting results from two recent studies (the "communication subspace" described by Semedo et al., 2019, and context dependent decision making described by Mante and Sussillo et al. 2013), and 3) use the measure and its decompositions to discover novel structure in an existing high dimensional olfactory dataset. The ENSD can be applied to data matrices sharing at least one dimension, reduces to the well-known participation ratio when both data sets are equivalent and has other robust and intuitive mathematical properties which we describe. We show that the ENSD allows for simple, model-free analysis of population-level metrics for multi-area by replicating the aforementioned experimental findings and using its constituent statistics to probe alignment in the parallel pathways of the fly olfactory system. This allows us to uncover a novel result, namely that there is an unexpected enrichment in the representation of food related odorants in the lateral horn that is directly related to the small structural bias of the mushroom body. This opens a new perspective on the interaction between innate and learned olfactory representations. Altogether, we show that the ENSD is an interpretable, computationally efficient and model-free dimensionality metric which promises to have applicability to a wide variety of datasets.