  A central question for neuroscience is how to characterize brain
representations of perceptual and cognitive content. An ideal characterization
should distinguish different functional regions with robustness to noise and
idiosyncrasies of individual brains that do not correspond to computational
differences. Previous studies have characterized brain representations by their
representational geometry, which is defined by the representational
dissimilarity matrix (RDM), a summary statistic that abstracts from the roles
of individual neurons (or responses channels) and characterizes the
discriminability of stimuli. Here we explore a further step of abstraction:
from the geometry to the topology of brain representations. We propose
topological representational similarity analysis (tRSA), an extension of
representational similarity analysis (RSA) that uses a family of
geo-topological summary statistics that generalizes the RDM to characterize the
topology while de-emphasizing the geometry. We evaluate this new family of
statistics in terms of the sensitivity and specificity for model selection
using both simulations and fMRI data. In the simulations, the ground truth is a
data-generating layer representation in a neural network model and the models
are the same and other layers in different model instances (trained from
different random seeds). In fMRI, the ground truth is a visual area and the
models are the same and other areas measured in different subjects. Results
show that topology-sensitive characterizations of population codes are robust
to noise and interindividual variability and maintain excellent sensitivity to
the unique representational signatures of different neural network layers and
brain regions. These methods enable researchers to calibrate comparisons among
representations in brains and models to be sensitive to the geometry, the
topology, or a combination of both.
