  Visual representations can be defined as the activations of neuronal
populations in response to images. The activation of a neuron as a function
over all image space has been described as a "tuning landscape". As a function
over a high-dimensional space, what is the structure of this landscape? In this
study, we characterize tuning landscapes through the lens of level sets and
Morse theory. A recent study measured the in vivo two-dimensional tuning maps
of neurons in different brain regions. Here, we developed a statistically
reliable signature for these maps based on the change of topology in level
sets. We found this topological signature changed progressively throughout the
cortical hierarchy, with similar trends found for units in convolutional neural
networks (CNNs). Further, we analyzed the geometry of level sets on the tuning
landscapes of CNN units. We advanced the hypothesis that higher-order units can
be locally regarded as isotropic radial basis functions, but not globally. This
shows the power of level sets as a conceptual tool to understand neuronal
activations over image space.
