On the Level Sets and Invariance of Neural Tuning Landscapes

Visual perception results from the activation of neuronal populations, a process mirrored by hidden units in artificial neural networks (ANNs). 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 robust signature for these maps based on the change of topology in level sets. We found this topological signature changes progressively throughout the cortical hierarchy. Further, we analyzed the tuning landscapes of ANN units. By measuring the geometry of level sets, we advance 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.