Abstract:
Ever since the seminal experiments by Hubel and Wiesel, cells in primary visual cortex are conceived as edge or line detectors. Based on the degree of invariance with respect to phase shift of their preferred stimulus, they are categorized as simple and complex cells. Their receptive fields have been shown to be selective for a variety of stimulus properties, e.g. for orientation and spatial frequency.
Recently, Berkes and Wiskott (Journal of Vision, 2005) demonstrated that the unsupervised learning principle of temporal slowness can account for a wide range of complex cell properties, including optimal stimuli, phase shift invariance and orientation and frequency selectivity. The structure of the simulated receptive fields was shown to crucially depend on the transformations present in the image sequences used for training while being largely independent of the statistics of natural images.
Using this observation as a starting point, we develop a mathematical framework for the simulations, which is based on the Lie group of the transformations in the training data. We show that the optimal receptive fields are the solutions of a partial differential eigenvalue equation, which can in certain cases be solved analytically. The properties of the resulting non-linear receptive fields are in agreement with those of simulated and physiological cells.
The theory demonstrates that the results of the simulations can be largely understood analytically and provides an intuitive explanation why the simulated receptive fields are optimal for temporal slowness learning.