Analysis of a correlation-based model for the development of orientation-selective receptive fields in the visual cortex

We analyse a model for the development of orientation-selective receptive fields of simple cells in a locally connected network of cortical neurons. The Hebbian learning rule that underlies the development is described by a linear differential equation. The structure of the emerging cortical map can be predicted by deriving the eigenfunctions corresponding to the leading eigenvalues of the associated matrix. We show that the receptive fields have the typical form of a wavelet. Mathematically, receptive fields are given by a Hermitian polynomial with Gaussian cut-off and a phase factor. Both the phase of the wavelet and the orientation are changing periodically along the surface of the cortical map as suggested by previous simulation studies and as also found in experiments. In order to get orientation-selective receptive fields, the spatial correlation function of the inputs that drive the development must have a zero crossing.