Mathematical analysis of a correlation-based model for orientation map formation

We consider a correlation-based model for the orientation map formation proposed by Miller [Journal of Neuroscience 14 (1994) 409] and study the formation mathematically. We perform the Fourier transform and compute the principal component of the model. With our analysis, the roles of functions are clarified and the result indicates that the developed orientation maps have the following properties. (1) Maps have oriented receptive fields. (2) Preferred orientations smoothly change on the cortical surface. (3) Periodicity of preferred orientations does not appear. (4) Periodicity of phases appears. (5) Singular points appear irregularly on the cortical surface. Our analytical results are justified by computer simulations.