Title : A Mathematical Analysis of Development of Oriented Receptive Fields in

Abstract. We consider Linsker’s neural network model, in particular, development of oriented receptive fields of cells on layer G, and investigate which features in this model are crucial for the development. We assume that the correlation function is a Mexican-hat shape function while we introduce the arbor function explicitly and discuss its role in the development. We compute the Fourier transform of the modification rule of synaptic weights to analyze its modification process. Through our mathematical analysis, we obtain the following observations. (1) the Fourier transform of the correlation function extracts a set of Fourier coefficients of a certain frequency, (2) the Fourier transform of the set of those coefficients consists of a finite number of its components, (3) among those components, the arbor function extracts only some of those and others degenerate, (4) thus, the shape of a receptive field can be described as a combination of a certain Fourier cosine components and Fourier sine components, (5) the arbor function determines one of the set of the Fourier cosine components and that of the Fourier sine components dominates the other. We consider two types of the arbor function and analyze when oriented (e.g., bi-lobed) receptive fields are developed. These observations are also justified by computer simulations. Finally we discuss the role of Linsker’s parameters k1 and k2.