NEURAL NETWORKS AS SET-VALUED DYNAMICAL-SYSTEMS AND THE UNIVERSALITY OF THE WINDOWED FOURIER-TRANSFORM

The visual pathway and other brain structures consist of a large numbe r of layers of neurons. At each point of a three-dimensional laminated structure there exists a direction ri that is perpendicular to the la yers. Assuming an information flow from top to bottom, the perceptive field of the neurons grows as one moves in the direction A. This enabl es the system to perform a multiscale analysis. Suppose that the densi ty of the connections between adjacent layers is distributed by a Gaus sian function and the autocorrelation Q of the input is of the form Q( x, x') = Q(\x - x'\) (i.e., shift invariant). Then it is shown that th e laminated system does indeed converge to some universal attractor. U nder certain conditions, the universal attractor takes the form of the Gabor filter (windowed Fourier transform). This enables the net to co mbine multiscale resolution with spectral analysis over a small portio n of the global receptive field. Under more general conditions the tra nsition from layer i to layer i + 1 is given by a set-valued dynamical system, and partial results on its global behavior are given.