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 number of layers of neurons. At each point of a three-dimensional laminated structure there exists a direction ri that is perpendicular to the layers. Assuming an information flow from top to bottom, the perceptive field of the neurons grows as one moves in the direction A. This enables the system to perform a multiscale analysis. Suppose that the density of the connections between adjacent layers is distributed by a Gaussian 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 the laminated system does indeed converge to some universal attractor. Under certain conditions, the universal attractor takes the form of the Gabor filter (windowed Fourier transform). This enables the net to combine multiscale resolution with spectral analysis over a small portion of the global receptive field. Under more general conditions the transition from layer i to layer i + 1 is given by a set-valued dynamical system, and partial results on its global behavior are given.