Abstract
This paper introduces the Membrain model describing a neural network architecture which is similar to the architecture underlying the class of cellular neural networks (CNNs). the main difference pertains to the characteristic processing equation, which is based on a wave equation instead of a heat equation. Within the CNN framework, a cellular Membrain model may be obtained by replacing the neuron output function by a first‐order state equation. Furthermore, the network‐cloning templates are chosen such that the CNN behaves like a system of coupled harmonical oscillators. Since the energy of such a system is bounded, the piecewise linear neuron characteristic function may be chosen such that it always operates in the linear regime. Our starting point is the analytical and general solution for forced vibrations with damping. This solution applies to a Membrain neural network whose functional architecture is based on the specialized solution for a network of coupled harmonic oscillators. In particular, we present a Membrain CNN (MCNN) having a toroidal connection structure such that the natural modes of vibration of the net are translation‐invariant. Moreover, depending on the point group of the network, some rotation invariance can also be obtained. Identifying the input of such a network with the initial state of the oscillators gives rise to an output which is in essence a transversally travelling wave made up of components which are coupled harmonic neuronal oscillators; that is, the wave is a superposition of natural modes of vibration of the network. the temporal wave pattern may be transformed into a one‐dimensional temporal signal which is invariant under translation of the initial deflection pattern of the MCNN. the amplitudes of the components in the temporal signal correspond to the power spectrum of the natural vibration modes in the MCNN. Interpreting the initial deflection pattern as a grey‐level image, the temporal signal can be viewed as a modulation of a translation‐invariant ‘fingerprint’ of the image. the signal may be sampled such that the modulated ‘fingerprint’ can be classified using some of the traditional neural network models. In particular we show that (1) a self‐organizing feature map clusters correlated images and (2) a back‐propagation neural network extracts position‐invariant features.