Existence of binary invariant sets in feedback neural networks with application to synthesis

New accurate and flexible design procedure for a stable KWTA continuous time network

The classical continuous time recurrent (Hopfield) network is considered and adapted to K -winner-take-all operation. The neurons are of sigmoidal type with a controllable gain G, an amplitude m and interconnected by the conductance p. The network is intended to process one by one a sequence of lists, each of them with N distinct elements, each of them squeezed to [0,I] admission interval, each of them having an imposed minimum separation between elements z(min). The network carries out: 1) a matching dynamic process between the order of list elements and the order of outputs, and 2) a binary type steady-state separation between K and K+1 outputs, the former surpassing a +ξ threshold and the later falling under the -ξ threshold. As a result, the machine will signal the ranks of the K largest elements of the list. To achieve 1), the initial condition of processing phase has to be placed in a computable θ -vicinity of zero-state. This requires a resetting procedure after each list. To achieve 2) the bias current M has to be within a certain interval computable from circuit parameters. In addition, the steady-state should be asymptotically stable. To these goals, we work with high gain and exploit the sigmoid properties and network symmetry. The various inequality type constraints between parameters are shown to be compatible and a neat synthesis procedure, simple and flexible, is given for the tanh sigmoid. It starts with the given parameters N, K, I, z(min), m and computes simple bounds of p, G, ξ, θ, and M. Numerical tests and comments reveal qualities and shortcomings of the method.

Invariant set of weight of perceptron trained by perceptron training algorithm

In this paper, an invariant set of the weight of the perceptron trained by the perceptron training algorithm is defined and characterized. The dynamic range of the steady-state values of the weight of the perceptron can be evaluated by finding the dynamic range of the weight of the perceptron inside the largest invariant set. In addition, the necessary and sufficient condition for the forward dynamics of the weight of the perceptron to be injective, as well as the condition for the invariant set of the weight of the perceptron to be attractive, is derived.