Stability Analysis of Complex- Valued Memristive Neural Networks with Delay-Dependent Discrete-Time Cases

Abstract

Theoretically, the memristor as the fourth basic circuit element, was firstly postulated by Chua based on the integral theory of fundamental circuit in 1971. It has the unique electrical characteristics relative to resistor, capacitor and inductor. In 2008, researchers at HP’s Laboratory implemented the physical model of the memristor, which means that it opens up new horizons for further development on circuit design. In 2009, the adaptive behavior of cells, which was similar to the property of the memristor, was proposed by means of the single-celled amoeba experiment. Based on the experimental verification, more research results show that artificial neural networks with variable weights constructed by the memristor can better simulate human brain like associative memory functions.

However, as an extension of RNNs, the main challenges we face are how to address the problems of complex-valued states and connection weights, especially complex-valued activation functions. Based on the Liouville’s theorem, the activation function in CNNs cannot be both bounded and analytic while it’s usually chosen to be a smooth bounded function in RNNs. The other way doesn’t need to divide into two parts but should satisfy the Lipschitz continuity. For example, some complex-valued activation functions can’t be divided into two parts, and some are discontinuous. As is known to us, when the system is discontinuous, it’s difficult to ensure the stability of system. On the other hand, it’s clear that the delay-dependent stability of neural networks(NNs) are less conservative than delay-independent ones, since time-delay phenomena are often encountered in various practical situations and may have negative effect on system stability.