Global exponential stability of competitive neural networks with different time scales

Abstract

The dynamics of cortical cognitive maps developed by self-organization must include the aspects of long and short-term memory. The behavior of such a neural network is characterized by an equation of neural activity as a fast phenomenon and an equation of synaptic modification as a slow part of the neural system. We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point.

Passivity Analysis of Dynamic Neural Networks with Different Time-scales

“…All of these conditions are about the stability of neural networks. The stability of the neural networks with different time-scales are key problems of patterns storage in the short-term and the long-term memories [19] and solving optimization problem [16]. For example, we can design a neural network as in (3) to store the short-term and long-term patterns, when the weights satisfy the condition (10), this neural network can work.…”

Complete Convergence of Competitive Neural Networks with Different Time Scales

“…This shows that (t) is a monotone decreasing function. By the assumption that the solution of network (1) is bounded, it follows that the solutions of (5), (8) Proof. Using Lemma 3, from (7) and (9), it follows that lim t→+∞ x (t), S (t) = 0.…”

“…This class of neural networks has attracted extensive interest, see for examples [1,3,[7][8][9] and the references in them. Many of these studies are around single equilibrium point of the networks.…”

“…In [8], a model of competitive neural networks with different time scales has been proposed. This model describes the temporal evolution of the short term memory(STM) and long term memory(LTM) states.…”

Synchronization of unknown chaotic delayed competitive neural networks with different time scales based on adaptive control and parameter identification

“…Recently, Meyer-Bäse et al [13][14][15] proposed the so-called competitive neural networks with different time scales. In the competitive neural networks model, there are two types of state variables: the short-term memory (STM) variable describing the fast neural activity, and the long-term memory (LTM) variable describing the slow unsupervised synaptic modifications.…”