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Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. (English) Zbl 0553.92009

Considered is a class of n-dimensional dynamical systems

${\stackrel{˙}{x}}_{i}={a}_{i}\left({x}_{i}\right)\left[{b}_{i}\left({x}_{i}\right)-\sum _{k=1}^{n}{c}_{ik}{d}_{k}\left({x}_{k}\right)\right],\phantom{\rule{1.em}{0ex}}i=1,2,···,n,$

where the matrix $C=\left[{c}_{ik}\right]$ is symmetric and the system as a whole is competitive. Several examples of applications of this type of equations are indicated as nonlinear neural networks and, in general, global pattern formation.

A global Lyapunov function for the system discussed is introduced. Its absolute stability with infinite but totally disconnected equilibrium points is studied by the LaSalle invariance principle. Decomposition of equilibria of the system into suprathreshold and subthreshold variables is also presented $\left({x}_{i}\left(t\right)$ is called suprathreshold at t if ${x}_{i}\left(t\right)>{{\Gamma }}_{i}^{-}$ where ${{\Gamma }}_{i}^{-}$ stands for inhibitory threshold of ${d}_{i}\right)$.

Reviewer: W.Pedrycz

##### MSC:
 92Cxx Physiological, cellular and medical topics 93D05 Lyapunov and other classical stabilities of control systems 92F05 Applications of mathematics to other natural sciences 37-99 Dynamic systems and ergodic theory (MSC2000) 93C15 Control systems governed by ODE