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

Cohen, Michael A.; Grossberg, Stephen

IEEE Trans. Syst. Man Cybern. 13, 815-826 (1983).

Considered is a class of n-dimensional dynamical systems \[ \dot x_ i=a_ i(x_ i)[b_ i(x_ i)-\sum^{n}_{k=1}c_{ik}d_ k(x_ k)],\quad i=1,2,...,n, \] where the matrix \(C=[c_{ik}]\) 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 \((x_ i(t)\) is called suprathreshold at t if \(x_ i(t)>\Gamma^-_ i\) where \(\Gamma^-_ i\) stands for inhibitory threshold of \(d_ i)\).

Reviewer: W.Pedrycz

MSC:

92Cxx	Physiological, cellular and medical topics
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
92F05	Other natural sciences (mathematical treatment)
37-XX	Dynamical systems and ergodic theory
93C15	Control/observation systems governed by ordinary differential equations

Keywords:

parallel memory storage; self-organizing networks; nonlinear neural networks; global pattern formation; global Lyapunov function; absolute stability; LaSalle invariance principle; Decomposition of equilibria; suprathreshold; subthreshold