##
**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
\[
\dot x_ i=a_ i(x_ i)[b_ i(x_ i)-\sum^{n}_{k=1}c_{ik}d_ k(x_ k)],\quad i=1,2,...,n,
\]
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 \(d_ i)\).

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 |