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Zbl 0553.92009
Cohen, Michael A.; Grossberg, Stephen
Absolute stability of global pattern formation and parallel memory storage by competitive neural networks.
(English)
[J] IEEE Trans. Syst. Man Cybern. 13, 815-826 (1983). ISSN 0018-9472

Considered is a class of n-dimensional dynamical systems $$\dot x\sb i=a\sb i(x\sb i)[b\sb i(x\sb i)-\sum\sp{n}\sb{k=1}c\sb{ik}d\sb k(x\sb k)],\quad i=1,2,...,n,$$ where the matrix $C=[c\sb{ik}]$ 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. \par 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 $(x\sb i(t)$ is called suprathreshold at t if $x\sb i(t)>\Gamma\sp-\sb i$ where $\Gamma\sp-\sb i$ stands for inhibitory threshold of $d\sb i)$.
[W.Pedrycz]
MSC 2000:
*92Cxx Medical topics etc.
93D05 Lyapunov and other classical stabilities of control systems
92F05 Appl. of mathematics to other natural sciences
37-99 Dynamic systems and ergodic theory
93C15 Control systems governed by ODE

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

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