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Multistability and convergence in delayed neural networks. (English) Zbl 1132.34058

The authors investigate a general delayed neural network
\[ \frac{dx_i(t)}{dt}=-\mu_i x_i(t)+\sum_{j=1}^n\alpha_{ij}g_j(x_j(t))+ \sum_{j=1}^n\beta_{ij}g_j(x_j(t-\tau_{ij}))+I_i, \]
where \(i=1,\dots,n\); \(\mu_i>0\); \(\alpha_{ij}\) and \(\beta_{ij}\) are connection weights from neuron \(j\) to neuron \(i\); \(g_j(\cdot)\) are activation functions; \(0\leq\tau_{ij}\leq\tau\) are time lags; \(I_i\) stands for an independent bias current source. Two classes of activation function are considered. Class \(A\) contains bounded smooth sigmoidal functions, and class \(B\) contains nondecreasing functions with saturations.
The authors derive conditions for the existence of \(3^n\) equilibria. The parameter conditions motivated by a geometrical observation. The basins of attraction for \(2^n\) stable stationary solutions are established. The theory is extended to the existence of \(2^n\) limit cycles for the \(n\)-dimensional network with time-periodic inputs. The strongly order preserving property and quasiconvergence are also discussed. The study illustrates distinct dynamical behaviors between systems with activation functions of different classes. Numerical simulations are presented to illustrate the presented theory.
This paper is an extention of the recent paper by authors [SIAM J. Appl. Math. 66, No. 4, 1301–1320 (2006; Zbl 1106.34048)].

MSC:

34K25 Asymptotic theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics

Citations:

Zbl 1106.34048
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References:

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