Multistability of Neural Networks with a Class of Activation Functions

Authors:
Lili Wang;Wenlian Lu;Tianping Chen
Affiliations:
Key Laboratory of Nonlinear Science of Chinese Ministry of Education, School of Mathematical Sciences, Fudan University, Shanghai, P.R.China 200433;Key Laboratory of Nonlinear Science of Chinese Ministry of Education, School of Mathematical Sciences, Fudan University, Shanghai, P.R.China 200433;Key Laboratory of Nonlinear Science of Chinese Ministry of Education, School of Mathematical Sciences, Fudan University, Shanghai, P.R.China 200433
Venue:
ISNN '09 Proceedings of the 6th International Symposium on Neural Networks on Advances in Neural Networks
Year:
2009

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Abstract

In this paper, we investigate the multistability of neural networks with a class of activation functions, which are nondecreasing piecewise linear with 2r (r *** 1) corner points. It shows that the n -neuron neural networks can have and only have (2r + 1) n equilibria under some conditions, (r + 1) n of which are locally exponentially stable and others are unstable. In addition, we discuss the attraction basins of the stable equilibria for the two-dimensional case and found out that under several conditions, the stable manifolds of the unstable equilibria precisely comprise of the bounds of each attractor.