Multiperiodicity and Attractivity of Delayed Recurrent Neural Networks With Unsaturating Piecewise Linear Transfer Functions

Multistability is a property necessary in neural networks in order to enable certain applications (e.g., decision making), where monostable networks can be computationally restrictive. This article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions. It deals fully with the three basic properties of a multistable network: boundedness, global attractivity, and complete convergence. This article makes the following contributions: conditions based on local inhibition are derived that guarantee boundedness of some multistable networks, conditions are established for global attractivity, bounds on global attractive sets are obtained, complete convergence conditions for the network are developed using novel energy-like functions, and simulation examples are employed to illustrate the theory thus developed.

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Qualitative analysis for recurrent neural networks with linear threshold transfer functions

IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications ◽

10.1109/tcsi.2005.846664 ◽

2005 ◽

Vol 52 (5) ◽

pp. 1003-1012 ◽

Cited By ~ 24

Author(s):

K.C. Tan ◽

HuaJin Tang ◽

Weinian Zhang

Keyword(s):

Neural Networks ◽

Qualitative Analysis ◽

Recurrent Neural Networks ◽

Transfer Functions ◽

Linear Threshold

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Dynamical Stability Conditions for Recurrent Neural Networks with Unsaturating Piecewise Linear Transfer Functions

Neural Computation ◽

10.1162/08997660152469350 ◽

2001 ◽

Vol 13 (8) ◽

pp. 1811-1825 ◽

Cited By ~ 73

Author(s):

Heiko Wersing ◽

Wolf-Jürgen Beyn ◽

Helge Ritter

Keyword(s):

Recurrent Neural Networks ◽

Transfer Functions ◽

Piecewise Linear ◽

Recurrent Networks ◽

Simple Interpretation ◽

Local Inhibition ◽

Linear Threshold ◽

Selectivity Model ◽

Inhibitory Interactions ◽

Symmetric Networks

We establish two conditions that ensure the nondivergence of additive recurrent networks with unsaturating piecewise linear transfer functions, also called linear threshold or semilinear transfer functions. As Hahn-loser, Sarpeshkar, Mahowald, Douglas, and Seung (2000) showed, networks of this type can be efficiently built in silicon and exhibit the coexistence of digital selection and analog amplification in a single circuit. To obtain this behavior, the network must be multistable and nondivergent, and our conditions allow determining the regimes where this can be achieved with maximal recurrent amplification. The first condition can be applied to nonsymmetric networks and has a simple interpretation of requiring that the strength of local inhibition match the sum over excitatory weights converging onto a neuron. The second condition is restricted to symmetric networks, but can also take into account the stabilizing effect of nonlocal inhibitory interactions. We demonstrate the application of the conditions on a simple example and the orientation-selectivity model of Ben-Yishai, Lev Bar-Or, and Sompolinsky (1995). We show that the conditions can be used to identify in their model regions of maximal orientation-selective amplification and symmetry breaking.

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