Multistability analysis of recurrent neural networks with a new type of activation functions
In this paper,the multistability of Hopfield neural network was studied based on a new type of activation function,the continuous sawtooth activation function.Firstly,the n-neural network model based on this activation function was proved to have at least 7n equilibrium points through interval partition method and Brouwer's fixed point theorem.By using the diagonally dominant matrix theory and local stability judgment theorem,4n equilibrium points were proved to be locally exponentially stable while the remaining equilibrium points to be unstable.Secondly,the activation function was generalized to a more general situation by adding the peak points of the sawtooth activation function.The n-neural network was found to have at least(2k+1)n equilibrium points in the continuous sawtooth activation function with k peak points,of which(k+1)n equilibrium points are local stable.Compared with the existing activation functions,the activation function designed in this paper can generate more stable equilibrium points and will not increase the computational complexity of the neural network in the process of increasing the number of peak points.Finally,two numerical examples were presented to demonstrate the validity of the results.
Hopfield neural networkmultistabilityequilibrium pointcontinuous sawtooth-type activation function
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