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We investigate the existence and dynamical behaviors of multiple equilibria for competitive neural networks with a class of general Mexican-hat-type activation functions. The Mexican-hat-type activation functions are not monotonously increasing, and the structure of neural networks with Mexican-hat-type activation functions is totally different from those with sigmoidal activation functions or nondecreasing saturated activation functions, which have been employed extensively in previous multistability papers. By tracking the dynamics of each state component and applying fixed point theorem and analysis method, some sufficient conditions are presented to study the multistability and instability, including the total number of equilibria, their locations, and local stability and instability. The obtained results extend and improve the very recent works. Two illustrative examples with their simulations are given to verify the theoretical analysis.

Multistability and Instability of Competitive Neural Networks with Mexican-Hat-Type Activation Functions