Asymptotic Stability of a Class of Neutral Delay Neuron System in a Critical Case

In this brief, the asymptotic stability properties of a neutral delay neuron system are studied mainly in a critical case when the exponential stability is not possible. If a critical value of the coefficient in the neutral delay neuron system is considered, then the difficulty for our investigation is caused by the fact that the spectrum of the linear operator is asymptotically approximated to the imaginary axis. It is obvious that, in such a case, the equation is not exponentially stable, and one needs more subtle methods in order to characterize this type of asymptotic stability. Hence, first, the local asymptotic stability for the neutral delay neuron system is studied, and the main tools involved are the asymptotic expansions of characteristic roots, Laplace transforms, and function series, and a complete analysis of the stability diagram is also presented. Then, based on the energy method, the globally asymptotic stability results for the neutral delay neuron system are derived.