On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays

In this paper, we first study the distribution of the zeros of a fourth degree exponential polynomial. Then we apply the obtained results to a neural network model consisting of four neurons with delays. By regarding the sum of the delays as a parameter, it is shown that under certain assumptions the steady state of the neural network model is absolutely stable. Under another set of conditions, there is a critical value of the delay, the steady state is stable when the parameter is less than the critical value and unstable when the parameter is greater than the critical value. Thus, oscillations via Hopf bifurcation occur at the steady state when the parameter passes through the critical value. Numerical simulations are presented to illustrate the results