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On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays. (English) Zbl 1098.37070
Summary: We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then, we apply the results obtained 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.
MSC:
37N25Dynamical systems in biology
92B20General theory of neural networks (mathematical biology)
34K20Stability theory of functional-differential equations
37G99Local and nonlocal bifurcation theory