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:
| 37N25 | Dynamical systems in biology |
| 92B20 | Neural networks for/in biological studies, artificial life and related topics |
| 34K20 | Stability theory of functional-differential equations |
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
Keywords:
distribution of the zeros; neural network model; delays; critical value; oscillations; Hopf bifurcationReferences:
| [1] | Campbell, S. A., Stability and bifurcation of a simple neural network with multiple time delays, Fields Inst Commun, 21, 65-79 (1999) · Zbl 0926.92003 |
| [2] | Campbell, S. A.; Ruan, S.; Wei, J., Qualitative and analysis of a neural network model with multiple time delays, Int J Bifurcat Chaos, 9, 1585-1995 (1999) · Zbl 1192.37115 |
| [3] | Cooke, K. L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J Math Anal Appl, 86, 592-627 (1982) · Zbl 0492.34064 |
| [4] | Dieudonné, J., Foundations of modern analysis (1960), Academic Press: Academic Press New York · Zbl 0100.04201 |
| [5] | Hale, J. K.; Infante, E. F.; Tsen, F.-S. P., Stability in linear delay equations, J Math Anal Appl, 105, 533-555 (1985) · Zbl 0569.34061 |
| [6] | Ruan, S.; Wei, J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn Continuous Discr Impuls Syst Ser A: Math Anal, 10, 863-874 (2003) · Zbl 1068.34072 |
| [7] | Ruan, S.; Wei, J., On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion, IMA J Math Appl Medic Biol, 18, 41-52 (2001) · Zbl 0982.92008 |
| [8] | Wu, J., Introduction to neural dynamics and signal transmission delay (2001), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 0977.34069 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
