Delay-induced Hopf bifurcation in a noise-driven excitable neuron model

Abstract

A FitzHugh–Nagumo neuron model with cubic nonlinearity and discrete delay is considered, in which the time delay is regarded as a parameter. The effect of time delay on the linear stability and Hopf bifurcation of the model is studied. The existence, stability and direction of the local and global Hopf bifurcation are derived. Some numerical simulations are employed to validate the main results of this work.

Keywords:

• FitzHugh–Nagumo model
• Hopf bifurcation
• normal forms
• centre manifold theory
• discrete delay

2010 AMS Subject Classifications :

• 37L10
• 37N25

Acknowledgements

The authors will be very grateful to the referees for their valuable suggestions, which help to improve the paper significantly. This work is supported partly by the China Postdoctoral Science Foundation (Grant No. 20100470783), National Natural Science Foundation of China (Grant No. 60904063) and Specialized Research Fund for Doctoral Programme of Higher Education from Ministry of Education of China (Grant No. 20090032110031).