Stability and bifurcation in a tri-neuron network model with discrete and distributed delays

In this paper, we consider a tri-neuron network model with discrete and distributed delays. After discussing the distributions of the roots of the characteristic equation local stability analysis of the steady-state solution leads to, we analyze the stability of network models with instantaneous feedback and neural interaction history, delayed neural feedback and no neural interaction history and delayed neural feedback and neural interaction history; we give sufficient conditions for the linear stability of the equilibrium solution; we also show that a Hopf bifurcation can occur when the delays take certain critical values; and we further prove that there is a positive integer K such that there are K switches from stability to instability and back to stability. Numerical simulations are finally performed to illustrate the obtained results.