Stability Switches and Hopf Bifurcation of a Neuron System with both Leakage and Distributed Delays

This paper focuses on the effect of leakage delays on the dynamics of nonlinear delay systems based on a neuron system with discrete and distributed delay. By introducing a virtual neuron, we can eliminate the impact of distributed delay and transform the original neuron system with mixed delays to an equivalent new system involving only discrete delay. This paper extends the existing works on neural networks to the case of leakage delays. We choose the leakage delay as the bifurcation parameter and analyze the role of leakage delays on the stability and Hopf bifurcation. Sufficient conditions for ensuring the neuron system to be stable, and undergoing the Hopf bifurcation are derived. Finally, by using the software package DDE-BIFTOOL, we provide the simulation results to substantiate our theoretical analysis, and give the relationship between the onset of Hopf bifurcation and the system parameters by drawing the bifurcation curves.