Stability, bifurcations and synchronization in a delayed neural network model of nn-identical neurons

This paper deals with dynamic behaviors of Hopfield type neural network model of n(≥3)n(≥3) identical neurons with two time-delayed connections coupled in a star configuration. Delay dependent as well as independent local stability conditions about trivial equilibrium is found. Considering synaptic weight and time delay as parameters Hopf-bifurcation, steady-state bifurcation and equivariant steady state bifurcation criteria are given. The criterion for the global stability of the system is presented by constructing a suitable Lyapunov functional. Also conditions for absolute synchronization about the trivial equilibrium are also shown. Using normal form method and the center manifold theory the direction of the Hopf-bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Numerical simulations are presented to verify the analytical results. The effect of synaptic weight and delay on different types of oscillations, e.g. in-phase, phase-locking, standing wave and oscillation death, has been shown numerically.