On a Planar System Modelling a Neuron Network with Memory

We consider a delay differential equation modelling a network of two neurons with memory. The situations without self-connections and two delays, and with self-connections and two equal delays, are addressed in terms of local stability and bifurcation analysis. In the first case, the dynamical behavior is studied by taking one of the delays as the bifurcating parameter. Namely, the Hopf bifurcation, whose existence was proven in former works, occurs as the delay crosses some critical values and is completely described. Conditions ensuring the stability of the periodic cycles are given. When self-connections are present, Hopf and pitchfork bifurcations co-exist and the equation describing the flow on the center manifold is derived.