Font Size: 

For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, taking into account fundamental properties of the characteristic function of the random variable with probability density function given by the distributed delay kernel. In the corresponding parameter plane, the bifurcation curves are determined, as well as the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed, by employing the method of multiple times scales. The theoretical results are exemplified in the framework of a simple neural model.