Neural fields equations describe the activity of neural populations at a mesoscopic level. Although the early derivation of these equations introduced space dependent delays coming from the finite speed of signal propagation along axons, there has been few studies concerning their role in shaping the dynamics of neural activity. This is mainly due to the lack of analytical tractable models. On the other hand, constant delays have to be introduced to model the synaptic transmission and the spike initiation dynamics. By incorporating the two kind of delays in the neural fields equations, we are able to find the Hopf bifurcation curves analytically as well as the normal forms of the main bifurcations. This allows an in-depth study of two different types of connectivity that reveals a surprisingly rich dynamical portrait because the connectivity and the space dependent delays couple dynamically. In particular, the shape of the connectivity strongly influences the spatiotemporal dynamics.
