From spiking neurons to brain waves
Networks of spiking neurons generate rhythms of the brain. They allow efficient simulation but are less suitable for analysis. In the early seventies of the previous century, Wilson and Cowan  formulated a mean field equation for neural tissue. Precise validity results are lacking, yet there are many examples that show that such a reduction is meaningful. We will discuss the reduction process and we will show several examples of spiking neurons networks and there companion neural fields. A mathematical framework will be presented that is suitable for local bifurcation analysis of the neural field equations. This talk is based on joint work with Koen Dijkstra, Sebastiaan Janssens, Yuri Kuznetsov and Sid Visser.
 Wilson, H., Cowan, J.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical journal 12(1), 1–24 (1972).