  Fifty years ago, Wilson and Cowan developed a mathematical model to describe
the activity of neural populations. In this seminal work, they divided the
cells in three groups: active, sensitive and refractory, and obtained a
dynamical system to describe the evolution of the average firing rates of the
populations. In the present work, we investigate the impact of the often
neglected refractory state and show that taking it into account can introduce
new dynamics. Starting from a continuous-time Markov chain, we perform a
rigorous derivation of a mean-field model that includes the refractory
fractions of populations as dynamical variables. Then, we perform bifurcation
analysis to explain the occurance of periodic solutions in cases where the
classical Wilson-Cowan does not predict oscillations. We also show that our
mean-field model is able to predict chaotic behavior in the dynamics of
networks with as little as two populations.
