  Mean-field systems have been recently derived that adequately predict the
behaviors of large networks of coupled integrate-and-fire neurons [14]. The
mean-field system for a network of neurons with spike frequency adaptation is
typically a pair of differential equations for the mean adaptation and mean
synaptic gating variable of the network. These differential equations are
non-smooth, and in particular are piecewise smooth continuous (PWSC). Here, we
analyze the smooth and non-smooth bifurcation structure of these equations and
show that the system is organized around a pair of co-dimension two
bifurcations that involve, respectively, the collision between a Hopf
equilibrium point and a switching manifold, and a saddle-node equilibrium point
and a switching manifold. These two co-dimension 2 bifurcations can coalesce
into a co-dimension 3 non-smooth bifurcation. As the mean-field system we study
is a non-generic piecewise smooth continuous system, we discuss possible
regularizations of this system and how the bifurcations which occur are related
to non-smooth bifurcations displayed by generic PWSC systems.
