  We consider a large network of Izhikevich neurons. Each neuron has a
quadratic integrate-and-fire type model with a recovery variable modelling
spike frequency adaptation (SFA). We introduce a biologically motivated
synaptic current expression and a delay in the synaptic transmission. Following
the Ott-Antonsen theory, we reduce the network model to a mean-field system of
delayed differential equations. Numerical bifurcation analysis allows us to
locate higher-codimension bifurcations and to identify the regions in the
parameter space where the network exhibits changes in the macroscopic dynamics,
including transitions between states where the individual neurons exhibit
asynchronous tonic firing and different types of synchronous bursting. We
investigate the impact of the heterogeneity of the quenched input current, the
SFA mechanism and the synaptic delay on macroscopic dynamics. In the limit that
the heterogeneity goes to zero, our perturbation and bifurcation analysis shows
that the behaviour of the mean-field model remains consistent, although this
limit breaks an assumption of the model reduction. For a single population of
neurons with SFA, the synaptic delay has little effect on the generation of COs
for weak coupling, but favours their emergence beyond that, and even induces
new macroscopic dynamics. In particular, Torus bifurcations may occur, and
these are a crucial mechanism for the emergence of population bursting with two
nested frequencies. We discuss how these solutions may relate to
cross-frequency coupling which is potentially relevant for understanding
healthy and pathological brain functions.
