  We consider large networks of theta neurons and use the Ott/Antonsen ansatz
to derive degree-based mean field equations governing the expected dynamics of
the networks. Assuming random connectivity we investigate the effects of
varying the widths of the in- and out-degree distributions on the dynamics of
excitatory or inhibitory synaptically coupled networks, and gap junction
coupled networks. For synaptically coupled networks, the dynamics are
independent of the out-degree distribution. Broadening the in-degree
distribution destroys oscillations in inhibitory networks and decreases the
range of bistability in excitatory networks. For gap junction coupled neurons,
broadening the degree distribution varies the values of parameters at which
there is an onset of collective oscillations. Many of the results are shown to
also occur in networks of more realistic neurons.
