  We study localized patterns in an exact mean-field description of a
spatially-extended network of quadratic integrate-and-fire (QIF) neurons. We
investigate conditions for the existence and stability of localized solutions,
so-called bumps, and give an analytic estimate for the parameter range where
these solutions exist in parameter space, when one or more microscopic network
parameters are varied. We develop Galerkin methods for the model equations,
which enable numerical bifurcation analysis of stationary and time-periodic
spatially-extended solutions. We study the emergence of patterns composed of
multiple bumps, which are arranged in a snake-and-ladder bifurcation structure
if a homogeneous or heterogeneous synaptic kernel is suitably chosen.
Furthermore, we examine time-periodic, spatially-localized solutions
(oscillons) in the presence of external forcing, and in autonomous, recurrently
coupled excitatory and inhibitory networks. In both cases we observe period
doubling cascades leading to chaotic oscillations.
