  Neural field equations are integro-differential systems describing the
macroscopic activity of spatially extended pieces of cortex. In such cortical
assemblies, the propagation of information and the transmission machinery
induce communication delays, due to the transport of information (propagation
delays) and to the synaptic machinery (constant delays). We investigate the
role of these delays on the formation of structured spatiotemporal patterns for
these systems in arbitrary dimensions. We focus on localized activity, either
induced by the presence of a localized stimulus (pulses) or by transitions
between two levels of activity (fronts). Linear stability analysis allows to
reveal the existence of Hopf bifurcation curves induced by the delays, along
different modes that may be symmetric or asymmetric. We show that instabilities
strongly depend on the dimension, and in particular may exhibit transversal
instabilities along invariant directions. These instabilities yield pulsatile
localized activity, and depending on the symmetry of the destabilized modes,
either produce spatiotemporal breathing or sloshing patterns.
