Neural field models describe the coarse-grained activity of populations of
interacting neurons. Because of the laminar structure of real cortical
tissue they are often studied in two spatial dimensions, where they are well
known to generate rich patterns of spatiotemporal activity. Such patterns
have been interpreted in a variety of contexts ranging from the
understanding of visual hallucinations to the generation of
electroencephalographic signals. Typical patterns include localised
solutions in the form of travelling spots, as well as intricate labyrinthine
structures. These patterns are naturally defined by the interface between
low and high states of neural activity. Here we derive the equations of
motion for such interfaces and show, for a Heaviside firing rate, that the
normal velocity of an interface is given in terms of a non-local Biot-Savart
type interaction over the boundaries of the high activity regions. This
exact, but dimensionally reduced, system of equations is solved numerically
and shown to be in excellent agreement with the full nonlinear integral
equation defining the neural field. We develop a linear stability analysis
for the interface dynamics that allows us to understand the mechanisms of
pattern formation that arise from instabilities of spots, rings, stripes and
fronts. We further show how to analyse neural field models with
linear adaptation currents, and determine the conditions for the dynamic
instability of spots that can give rise to breathers and travelling waves.
We end with a discussion of amplitude equations for analysing behaviour in
the vicinity of a bifurcation point (for smooth firing rates). The condition
for a drift instability is derived and a center manifold reduction is used
to describe a slowly moving spot in the vicinity of this bifurcation. This
analysis is extended to cover the case of two slowly moving spots, and
establishes that these will reflect from each other in a head-on collision.