  We study localised activity patterns in neural field equations posed on the
Euclidean plane; such models are commonly used to describe the coarse-grained
activity of large ensembles of cortical neurons in a spatially continuous way.
We employ matrix-free Newton-Krylov solvers and perform numerical continuation
of localised patterns directly on the integral form of the equation. This opens
up the possibility to study systems whose synaptic kernel does not lead to an
equivalent PDE formulation. We present a numerical bifurcation study of
localised states and show that the proposed models support patterns of activity
with varying spatial extent through the mechanism of homoclinic snaking. The
regular organisation of these patterns is due to spatial interactions at a
specific scale associated with the separation of excitation peaks in the chosen
connectivity function. The results presented form a basis for the general study
of localised cortical activity with inputs and, more specifically, for
investigating the localised spread of orientation selective activity that has
been observed in the primary visual cortex with local visual input.
