  Bump attractors are wandering localised patterns observed in in vivo
experiments of spatially-extended neurobiological networks. They are important
for the brain's navigational system and specific memory tasks. A bump attractor
is characterised by a core in which neurons fire frequently, while those away
from the core do not fire. We uncover a relationship between bump attractors
and travelling waves in a classical network of excitable, leaky
integrate-and-fire neurons. This relationship bears strong similarities to the
one between complex spatiotemporal patterns and waves at the onset of pipe
turbulence. Waves in the spiking network are determined by a firing set, that
is, the collection of times at which neurons reach a threshold and fire as the
wave propagates. We define and study analytical properties of the voltage
mapping, an operator transforming a solution's firing set into its
spatiotemporal profile. This operator allows us to construct localised
travelling waves with an arbitrary number of spikes at the core, and to study
their linear stability. A homogeneous "laminar" state exists in the network,
and it is linearly stable for all values of the principal control parameter.
Sufficiently wide disturbances to the homogeneous state elicit the bump
attractor. We show that one can construct waves with a seemingly arbitrary
number of spikes at the core; the higher the number of spikes, the slower the
wave, and the more its profile resembles a stationary bump. As in the
fluid-dynamical analogy, such waves coexist with the homogeneous state, are
unstable, and the solution branches to which they belong are disconnected from
the laminar state; we provide evidence that the dynamics of the bump attractor
displays echoes of the unstable waves, which form its building blocks.
