  We study a deterministic version of a one- and two-dimensional attractor
neural network model of hippocampal activity first studied by Itskov et al
2011. We analyze the dynamics of the system on the ring and torus domain with
an even periodized weight matrix, assum- ing weak and slow spike frequency
adaptation and a weak stationary input current. On these domains, we find
transitions from spatially localized stationary solutions ("bumps") to
(periodically modulated) solutions ("sloshers"), as well as constant and
non-constant velocity traveling bumps depending on the relative strength of
external input current and adaptation. The weak and slow adaptation allows for
a reduction of the system from a distributed partial integro-differential
equation to a system of scalar Volterra integro-differential equations
describing the movement of the centroid of the bump solution. Using this
reduction, we show that on both domains, sloshing solutions arise through an
Andronov-Hopf bifurcation and derive a normal form for the Hopf bifurcation on
the ring. We also show existence and stability of constant velocity solutions
on both domains using Evans functions. In contrast to existing studies, we
assume a general weight matrix of Mexican-hat type in addition to a smooth
firing rate function.
