Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

Abstract

We study a deterministic version of a one- and two-dimensional attractor neural network model of hippocampal activity first studied by Itskov et al. [J. Neurosci., 31 (2011), pp. 2828--2834]. We analyze the dynamics of the system on the ring and torus domains with an even periodized weight matrix, assuming 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 nonconstant 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--H...