Scalar Reduction of a Neural Field Model with Spike Frequency Adaptation

TL;DR

This paper analyzes a neural field model with spike frequency adaptation, reducing it to scalar equations to study solution dynamics, bifurcations, and stability on ring and torus domains, considering general weight matrices.

Contribution

It introduces a scalar reduction method for a neural field model with adaptation, enabling detailed analysis of solution behaviors and bifurcations on complex domains.

Findings

Bumps can transition to sloshing solutions via Hopf bifurcation.

Constant velocity bumps are shown to be stable under certain conditions.

The reduction simplifies analysis of complex neural dynamics.

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 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.