Compression and Model Reduction: A Case Study

Abstract

We discuss a method by which the dynamics of a network of coupled neurons can be captured in a one-dimensional map. The network used as an example of this technique consists of a pair of neurons, one of which is an endogenous burster and the other excitable, but not bursting in the absence of phasic input. The reduction is accomplished by decomposing the flow into fast and slow subsystems, each operating on a distinct time scale. A "map of knees" is constructed using singular perturbation techniques. A concise expression for this map is developed by introducing time coordinates to each stable branch of the slow manifold. The compression associated with the fast subsystem is used to determine the qualitative properties of the map. 1 Introduction In this paper we illustrate, by way of example, a method to capture the dynamics of a network of voltage-gated conductance equations in a low-dimensional map. This technique utilizes ideas from geometric singular perturbation theory to decompos..