Existence and Wandering of Bumps in a Spiking Neural Network Model

Carson C Chow,S Coombes

doi:10.1137/060654347

Abstract

We study spatially localized states of a spiking neuronal network populated by a pulse‐coupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For nonslow synaptic connections we are able to go beyond the standard firing rate analysis of localized solutions, allowing us to explicitly construct a family of coexisting one‐bump solutions and then track bump width and firing pattern as a function of system parameters. We also present an analysis of the model on a discrete lattice. We show that multiple width bump states can coexist, and uncover a mechanism for bump wandering linked to the speed of synaptic processing. Moreover, beyond a wandering transition point we show that the bump undergoes an effective random walk with a diffusion coefficient that scales exponentially with the rate of synaptic processing and linearly with the lattice spacing.

Highlights

A goal of theoretical neuroscience is to develop a tractable model of a spiking neuronal network
We show that in the limit of slow synaptic interactions it reduces to the classic Wilson-Cowan and Amari firing rate models
It displays some of the complex properties that have been observed in simulations of spiking neuronal networks

Summary

Introduction

A goal of theoretical neuroscience is to develop a tractable model of a spiking neuronal network. A detailed investigation into the network dynamics of the lighthouse model may pave the way to the development of a specific soluble spiking neurodynamics With this in mind we turn our attention to spatially localized bumps of persistent activity, which have been linked to working memory (the temporary storage of information within the brain) [12, 13, 14]. The most popular mathematical formulations of such models assume long-range inhibition with local recurrent excitation and invoke a population level description in terms of a rate model (see for example [15, 16]) Interesting in their own right and s 6.–*/ed by a number of authors (surveyed in [17]), such models are only useful for describing systems with slow synaptic interactions.

C C CHOW AND S COOMBES

Discussion