Existence and Wandering of Bumps in a Spiking Neural Network Model

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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.