## Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

preprint

posted on 15.02.2006, 10:36 by P.C. Bressloff#### preprint

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We study the existence and stability of traveling waves and pulses in a one-dimensional
network of integrate-and-fire neurons with synaptic coupling. This provides a simple model
of excitable neural tissue. We first derive a self-consistency condition for the existence of
traveling waves, which generates a dispersion relation between velocity and wavelength. We
use this to investigate how wave-propagation depends on various parameters that characterize
neuronal interactions such as synaptic and axonal delays, and the passive membrane
properties of dendritic cables. We also establish that excitable networks support the propagation
of solitary pulses in the long-wavelength limit. We then derive a general condition
for the (local) asymptotic stability of traveling waves in terms of the characteristic equation
of the linearized firing time map, which takes the form of an integro-difference equation of
infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength
limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution
stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the
stable wave can itself destabilize via a Hopf bifurcation.

### History

#### School

- Science

#### Department

- Mathematical Sciences