Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity

David Golomb*,† and
G. Bard Ermentrout‡

^*Zlotowski Center for Neuroscience and Department of Physiology, Faculty of Health Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel; and ^‡Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

Edited by Nancy J. Kopell, Boston University, Boston, MA, and approved August 20, 1999 (received for review May 18, 1999)

Abstract

Propagation of discharges in cortical and thalamic systems, which is used as a probe for examining network circuitry, is studied by constructing a one-dimensional model of integrate-and-fire neurons that are coupled by excitatory synapses with delay. Each neuron fires only one spike. The velocity and stability of propagating continuous pulses are calculated analytically. Above a certain critical value of the constant delay, these pulses lose stability. Instead, lurching pulses propagate with discontinuous and periodic spatio-temporal characteristics. The parameter regime for which lurching occurs is strongly affected by the footprint (connectivity) shape; bistability may occur with a square footprint shape but not with an exponential footprint shape. For strong synaptic coupling, the velocity of both continuous and lurching pulses increases logarithmically with the synaptic coupling strength g _syn for an exponential footprint shape, and it is bounded for a step footprint shape. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysmal discharges stem from their different effective delay; in thalamic networks, large effective delay between inhibitory neurons arises from their effective interaction via the excitatory cells which display postinhibitory rebound.

Footnotes

↵ † To whom reprint requests should be addressed at: Department of Physiology, Faculty of Health Sciences, P.O.B. 653, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. E-mail: golomb{at}bgumail.bgu.ac.il.
This paper was submitted directly (Track II) to the PNAS office.
↵ § The word “bistability” is used here in a broad sense, despite the fact that the system dynamics do not converge to an attractor.
↵ ¶ For g _syn > g _syn,min, there are two branches of solutions to Eq. 14. At the slow branch, ν decreases with g _syn (3, 6), and therefore this solution is not meaningful.
Abbreviations:

TC,

thalamocortical;

RE,

reticular;

EPSC,

excitatory postsynaptic conductance