Effects of delay on the type and velocity of travelling pulses in neuronalnetworks with spatially decaying connectivity

We study a one-dimensional model of integrate-and-fire neurons that are all owed to fire only one spike, and are coupled by excitatory synapses with de lay. At small delay values, this model describes a disinhibited cortical sl ice. At large delay values, the model is a reduction of a model of thalamic networks composed of excitatory and inhibitory neurons, in which the excit atory neurons show the post-inhibitory rebound mechanism. The velocity and stability of propagating continuous pulses are calculated analytically. Two pulses with different velocities exist if the synaptic coupling is larger than a minimal value; the pulse with the lower velocity is always unstable. Above a certain critical value of the constant delay, continuous pulses lo se stability via a Hopf bifurcation, and lurching pulses emerge. The parame ter regime for which lurching occurs is strongly affected by the synaptic f ootprint (connectivity) shape. A bistable regime, in which both continuous and lurching pulses can propagate, may occur with square or Gaussian footpr int shapes but not with an exponential footprint shape. A perturbation calc ulation is used in order to calculate the spatial lurching period and the v elocity of lurching pulses at large delay values. For strong synaptic coupl ing, the velocity of the lurching pulse is governed by the tail of the syna ptic footprint shape. Moreover, the velocities of continuous and lurching p ulses have the same functional dependencies on the strength of the synaptic coupling strength g(syn): they increase logarithmically with g(syn) for an exponential footprint shape, they scale like (1n g(syn))(1/2) for a Gaussi an footprint shape, and they are bounded for a square footprint shape or an y shape with a finite support. We find analytically how the axonal propagat ion velocity reduces the velocity of continuous pulses; it does not affect the critical delay. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysma l discharges stem from their different effective delays.