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

We study a one-dimensional model of integrate-and-fire neurons that are allowed to fire only one spike, and are coupled by excitatory synapses with delay. At small delay values, this model describes a disinhibited cortical slice. At large delay values, the model is a reduction of a model of thalamic networks composed of excitatory and inhibitory neurons, in which the excitatory 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 lose stability via a Hopf bifurcation, and lurching pulses emerge. The parameter regime for which lurching occurs is strongly affected by the synaptic footprint (connectivity) shape. A bistable regime, in which both continuous and lurching pulses can propagate, may occur with square or Gaussian footprint shapes but not with an exponential footprint shape. A perturbation calculation is used in order to calculate the spatial lurching period and the velocity of lurching pulses at large delay values. For strong synaptic coupling, the velocity of the lurching pulse is governed by the tail of the synaptic footprint shape. Moreover, the velocities of continuous and lurching pulses 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 (ln g_syn)^1/2 for a Gaussian footprint shape, and they are bounded for a square footprint shape or any shape with a finite support. We find analytically how the axonal propagation 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 paroxysmal discharges stem from their different effective delays.