  Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic
insights into how neurons self-organize to produce coherent collective
behavior. These networks use simple spiking neuron models, such as the
$\theta$-neuron or the quadratic integrate-and-fire (QIF) neuron, that
replicate the essential features of real neural dynamics. Interactions between
neurons are modeled with infinitely narrow pulses, or spikes, rather than the
more complex dynamics of real synapses. To make these networks biologically
more plausible, it has been proposed that they must also account for the finite
width of the pulses, which can have a significant impact on the network
dynamics. However, the derivation and interpretation of these pulses is
contradictory and the impact of the pulse shape on the network dynamics is
largely unexplored. Here, I take a comprehensive approach to pulse-coupling in
networks of QIF and $\theta$-neurons. I argue that narrow pulses activate
voltage-dependent synaptic conductances and show how to implement them in QIF
neurons such that their effect can last through the phase after the spike.
Using an exact low-dimensional description for networks of globally coupled
spiking neurons, I prove for instantaneous interactions that collective
oscillations emerge due to an effective coupling through the mean voltage. I
analyze the impact of the pulse shape by means of a family of smooth pulse
functions with arbitrary finite width and symmetric or asymmetric shapes. For
symmetric pulses, the resulting voltage-coupling is not very effective in
synchronizing neurons, but pulses that are slightly skewed to the phase after
the spike readily generate collective oscillations. The results unveil a
voltage-dependent spike synchronization mechanism in neural networks, which is
facilitated by pulses of finite width and complementary to traditional synaptic
transmission.
