The response of a neuron to synaptic input strongly depends on whether or not the neuron has just emitted a spike. We propose a neuron model that after spike emission exhibits a partial response to residual input charges and study its collective network dynamics analytically. We uncover a desynchronization mechanism that causes a sequential desynchronization transition: In globally coupled neurons an increase in the strength of the partial response induces a sequence of bifurcations from states with large clusters of synchronously firing neurons, through states with smaller clusters to completely asynchronous spiking. We briefly discuss key consequences of this mechanism for more general networks of biophysical neurons.