I just had a quick look at the MATLAB code on the linked page, and the model has some peculiarities. It uses a dt of 1ms, but with two 0.5ms update steps for v, and the synaptic input gets reset to 0 at every time step (so in a way it is a delta synapse implementation, but since the time step is very long one should probably rather think of it as a 1ms rectangular step current). There is no refractory period in that model.
Just in case you are not aware of it: some of Izhikevhich’s results have been a bit hard to replicate, and in particular the observed oscillations seem to depend on the details of the integration method and the integration time step. See e.g. the comment in the NEST documentation and this paper replicating the polychronization network.