In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human EEG activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from healthy activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of DDEs. We observe the the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures