Increasing evidence has shown that excitatory neurons in the brain play a significant role in seizure generation. However, spiny stellate cells are cortical excitatory non-pyramidal neurons in the brain which their basic role in seizure occurrence is not well understood. In the present research, we study the critical role of spiny stellate cells or the excitatory interneurons (EI), for the first time, in epileptic seizure generation using an extended neural mass model introduced originally by Taylor and colleagues in 2014. Applying bifurcation analysis on this modified model, we investigated the rich dynamics corresponding to the epileptic seizure onset and transition between interictal and ictal states due to the EI. Our results indicate that the transition is described by a supercritical Hopf bifurcation which shapes the preictal activity in the model and suggests why before seizure onset, the amplitude and frequency of neural activities increase gradually. Moreover, we showed that 1) the altered function of GABAergic and glutamatergic receptors of EI can cause seizure, and 2) the pathway between the thalamic relay nucleus and EI facilitates the transition from interictal to the ictal activity by decreasing the preictal period. Thereafter, we considered both sensory and cortical periodic inputs to drive the model responses to various harmonic stimulations. Our results from the bifurcation analysis of the model suggest that the initial stage of the brain might be the main cause for the transition between interictal and ictal states as the stimulus frequency changes. The extended thalamocortical model shows also that the amplitude jump phenomenon and nonlinear resonance behavior result from the preictal stage of the brain. These results can be considered as a step forward to a deeper understanding of the mechanisms underlying the transition from brain normal activities to epileptic activities.