Neuronal activity occurs on two separate time scales: a fast scale on the order of 10^{-3} seconds, associated with action potentials and channel gating, and a slow scale on the order of 10 seconds, associated with ionic dynamics [Cressman_1, Mardal, Cressman_2]. Interactions between these fast and slow mechanisms can cause oscillation that are seen in epileptic seizures. In this study, we investigate how slow ion dynamic contribute to bursting behavior and transition to seizures with the help of bifurcation analysis. Nullcline analysis shows that stronger pump strength (\rho) causes the neuron to have three different unstable equilibria, which may collide or disappear through saddle-node bifurcations, whereas for weak "\rho" only a single equilibrium exists [Kumar]. Presence of multiple unstable equilibria cause the neurons to keep oscillating around them. These finding motivate us to perform bifurcation analysis of neural dynamics at elevated pump strength. In addition, extracellular potassium concentration ( K_{bath} ) also plays an important role in neuronal firing and excitability [Wu]. Bifurcation analysis with respect to "\rho" and "K_{bath}" reveals critical transitions, including the birth of oscillations through Hopf bifurcations. For higher pump strength, codim-1 bifurcation with respect to "K_{bath}" reveals two limit points; however, the neurons bypass them and immediately transition to a globally stable limit cycle. Extending the bifurcation analysis to the ( K_{bath} - \rho) parameter space reveals higher-order bifurcations, such as cusp and generalized Hopf points, which help us to delineate the regions of multistability, bursting activity, and transitions between normal and seizure-like states [Kumar]. Together, these results provide biological insight into how impaired potassium homeostasis and pump function can promote seizure dynamics and help define parameter regimes associated with stable, non-pathological neuronal behavior.