At the onset of Alzheimer's disease (AD), accumulation of oligomeric amyloid-β (Aβ) correlates with local hyperexcitability, possibly due to dyshomeostasis of extracellular glutamate. However, the interplay between Aβ oligomers and extracellular glutamate accumulation remains unresolved. We use an in silico approach to characterize the putative underpinnings of Aβ-dependent modulations of extracellular glutamate. We first consider how Aβ modifies the expression of astrocytic glutamate transporters (GLT1) and how it impacts extracellular glutamate time course. Accordingly, we develop a mathematical model for glutamate diffusion and uptake at synaptic terminals and their surroundings. Our model predicts that above a threshold Aβ concentration, astrocytic GLT1 transporters cannot prevent the accumulation of extracellular glutamate. This, in turn, promotes a positive feedback on synaptic glutamate release that favors excitotoxicity and neuronal hyperactivity. Next, we complement our model including calcium-dependent Aβ production and glutamate release. The interaction of multiple positive feedback loops of signaling can account for different tissue states, overcoming the traditional dichotomy of healthy and excitotoxic conditions. Specifically, changes in the basal firing activity can promote intermediate conditions useful to either predict AD's degeneration or offer new therapeutic perspectives. Our results support the notion of AD as a multistage pathology where transitions can follow multiple pathways, such as Aβ accumulation, excitotoxicity, or calcium dysregulation.