Recent research highlights that calcium dysfunction, emerging from impaired neuron–astrocyte interactions plays a crucial role in the pathogenesis of Alzheimer’s disease (AD). In our current study, we investigate this through a computational model of bidirectional coupling between a neuron and an astrocyte within a tripartite synapse framework. Individually, neuron is designed to exhibit neuronal firing dynamics, while the astrocyte captures amyloid- \(\beta \) -mediated calcium signaling. In particular, we consider the spatiotemporal version of the model across three scenarios: (i) no diffusion; (ii) diffusion in either neurons or astrocytes; and (iii) diffusion in both. Without diffusion, bifurcation analysis reveals that astrocytic feedback can trigger neuronal firing via a supercritical Andronov-Hopf bifurcation, emphasizing astrocytic regulation of excitability. With diffusion only in neurons, complex Ginzburg–Landau analysis (CGLE) reveals spiral and antispiral wave patterns. When only astrocytic diffusion is present, regular and distorted hexagonal patterns emerge. The third scenario yields Turing-like structures. We further extend the model to a spatial network to explore collective dynamics and synchronization behaviors, where stronger coupling leads to partially synchronized neuronal activity. These findings demonstrate how astrocyte–neuron crosstalk and diffusion-driven instabilities contribute to emergent wave-like activity, shedding light on spatial mechanisms in AD progression.
