Unravelling the spatiotemporal dynamics of amyloid- β -induced astrocyte-neuron network model in Alzheimer's disease.

-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.