Deciphering the calcium dynamics of a fractional order Alzheimer’s disease model in astrocytes and their networks

Abstract

Exposure of astrocytes to amyloid beta ($A_{\beta }$) is believed to trigger the dysregulation of intracellular calcium (${\left[C\right]}_i$) oscillations. This study explores a fractional-order model of $A_{\beta }$-directed astrocytic ${\left[C\right]}_i$ dynamics, focusing on multi-pathway flux contributions, stability, and solution properties. The complex behaviors of the model system are explained under three parameter regimes by incorporating an additional external current $\left(I_{ext}\right)$ in voltage-gated calcium channels. These behaviors include quiescent states, periodic spiking, and mixed-mode oscillations (MMOs), describing the memory effect of aberrant calcium signaling. Additionally, the bifurcation analysis of ${\left[C\right]}_i$ with $A_{\beta }$ level and $I_{ext}$ reveals that amyloid beta substantially induces the calcium oscillation and provides insight into why spontaneous astrocytic ${\left[C\right]}_i$ oscillations appear and disappear. Further, a coupled model of astrocytes with fractional order is introduced through linear and nonlinear gap junctions in intercellular diffusion of IP3 and then extended to a network model to study the synchronized firing activities. We demonstrate that for a coupled system with differing fractional-order coefficients $(\alpha \ne \beta )$, as the coupling strength $F$ increases, the system with linear coupling can achieve complete synchronization for a higher value of $F$. In contrast, nonlinear coupling fails to synchronize at the same strength, indicating the complexity of the underlying system dynamics. For a network model with linear coupling, we observe that at $F=20$ and 110, the system displays evidence of synchronized behavior. These results highlight that linear gap junctions with weak coupling could potentially explain the intricate intracellular oscillations observed during wave propagation in astrocyte networks, which has implications for neurodegenerative diseases like Alzheimer’s.