Unraveling the fractional dynamics of calcium ions for the ER and membrane in the presence of amyloid beta pore

Calcium is a substantial second messenger for neuronal multifold transduction pathways. The molecular gating of different channels triggers the influx of calcium neuronal signaling. The calcium intracellular sequestering prominent entity the Endoplasmic Reticulum (ER) mediated flux with SERCA, IPR, and Leak channel mechanism, plays an important role in controlling the calcium signaling process. When calcium dynamics aberrate, it actuates intrinsic stress, cell damage, and apoptosis. In the present work, an attempt has been made with a Caputo fractional time derivative two-dimensional model for different physiological parameters to investigate the role of nonlocal free calcium ions with amyloid beta pores, Orai channels, and ER fluxes. The analytical solution is obtained by the hybrid integral transform method. The closed form is obtained in terms of Mainardi and Wright's function by using Green's function method. Fundamental robustness is shown by proving the uniqueness and existence of the solution. The behavior of free calcium ions is captured with memory in space and time. Spatiotemporal dynamics with buffers and fluxes show a compelling relation with neuronal calcium. Impaired neuronal conditions were observed with various simulations in 2D space. Nonlocal and memory impact on $ Ca^{2+} $ simulated for buffer and different parameters. Results clearly show the concern that the different parameters could lead to atypical calcium levels in dementia. Simulations help to understand buffer reactions and calcium levels in neurons.