Generalised Neuronal Calcium Dynamics of Membrane and ER in the Polar Dimension

Calcium ions are the second messenger playing as regulators for various cellular activities. Its spatiotemporal control is critical for various brain functions, including neuroplasticity, apoptosis, and cell death. The Endoplasmic Reticulum (ER) plays an important role in determining these spatiotemporal calcium dynamics. Stromal interaction molecule (STIM) - Orai channel on the membrane generates additional calcium flow, whereas other membrane fluxes contribute to cytosolic flux. Due to their anomalous character, we used the Caputo fractional differential operator to mimic these interactions in polar coordinates. Solutions were generated using hybrid integral transform methods to control the analytical approach. Using Green's function yielded a closed-form solution for Mittag-Leffler-type functions. This work emphasizes the significant relationship between calcium and various buffer levels in neurons. The differential transition simulation of a time derivative with space across different parameters indicated a decrease in calcium concentration. Anomalously low buffer levels exhibited the impact of Alzheimer's disease on calcium higher concentration, leading to the death of neurons. Additionally, the research introduces a method involving S100B, BAPTA, and calmodulin buffers to uphold optimal calcium levels within the neuronal cytosol. The applicability of this model with different buffer properties and parameters and memory impacts the calcium concentration with the neurological disorder.