Analyzing fuzzy boundary value problems: a study on the influence of mitochondria and ER fluxes on calcium ions in neuron cells

Cytosolic-free calcium ions play an important role in various physical and physiological processes. A vital component of neural signaling is the free calcium ion concentration often known as the second messenger. There are many parameters that effect the cytosolic free calcium concentration like buffer, voltage-gated ion channels, Endoplasmic reticulum, Mitochondria, etc. Mitochondria are small organelles located within the nervous system that are involved in processes within cells such as calcium homeostasis management, energy generation, response to stress, and cell demise pathways. In this work, a mathematical model with fuzzy boundary values has been developed to study the effect of Mitochondria and ER fluxes on free Calcium ions. The intended findings are displayed utilizing the physiological understanding that amyloid beta plaques and tangles of neurofibrillary fibers have been identified as the two main causes of AD. The key conclusion of the work is the investigation of \({Ca}^{2+}\) for healthy cells and cells affected by Alzheimer's disease, which may aid in the study of such processes for computational scientists and medical practitioners. Also, it has been shown that when a unique solution is found for a specific precise problem, it also successfully deals with any underlying ambiguity within the problem by utilizing a technique based on the principles of linear transformation. Furthermore, the comparison between the analytical approach and the generalized hukuhara derivative approach is shown here, which illustrates the benefits of the analytical approach. The simulation is carried out in MATLAB.