Calcium ions (Ca2+), 1, 4, 5-inositol trisphosphate (IP3), and nitric oxide (NO) function as signaling molecules and are crucial for controlling several physiological mechanisms in neuronal cells. Nevertheless, there is a lack of research investigating the interplay between Ca2+, IP3 and NO in neuronal cells, in terms of fractional-order dynamics. The investigation of fractional-order interacting dynamical systems including calcium, IP3, and NO is of crucial significance as it sheds light on several phenomena such as superdiffusion and the emergence of cell memory exhibiting Brownian motion (BM) in neurons. These aspects have not yet been thoroughly examined in neurons. The theoretical framework employed in this study is a fractional model that describes the interacting calcium, IP3, and NO systems incorporating the nonlinear reaction-diffusion equations, fractional diffusion processes, and memory effects. The Crank-Nicholson (CN) method with the Grunwald technique is utilized to address the fractional-order space derivatives, while the L1 technique is applied to solve the fractional-order temporal derivatives. The Gauss-Seidel (GS) procedure is employed to solve the system of nonlinear equations governing the dynamics of Ca2+, IP3, and NO. The previously unexplored bidirectional feedback mechanisms between calcium and NO and calcium and IP3 have been explored. The current study includes the incorporation of several critical parameters, including the voltage-gated calcium channel (VGCC), Orai channel, IP3-receptor (IP3R), Sarcoendoplasmic Reticulum Calcium ATPase (SERCA) pump, leak, plasma membrane Ca2+ ATPase (PMCA) channel, sodium-calcium exchanger (NCX), ryanodine receptor (RyR), and [NO]-dependent Ca2+ flux. The numerical findings show that superdiffusion and cell memory with several cellular mechanisms significantly affect the regulation of the interacting nonlinear calcium, IP3, and NO dynamical systems in neuronal cells. Neurotoxic events potentially contributing to disease-related conditions such as Alzheimer’s may result from dysfunction in the superdiffusion and memory of signaling ions and molecules, as well as the cellular mechanisms within neurons.
