Fractional-order reaction-diffusion model to study the dysregulatory impacts of superdiffusion and memory on neuronal calcium and IP3 dynamics

The models for the dynamics of single systems like calcium ([Ca2+]) in a neuron cell are able to provide information about factors affecting the calcium regulation in the neuron cell. But in realistic conditions, the dynamics of calcium also depends on the dynamics of other chemical molecules like inositol 1, 4, 5-trisphophate (IP3) in the nerve cell. Some research workers explored the interdependent [Ca2+] and IP3 signaling systems in neurons, but that too for integer-order systems only. The studies on integer-order system dynamics are not able to generate insights about the superdiffusion mechanisms, cell memory causing Brownian motion of signaling molecules in neurons. Few attempts to investigate fractional reaction-diffusion models of individual [Ca2+] dynamics have been reported in neurons. However, no attempt is noticed in the literature to analyze the interdependent fractional processes of [Ca2+] and IP3 dynamics in neurons. In the present paper, a mathematical model of nonlinear interdependent spatiotemporal [Ca2+] and IP3 signaling systems is proposed incorporating fractional diffusion processes and cell memory of [Ca2+] and IP3 causing Brownian motion effects in neurons. The numerical solution is obtained by the Crank-Nicolson scheme (CNS) with Grunwald estimation for fractional derivatives along spatial dimensions and L1 scheme for fractional derivatives along temporal dimensions with Gauss-Seidel (GS) iterations. The influences of the superdiffusion mechanism and cell memory on the interdependent [Ca2+] and IP3 signaling systems in neurons are analyzed with the help of numerical results. A significant difference in results is observed between fractional-order and integer-order dynamics. It is concluded that the cell can use memory effects and superdiffusion mechanisms to regulate the calcium and IP3 concentrations at appropriate levels. Further, the fractional-order processes like superdiffusion and cell memory can also be a cause of dysregulation of calcium and IP3 dynamics leading to neurological disorders like Parkinson's, etc.