Fractional order interdependent nonlinear chaotic spatiotemporal calcium and Aβ dynamics in a neuron cell

Abstract

The formation of β-amyloid (Aβ) and its accumulation depend on the calcium ([Ca2+]) signaling in neurons. The individual and independent dynamics of calcium and β-amyloid give very limited information about different cellular mechanisms. Some researchers have explored the interdependent system dynamics of integer-order calcium and β-amyloid, which provides some more crucial information on different regulatory and dysregulatory processes in neurons. However, these integer-order systems are not capable of generating the information on the superdiffusion, cell memory and Brownian motion effects in neuron cells. A nonlinear mathematical model has been framed to explore the fractional-order interdependent chaotic spatiotemporal [Ca2+] and Aβ dynamics in neurons. The proposed model integrates the two-way feedback mechanism between [Ca2+] and Aβ dynamics in neurons. The Crank-Nicolson scheme with the Grunwald approximation is employed for space fractional derivatives and the L1 formula is employed for time fractional derivatives. The Gauss-Seidel iterations are utilized to solve the resultant system of nonlinear algebraic equations. The effects of cell memory, Brownian motion and superdiffusion phenomena with different crucial mechanisms like buffer, source influx, ryanodine receptor, etc on the spatiotemporal interdependent [Ca2+] and Aβ dynamics have been explored in neurons. The numerical findings give novel insights on the regulatory and dysregulatory effects of cell memory, Brownian motion and superdiffusion on the system dynamics of [Ca2+] and Aβ in neuron cells and the conditions that may cause the different neurodegenerative illnesses like Alzheimer’s disease.