What is ECS and why is it important?Brain tissue is essentially composed of two regions: cellular elements (neurons and glial cells), and the gap between the elements, which is known as the extracellular SPACE (ECS; Figure 1) (Sykova & Nicholson, 2008). The ECS resembles the water phase of a foam and remains a highly connected domain even though it is convoluted in shape and may form dead-SPACE microdomains (e.g. local expansions or voids) (Hrabetova, Hrabe, & Nicholson, 2003). The width of the ECS is about 20-60 nm (Thorne & Nicholson, 2006), nevertheless in totality it occupies approximately 20% of the entire tissue volume (Sykova & Nicholson, 2008) The surprisingly large relative volume of the ECS makes it an important area for neuroscience research.

‎This paper deals with a time-SPACE fractional Schrödinger equation with homogeneous Dirichlet boundary conditions‎. ‎A common strategy for discretizing time-fractional operators is finite difference schemes‎. ‎In these methods‎, ‎the time-step size should usually be chosen sufficiently small‎, ‎and subsequently‎, ‎too many iterations are required which may be time-consuming‎.‎To avoid this issue‎, ‎we utilize the Laplace transform method in the present work to discretize time-fractional operators‎. ‎By using the Laplace transform‎, ‎the equation is converted to some time-independent problems‎. ‎To solve these problems‎, ‎matrix transformation and improved matrix transformation techniques are used to approximate the spatial derivative terms which are defined by the spectral fractional Laplacian operator‎. ‎After solving these stationary equations‎, ‎the numerical inversion of the Laplace transform is used to obtain the solution of the original equation‎. ‎The combination of finite difference schemes and the Laplace transform creates an efficient and easy-to-implement method for time-SPACE fractional Schrödinger equations‎. ‎Finally‎, ‎some numerical experiments are presented and show the applicability and accuracy of this approach‎.