Accurate numerical simulation of electrodiffusion and water movement in brain tissue

Abstract

Mathematical modelling of ionic electrodiffusion and water movement is emerging as a powerful avenue of investigation to provide a new physiological insight into brain homeostasis. However, in order to provide solid answers and resolve controversies, the accuracy of the predictions is essential. Ionic electrodiffusion models typically comprise non-trivial systems of non-linear and highly coupled partial and ordinary differential equations that govern phenomena on disparate time scales. Here, we study numerical challenges related to approximating these systems. We consider a homogenized model for electrodiffusion and osmosis in brain tissue and present and evaluate different associated finite element-based splitting schemes in terms of their numerical properties, including accuracy, convergence and computational efficiency for both idealized scenarios and for the physiologically relevant setting of cortical spreading depression (CSD). We find that the schemes display optimal convergence rates in space for problems with smooth manufactured solutions. However, the physiological CSD setting is challenging: we find that the accurate computation of CSD wave characteristics (wave speed and wave width) requires a very fine spatial and fine temporal resolution.

Validating a computational framework for ionic electrodiffusion with cortical spreading depression as a case study

“…Numerical verification tests have been conducted to ensure convergence of solutions. The numerical error in the calculated mean wave propagation speed is estimated to be <1.5%, whereas for the other quantities of interest we expect the numerical errors to be negligible ( Ellingsrud et al, 2021 ).…”

“…We model ionic electrodiffusion and osmotic water flow in brain tissue via the Mori framework, as introduced by Mori (2015) , studied numerically by Ellingsrud et al (2021) , and applied by O’Connell and Mori (2016) and Tuttle et al (2019) . This framework describes tissue dynamics in an arbitrary number of cellular compartments and the ECS via coupled ordinary differential equations and partial differential equations in a model domain.…”

“…We apply the solution algorithm presented previously ( Ellingsrud et al, 2021 ) and approximate the mathematical model numerically by a finite element method in space, a BDF2 scheme in time and a Strang splitting method. We use spatial and temporal discretization sizes of Δ x = 1.25 μm and Δ x = 3.125 ms, respectively.…”

Implementation of the Hindmarsh–Rose Model Using Stochastic Computing

“…The brain being the most complex structure in the known universe, problems inherent to this goal are many and different [1][2][3][4]. The current proposals cover many different approaches from quantum computing [5] to neural networks [6], including simulation at the level of tissues [7], interactions with different parts [8], or descriptions of the communications at a higher level [9].…”

Accurate numerical simulation of electrodiffusion and water movement in brain tissue

“…Finally, Section 9 contains a discussion and concluding remarks. The code used to obtain the simulation results presented within this work is based on FEniCS 23 , and is publicly available 9 .…”