Abstract
In neural structures with complex geometries, numerical resolution of the Poisson-Nernst-Planck (PNP) equations is necessary to accurately model electrodiffusion. This formalism allows one to describe ionic concentrations and the electric field (even away from the membrane) with arbitrary spatial and temporal resolution which is impossible to achieve with models relying on cable theory. However, solving the PNP equations on complex geometries involves handling intricate numerical difficulties related either to the spatial discretization, temporal discretization or the resolution of the linearized systems, often requiring large computational resources which have limited the use of this approach. In the present paper, we investigate the best ways to use the finite elements method (FEM) to solve the PNP equations on domains with discontinuous properties (such as occur at the membrane-cytoplasm interface). 1) Using a simple 2D geometry to allow comparison with analytical solution, we show that mesh adaptation is a very (if not the most) efficient way to obtain accurate solutions while limiting the computational efforts, 2) We use mesh adaptation in a 3D model of a node of Ranvier to reveal details of the solution which are nearly impossible to resolve with other modelling techniques. For instance, we exhibit a non linear distribution of the electric potential within the membrane due to the non uniform width of the myelin and investigate its impact on the spatial profile of the electric field in the Debye layer.
Highlights
Since the pioneer work of Hodgkin and Huxley [1], mathematical modelling of the electric activity of neurons has become an important tool to investigate the nervous system
While most models have relied on the cable theory formalism based on the strong analogy between neurons and electric circuits, limits of this formalism include its incapacity to account for fluctuations of ionic concentrations or to describe the electric field beyond the membrane
We show that the technique of the mesh adaptation method, which has a proven track record in the field of industrial mathematics, can be applied to describe electrodiffusion by solving the PNP equations on complex three dimensional geometries in neuroscience
Summary
Since the pioneer work of Hodgkin and Huxley [1], mathematical modelling of the electric activity of neurons has become an important tool to investigate the nervous system. Solving the Poisson-Nernst-Planck (PNP) partial differential equations is a promising approach to overcome these limitations and model the evolution of ionic concentrations and of the electric field in neural structures such as axons, nodes of Ranvier, dendritic spines or the synaptic cleft ([5,6,7,8,9]). This strategy, not relying on oversimplifying assumptions such as the charge difference between intra and extracellular media being localized at the membrane, offers a potential spatial resolution in the nanometer range [7]
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