A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons
Authors: Solbrå Andreas , Bergersen Aslak Wigdahl , van den Brink Jonas , Malthe-Sørenssen Anders , Einevoll Gaute T. , Halnes Geir 
Published: February 2018

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Many pathological conditions, such as seizures, stroke, and spreading depression, are associated with substantial changes in ion concentrations in the extracellular space (ECS) of the brain. An understanding of the mechanisms that govern ECS concentration dynamics may be a prerequisite for understanding such pathologies. To estimate the transport of ions due to electrodiffusive effects, one must keep track of both the ion concentrations and the electric potential simultaneously in the relevant regions of the brain. Although this is currently unfeasible experimentally, it is in principle achievable with computational models based on biophysical principles and constraints. Previous computational models of extracellular ion-concentration dynamics have required extensive computing power, and therefore have been limited to either phenomena on very small spatiotemporal scales (micrometers and milliseconds), or simplified and idealized 1-dimensional (1-D) transport processes on a larger scale. Here, we present the 3-D Kirchhoff-Nernst-Planck (KNP) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. By assuming electroneutrality, the KNP-framework circumvents charge-relaxation processes on the spatiotemporal scales of nanometers and nanoseconds, and makes it feasible to run simulations on the spatiotemporal scales of millimeters and seconds on a standard desktop computer. In the present work, we use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell. In addition to elucidating the single neuron contribution to electrodiffusive effects in the ECS, the simulation demonstrates the efficiency of the 3-D KNP framework. We envision that future applications of the framework to more complex and biologically realistic systems will be useful in exploring pathological conditions associated with large concentration variations in the ECS.

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Author summary

Many pathological conditions, such as epilepsy and cortical spreading depression, are linked to abnormal extracellular ion concentrations in the brain. Understanding the underlying principles of such conditions may prove important in developing treatments for these illnesses, which incur societal costs of tens of billions annually. In order to investigate the role of ion-concentration dynamics in the pathological conditions, one must measure the spatial distribution of all ion concentrations over time. This remains challenging experimentally, which makes computational modeling an attractive tool. We have previously introduced the Kirchhoff-Nernst-Planck framework, an efficient framework for modeling electrodiffusion. In this study, we introduce a 3-dimensional version of this framework and use it to model the electrodiffusion of ions surrounding a morphologically detailed neuron. The simulation covered a 1 mm3cylinder of tissue for over a minute and was performed in less than a day on a standard desktop computer, demonstrating the framework’s efficiency. We believe this to be an important step on the way to understanding phenomena involving ion concentration shifts at the tissue level.