# A Kirchhoff-Nernst-Planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons

TL;DR: A 3-dimensional version of the Kirchhoff-Nernst-Planck framework is introduced and used to model the electrodiffusion of ions surrounding a morphologically detailed neuron and demonstrates the efficiency of the 3-D KNP framework.

Abstract: 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.

## Summary (3 min read)

### Introduction

- The brain mainly consists of a dense packing of neurons and neuroglia, submerged in the cerebrospinal fluid which fills the extracellular space (ECS).
- A better understanding of the electrodiffusive interplay between ECS ion dynamics and ECS potentials may be a prerequisite for understanding the mechanisms behind many pathological conditions linked to substantial concentration shifts in the ECS, such as epilepsy and spreading depression [3, 5–7].
- In most computational models in neuroscience ionconcentration dynamics are only partially modeled, or are ignored altogether.
- In these schemes, concentration dynamics are simulated under the simplifying assumption that ions move due to diffusion only.
- The authors compare results obtained with the VC- and KNP-schemes to highlight their similarities and differences.

### Materials and methods

- This section is thematically split into three parts.
- The authors begin by explaining the necessary physical theory, stating and deriving the equations which they implemented.
- Finally, the authors give the specific details on each of the three applications used in the study.
- The source code can be found online, at https://github.com/CINPLA/.
- KNPsim, and the results in this study can be reproduced by checking out the tag PLoS.

### Theory

- The diffusion constants of the ion species are modified as ~Dk ¼ Dk l 2 ; ð6Þ where λ is the tortuosity, which accounts for various hindrances to free diffusion and electrical migration through the ECS.
- Several frameworks that assume the system to be electroneutral at all interior points have been developed to overcome the limitations of the PNP framework [23, 36–41].
- October 4, 2018 6 / 26 exclusively stems from the capacitive current across a neuronal membrane.

### Implementation

- The solver for the above modeling schemes was implemented utilizing FEniCS, an opensource platform for solving partial differential equations using the finite element method [51].
- The authors chose to use this method as the PNP equations are highly unstable, and the implicit Euler step offers superior stability to other methods [33].
- For the second and third application, the authors assumed a concentration-clamp boundary condi- tion, ck ¼ ck;out; at @O; ð22Þ where ck,out was set to typical ECS baseline concentrations, see Table 2.
- This can be interpreted as their system interacting with a larger reservoir of ions (the rest of the brain).
- In order to compare the PNP- and KNP-schemes directly, the authors chose the same boundary condition for the potential in the PNP-scheme.

### Application details

- In the first application, which was implemented using the PNP-, KNP-, and DO-schemes, the authors used an idealized 1-D mesh with a resolution sufficiently fine for PNP to be stable.
- Ij denotes the sum of all ionic and capacitive membrane current at compartment j, Ij ¼ Ijcap þ X k2ions Ijk: ð41Þ.
- The NEURON simulation was nearly identical to that used by us previously, and the authors refer to the original implementation for further details [4].
- Two measurement points were chosen for creating time series of the concentrations.

### Results

- Below the authors present the results obtained with the three different applications listed in the introduction and methodology.
- For this final application, the authors also compared the predictions of the KNP-scheme to those of the simpler DO- and VC-schemes, and analyzed their differences.
- As the simulation progressed, the local build-up of ion concentrations evoked a shift in the KNP-simulated potential ϕ, but left ϕVC unaffected.
- The diffusion component had a sort of screening effect, reducing the potential difference between the source and sink compared to what the authors would predict in the absence of diffusion (i.e. |ϕ|< |ϕVC|).

### Discussion

- In the current work the authors presented a 3-D version of the electrodiffusive KNP-scheme, and used it to simultaneously simulate the dynamics of ion concentrations and the electrical potential in the ECS of a piece of tissue containing a morphologically realistic distribution of neuronal current sources/sinks.
- The authors demonstrated the applicability of this simulation framework by comparing it to the more physically detailed, but more computationally demanding PNP-scheme PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510.
- To their knowledge, the presented model is the first in the field of computational neurosci- ence that can handle electrodiffusive processes in 3-D on spatiotemporal scales spanning over millimeters and seconds without demanding an insurmountable amount of computer power.
- Even the most resource-demanding simulations presented here could be performed in about 15 hours on a normal stationary computer, and the authors believe that simulation efficiency can be improved even further if they select an optimal numerical scheme for KNP.
- This choice was mainly based on the requirements of the PNP-scheme, which requires an implicit scheme in order to not become unstable.

### Model limitations

- The presented implementation of the 3-D KNP-scheme was limited to a relatively small piece of neural tissue, which included only a single pyramidal neuron modeled with the NEURON simulator.
- Thirdly, the neuron model used in their study did not include the Na+/K+-exchanger pump [43].
- For a biophysical modeling scheme that derives the transmembrane transport through channels and pumps from first principles, see e.g., [57, 58].
- In the presence of such mechanisms, the single-neuron contribution to ECS concentration shifts would likely be smaller than in the simulations presented here, or would require a higher neural activity level in order to occur.
- The model limitations mentioned above were also present in the previous 1-D implementation of the KNP-scheme, and the authors refer to this previous work for a more thorough discussion [4].

### Previous models of ECS electrodiffusion

- Several previous studies have explored ECS electrodiffusion on small spatiotemporal scales [23, 28–34, 36–39].
- The authors have previously used a 1-D implementation of the KNP-scheme to explore the effect of diffusive currents on ECS potentials [4].
- Clearly, this is would not apply to most biological scenarios, which means that the results obtained with the 3-D implementation are generally more reliable.
- These diffusion potentials are unrelated to filtering effects hypothesized to arise due to diffusion in the vicinity of the membrane when electric charge is transferred from the intracellular to the extracellular space [60, 61].

### Outlook

- The presented version of the KNP-scheme was developed for use in a hybrid simulation setup where the dynamics of ion concentrations and the electrical potential in the ECS were computed with KNP, while the neurodynamics was computed with the NEURON simulator tool.
- By necessity, this scheme shares the limitations of the NEURON simulator in terms of handling intracellular ion dynamics, which by default is not electrodiffusive in the NEURON environment [14].
- This being said, the hybrid KNP/NEURON version presented here is valuable in its own right, since it allows the KNP framework to be combined with the many models that are already available in the NEURON software.
- In such applications, however, the hybrid KNP framework would need to be expanded to also account for the effect of ECS concentration dynamics on neuronal reversal potentials.

### Author Contributions

- Andreas Solbrå, Gaute T. Einevoll, Geir Halnes, also known as Conceptualization.
- Andreas Solbrå, Geir Halnes, also known as Formal analysis.
- Andreas Solbrå, Geir Halnes, also known as Investigation.
- Andreas Solbrå, Aslak Wigdahl Bergersen, Jonas van den Brink, also known as Software.
- Anders Malthe-Sørenssen, Gaute T. Einevoll, Geir Halnes, also known as Supervision.

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### Cites background or methods from "A Kirchhoff-Nernst-Planck framework..."

...On this background, a series of electroneutral models for ionic electrodiffusion have been developed, both for homogenized domains (Mori et al., 2008; Halnes et al., 2013, 2016, 2017; Niederer, 2013; Pods, 2017; Solbrå et al., 2018), and for domains including an explicit geometrical representation of the cells and of the extracellular space (Mori and Peskin, 2009)....

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...In addition, diffusion along extracellular ion concentration gradients can generate so-called diffusion potentials (Halnes et al., 2016; Savtchenko et al., 2017; Solbrå et al., 2018), which may constitute an additional ephaptic effect on membrane potentials....

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...The framework can be viewed as a combination of the EMI framework and the electroneutralKirchhoff-Nernst-Planck (KNP) framework (Solbrå et al., 2018), and will henceforth be referred to as the KNP-EMI framework....

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..., 2020), or components within a hybridmodeling scheme to compute extracellular dynamics (Halnes et al., 2016, 2017; Solbrå et al., 2018)....

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...The solver for the above modeling schemes was implemented utilizing FEniCS, an open-source platform 191 for solving partial differential equations using the finite element method [39]....

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