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

07 Feb 2018-bioRxiv (Cold Spring Harbor Laboratory)-pp 261107

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.

AbstractMany 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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RESEARCH ARTICLE
A Kirchhoff-Nernst-Planck framework for
modeling large scale extracellular
electrodiffusion surrounding morphologically
detailed neurons
Andreas Solbrå
ID
1,2
, Aslak Wigdahl Bergersen
3
, Jonas van den Brink
3
,
Anders Malthe-Sørenssen
ID
1,2
, Gaute T. Einevoll
ID
1,2,4
, Geir Halnes
ID
4
*
1 Center for Integrative Neuroplasticity, University of Oslo, Oslo, Norway, 2 Department of Physics, University
of Oslo, Oslo, Norway, 3 Simula Research Laboratory, Fornebu, Norway, 4 Department of Mathematical
Sciences and Technology, Norwegian University of Life Sciences, Ås, Norway
* geir.halnes@nmbu.no
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 cur-
rently unfeasible experimentally, it is in principle achievable with computational models
based on biophysical principles and constraints. Previous computational models of extracel-
lular 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 electroneu-
trality, 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 concentra-
tions 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.
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510 October 4, 2018 1 / 26
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OPEN ACCESS
Citation: Solbrå A, Bergersen AW, van den Brink J,
Malthe-Sørenssen A, Einevoll GT, Halnes G (2018)
A Kirchhoff-Nernst-Planck framework for modeling
large scale extracellular electrodiffusion
surrounding morphologically detailed neurons.
PLoS Comput Biol 14(10): e1006510. https://doi.
org/10.1371/journal.pcbi.1006510
Editor: Ernest Barreto, George Mason University,
UNITED STATES
Received: February 6, 2018
Accepted: September 12, 2018
Published: October 4, 2018
Copyright: © 2018 Solbra
˚
et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All model code will be
available at https://github.com/CINPLA/KNPsim.
Funding: This work was funded by the Research
Council of Norway (BIOTEK2021 Digital Life project
‘DigiBrain’, project 248828). The funders had no
role in study design, data collection and analysis,
decision to publish, or preparation of the
manuscript.
Competing interests: The authors have declared
that no competing interests exist.

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 inves-
tigate the role of ion-concentration dynamics in the pathological conditions, one must
measure the spatial distribution of all ion concentrations over time. This remains chal-
lenging 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 mm
3
cylinder 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 understand-
ing phenomena involving ion concentration shifts at the tissue level.
Introduction
The brain mainly consists of a dense packing of neurons and neuroglia, submerged in the
cerebrospinal fluid which fills the extracellular space (ECS). Neurons generate their electrical
signals by exchanging ions with the ECS through ion-selective channels in their plasma
membranes. During normal signaling, this does not lead to significant changes in local ion
concentrations, as neuronal and glial transport mechanisms work towards maintaining ion
concentrations close to baseline levels. However, endured periods of enhanced neuronal activ-
ity or aberrant ion transport may lead to changes in ECS ion concentrations. Local concentra-
tion changes often coincide with slow shifts in the ECS potential [13], which may be partly
evoked by diffusive electrical currents, i.e., currents carried by charged ions moving along ECS
concentration gradients [2, 4]. While concentration gradients can influence electrical fields,
the reverse is also true, since ions move not only by diffusion but also by electric drift. 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 depres-
sion [3, 57].
A simultaneous and accurate knowledge of the concentration of all ion species is needed to
make reliable estimates of electrodiffusive effects in the ECS. Although this is currently unfea-
sible experimentally, it is in principle achievable with computational models based on biophys-
ical principles and constraints. However, in most computational models in neuroscience ion-
concentration dynamics are only partially modeled, or are ignored altogether. One reason for
this is the challenge involved in keeping track of all ion concentrations and their spatiotempo-
ral dynamics. Another reason may be the strong focus within the community on modeling the
neuronal membrane dynamics at short timescales, during which both intra- and extracellular
concentration changes are relatively small and putatively negligible. Although there exist mod-
els that account for ion concentration shifts and their effects on neuronal and glial reversal
potentials [811], the most common computational models for excitable cells, the multi-com-
partmental models and the cable equation, are based on the assumptions that (i) the ECS
potential is constant (ground), and (ii) the ion concentrations are constant [12, 13]. The
NEURON simulator [14, 15] is based on these assumptions, and although they are physically
Modelling large scale electrodiffusion surrounding morphologically detailed neurons
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510 October 4, 2018 2 / 26

incorrect, they still allow for efficient and fairly accurate predictions of the membrane-poten-
tial dynamics.
Because of assumption (i), multi-compartmental models are unsuited for modeling ECS
dynamics, and several approaches have been taken to construct models which include ECS
effects. A majority of computational studies of ECS potentials are based on volume conductor
(VC) theory [1621]. VC-schemes link neuronal membrane dynamics to its signatures in the
ECS potential. In contrast to the multi-compartmental models, VC-schemes are derived by
allowing the ECS potential to vary, but still assuming that the ion concentrations are constant.
VC-schemes are attractive, because they offer closed-form solutions, and allow the calculation
of the electric field for arbitrarily large systems. Although it may be reasonable to neglect varia-
tions in ECS ion concentrations on short timescales, the accumulative effects of endured neu-
ronal activity may lead to significant concentration changes in the ECS, which are related to
the aforementioned pathological conditions. Naturally, models that do not include ion-con-
centration dynamics are not applicable for exploring such pathologies. Furthermore, VC-
schemes neglect the effects from diffusive currents on the ECS potentials [4, 22, 23], and in
previous computational studies we have found the low-frequency components of the ECS
potential to be dominated by diffusion effects [4, 24].
A simplified approach to modeling concentration dynamics in brain tissue, is to use reac-
tion-diffusion schemes (see e.g., [2527]). In these schemes, concentration dynamics are simu-
lated under the simplifying assumption that ions move due to diffusion only. This approach
has been used for many specific applications, giving results in close agreement with experi-
mental data [26]. However, the net transport of abundant charge carriers such as Na
+
, K
+
,
Ca
2+
, and Cl
, is also influenced by electric forces, which is not incorporated in diffusion only
(DO)-schemes. Furthermore, DO-schemes do not include the influence that diffusing ions can
have on the electrical potential.
To account for the electric interactions between the different ion species, as well as the effect
of such electric forces on the ECS potential, an electrodiffusive modeling framework is needed.
The most detailed modeling scheme for electrodiffusion is the Poisson-Nernst-Planck (PNP)
scheme [2834]. The PNP-scheme explicitly models charge-relaxation processes, that is, tiny
deviations from electroneutrality involving only about 10
—9
of the total ionic concentration
[35]. This requires a prohibitively high spatiotemporal resolution, which makes the PNP-
scheme too computationally expensive for modeling the ECS on the tissue scale. Even the
state-of-the-art simulations in the literature are on the order of milliseconds on computational
domains of micrometers. The PNP-scheme is therefore not suited for simulating processes tak-
ing place at the tissue scale [23].
A series of modeling schemes have been developed that circumvent the brief charge-relaxa-
tion processes, and solve directly for the ECS potential when the system is in a quasi-steady
state [4, 23, 3642]. Circumventing charge-relaxation allows for simulations on spatiotemporal
scales which are larger, compared to what is possible with the PNP-scheme, by several orders
of magnitude. The charge-relaxation can be bypassed by replacing Poisson’s equation with the
constraint that the bulk solution is electroneutral. These schemes have been shown to deviate
from the PNP-scheme very close to the cell membrane (less than 5 I
ˆ
¼m), but to give a good
agreement in the bulk solution [23]. The simplest electroneutral modeling scheme is the
Kirchhoff-Nernst-Planck (KNP) scheme, previously developed in our group [41, 42]. A similar
scheme was developed in parallel in the heart cell community [40].
The KNP-scheme has previously been used to study electrodiffusive phenomena such as
spatial K
+
buffering by astrocytes [41], effects of ECS diffusion on the local field potential [4],
and the implication for current-source density analysis [24]. For simplicity, these previous
applications were limited to idealized 1-D setups with a relatively coarse spatial resolution.
Modelling large scale electrodiffusion surrounding morphologically detailed neurons
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510 October 4, 2018 3 / 26

Furthermore, a comparison between the KNP framework and other simulation frameworks
was not included in previous studies.
In the present study, we introduce a 3-D version of the KNP framework which can be used
to simulate the electrodiffusive dynamics of ion-concentrations and the electrical potential
in the ECS on large spatiotemporal scales. We establish in which situations the assumptions
used in the KNP scheme are warranted by comparing it to the more physically detailed PNP
scheme. Furthermore, we identify the conditions under which an electrodiffusive formalism is
needed by comparing the KNP scheme to the VC and DO schemes. The simplified schemes
can be derived from the KNP scheme by assuming, respectively, that (for VC) diffusive effects
on the membrane potential and (for DO) migratory effects on the concentration dynamics are
negligible. Accordingly, the accuracy of the simplifying assumptions can be assessed by com-
paring how close their predictions come to the KNP scheme.
We present the results of three distinct simulation setups, which we will refer to as Applica-
tion 1, Application 2, and Application 3 for the remainder of this study:
In Application 1, we consider an idealized 1-D domain filled with a salt solution, starting
with a nonzero ion concentration gradient. We solve the system using the PNP-scheme, the
KNP-scheme, and a DO-scheme. We compare results on short and long timescales (nanosec-
onds and seconds), to highlight the similarities and differences between the schemes.
In Application 2, we consider a 3-D domain with an ion concentration point source and a
point sink, of equal magnitude, embedded in a standard ECS ion solution. We compare results
obtained with the VC- and KNP-schemes to highlight their similarities and differences.
In Application 3, we consider a morphologically realistic pyramidal neuron model [43])
embedded in a 3-D ECS solution. The neuronal morphology is inserted as a 1-D branching
tree, which means that it does not occupy any volume, but gives rise to a morphologically real-
istic spatial distribution of neuronal membrane current sources or sinks. The ECS dynamics is
computed using the KNP-scheme, and show how concentration gradients gradually build up
in the ECS due to the neural activity, and how this influences the local potential in the ECS.
We compare results obtained with the VC-, DO-, and KNP-schemes to highlight their similari-
ties and differences.
The first two applications are simplified simulation setups, used to better understand the
differences between the schemes introduced above, while the third application is the main
result of this study, as it illustrates the scales at which the KNP-scheme can be used.
To our knowledge, the KNP-scheme is the first simulation framework which can handle
3-D electrodiffusion in neuronal tissue at relatively large spatiotemporal scales without
demanding an insurmountable amount of computer power. For Application 3, the long-term
ECS ion-concentration dynamics (about 100 s) in a spatial region of about 1 mm
3
was run on
a standard desktop computer within a day. We expect that the presented simulation frame-
work will be of great use for future studies, especially for modeling tissue dynamics in the con-
text of exploring pathological conditions associated with large shifts in ECS ion concentrations
[3, 57].
Materials and methods
This section is thematically split into three parts. We begin by explaining the necessary physi-
cal theory, stating and deriving the equations which we implemented. Then, we explain in
more detail how the models were implemented, including details such as numerical schemes
and boundary conditions. Finally, we 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.
Modelling large scale electrodiffusion surrounding morphologically detailed neurons
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510 October 4, 2018 4 / 26

Theory
The Nernst-Planck equation for electrodiffusion. The ion concentration dynamics of an
ion species in a solution is described by the continuity equation:
@c
k
@t
¼ r J
k
þ f
k
; in O;
ð1Þ
where c
k
is the concentration of ion species k, f
k
represent any source terms in the system, O is
the domain for which the concentrations are defined, and J
k
is the concentration flux of ion
species k. In the applications in this study, f
k
is implemented as a set of point sources at speci-
fied coordinates in the ECS. In the Nernst-Planck equation, J
k
consists of a diffusive and an
electric component:
J
k
¼ J
diff
k
þ J
field
k
: ð2Þ
The diffusive component is given by Fick’s first law,
J
diff
k
¼ D
k
rc
k
; ð3Þ
where D
k
is the diffusion coefficient of ion species k. The electric component is
J
field
k
¼
D
k
z
k
c
k
c
r;
ð4Þ
where ϕ is the electric potential, z
k
is the valency of ion species k, and ψ = RT/F is defined by
the gas constant (R), Faraday’s constant (F) and the temperature (T) which we assume to be
constant (cf. Table 1). Inserting Eqs 24 into Eq 1, yields the time evolution of the concentra-
tion of ion species k:
@c
k
@t
¼ r D
k
rc
k
þ
D
k
z
k
c
k
c
r
þ f
k
; in O: ð5Þ
We model the ECS as a continuous medium, while in reality, the ECS only takes up roughly
20% of the tissue volume [44] in the brain. To compensate for this, we use the porous medium
approximation [45]. This involves two changes to the model. The diffusion constants of the
ion species are modified as
~
D
k
¼
D
k
l
2
;
ð6Þ
where λ is the tortuosity, which accounts for various hindrances to free diffusion and electrical
migration through the ECS. We used the value λ = 1.6 [46]. We denote the fraction of tissue
volume belonging to the ECS by α, and set the value α = 0.2. The sources in the system are
Table 1. The physical parameters used in the simulations.
symbol explanation value
R gas constant 8.314 J/(K mol)
T temperature 300 K
F Faraday’s constant 9.648 × 10
4
C/mol
0
vacuum permittivity 8.854 × 10
12
F/m
r
relative permittivity 80
https://doi.org/10.1371/journal.pcbi.1006510.t001
Modelling large scale electrodiffusion surrounding morphologically detailed neurons
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1006510 October 4, 2018 5 / 26

Figures (9)
Citations
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Journal ArticleDOI
TL;DR: This paper introduces and numerically evaluates a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree and studies ephaptic coupling induced in an unmyelinated axon bundle.
Abstract: Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modeling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting.

15 citations


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)....

    [...]

  • ...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....

    [...]

  • ...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....

    [...]

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

    [...]


Journal ArticleDOI
TL;DR: The creation of this NEURON interface provides a pathway for interoperability that can be used to automatically export this class of models into complex intracellular/extracellular simulations and future cross-simulator standardization.
Abstract: Development of credible clinically-relevant brain simulations has been slowed due to a focus on electrophysiology in computational neuroscience, neglecting the multiscale whole-tissue modeling approach used for simulation in most other organ systems. We have now begun to extend the NEURON simulation platform in this direction by adding extracellular modeling. The extracellular medium of neural tissue is an active medium of neuromodulators, ions, inflammatory cells, oxygen, NO and other gases, with additional physiological, pharmacological and pathological agents. These extracellular agents influence, and are influenced by, cellular electrophysiology, and cellular chemophysiology-the complex internal cellular milieu of second-messenger signaling and cascades. NEURON's extracellular reaction-diffusion is supported by an intuitive Python-based where/who/what command sequence, derived from that used for intracellular reaction diffusion, to support coarse-grained macroscopic extracellular models. This simulation specification separates the expression of the conceptual model and parameters from the underlying numerical methods. In the volume-averaging approach used, the macroscopic model of tissue is characterized by free volume fraction-the proportion of space in which species are able to diffuse, and tortuosity-the average increase in path length due to obstacles. These tissue characteristics can be defined within particular spatial regions, enabling the modeler to account for regional differences, due either to intrinsic organization, particularly gray vs. white matter, or to pathology such as edema. We illustrate simulation development using spreading depression, a pathological phenomenon thought to play roles in migraine, epilepsy and stroke. Simulation results were verified against analytic results and against the extracellular portion of the simulation run under FiPy. The creation of this NEURON interface provides a pathway for interoperability that can be used to automatically export this class of models into complex intracellular/extracellular simulations and future cross-simulator standardization.

10 citations


References
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Journal ArticleDOI
TL;DR: This work presents the basic ideas that would help informed users make the most efficient use of NEURON, the powerful and flexible environment for implementing models of individual neurons and small networks of neurons.
Abstract: The moment-to-moment processing of information by the nervous system involves the propagation and interaction of electrical and chemical signals that are distributed in space and time. Biologically realistic modeling is needed to test hypotheses about the mechanisms that govern these signals and how nervous system function emerges from the operation of these mechanisms. The NEURON simulation program provides a powerful and flexible environment for implementing such models of individual neurons and small networks of neurons. It is particularly useful when membrane potential is nonuniform and membrane currents are complex. We present the basic ideas that would help informed users make the most efficient use of NEURON.

2,410 citations


Book
24 Feb 2012
TL;DR: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software.
Abstract: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

1,982 citations


"A Kirchhoff-Nernst-Planck framework..." refers methods in this paper

  • ...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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BookDOI
12 Nov 1998
TL;DR: This work presents simplified models of individual neurons, and unconventional coupling, of action-potential generation and phase space analysis of neuronal excitability in response to the Hodgkin-Huxley model.
Abstract: 1. The membrane equation 2. Linear cable theory 3. Passive dendritic trees 4. Synaptic input 5. Synaptic interactions in a passive dendritic tree 6. The Hodgkin-Huxley model of action-potential generation 7. Phase space analysis of neuronal excitability 8. Ionic channels 9. Beyond Hodgkin and Huxley: calcium, and calcium-dependent potassium currents 10. Linearizing voltage-dependent currents 11. Diffusion, buffering, and binding 12. Dendritic spines 13. Synaptic plasticity 14. Simplified models of individual neurons 15. Stochastic models of single cells 16. Bursting cells 17. Input resistance, time constants, and spike initiation 18. Synaptic input to a passive tree 19. Voltage-dependent events in the dendritic tree 20. Unconventional coupling 21. Computing with neurons - a summary

1,351 citations


DOI
07 Dec 2015
TL;DR: The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods.
Abstract: The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5.

1,200 citations


"A Kirchhoff-Nernst-Planck framework..." refers methods in this paper

  • ...The mesh was automatically generated using mshr [49], yielding a mesh with 53619 263 linear tetrahedral cells....

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Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "A kirchhoff-nernst-planck framework for modeling large scale extracellular electrodiffusion surrounding morphologically detailed neurons" ?

Here, the authors present the 3-D Kirchhoff-Nernst-Planck ( KNP ) framework, tailored to explore electrodiffusive effects on large spatiotemporal scales. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. This work was funded by the Research Council of Norway ( BIOTEK2021 Digital Life project ‘ DigiBrain ’, project 248828 ). The authors have declared that no competing interests exist. In the present work, the authors use the 3-D KNP framework to simulate the dynamics of ion concentrations and the electrical potential surrounding a morphologically detailed pyramidal cell.