#  A Tridomain Model for Potassium Clearance in Optic Nerve of Necturus

Yi Zhu  Department of Mathematics and Statistics, York University, Toronto,
Ontario, Canada. Shixin Xu Corresponding author<EMAIL_ADDRESS>Duke Kunshan University, 8 Duke Ave, Kunshan, Jiangsu, China.  Robert.S.
Eisenberg  Department of Applied Mathematics, Illinois Institute of
Technology, Chicago IL 60616 USA.  Huaxiong Huang  Joint Mathematical Research
Centre of Beijing Normal University and BNU-HKBU United International College,
Zhuhai, China  Department of Mathematics and Statistics, York University,
Toronto, Ontario, Canada. Division of Science and Technology, BNU- HKBU United
International College, Zhuhai, 519087, China

## Abstract

The accumulation of potassium in the narrow space outside nerve cells is a
classical subject of biophysics that has received much attention recently. It
may be involved in potassium accumulation including spreading depression,
perhaps migraine and some kinds of epilepsy, even (speculatively) learning.
Quantitative analysis is likely to help evaluate the role of potassium
clearance from the extracellular space after a train of action potentials.
Clearance involves three structures that extend down the length of the nerve:
glia, extracellular space, and axon and so need to be described as systems
distributed in space in the tradition used for electrical potential in the
‘cable equations’ of nerve since the work of Hodgkin in 1937. A three-
compartment model is proposed here for the optic nerve and is used to study
the accumulation of potassium and its clearance. The model allows the
convection, diffusion, and electrical migration of water and ions. We depend
on the data of Orkand et al to ensure the relevance of our model and align its
parameters with the anatomy and properties of membranes, channels, and
transporters: our model fits their experimental data quite well. The aligned
model shows that glia has an important role in buffering potassium, as
expected. The model shows that potassium is cleared mostly by convective flow
through the syncytia of glia driven by osmotic pressure differences. A
simplified model might be possible, but it must involve flow down the length
of the optic nerve. It is easy for compartment models to neglect this flow.
Our model can be used for structures quite different from the optic nerve that
might have different distributions of channels and transporters in its three
compartments. It can be generalized to include a fourth (distributed)
compartment representing blood vessels to deal with the glymphatic flow into
the circulatory system.

##  1 Introduction

The now classical experiments of Hodgkin, Huxley, and Katz [27, 25] were
designed to avoid the artifact of concentration polarization, the
(significant) change of concentration of ions as the currents maintaining
their voltage clamp flowed across electrodes inside the axon and its
membranes. Change of concentration of potassium was seen nonetheless after a
few milliseconds of outward potassium currents ([28]: p. 482, 485, 489, 494;
[17]), but the change was slow enough to be ignored in their calculations of
the action potential [26].

The Kuffler group at Harvard was interested in the role of glia [36, 37] in
the central nervous system and showed that the membrane potential of the optic
nerve of Necturus reported the potassium concentration in the narrow
extracellular space just outside axons, between glia and axon. Orkand et al
[53] used this membrane potential of glia to report the change in potassium
concentration—the polarization of concentration in the narrow extracellular
space—as it accumulated during a train of nerve action potentials. Earlier
work [28, 17] had inferred this concentration change. Orkand et al [37]
measured it quite directly. The artifact of concentration polarization that so
worried Hodgkin and Huxley became the experimental reality of potassium
accumulation [28] in the central nervous system [53], that interested the
Kuffler group. Interest has only grown in the following fifty some odd years.

Potassium accumulation and flow in the extracellular space have been shown to
have important roles in many papers focused on aging, Alzheimers disease,
anesthesia, dementia, diabetes, epilepsy, migraine, sleep, stroke and
traumatic brain injury. Potassium accumulation and flow play an important role
in the biology of the central nervous system [49] normal, and abnormal [52,
66, 2, 73, 67, 48, 46]. The glymphatic model has received much interest in the
last months. It links potassium accumulation, flows in the extracellular
space—particularly in sleep and diseases of aging—with general disposal of
waste through glial pathways to the circulatory system [49, 18, 32, 1, 31,
45].

The accumulation of ions in a narrow extracellular space, like that between
nerve and glia, or Schwann cell [17], depends on the diffusion, convection,
and migration of ions in an electric field. Convection is likely to be
important. Evolution uses the circulatory system to provide convective
transport close to nearly every cell in a mammal, the lens of the eye being a
notable exception, so the delays involved in electrodiffusion are overcome.
Convection provides what diffusion denies: speed.

The optic nerve, and the mammalian central nervous system in general, involves
nerve, glia and narrow extracellular space. It involves three types of flow,
convection, diffusion, and migration in the electric field in radial and
longitudinal directions of a cylindrical structure. It involves (chiefly)
three ions (K, Na, Cl), and a number of different types of channels and pumps
(voltage activated Na channels and at least two types of K channels and the
Na/K pump). The description and analysis, not to say the numerical computation
of the optic nerve, must deal with what is actually in the optic nerve. It
must deal with what evolution has actually built in the central nervous system
in general. Thus, the analysis must involve many forces, flows, structures,
and channels and transporters. Here we report the subset of our work that
deals with the accumulation of potassium in the narrow extracellular space of
optic nerve.

The interactions between neuronal cells and glial cells have been included in
models of the important phenomenon of spreading depression [65, 58, 7, 75]
thought to be related to epilepsy and migraine. Some two-compartment models
for potassium clearance (or spatial buffering) include interactions between
neuron cells and extracellular space [4] or interactions between glial cells
and extracellular space [8, 60, 54]. A tri-compartment model using ordinary
differential equations (ODEs) was introduced by Sibille [62] to study the role
of $\mathrm{K_{ir4}}$ channels. It shows that the flows play an important role
in the central nerve system [5] via influx and efflux routes to help waste
clearance, which has been called (with understandable enthusiasm if not
hyperbole) a final frontier of neuroscience [51]. Some models, including flow
but not electrodiffusion, were introduced to study the pressure effect on the
flow [29, 47, 34, 3]. Mori [48] proposed a multidomain model for cortical
spreading depression, where ionic electrodiffusion and osmosis between
different compartments are considered. In this paper, we first extend those
results and the two-compartment structural analysis of the spherical lens [76]
to the three-compartment cylindrical optic nerve fiber of Necturus. Then we
present some general conclusions based on the analysis of a specific set of
experiments Orkand et al [37, 53] using a model distributed in space in both
radial and longitudinal directions and that involves action potentials
generated by Hodgkin Huxley equations.

This paper is organized as follows. The full model for microcirculation of
water and ions are proposed based on conservation laws in Section 2. Then the
model is calibrated and aligned with the Orkand experiment results in Section
3. In Section 4, the calibrated model is used to study the flow and ion
microcirculations during Potassium clearance. A discussion on the parameters
is presented in Section 5. Then the conclusions and future work are given in
Section 6.

##  2 Mathematical model

In this section, we present a tridomain model for microcirculation of the
optic nerve. The model deals with two types of flow: the circulation of water
(hydrodynamics) and the circulation of ions (electrodynamics) in the

  * •

glial compartment $(\Omega_{gl})$;

  * •

axon compartment $(\Omega_{ax})$;

  * •

extracellular space $(\Omega_{ex})$.

The glial compartment and axon compartment exist only in the optic nerve,
while extracellular space exists both in the optic nerve $\Omega^{OP}_{ex}$
and subarachnoid space $\Omega^{SAS}_{ex}$ (See Fig.1)

$\Omega_{OP}=\Omega_{ax}\cup\Omega_{gl}\cup\Omega_{ex}^{OP},\quad\Omega_{SAS}=\Omega_{ex}^{SAS}.$

The model is mainly derived from laws of conservation of ions and water for
flow through membranes between intracellular compartments and extracellular
space [50].

Figure 1: Optic nerve structure. a: Key features of the optic nerve region and
subarachnoid space (SAS); b: Longitudinal section of the optic nerve; c: cross
section of the optic nerve. Figure 2: Domain of the axial symmetry model. The
optic nerve $\Omega_{OP}$ consist of axon compartment $\Omega_{ax}$, glial
compartment $\Omega_{gl}$ and extracellular space $\Omega_{ex}^{OP}$. The
subarachnoid space only has extracellular space $\Omega_{ex}^{SAS}$.
$R_{a}={\color[rgb]{0,0,0}48}\ \mu m$ is the radius of optic nerve and
$R_{b}={\color[rgb]{0,0,0}60}\ \mu m$ is the radius from optical nerve center
to the dura mater.

###  2.1 Notations and Assumptions

We first introduce the following notations used in the paper, where
$i=\mathrm{Na^{+},K^{+},Cl^{-}}$ for ion species, $l=ex,gl,ax$ for
extracellular space, glial compartment and axon compartment, and $k=gl,ax$ for
glial or axon membrane in the optic nerve.

Fig.1 b shows the model with the whole domain $\Omega$ that consists of the
subarachnoid space (SAS) region $\Omega_{SAS}$ and optic nerve region
$\Omega_{OP}$ (see Fig.2),

$\Omega=\Omega_{SAS}\cup\Omega_{OP}.$

In the optic nerve region $\Omega_{OP}$, the glial membrane and axon membrane
separate domains $\Omega_{gl}$, $\Omega_{ax}$ from the extracellular region
$\Omega_{ex}^{OP}$, respectively (also see Fig. 1). Based on the structure of
the optic nerve, we have the following global assumptions for the model:

•Charge neutrality: In each domain, we assume that there is electroneutrality

$\displaystyle\eta_{gl}\sum_{i}z^{i}C_{gl}^{i}+z^{gl}\eta_{gl}^{re}A_{gl}$
$\displaystyle=0,$ (1a)
$\displaystyle\eta_{ax}\sum_{i}z^{i}C_{ax}^{i}+z^{ax}\eta_{ax}^{re}A_{ax}$
$\displaystyle=0,$ (1b) $\displaystyle\sum_{i}z^{i}C_{ex}^{i}$
$\displaystyle=0,$ (1c)

where $A_{l}>0$ with $l=ax,gl$ is the density of proteins in axons or glial
cells. The proteins are negatively charged, but the charge density is
customarily described by a positive number. The $\eta_{ax}$ and $\eta_{gl}$
are the volume fraction of axon and glial compartments in the optic nerve and
$\eta_{ax}^{re}$ and $\eta_{gl}^{re}$ are the resting state volume fractions.

•Axial symmetry: For simplicity, axial symmetry is assumed. The model can be
straightforwardly extended to three dimensions when data and needs justify the
considerable extra computational resources needed to analyze such models.

•Isotropy of glial compartment and extracellular space:

  * 1)

The extracellular space forms a narrow structure of branching clefts
surrounding the glial cells and nerve axons.

  * 2)

The glial cells are connected to each other by connexins and form a syncytium.

  * 3)

The extracellular space is continuous and forms a syncytium.

Both syncytia are assumed isotropic here, until we know better. The axons are
not connected to each other. For $l=gl,ex,$ and
$i=\mathrm{Na^{+},K^{+},Cl^{-}}$, the ion flux and water flow velocity are in
the following forms

$\displaystyle\boldsymbol{j}_{l}^{i}$
$\displaystyle=j_{l,r}^{i}\hat{\boldsymbol{r}}+j_{l,z}^{i}\hat{\boldsymbol{z}},$
(2) $\displaystyle\boldsymbol{u}_{l}$
$\displaystyle=u_{l}^{r}\hat{\boldsymbol{r}}+u_{l}^{z}\hat{\boldsymbol{z}}.$
(3)

•Anisotropy of axon compartment: The axons are separated, more or less
parallel cylindrical cells that do not form a syncytium. For
$i=\mathrm{Na^{+},K^{+},Cl^{-}}$, the ion flux and water flow velocity are in
the following forms

$\displaystyle\boldsymbol{j}_{ax}^{i}$
$\displaystyle=j_{ax,z}^{i}\hat{\boldsymbol{z}},$ (4)
$\displaystyle\boldsymbol{u}_{ax}$
$\displaystyle=u_{ax}^{z}\hat{\boldsymbol{z}}.$ (5)

•Communications between compartments: The communications between intracellular
compartments and extracellular space are through membranes. There is no direct
interaction between glial and axon compartments. Interactions occur only
through changes in concentration, electrical potential, and flows in the
extracellular space [62].

The interface of optic nerve and SAS is the pia mater denoted by $\Gamma_{7}$,

$\Omega_{OP}\cap\Omega_{SAS}=\Gamma_{7}.$

In our model, both pia mater $\Gamma_{7}$ and dura mater $\Gamma_{4}$ are
modeled as macroscopic membranes and appropriate boundary conditions. The
transmembrane water flow through pia mater [15] depends on hydrostatic
pressure, osmotic pressure and electric potential, while that through dura
mater only depends on the hydrostatic pressure.

For the domain boundaries, $\Gamma_{1}$ is the radius center of the optic
nerve; $\Gamma_{2}$ and $\Gamma_{3}$ are the far end (away from the eyeball)
of the optic nerve which is connected to optic canal region [21]. $\Gamma_{5}$
is used to model the dura mater connected to the sclera (the white matter of
the eye) and assumed to be non-permeable [22]. $\Gamma_{6}$ is used to denote
the lamina cribrosa where the optic nerve head exits the eye posteriorly
through pores of the lamina cribrosa [33].

###  2.2 Water circulation

We model water circulation with the following assumptions

  * •

the loss or gain of water in axons and glial cells is only through membranes
flowing into or out of the extracellular space.

  * •

the transmembrane water flux is proportional to the intra/extra-cellular
hydrostatic pressure and osmotic pressure differences.

  * •

the glial cell and axons can swell and shrink due to the water inflows and
outflows.

We use $\eta_{l}(r,z,t)$ to describe the volume fraction of $l$ region
$(l=gl,ax,ex)$, which varies over time and space due to transmembrane water
flows. The conservation of water in each domain yields

$\displaystyle\frac{\partial\eta_{gl}}{\partial
t}+\mathcal{M}_{gl}U^{m}_{gl}+\nabla\cdot\left(\eta_{gl}\boldsymbol{u}_{gl}\right)=0,$
(6a) $\displaystyle\frac{\partial\eta_{ax}}{\partial
t}+\mathcal{M}_{ax}U^{m}_{ax}+\frac{\partial}{\partial
z}\left(\eta_{ax}u_{ax}^{z}\right)=0,$ (6b)
$\displaystyle\nabla\cdot\left(\eta_{gl}\boldsymbol{u}_{gl}\right)+\nabla\cdot\left(\eta_{ex}\boldsymbol{u}_{ex}\right)+\frac{\partial}{\partial
z}\left(\eta_{ax}u_{ax}^{z}\right)=0,$ (6c)

where we use the fact that density of water is constant. Here
$\boldsymbol{u}_{l}$ with $l=gl,ax,ex$ is the velocity in the glial cells,
axons, and extracellular space, respectively. The transmembrane water flow
$U^{m}_{k}$ with $k=gl,ax$ follows the Starling’s law on the $k$th membrane,

$\displaystyle U^{m}_{gl}$
$\displaystyle=L_{gl}^{m}\left(P_{gl}-P_{ex}-\gamma_{gl}k_{B}T\left(O_{gl}-O_{ex}\right)\right),$
$\displaystyle U^{m}_{ax}$
$\displaystyle=L_{ax}^{m}\left(P_{ax}-P_{ex}-\gamma_{ax}k_{B}T\left(O_{ax}-O_{ex}\right)\right).$

The $P_{l}$ with $l=gl,ax,ex$ are the hydrostatic pressure in the glial cells,
axons, and extracellular space, respectively. And $k_{B}TO_{l}$ is the osmotic
pressure [74, 76] defined by

$O_{ex}=\sum_{i}C_{ex}^{i},\quad
O_{l}=\sum_{i}C_{l}^{i}+A_{l}\frac{\eta_{j}^{re}}{\eta_{l}},\quad l=gl,ax,$

where $A_{l}\frac{\eta_{l}^{re}}{\eta_{l}}>0$ is the density of the permanent
negatively charged protein in glial cells and axons that varies with the
volume (fraction) of the region. In this paper, we assume the permanent
negatively charged protein is uniformly distributed within glial cells and
axons and has valence $z^{l}$, $l=gl,ax$. The $\mathcal{M}_{k}$ and
$\gamma_{k}$, $k=gl,ax$ are the glial cells (or axons) membrane area per unit
volume and membrane reflection coefficient [13] respectively. $L_{gl}^{m}$ and
$L_{ax}^{m}$ are the surface membrane hydraulic permeabilities of glial cells
and axons. The membrane reflection coefficient $\gamma_{k}\ (k=gl,ax)$ is the
ratio between the observed osmotic pressure and theoretical osmotic pressure.
$k_{B}$ is Boltzmann constant, and $T$ is temperature.

For the volume fraction $\eta_{l},\ l=gl,ax,ex$, we have

$\eta_{gl}+\eta_{ax}+\eta_{ex}=1,\quad\text{ in }\Omega.$ (7)

######  Remark 2.1.

Note the glial cells and axons are found only in the $\Omega_{OP}$ region. In
other words, $\eta_{ax}=\eta_{gl}\equiv 0$ and $\eta_{ex}\equiv 1$ are fixed
in $\Omega_{SAS}$. Therefore, the solution is incompressible in the
$\Omega_{SAS}$, and we have

$\nabla\cdot\boldsymbol{u}_{ex}=0,\quad\text{ in }~{}\Omega_{SAS}.$ (8)

The relation between the hydrostatic pressure $P_{l}$ and volume fraction
$\eta_{l}\ (l=gl,ax,ex)$ is connected by the force balance on the membrane
$\mathcal{M}_{k}\ (k=gl,ax)$.  The membrane force is balanced with the
hydrostatic pressure difference on both sides of the semipermeable membrane
[74, 48]. Then the variation of volume fraction from the resting state is
proportional to the variation of hydrostatic pressure difference from the
resting state.

$\displaystyle K_{gl}\left(\eta_{gl}-\eta_{gl}^{re}\right)$
$\displaystyle=P_{gl}-P_{ex}-\left(P_{gl}^{re}-P_{ex}^{re}\right),$ (9a)
$\displaystyle K_{ax}\left(\eta_{ax}-\eta_{ax}^{re}\right)$
$\displaystyle=P_{ax}-P_{ex}-\left(P_{ax}^{re}-P_{ex}^{re}\right),$ (9b)

where $K_{gl,ax}$ is the stiffness constant and $\eta_{l}^{re}$ and
$P_{l}^{re}\ (l=gl,ax,ex)$ are the resting state volume fraction and
hydrostatic pressure.

######  Remark 2.2.

The osmotic pressure influences the volume fraction in an implicit way. Due to
Eq. (6) and the definition of transmembrane water flow $U^{m}_{gl}$, the
osmostic pressure difference induces the transmembrane water flow going in and
out of compartments, and the hydrostatic pressure changes correspondingly.

######  Remark 2.3.

In reality, both dura mater $\Gamma_{7}$ and pia mater $\Gamma_{4}$ are
deformable. However, the stiffness of them are much larger than glial cells’
[30, 39, 14]. For the sake of simplicity, we neglect the deformation of dura
mater and pia mater so the system could be solved in fixed domains $\Omega$
and $\Omega^{OP}$. We are aware that in some important clinical applications
the deformation is of great importance and that our model will need to be
extended to deal with them. The extensions do not produce analytical
challenges but they are likely to add significant complexity to the numerical
methods and make the computations much more involved and longer.

Note that the concentrations of ions and effective concentration of water vary
a great deal and so are described by equations in which the number density of
ions and effective number density of water vary in both the radial and
longitudinal directions according to conservation laws, without using
compartments that may not have unique definitions or relations to anatomical
structures. Indeed, the variation of concentration is one of the main
determinants of the properties of ionic solutions. The solution is
incompressible, the components are not [9].

Next, we define the velocity in each domain.

Water Velocities in the Glial Compartment. As we mentioned before, the glial
cells are connected through the narrow connections on the membranes and form a
syncytium. The fluid velocity inside the compartment is limited by the
transmembrane fluid velocity, which is determined by the difference of hydro
pressure and osmotic pressure on both sides of the membrane [74, 48]. On the
macro-scale, by using averaging process [76, 40, 41], the glial compartment is
treated with membranes everywhere. So the velocity of fluid in glial syncytium
$\boldsymbol{u}_{gl}$ depends on the gradients of hydrostatic pressure and
osmotic pressure:

$\displaystyle u_{gl}^{r}$
$\displaystyle=-\frac{\kappa_{gl}\tau_{gl}}{\mu}\left(\frac{\partial
P_{gl}}{\partial r}-\gamma_{gl}k_{B}T\frac{\partial O_{gl}}{\partial
r}\right),$ (10a) $\displaystyle u_{gl}^{z}$
$\displaystyle=-\frac{\kappa_{gl}\tau_{gl}}{\mu}\left(\frac{\partial
P_{gl}}{\partial z}-\gamma_{gl}k_{B}T\frac{\partial O_{gl}}{\partial
z}\right).$ (10b)

Substituting Eq. (10a) into the incompressibility (6a) yields a Poisson
equation for pressure in the glial compartment. The boundary conditions in the
glial syncytium are as follows

$\left\\{\begin{aligned} &\frac{\partial P_{gl}}{\partial r}=0,&\text{ on
}\Gamma_{1}\cup\Gamma_{7},\\\ &\frac{\partial P_{gl}}{\partial z}=0,&\text{ on
}\Gamma_{2}\cup\Gamma_{6},\end{aligned}\right.$ (11)

where homogeneous Neumann boundary condition is applied at all four of these
boundaries.

Water Velocity in the Axon Compartment. Axons are arranged parallel in the
longitudinal direction and are isolated from each other. The fluid velocity in
the axon compartment is defined along $z$ direction as

$\displaystyle u_{ax}^{r}$ $\displaystyle=0,$ (12a) $\displaystyle u_{ax}^{z}$
$\displaystyle=-\frac{\kappa_{ax}}{\mu}\frac{\partial P_{ax}}{\partial z}.$
(12b)

Similarly, substituting above velocity into the incompressibility equation
gives the Poisson equation for pressure in the axon compartment. A homogeneous
Neumann condition for pressure is used on the left and right boundaries of the
axon compartment

$\frac{\partial P_{ax}}{\partial z}=0,~{}~{}\text{ on
}\Gamma_{2}\cup\Gamma_{6}.$ (13)

Velocity in the Extracellular Space. The extracellular space is a narrow
connected domain, where the electro-osmotic effect needs to be considered [76,
71, 68]. The extracellular velocity is determined by the gradients of
hydrostatic pressure and electric field

$\displaystyle u_{ex}^{r}$
$\displaystyle=-\frac{\kappa_{ex}\tau_{ex}}{\mu}\frac{\partial
P_{ex}}{\partial r}-k_{e}\tau_{ex}\frac{\partial\phi_{ex}}{\partial r},$ (14a)
$\displaystyle u_{ex}^{z}$
$\displaystyle=-\frac{\kappa_{ex}\tau_{ex}}{\mu}\frac{\partial
P_{ex}}{\partial z}-k_{e}\tau_{ex}\frac{\partial\phi_{ex}}{\partial z},$ (14b)

where $\phi_{ex}$ is the electric potential in the extracellular space,
$\tau_{ex}$ is the tortuosity of extracellular region [50, 56] and $\mu$ is
the viscosity of water, $k_{e}$ describes the effect of electro-osmotic flow
[44, 68, 71], $\kappa_{ex}$ is the permeability of the extracellular space.
Here the hydro permeability $\kappa_{ex}$, tortuosity $\tau_{ex}$ and
electric-osmotic parameter $k_{e}$ have two distinguished values in the region
$\Omega_{ex}^{OP}$ and $\Omega_{ex}^{SAS}$,

$\displaystyle\kappa_{ex}=\left\\{\begin{array}[]{l}\kappa_{ex}^{OP},\text{ in
}\Omega_{OP},\\\ \kappa_{ex}^{SAS},\text{ in
}\Omega_{SAS},\end{array}\right.\tau_{ex}=\left\\{\begin{array}[]{l}\tau_{ex}^{OP},\text{
in }\Omega_{OP},\\\ \tau_{ex}^{SAS},\text{ in
}\Omega_{SAS},\end{array}\right.$ (15) $\displaystyle
k_{e}=\left\\{\begin{array}[]{l}k_{e}^{OP},\text{ in }\Omega_{OP},\\\
k_{e}^{SAS},\text{ in }\Omega_{SAS},\end{array}\right.$

######  Remark 2.4.

By substituting the definitions of velocities Eqs. (12a), (10a) and (14a), and
volume fractions Eq.(9a) into the mass conservation law Eq. (6), it yields the
evolution equations of hydrostatic pressure in different compartments, which
are solved during the simulations in Session 4.

Since $\Gamma_{2}\cup\Gamma_{3}$ are the far end of optic nerve away from
eyeball and connect to the optic canal, we assume the hydrostatic pressure of
extracellular fluid is equal to the cerebrospinal fluid pressure. On the other
hand, the intraocular pressure (IOP) is imposed at $\Gamma_{6}$ where the
extracellular space is connected to the retina. At the boundary $\Gamma_{5}$,
we assume a non-permeable boundary. We are aware of the significance of the
pressures and flows at these boundaries for clinical phenomena including
glaucoma [3, 20, 55] and will return to that subject in later publications.

The water flow across the semi-permeable membrane $\Gamma_{4}$ is produced by
the lymphatic drainage on the dura membrane, which depends on the difference
between extracellular pressure and orbital pressure (OBP). We assume the
velocity across the pia membrane $\Gamma_{4}$, is continuous and determined by
the hydrostatic pressure and osmotic pressure. To summarize, the boundary
conditions of the extracellular fluid are

$\left\\{\begin{aligned}
&\boldsymbol{u}_{ex}\cdot\hat{\boldsymbol{r}}=0,&&\mbox{on $\Gamma_{1}$},\\\
&P_{ex}=P_{CSF},&&\mbox{on $\Gamma_{2}\cup\Gamma_{3}$},\\\
&\boldsymbol{u}^{SAS}_{ex}\cdot\
\hat{\boldsymbol{r}}=L^{m}_{dr}\left(P^{SAS}_{ex}-P_{OBP}\right),&&\mbox{on
$\Gamma_{4}$},\\\ &\boldsymbol{u}_{ex}\cdot\hat{\boldsymbol{z}}=0,&&\mbox{on
$\Gamma_{5}$},\\\ &P_{ex}=P_{IOP},&&\mbox{on $\Gamma_{6}$},\\\
&\boldsymbol{u}^{OP}_{ex}\cdot\hat{\boldsymbol{r}}=\boldsymbol{u}^{SAS}_{ex}\cdot\hat{\boldsymbol{r}}&&\\\
&=L^{m}_{pia}\left(P^{OP}_{ex}-P^{SAS}_{ex}-\gamma_{pia}k_{B}T\left(O^{OP}_{ex}-O^{SAS}_{ex}\right)\right),&&\mbox{on
$\Gamma_{7}$},\end{aligned}\right.$ (16)

where $P_{CSF}$ is the cerebrospinal fluid pressure [3] and $P_{IOP}$ is the
intraocular pressure and $P_{OBP}$ is the orbital pressure on the dura mater.

###  2.3 Ion Transport

For ion circulation, we assume

  * •

only three types of ions are considered: $\mathrm{Na^{+},K^{+}}$ and
$\mathrm{Cl^{-}}$.

  * •

the sodium-potassium ATP pump is present on both glial and axon membranes.

  * •

ion channel conductance on glial cell membranes is a fixed constant,
independent of the voltage and time. The sodium conductance is assumed small,
and its channel origin unknown. The potassium conductance is large and comes
from the $K_{ir4}$ channels [5, 63]. When experimental evidence is available,
other types of pumps and channels can be added to the model.

  * •

sodium channel and potassium channel conductance on axons are voltage-gated,
while the conductance of chloride channel is fixed.

The conservation of ions implies the following system of partial differential
equations to describe the dynamics of ions in each region, for
$i=\mathrm{Na^{+},K^{+},Cl^{-}}$

$\displaystyle\frac{\partial\left(\eta_{gl}C_{gl}^{i}\right)}{\partial
t}+\mathcal{M}_{gl}J^{m,i}_{gl}+\nabla\cdot\left(\eta_{gl}\boldsymbol{j}_{gl}^{i}\right)=0,$
(17a) $\displaystyle\frac{\partial\left(\eta_{ax}C_{ax}^{i}\right)}{\partial
t}+\mathcal{M}_{ax}J^{m,i}_{ax}+\frac{\partial}{\partial
z}\left(\eta_{ax}j_{ax,z}^{i}\right)=0,$ (17b)
$\displaystyle\frac{\partial\left(\eta_{ex}C_{ex}^{i}\right)}{\partial
t}-\mathcal{M}_{ax}J^{m,i}_{ax}-\mathcal{M}_{gl}J^{m,i}_{gl}+\nabla\cdot\left(\eta_{ex}\boldsymbol{j}_{ex}^{i}\right)=0,$
(17c)

where the last equation reduces to the following in the $\Omega_{SAS}$ region,

$\frac{\partial C_{ex}^{i,SAS}}{\partial
t}+\nabla\cdot\boldsymbol{j}_{ex}^{i,SAS}=0.$ (18)

The transmembrane ion flux $J_{k}^{m,i}\ (k=gl,ax)$ consists of active ion
pump source $a_{k}^{i}$ and passive ion channel source $b_{k}^{i}$, for ion
$i$ on the axons $(k=ax)$ or glial cells membranes $(k=gl)$. In the glial cell
membranes,

$J_{k}^{m,i}=a_{k}^{i}+b_{k}^{i},\quad k=gl,ax,\quad
i=\mathrm{Na}^{+},\mathrm{K}^{+},\mathrm{Cl}^{-}.$

In the glial cell membranes, $b_{gl}^{i}$ is defined as

$b_{gl}^{i}=\frac{g_{gl}^{i}}{z^{i}e}\left(\phi_{gl}-\phi_{ex}-E_{gl}^{i}\right),$
(19)

where the Nernst potential is used to describe the gradient of chemical
potential
$E_{gl}^{i}=\frac{k_{B}T}{ez^{i}}\log\left(\frac{C_{ex}^{i}}{C_{gl}^{i}}\right)$
and the conductance $g_{gl}^{i}$ for each ion on the glial membrane is a fixed
constant, independent of voltage and time. On the axon’s membrane,
$b_{ax}^{i}$ is defined as

$b_{ax}^{i}=\frac{g_{ax}^{i}}{z^{i}e}\left(\phi_{ax}-\phi_{ex}-E_{ax}^{i}\right),$
(20)

where

$\begin{array}[]{l}g_{ax}^{Na}=\bar{g}^{Na}m^{3}h+g_{leak}^{Na},\quad
g_{ax}^{K}=\bar{g}^{K}n^{4}+g_{leak}^{K},\\\
E_{ax}^{i}=\frac{k_{B}T}{ez^{i}}\log\left(\frac{C_{ex}^{i}}{C_{ax}^{i}}\right).\end{array}$

The time-dependent dynamic of open probability, often loosely called ‘gating’
is governed by the Hodgkin-Huxley model [64, 16]

$\displaystyle\frac{dn}{dt}$ $\displaystyle=\alpha_{n}(1-n)-\beta_{n}n,$ (21a)
$\displaystyle\frac{dm}{dt}$ $\displaystyle=\alpha_{m}(1-m)-\beta_{m}m,$ (21b)
$\displaystyle\frac{dh}{dt}$ $\displaystyle=\alpha_{h}(1-h)-\beta_{h}h.$ (21c)

We assume that the only pump is the Na/K active transporter. We are more than
aware that other active transport systems can and likely do move ions and
water in this system. They will be included as experimental information
becomes available.

In the case of the Na/K pump $a_{l}^{i},l=ax,gl$, the strength of the pump
depends on the concentration in the intracellular and extracellular space [19,
16], i.e.

$a_{k}^{Na}=\frac{3I_{k}}{e},\ a_{k}^{K}=-\frac{2I_{k}}{e},\ a_{k}^{Cl}=0,\
k=gl,ax,$ (22)

where

$\displaystyle I_{k}$
$\displaystyle=I_{k,1}\left(\frac{c_{k}^{Na}}{c_{k}^{Na}+K_{Na1}}\right)^{3}\left(\frac{c_{ex}^{K}}{c_{ex}^{K}+K_{K1}}\right)^{2}$
$\displaystyle+I_{k,2}\left(\frac{c_{k}^{Na}}{c_{k}^{Na}+K_{Na2}}\right)^{3}\left(\frac{c_{ex}^{K}}{c_{ex}^{K}+K_{K2}}\right)^{2},$

$I_{k,1}$ and $I_{k,2}$ are related to $\alpha_{1}-$ and $\alpha_{2}-$ isoform
of $\mathrm{Na/K}$ pump.

The definitions of ion flux in each domain are as follows, for
$i=\mathrm{Na^{+},K^{+},Cl^{-}}$,

$\displaystyle\boldsymbol{j}_{l}^{i}=C_{l}^{i}\boldsymbol{u}_{l}-D_{l}^{i}\tau_{l}\left(\nabla
C_{l}^{i}+\frac{z^{i}e}{k_{B}T}C_{l}^{i}\nabla\phi_{l}\right),\ l=gl,ex,$
(23a) $\displaystyle
j_{ax,z}^{i}=C_{ax}^{i}u_{ax}^{z}-D_{ax}^{i}\left(\frac{\partial
C_{ax}^{i}}{\partial
z}+\frac{z^{i}e}{k_{B}T}C_{ax}^{i}\frac{\partial\phi_{ax}}{\partial
z}\right).$ (23b)

For the axon compartment boundary condition, we have

$C_{ax}^{i}=C_{ax}^{i,re},\quad\text{ on }\ \Gamma_{2}\cup\Gamma_{6},$

and

$\left\\{\begin{array}[]{ll}{\color[rgb]{0,0,0}\frac{\partial
C_{gl}^{i}}{\partial r}}=0,&\text{ on }\Gamma_{1},\\\
C_{gl}^{i}=C_{gl}^{i,re},&\text{ on }\Gamma_{2}\cup\Gamma_{6},\\\
\boldsymbol{j}_{gl}^{i}\cdot\hat{\boldsymbol{r}}=0,&\text{ on
}\Gamma_{7},\end{array}\right.$

where the Dirichlet boundary conditions are used at locations
$\Gamma_{2}\cup\Gamma_{6}$ for axons and glial cell, and a non-flux boundary
condition is used for glial cells ions flux on the radius center $\Gamma_{1}$
and pia mater $\Gamma_{7}$ .

For the extracellular space boundary condition, similar boundary conditions
are imposed except on the pia mater $\Gamma_{7}$. The flux across the pia
mater is assumed continuous and Ohm’s law is used [76]. Additionally, a non-
permeable boundary condition is used at location $\Gamma_{5}$ and a
homogeneous Neumann boundary condition is applied at the location of the dura
mater $\Gamma_{4}$,

$\left\\{\begin{array}[]{ll}{\color[rgb]{0,0,0}\frac{\partial
C_{ex}^{i}}{\partial r}}=0,&\text{ on }\Gamma_{1},\\\
C_{ex}^{i}=C_{csf}^{i},&\text{ on }\Gamma_{2}\cup\Gamma_{3},\\\
{\color[rgb]{0,0,0}\frac{\partial C_{ex}^{i}}{\partial r}}=0,&\text{ on
}\Gamma_{4},\\\ \boldsymbol{j}_{ex}^{i}\cdot\hat{\boldsymbol{z}}=0,&\text{ on
}\Gamma_{5},\\\ C_{ex}^{i}=C_{IOP}^{i},&\text{ on }\Gamma_{6},\\\
\boldsymbol{j}_{ex}^{i,OP}\cdot\hat{\boldsymbol{r}}=\boldsymbol{j}_{ex}^{i,SAS}\cdot\hat{\boldsymbol{r}}=\frac{G_{pia}^{i}}{z^{i}e}\left(\phi_{ex}^{OP}-\phi_{ex}^{SAS}-E_{pia}^{i}\right),&\text{
on }\Gamma_{7}.\end{array}\right.$ (24)

Multiplying equations in (17a-17c) with $z_{i}e$ respectively, summing up, and
using equation (1a-1c) and equations (23), we have following system for the
electric potential in $ax,gl,ex$

$\displaystyle\sum_{i}z^{i}e\mathcal{M}_{gl}J^{m,i}_{gl}+\sum_{i}\nabla\cdot\left(z^{i}e\eta_{g}\boldsymbol{j}_{gl}^{i}\right)=0,$
(25a)
$\displaystyle\sum_{i}z^{i}e\mathcal{M}_{ax}J^{m,i}_{ax}+\sum_{i}\frac{\partial}{\partial
z}\left(z^{i}e\eta_{ax}j_{ax,z}^{i}\right)=0,$ (25b)
$\displaystyle\sum_{i}\nabla\cdot\left(z^{i}e\eta_{gl}\boldsymbol{j}_{gl}^{i}\right)+\sum_{i}\frac{\partial}{\partial
z}\left(z^{i}e\eta_{ax}j_{ax,z}^{i}\right)+\sum_{i}\nabla\cdot\left(z^{i}e\eta_{ex}\boldsymbol{j}_{ex}^{i}\right)=0,$
(25c)

which describe the spatial distributions of electric potentials in three
compartments.

In the subarachnoid space $\Omega_{SAS}$, the governing equation for
extracellular electric potential reduces to

$\nabla\cdot\left(\sum_{i}z^{i}e\boldsymbol{j}_{ex}^{i,SAS}\right)=0.$ (26)

The boundary conditions for electric fields $\phi_{ax}$, $\phi_{gl}$ and
$\phi_{ex}$ are given below.
In the axon compartment:

$\left\\{\begin{aligned} {\color[rgb]{0,0,0}\frac{\partial\phi_{ax}}{\partial
z}}=0,&\text{ on }\Gamma_{2},\\\
{\color[rgb]{0,0,0}\frac{\partial\phi_{ax}}{\partial z}}=0,&\text{ on
}\Gamma_{6},\end{aligned}\right.$

In the glial compartment:

$\left\\{\begin{aligned} &{\color[rgb]{0,0,0}\frac{\partial\phi_{gl}}{\partial
r}}=0,&\text{ on }\Gamma_{1},\\\
&{\color[rgb]{0,0,0}\frac{\partial\phi_{gl}}{\partial z}}=0,&\text{ on
}\Gamma_{2},\\\ &{\color[rgb]{0,0,0}\frac{\partial\phi_{gl}}{\partial
z}}=0,&\text{ on }\Gamma_{6},\\\
&{\color[rgb]{0,0,0}\frac{\partial\phi_{gl}}{\partial r}}=0,&\text{ on
}\Gamma_{7},\end{aligned}\right.$

and in the extracellular space:

$\left\\{\begin{aligned} &{\color[rgb]{0,0,0}\frac{\partial\phi_{ex}}{\partial
r}}=0,&&\text{on }\Gamma_{1},\\\
&{\color[rgb]{0,0,0}\frac{\partial\phi_{ex}}{\partial z}}=0,&&\text{on
}\Gamma_{2}\cup\Gamma_{3},\\\
&{\color[rgb]{0,0,0}\frac{\partial\phi_{ex}}{\partial r}}=0,&&\text{on
}\Gamma_{4},\\\ &{\color[rgb]{0,0,0}\frac{\partial\phi_{ex}}{\partial
z}}=0,&&\text{on }\Gamma_{5},\\\
&{\color[rgb]{0,0,0}\frac{\partial\phi_{ex}}{\partial z}}=0,&&\text{on
}\Gamma_{6},\\\
&\sum_{i}z^{i}e\boldsymbol{j}_{ex}^{i,OP}\cdot\hat{\boldsymbol{r}}=\sum_{i}z^{i}e\boldsymbol{j}_{ex}^{i,SAS}\cdot\hat{\boldsymbol{r}}&&\\\
&=\sum_{i}G_{pia}^{i}\left(\phi_{ex}^{OP}-\phi_{ex}^{SAS}-E_{pia}^{i}\right),&&\text{on
}\Gamma_{7}.\end{aligned}\right.$ (27)

##  3 Model calibration

Our work is possible because of, and was motivated by the paper of Orkand et
al [53, 37] that measured the accumulation of potassium in the narrow
extracellular space ($\Omega^{OP}_{ex}$) of the optic nerve of the amphibian
salamander Necturus, in the spirit of the original work of Frankenhaeuser and
Hodgkin [17], that first identified and analyzed accumulation of potassium
outside a nerve fiber. The existence and qualitative properties of that
accumulation of potassium were known to, and a cause for concern for Hodgkin,
from his first work on the voltage clamp [27, 25], if not earlier. Hodgkin
described the phenomena as part of what was called [concentration]
‘polarization’ and discussed it extensively with students sometime later
(Eisenberg, personal communication, 1962)

The key experiment in the Orkand paper [53] measures the change in potential
across the glial membrane produced by a train of action potentials. The glial
membrane potential is used to estimate and report the potassium concentration
in the narrow extracellular space, because the glial membrane is populated
with more or less voltage independent potassium channels and not much else.

In the experiment, optic nerve has been put in three different
$\mathrm{K}^{+}$ concentrations (1.5 mM, 3 mM, 4.5 mM) in the bathing solution
to change the resting potential across the glia membrane. Then the axon was
stimulated to give a train of action potentials. The action potentials
increased $\mathrm{K}^{+}$ in $\Omega_{ex}^{OP}$. The accumulated
$\mathrm{K}^{+}$ then made the glia membrane potential more positive. Stimuli
were applied at both ends of a region of the optic nerve thereby producing a
more uniform (in space) potential within that region.

The model of this system is solved by using the Finite Volume Method with mesh
size $h=1/20$ and temporal size $t=1/10$ in dimensionless units. The code is
written and executed in the Matlab environment. The flowchart for the
simulation is shown in Fig. 3. In the first step, we obtain the resting state
of the system by iteration and fixed volume fraction [37]

$\eta^{re}_{ax}=0.5,\ \eta^{re}_{gl}=0.4,\ \eta^{re}_{ex}=0.1.$

In the dynamic process, the resting state values are taken as initial values.
Then, we first solve the concentration governing equations, followed by
electrical potentials equation and pressure equations. We update the volume
fraction by using Eqs. (9a) and (9b).

Figure 3: Flowchart for simulation process

In our simulation study, we first set the ECS ($\Omega_{ex}^{OP}$)
concentration of $\mathrm{K^{+}}$ to be $3\ \mathrm{mM}$ and obtained a
resting potential across the glial membrane $(\sim-89\ \mathrm{mV})$. In
Orkand’s work [53, 37], suction electrodes were used for stimulating; The two
ends of the optic nerve were placed in suction electrodes as described in
their Methods Section. We modeled the suction electrodes by applying a train
of rectangular function (sometimes called a ‘box car’ stimulus in the
engineering literature) currents through the axon membrane at $z=2.25\
\mathrm{mm},13.5\ \mathrm{mm},\ \mathrm{and}\ 0<r<R_{a}=48\ \mu m$. Each
stimulus in the train lasted $3\ \mathrm{ms}$ (as Orkand’s paper indicated)
with current strength $3\ \mathrm{mA/m^{2}}$. The stimulus was large enough to
exceed the threshold and generate action potentials. After a train of stimuli
with a frequency of $17/\mathrm{s}$ for 1 $\mathrm{s}$, the first panel of
Figure 4 shows the train of axon membrane action potentials and its return to
the initial level $(\sim-89\ \mathrm{mV})$ after $1\ \mathrm{s}$. The profiles
of the first action potential and the last action potential in the train are
presented in the second panel. The third and fourth panels are used to
illustrate the increase in glial cell membrane potential and extracellular
potassium concentration during and after the train of stimuli. The fourth
panel of Figure 4 shows that during stimulus, the $\mathrm{K^{+}}$
concentration in $\Omega_{ex}^{OP}$ keeps increasing due to the opening of the
voltage-gated potassium channel of the axon membrane. As a result of the
accumulated $\mathrm{K^{+}}$ in $\Omega_{ex}^{OP}$, the membrane potential of
glial cells also continues to increase until the stimulus stops.

Figure 4:  Recording axon membrane potential, glial membrane potential and
extracellular $\mathrm{K^{+}}$ at center axis point (where $r=0$ and $z=L/2$)
when the extracellular solution with $3\ \mathrm{mM}$ $\mathrm{K^{+}}$.

Then, we vary the $\mathrm{K^{+}}$ to be 1.5 mM, 3 mM, 4.5 mM in
$\Omega_{ex}^{OP}$ and record the magnitude of the maximum glial membrane
depolarized potential in each case as in the Fig. 5. The black symbols are
used for experimental data, red ones are the simulations results of our model,
respectively. Fig. 5 shows that our model could match the experimental resting
potentials (solid symbols) and depolarization potentials (open symbols) very
well with different $\mathrm{K^{+}}$ concentrations in $\Omega_{ex}^{OP}$.

Our work is limited by the lack of other types of data for calibration,
although it is important to remember that the structural parameters, values of
membrane capacitance, many conductance variables, and resistivities of bulk
solutions are known quite well because of the work of generations of
anatomists, physiologists, and biophysicists.

Figure 5: Comparison between the experiment in [53] and simulation on the
effect of nerve impulses on the membrane potential of glial cells. The solid
symbols are resting potentials and the open symbols are depolarization
potentials with different $\mathrm{K^{+}}$ concentrations in
$\Omega_{ex}^{OP}$ .

##  4 Potassium clearance

In this section, we compare the potassium clearance under various conditions
based on the full model. In Section 4.1, by applying the stimuli at
alternative locations on the axon, we show how the interaction between the
extracellular pathway and glial transmembrane pathway helps potassium
clearance. The microcirculation patterns of water and ions between glial
compartment and extracellular space is presented. In Section 4.2, we have a
glimpse of the effect of the glial membrane conductance changes as well as
variations in the pia mater boundary conditions. We introduce the NKCC channel
into the glial membrane and a non-selective pathway on the pia mater and
compare the potassium clearance with the baseline model. The NKCC channels on
the glial membrane increase potassium clearance efficiency. An additional non-
selective pathway in the pia mater does not have significant effect on
potassium clearance due to the limited surface area of pia mater.

###  4.1 Alternative Distribution of Stimulus Location

In this section, current is applied at several different radial locations. We
wondered whether the choice of radial location would change our calculations
of potassium clearance and fluid velocity. This approach is used for many
variables to crudely estimate the sensitivity of our results to assumptions.

To facilitate the discussion of the interaction between the extracellular
pathway and the glial transmembrane pathway for $\mathrm{K}^{+}$ clearance, we
define the following regions which potassium flux could pass through,

  * •

$M_{S}$: Glial membrane in stimulated region,

  * •

$E_{T}$: Extracellular pathway on transition interface.

  * •

$M_{NS}$: Glial transmembrane in non-stimulated region.

  * •

$G_{T}$: Glial pathway on transition interface.

The normal directions on $M_{S}$ and $M_{NS}$ point to extracellular space and
the normal direction of $E_{T}$ and $G_{T}$ is the radial direction. We mainly
focus on the two particular periods of time, (1) during a train of axon firing
$([0,T_{sti}])$ (2) after axon firing $([T_{sti},T_{af}])$ after the axon
stimulation stops. In the simulations below, we take $T_{sti}=0.2\ \mathrm{s}$
and $T_{af}=10\ \mathrm{s}$. The frequency of the stimuli is $50\ \mathrm{Hz}\
(T=0.02\ \mathrm{s})$ and each single stimulus has current strength
$I_{sti}=3\times 10^{-3}\ \mathrm{A/m^{2}}$ with duration $3\ \mathrm{ms}$.

####  4.1.1 Inner and Outer radial regions stimulated

We first make a comparison between the inner radial region stimulated case in
which the current is applied at $z=z_{0}(=2.25\ \mathrm{mm})$ cross-section:
$S_{sti}^{in}\left(=\\{(r,z)|r<\frac{R_{a}}{2},\ z=z_{0}\\}\right)$ and outer
radial region stimulated case in which
$S_{sti}^{out}\left(=\\{(r,z)|\frac{R_{a}}{2}<r<R_{a},\ z=z_{0}\\}\right)$.
Since the axon signal propagates in the $z$ direction, for the inner radial
region stimulated case, the stimulated region is
$V_{S}^{in}=\\{(r,z)|r<\frac{R_{a}}{2},z\in[0,L]\\}$ and the non-stimulated
region is $V_{NS}^{in}=\\{(r,z)|\frac{R_{a}}{2}<r<R_{a},\ z\in[0,L]\\}$; for
the outer radial region stimulated case, $V_{S}^{out}=V_{NS}^{in}$ and
$V_{ns}^{out}=V_{s}^{in}$. The transition interface
$S_{sti}=\\{(r,z)|r=\frac{R_{a}}{2},\ z\in[0,L]\\}$ is the same for both
cases.

(a) During a train of neuron firing
In Fig. 6, we show the total potassium flux (potassium flux density integrated
over area) and cumulative potassium flux (total potassium flux integrated over
time) during the axon firing period $[0,T_{sti}]$. In both cases, the figures
show that the transmembrane flow from extracellular to glial and the
communication inside the extracellular act together to help the potassium
clearance. The strength of fluxes is gradually increased during axon firing
period as in Fig. 6a$\&$b. The results confirm that the potassium flux during
stimulus flows from the stimulus region to the non-stimulus region, in both
the extracellular space and glial compartment. The cumulative potassium flux
through the glial membrane $(M_{S})$ is twice as large as that through the
extracellular pathway in the transition interface $(E_{T})$ as shown in Fig.
6e&f.

Figure 6: a-d: Potassium flux through $M_{S},E_{T},M_{NS}$ and $G_{T}$ during
a train of axon firing. e-h: cumulative potassium flux during axon firing
period $[0,T_{sti}]$.

The glial compartment serves as an important and quick potassium transport
device to remove potassium accumulated while the neuron fires action
potentials. In the stimulated region, the accumulated potassium makes the
potassium Nernst potential more positive. The change in the potassium Nernst
potential induces potassium movement into the glial compartment from the
extracellular space (Fig. 6a). This inflow makes the glial compartment
electric potential more positive and moves potassium ions from the stimulated
region to the unstimulated region (Fig. 6d). In the unstimulated region, the
glial membrane potential also becomes more positive as it does in the
stimulated region, because the glia is an electrical syncytium in the
longitudinal and radial directions. However, the glial potassium Nernst
potential in the unstimulated region is not very different from that in the
resting state. These potentials produce the outward potassium flux from the
glial compartment in the unstimulated region (Fig. 6c). Interacting regions of
this sort depend on spatial variables and the properties of the glia as a
syncytium in the longitudinal as well as radial directions. It is difficult to
capture these effects in models that do not include radial and longitudinal
directions as independent variables. Compartment models are possible but it is
very difficult to uniquely define invariant parameters over the range of
conditions of interest. If additional information becomes available
experimentally, the ‘new’ conductances and pumps cannot be introduced in a
unique way, without much thought. As the model is adapted to other structures
in the brain, the parameters of a compartment model become quite difficult to
specify. The distributed model uses structurally defined parameters with
structural or specific biophysical meaning. These can be adjusted in a
reasonably specific way in different versions of this model, appropriate for
different systems in the brain. The schematic graph of potassium circulation
in the optic nerve is summarized in the Fig. 10a.

The spatial distributions of $\mathrm{K}^{+}$ concentration changes from
resting state over time are shown in the Fig. 7 and Fig. 8. Fig. 7 shows that
the $\mathrm{K}^{+}$ concentration varies in the stimulus region along the
longitudinal direction during one action potential, while there is no change
of $\mathrm{K}^{+}$ in the non-stimulus region. Fig. 8 a$\&$b show there an
obvious potassium concentration difference in the radial direction after a
train of stimuli over time $T_{sti}=0.2s$. In Fig. 8c, the potassium
concentration difference vanishes due to the communication of the
extracellular space and glial compartment shown in Fig 11fg.

Figure 7: Spatial distribution of potassium changes from the resting state

during an action potential.

Figure 8: Spatial distribution of potassium changes from the resting state
during and after a train of stimuli. Figure 9: a-d: Average water velocity
through $M_{S},E_{T},M_{NS}$ and $G_{T}$ during a train of axon firing period
$[0,T_{sti}]$. e-f: the extracellular volume fraction variation in the
stimulated region and non-stimulated region. Figure 10: a: Schematic graph of
the potassium flux when axon is stimulated. In the stimulated region, the
potassium moves through the extracellular pathway and through the glial
compartment by way of the glial membrane. In the non-stimulated region, the
potassium leaks out to the extracellular space through the glial membrane. b:
Schematic graph of the water circulation when the inner part of the axon is
stimulated. In the stimulated region, the glial transmembrane water flow goes
from extracellular space into glial compartment as the effect of osmosis
difference. In the extracellular space, water goes from non-stimulated region
to stimulated region in the radial direction. In the glia compartment goes in
the opposite direction. Note these graphs summarize outputs of large numbers
of calculations solving partial differential equations in longitudinal and
radial spatial directions and time. They do not represent a compartmental
model. They are the output of a model distributed in space.

The water circulation in the optic nerve is driven by the gradient of osmotic
pressure in the stimulated region. In the stimulated region, the extracellular
osmotic pressure $k_{B}TO_{ex}$ decreased, and glial compartment osmotic flow
of water is increased. This is because there is more potassium flux moving
into the glial compartment through the glial membrane, but a smaller amount of
sodium flux out of the glial compartment because of the ion channel
conductance difference between the potassium and sodium channels in the glial
membrane. In the glial compartment in the stimulated region of the optic nerve
fiber, there is an increase in the water flux into the glial compartment from
the extracellular space (Fig. 9a). The volume fraction of the glial
compartment and the hydrostatic pressure have also increased. The increased
hydrostatic pressure in the stimulated region also raises the hydrostatic
pressure in locations far away from the stimulated region. The increased
pressure also drives the flow from stimulated region to the unstimulated
region because the glial compartment is a connected space (Fig. 9d), a
longitudinal syncytium. In the unstimulated region, the water flows out of the
glial compartment into the unstimulated region because of the increased
hydrostatic pressure in the glial compartment (Fig 9c). Then, because the
fluid is incompressible, the fluid in the unstimulated region flows back to
the stimulated region (Fig. 9b). The schematic graph of water circulation in
the optic nerve is summarized in Fig. 10b. This is a summary of our results.
It is not a compartment model. Our models are distributed.

In Fig. 9e&f, we show the volume change of extracellular space. In the
stimulated region, the extracellular space decreases because the water flows
into the glial cell; while in the unstimulated region, the extracellular space
swells because of spatial buffering water flow [35].

In sum, during a train of action potentials in the axon, the potassium flux
transport pattern and potassium flux strength across the glia membrane and
through the extracellular pathway are the same for both radial regions, inner
and outer. The glial compartment pathway is the dominant clearance mechanism
of the potassium accumulated in the extracellular stimulated region, in both
cases.

(b) After axon firing period
After the stimulus period, the main potassium clearance mechanism is the
passive flow from extracellular space to glial compartment through the glial
membrane. The potassium flux in extracellular region and glial compartment is
negligible. In these calculations, the extracellular region and the glial
compartment could be approximated as a single compartment. We show a schematic
figure of potassium flux pattern in Fig. 13a. This is a summary and sketch of
our results. We did not use a compartment model. Our model is distributed.

Figure 11: a-d: potassium flux through $M_{S},E_{T},M_{NS}$, and $G_{T}$ after
a train action potentials. e-h: cumulative potassium flux after axon firing.

In the Fig. 11, we show the total potassium flux and cumulative potassium flux
through $M_{S},E_{T},M_{NS}$, and $G_{T}$ after axon firing period
$[T_{sti},T_{af}]$. For both cases, the strength of potassium fluxes through
the glial transmembrane pathway $(M_{S})$ and extracellular pathway have
dramatically decreased after the axon stopped firing.

Fig. 11e shows that in both cases (inner stimulus and outer stimulus), the
potassium flows into the stimulated glial compartment after the axon firing
period. Fig. 11g shows that the potassium flux through the glial membrane in
the non-stimulated region reverses its direction for a short time after axon
stop firing. This occurs because the extracellular potassium concentration
becomes evenly distributed in both the stimulated and non-stimulated
extracellular space.

Figure 12: a-d: Average water velocity through $M_{S},E_{T},M_{NS}$ and
$G_{T}$ after a train of axon firing. Figure 13: a: Schematic graph of the
potassium flux after the axon was stimulated. The potassium flux leaks into
the glial compartment from the extracellular space through the glial membrane
in both stimulated and unstimulated regions. The potassium flux in the
extracellular space and glial compartment is negligible. b: Schematic graph of
the water flux after the axon was stimulated. Note these graphs are graphs
that summarize outputs of large numbers of calculations solving partial
differential equations in longitudinal and radial spatial directions and time.
They do not represent a compartmental model. They are the output of a model
distributed in space.

In the Fig. 11f, in the extracellular space, potassium flows back to the
stimulus region from the non-stimulus region via the extracellular pathway (in
both cases). Accordingly, Fig. 12 shows that after the axon stops firing, the
water flow inside the compartments becomes almost zero. The water flows
through both stimulated and unstimulated glial membrane into the glial
compartment, which is the same as the schematic graph Fig. 13b.

In sum, after the axon stops firing action potentials, the extracellular
potassium concentration is quickly transported in the glial compartment and
extracellular pathway. The main clearance of potassium is through the glia
membrane in both stimulated and unstimulated regions. The potassium flux
through the extracellular pathway becomes weaker after the axon stops firing
action potentials.

Fig. 14a&b, we show the variation of the potassium concentration in the
extracellular stimulated and unstimulated regions, respectively. The peak
potassium concentration in the stimulated region is higher in the outer
stimulated case compared to the inner stimulated case. As discussed
previously, the strength of potassium clearance is the same for both cases,
while in the outer radial stimulated case, there is three times as much
potassium in the extracellular space during the axon firing as in the inner
radial stimulated case.

The equality of potassium clearance in both cases also explains why the
potassium concentration drops faster in the inner radial stimulated case than
the outer one after axon stop firing. We provide a decay timetable in the
appendix.

Figure 14: a-b: potassium concentration variation in the extracellular
stimulated region and non-stimulated region.

####  4.1.2 Randomly distributed stimulation

Spatially uniform patterns of stimulation might produce systematic artifacts
as occur in Moire patterns and aliasing. In this section, we study whether
random stimulation patterns in space differ from spatially uniform patterns.
We apply a train of stimuli to four randomly distributed stimulated regions in
the radial direction. The strength and duration of the stimulus is same as in
the section 4.1.1 and the current is applied at the same longitudinal location
$(z=z_{0})$. The details of the radial stimulated location in each case is
shown in Fig. 15.

Figure 15: The stimulated radial segments in each case. The intervals with
value 1 are stimulated segments and the intervals with value 0 are
unstimulated segments.

(a) During a train of axon firing
We compare the spatially random stimulated case (case 1, the rest of the cases
are similar and shown in an appendix) with the inner radial region stimulated
case in the Fig. 16a-d. During a train of axon firing, Fig. 16a&b, shows, in
the randomly stimulated case, that the potassium clearance through the glial
transmembrane $(M_{S})$ has been reduced while the potassium flux through
extracellular pathway in transition interface $(E_{T})$ has dramatically
increased. Fig. 16e&f shows that the major clearance pathway in the randomly
stimulated cases becomes the extracellular pathway in transition interface
$(E_{T})$. More potassium flux goes though the extracellular path in the
transition interface in comparison to how much goes through glial membrane in
the stimulated region. This differs from the outer and inner stimulated cases,
where the glial transmembrane $(M_{S})$ dominates as seen in Fig. 6e&f.

Figure 16: a-d: the comparison between the spatially random stimulated case
with the spatially uniform radial (inner) case during a train of axon firing.
e-f: the cumulative flux comparison in $M_{S},E_{T},M_{NS}$, and $G_{T}$
during a train of axon action potentials.

Fig. 16a-d shows that in the inner stimulated case (blue line), the potassium
flux strength gradually increased with small oscillation in each axon stimulus
time period, while in the spatially random stimulated case (red line), the
potassium flux strength shows a periodic pattern in time with larger
oscillation in each stimulus period. The reason for this quite different
potassium flux pattern is that the extra potassium in the extracellular
stimulated region has been cleared quicker in the random selected case. So, in
each stimulus period, the potassium flux decreases dramatically since the
potassium goes back to its resting state. While in the inner stimulus case,
for each stimulus period, the clearance of potassium is slower and there is
accumulation of potassium in the stimulated extracellular space.

In sum, Fig. 16e&f, during a train of axon firing, much more potassium flux
goes through the extracellular pathway through transition interface $(E_{T})$,
which reduces the effect of glial compartment pathway in the stimulated region
$(M_{S})$.

(b) After axon firing period
After the axon stops firing action potentials, both the spatially random case
and the uniform inner case have similar reduced potassium fluxes, See
$M_{S},E_{T},M_{NS}$, and $G_{T}$ In Fig. 17e&g shows that the main pathway
for potassium clearance (after the action potentials cease) is through the
glial transmembrane pathway.

Figure 17: a-d: the comparison between the spatially random case with the
uniform inner radial case after a train of action potentials Vector directions
as defined previously. e-f: the cumulative flux comparison in
$M_{S},E_{T},M_{NS}$, and $G_{T}$ after a train of action potentials.

Fig. 17h shows a large difference between the spatially random and the
spatially uniform case in some properties as we feared might occur. The
cumulative transport of potassium flux through the glial compartment
transition interface in the spatially random cases is much smaller than it is
in the uniform inner case. This is because during axon firing period, the
extracellular potassium in the stimulated region has been quickly removed in
the random cases. The potassium concentration became more homogeneous and
there was less difference between the stimulated region and non-stimulated
regions. As a result, the glial compartment electric potential $\phi_{gl}$
becomes homogeneous in entire glial compartment and the electric drift flow in
the glial transition interface was reduced for potassium. These results show
the importance of checking for artifacts produced by artificial assumptions of
spatial uniformity or periodicity.

(c) Potassium clearance and fluid velocity in the extracellular space and
glial compartment

Since the potassium goes less through the glial transmembrane pathway in the
stimulated region, the osmosis in the extracellular $k_{B}TO_{ex}$ and in
glial $k_{B}TO_{gl}$, and variation in the stimulated region has been reduced.
Therefore, the strength of the velocities decreases in both glia and
extracellular compartment since the decrease of the glial transmembrane water
flow as shown in Fig. 18c&d. The average potassium concentration changes in
both stimulated region and non-stimulated region as shown in the Fig. 18a&b
and its decay time in each case is in the appendix.

Figure 18: a-b: variation of potassium concentration in the extracellular
simulated region and unstimulated regions. c-d: average glial compartment
radial absolute velocity and extracellular space radial absolute velocity.

###  4.2 Effect of NKCC and non-selective pathway

In this section, we consider the effect of glial membrane conductance and a
non-selective pathway on the pia boundary on the potassium clearance process.
We first introduce the NKCC channel into the glial membrane, as widely studied
in the literature. We compare simulation results between the model with NKCC
and without NKCC. In the second part, we consider the effect of a non-
selective pathway in the pia mater boundary. That pathway allows the
convection flux through an extracellular pathway out of the optic nerve. We
investigate how these factors affect the potassium clearance. While the
contribution of the NKCC channel is significant, it is important to realize
that in the present state of knowledge, other relevant channels may not yet
have revealed themselves. Some channels become activated only under special
conditions that are hard to find. Some channels may become activated only to
protect systems under severe stress, as in clinical situations like oxygen
deprivation, swelling, and so on. It will be important to identify such
channels as the model is applied to clinical situations, some of which are of
considerable importance.

####  4.2.1 Effect of NKCC on the glial membrane

We introduce a model of NKCC channels in the glial membrane as in the [54,
38], We describe the $\mathrm{K^{+}}$,$\mathrm{Na^{+}}$and $\mathrm{Cl^{-}}$
flux through the NKCC channel in the glial membrane as in [54, 38]

$\begin{array}[]{l}J_{NKCC}^{K}=-\frac{I_{\max}^{NKCC}}{ez^{K}}\log\left(\frac{C_{ex}^{K}}{C_{gl}^{K}}\frac{C_{ex}^{Na}}{C_{gl}^{Na}}\left(\frac{C_{ex}^{Cl}}{C_{gl}^{Cl}}\right)^{2}\right),\\\
J_{NKCC}^{Na}=-\frac{I_{\max}^{NKCC}}{ez^{Na}}\log\left(\frac{C_{ex}^{K}}{C_{gl}^{K}}\frac{C_{ex}^{Na}}{C_{gl}^{Na}}\left(\frac{C_{ex}^{Cl}}{C_{gl}^{Cl}}\right)^{2}\right),\\\
J_{NKCC}^{Cl}=2\frac{I_{\max}^{NKCC}}{ez^{Cl}}\log\left(\frac{C_{ex}^{K}}{C_{gl}^{K}}\frac{C_{ex}^{Na}}{C_{gl}^{Na}}\left(\frac{C_{ex}^{Cl}}{C_{gl}^{Cl}}\right)^{2}\right).\end{array}$
(28)

To compare the effect of NKCC without NKCC, we keep the potassium and sodium
concentration as well as the resting electric potential in the glial
compartment and extracellular space the same. In the resting state, we set
sodium and potassium current through NKCC channel to be comparable with
respect to the Na/K pump current as in the paper [54, 38]. In the appendix, we
provide the two sets of parameters (NKCCa and NKCCb) that balance the
additional NKCC current through the glial membrane.

In the simulation below, we compare the models with and without NKCC. The
stimulated region Fig. 19, shows the cumulative potassium flux through the
$M_{S},E_{T},M_{NS}$, and $G_{T}$ during and after stimuli.

Figure 19: a-d: the cumulative flux comparison in $M_{S},E_{T},M_{NS}$, and
$G_{T}$ during a train of action potentials. e-h: the cumulative flux
comparison in $M_{S},E_{T},M_{NS}$, and $G_{T}$ after a train of action
potentials.

Fig. 19a shows that more potassium goes through the glial membrane in the
simulated region during axon firing when NKCC is present, which is hardly
surprising. The NKCC channel has enhanced the transport of potassium through
the glial membrane after the axon stopped firing as in Fig. 19e&g.

Fig. 4.13a&b shows the variation of potassium concentration in the
extracellular space in the stimulated and unstimulated cases. The potassium
decay in much faster when NKCC is present presumably because NKCC allows
larger movement of potassium into the glial compartment. We provide the decay
timetable in the appendix.

The quicker potassium is taken into glial compartment by the NKCC, the slower
the return of potassium concentration back to resting state. Fig. 4.13c, shows
the variation of the potassium concentration in the stimulated region. After
action potentials cease, the potassium movement back to the axon compartment
is reduced in the model with NKCC channel. Fig. 4.13c, shows that the average
potassium concentration in the baseline model increases faster after the axon
stops firing than in the model with NKCC.

Figure 20: a-b: extracellular potassium concentration variation comparison
between the model with NKCC and baseline model (without NKCC). c: average
potassium variation in the axon stimulated region.

####  4.2.2 Non-selective pathway through the pia matter

In this section, we consider the effect of a non-selective pathway across the
pia boundary. The nonselective pathway allows both ion and water fluid
transport through the cleft between the cells in the pia mater. We assume that
the water fluid goes through the non-selective pathway only depends on the
hydrostatic pressure difference. Therefore, the fluid condition on pia
boundary $(\Gamma_{7})$ in (16) become

$\boldsymbol{u}_{ex}^{OP}\cdot\hat{\boldsymbol{r}}=\boldsymbol{u}_{ex}^{SAS}\cdot\hat{\boldsymbol{r}}=u_{pia}^{m}+u_{pia}^{ns},\quad\text{
on }\Gamma_{7},$

where

$\displaystyle u_{pia}^{m}$
$\displaystyle=L_{pia}^{m}\left(p_{ex}^{OP}-p_{ex}^{SAS}-\gamma_{pia}k_{B}T\left(O_{ex}^{OP}-O_{ex}^{SAS}\right)\right),$
$\displaystyle u_{pia}^{ns}$
$\displaystyle=L_{pia}^{ns}\left(p_{ex}^{OP}-p_{ex}^{SAS}\right).$

The non-selective pathway between the cell clefts provides an additional
pathway for diffusion ($\nabla C_{l}^{i}$), electric drift
($C_{l}^{i}\nabla\phi_{l}$) as well as convection for ions
($C_{l}^{i}\boldsymbol{u}_{l}$). We modify the boundary condition (24) for ion
on pia boundary $(\Gamma_{7})$ as

$\displaystyle\boldsymbol{j}_{ex}^{i,OP}\cdot\hat{\boldsymbol{r}}=\boldsymbol{j}_{ex}^{i,SAS}\cdot\hat{\boldsymbol{r}}$
$\displaystyle=\frac{G_{pia}^{i}+G^{ns}}{z^{i}e}\left(\phi_{ex}^{OP}-\phi_{ex}^{SAS}-E_{pia}^{i}\right)+c_{ex}^{i}u_{pia}^{ns},\
\text{ on }\Gamma_{7},$

Here $G^{ns}$ is the additional conductance due to the non-selective pathway
and $C_{ex}^{i}u_{pia}^{ns}$ is the convection flux through the non-selective
pathway on pia boundary.

In the simulation below, we compare the model with non-selective pathway with
the model without the non-selective pathway. We choose the comparison
parameter to be

$\frac{G^{ns}}{G_{pia}^{K}}=\frac{L_{pia}^{ns}}{L_{pia}^{m}}=10.$

In the Fig. 21, we show the potassium variation in extracellular space and the
cumulative potassium flux through the pia mater and glial membrane.

Figure 21: a-b: extracellular potassium concentration variation comparison
between the model with non-selective pathway on pia boundary and baseline
model (without non-selective pathway).c: total cumulative potassium flux
through glial membrane. d: total cumulative potassium flux though pia
boundary.

The amount of potassium leak out of the optic nerve through the pia boundary
is dramatically increased when the pathway is present in Fig. 21d. However,
the dominant pathway of potassium clearance is still through the glial
membrane as previously in Fig. 21c. This is because the total glial membrane
area is much larger than the surface area on pia boundary,

$\frac{2\pi R_{a}L}{V_{op}\mathcal{M}_{gl}}=O\left(10^{-3}\right),$

where the $V_{o}n$ is the total optic nerve volume and $M_{g}l$ is the glial
membrane per unit volume. Fig. 21a&b shows that with the non-selective pathway
on the pia boundary does not accelerate the potassium clearance rate very
much.

##  5 Historical Comments and Discussion

There may be a concern about the large number of parameters in this and
similar models. Models that contain large numbers of parameters can be
difficult to compare and thus difficult to use in the scientific process of
conjecture and refutation needed to understand these complex systems.

The need to deal with the complexity of the system as it is presented to us by
evolution and used by animals should be clearly understood. Perhaps it is best
understood by comparison with engineering systems. Engineering systems use
enormous complexity—consider the $10^{13}$ components in our computers and
their connections—to perform specific functions. If the complexity is not
described in a model, the model cannot describe the functions. Biological
systems perform definite functions and use structures to do so. They are often
devices in the exact engineering sense of the word. Complexity is needed to
define these systems because without that complexity, the systems do not
function.

Of course, the complexity cannot be uniquely defined in either engineering or
biological systems. One can always appeal to an atomic description in the
desperation of ignorance, but it is not at all clear that much detail helps,
and the problems of dealing with irrelevant thermal motion, and difficulties
in computation, have made such an approach so difficult that it is rarely used
in engineering. Here we do the best we can by choosing structures and
parameters that are needed considering their role in other similar biological
systems.

The work here is done in the spirit of the structural analysis of
physiological problems started by Falk and Fatt [12], continued by their
student Eisenberg and his colleagues [61, 69] and applied to the lens of the
eye with Rae and Mathias. This approach uses measured anatomical parameters
(best by application of statistical sampling methods to biological systems,
which was pioneered by Eisenberg and then extended generally by Brenda
Eisenberg [10]), and impedance spectroscopy [11], to determine parameters
exploiting the invariance of the capacitance (per unit area) of biological
membranes. Mathias and his group showed how to extend these methods to include
water flow [42, 43, 59] in a bidomain tissue, the lens of the eye. We extend
that structural approach here to a tridomain model of an optic nerve.

In the electrical case, the parameters of a structural model are quite well
specified by this approach. When dealing with water flow, it is important to
include measurements of flow and pressure. In our situation the electrical
information is available from the enormous knowledge of the properties of
nerve fibers and action potential conduction developed since Hodgkin showed
[24] that nerve conduction is electrical and not chemical [23]. Measurements
of pressure and flow are not available and they are surely needed if the model
is to be further refined or extended to other analogous systems of the central
nervous system, of great clinical importance. Measurements of a crucial
property modified by water flow are possible thanks to the work of the Harvard
group (Orkand et al). That work allows us to define our system as well as we
have, but surely not well enough.

As this work is developed to become a model of the important phenomena of the
recently discovered glymphatic system [31, 45], measurements of flow and
pressure, as well as stereological measurements of structure and biophysical
measurements of channel and pump distribution will assume crucial importance,
in our view. Extensions of our theory and appropriate simplifications will
also be helpful.

##  6 Conclusion

In this work, we propose a tridomain model to study potassium clearance in the
optic nerve of Necturus in a series of experiments from Richard Orkand and the
Harvard group [53, 37]. Our model, analysis, and simulations provide a
detailed picture of the role of glial cells in buffering the concentration of
ions, mostly in the narrow extracellular space. While the nerve axons are
being stimulated, both the extracellular space and glial cells play important
roles. They both clear extra potassium from the narrow extracellular space
while the axon is firing action potentials. After the action potentials stop,
the potassium remaining in the extracellular space is cleared by the glial
compartment.

Our model shows that the longitudinal electrical syncytium of the glial cells
is critical for clearing potassium (from the extracellular space) when the
neuron fires. The inward glial transmembrane potassium flux in the stimulated
region is almost the same as the outward potassium flux out to the
extracellular space in the non-stimulated region, in response to the change in
potassium concentration between the extracellular space in the stimulated and
unstimulated regions. This is because the electric potential spreads through
the connected cells in the glial compartment. The glial electric potential in
the unstimulated region becomes more positive in response to the
depolarization of the glial electric potential in the stimulus region. The
‘syncytial properties’ of the glial compartment are a feature that separates
our field model with its partial differential equations from the compartment
models in the literature that use ordinary differential equations. Compartment
models lack the combination of space and time dependence needed to describe
the temporal and spatial spread of potential. The combined temporal and
spatial spread of potential is a crucial property of neurons. Here we show
that the combined temporal and spatial spread of potential is a crucial
property of the glia, and the narrow extracellular space between glia and
neurons. In a sense, we extend the electrical cable theory used to describe
the spatial spread of the electrical potential of neurons to a theory of
spatial distribution of flow in neuron, glia, and the narrow extracellular
space between them.

We discuss the effect of enhanced potassium conductance in the glial membrane
and nerve membranes of the pia mater. On the one hand, incorporating NKCC
channels into the glial membrane increases potassium clearance. Potassium
clearance time is much shorter than that predicted by the baseline model
(without NKCC channels in the glial membrane). On the other hand, an
additional non-selective pathway in another location—in the pia mater—does not
have significant effect on potassium clearance. This is not surprising since
the total membrane area of the glial membrane in the optic nerve is much
greater than the effective surface membrane of nerves in the pia mater.

Finally, our analysis of the model for the optic nerve is just a first small
step towards the understanding of the mechanisms of the glial compartment
buffering effect during the potassium clearance and microcirculation patterns
of water and ions. The axons considered here is without myelin sheath. Myelin
can be included by combining the model of myelinated nerve proposed in [64] as
a first next step. Our distributed model and its partial differential
equations can be generalized to describe ionic and water transport in tissues
with more complicated and heterogeneous structures and with glymphatic
pathways connected to the circulatory system. We expect that the spatially
nonuniform distribution of ion and water channels and transporters will be
used in many structures in the central nervous system to control flow.
Obviously, at higher resolution much more detailed experimental observations
and structure measurements ( like pump distributions and membrane
permeabilities) will be needed.

Author Contributions Y.Z., S.X., and H.H. did the model derivations and
carried out the numerical simulations. R.S.E. and H.H. designed the study,
coordinated the study, and commented on the manuscript. All authors gave final
approval for publication.

Acknowledgments This research is supported in part by National Natural Science
Foundation of China 12071190 and the Fundamental Research Funds for the
Central Universities (S.X), the Fields Institute for Research in mathematical
Science (S.X., R.S.E., and H.H.) and the Natural Sciences and Engineering
Research Council of Canada (H.H.).

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##  Appendix A Supporting Information

###  A.1 Randomly distributed stimulation

Table 1 compares the decay time between the spatially uniform radial (inner
and outer) and spatially random stimulated case (Case 1 – Case 4). The
stimulated volume ratio is the ratio of the volume of stimulated axons divided
by the total axon volume. We calculate how much time the potassium
concentration takes to decay (50%, 70% and 90%) from its maximum in each case.

Table 1: Stimulated Volume Ratio and Potassium Decay Time | Stimulated Volume Ratio | 50% decay | 70% decay | 90% decay
---|---|---|---|---
Inner case | 0.25 |  0.2 $\mathrm{~{}s}$ |  1.0 $\mathrm{~{}s}$ |  4.5 $\mathrm{~{}s}$
Outer case | 0.75 |  1.7 $\mathrm{~{}s}$ |  3.4 $\mathrm{~{}s}$ |  7.5 $\mathrm{~{}s}$
Random case 1 | 0.55 |  2.0 $\mathrm{~{}s}$ |  3.6 $\mathrm{~{}s}$ |  7.8 $\mathrm{~{}s}$
Random case 2 | 0.30 |  1.5 $\mathrm{~{}s}$ |  3.1 $\mathrm{~{}s}$ |  6.9 $\mathrm{~{}s}$
Random case 3 | 0.73 |  2.0 $\mathrm{~{}s}$ |  3.7 $\mathrm{~{}s}$ |  7.9 $\mathrm{~{}s}$
Random case 4 | 0.70 |  2.0 $\mathrm{~{}s}$ |  3.7 $\mathrm{~{}s}$ |  7.6 $\mathrm{~{}s}$

Fig. 22-24 shows the potassium flux through $M_{S},E_{T},M_{NS}$ and $G_{T}$
in the spatially random stimulated case (Case 2 – Case 4) during a train of
axon firing and after the axon stops firing. When the axon is firing, the
pattern of periodic oscillation is hardly changed but the magnitude of the
oscillation is lager than in the inner stimulated case.

Figure 22: Comparison between spatially randomly stimulated case 2 with the
spatially uniform (inner) case.

Figure 23: Comparison between spatially randomly stimulated case 3 with the
spatially uniform (inner) case.

Figure 24: Comparison between spatially randomly stimulated case 4 with the
spatially uniform (inner) case.

###  A.2 NKCC Parameters

In the table 2 below, we provide the parameters of the glial membrane when the
NKCC channel is present. In resting state, the concentration of potassium and
sodium and the electric potential in both glial compartment and extracellular
space is the same in both cases, NKCC and baseline.

Table 2: NKCC Parameters | $I_{\max}^{NKCC}$ | $g_{gl}^{K}$ | $g_{gl}^{Na}$ | $I_{gl,1}$ | $A_{gl}$
---|---|---|---|---|---
Baseline | $0\quad\mathrm{~{}A}/\mathrm{m}^{2}$ | $2.1\mathrm{~{}S}/\mathrm{m}^{2}$ | $2.2\times 10^{-3}\mathrm{~{}S}/\mathrm{m}^{2}$ | $4.78\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ | $105\mathrm{mM}$
NKCCa | $2\times 10^{-3}\mathrm{~{}A}/\mathrm{m}^{2}$ | $2.88\mathrm{~{}S}/\mathrm{m}^{2}$ | $1.65\times 10^{-3}\mathrm{~{}S}/\mathrm{m}^{2}$ | $4.78\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ | $34.7\mathrm{mM}$
NKCCb | $2\times 10^{-3}\mathrm{~{}A}/\mathrm{m}^{2}$ | $2.1\mathrm{~{}S}/\mathrm{m}^{2}$ | $8.36\times 10^{-4}\mathrm{~{}S}/\mathrm{m}^{2}$ | $2.49\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ | $34.7\mathrm{mM}$

Table 3 shows the potassium concentration decay time in the extracellular
stimulated region in the model with/without NKCC. The model with the NKCC
channel in the glial membrane produces a faster decay of the potassium
concentration in the extracellular space.

Table 3: Potassium Concentration Decay Time with/without NKCC | 50% decay | 70% decay | 90% decay
---|---|---|---
Baseline | $1.95\mathrm{~{}s}$ | $3.58\mathrm{~{}s}$ | $7.83\mathrm{~{}s}$
NKCCa | $0.60\mathrm{~{}s}$ | $1.11\mathrm{~{}s}$ | $2.41\mathrm{~{}s}$
NKCCb | $0.63\mathrm{~{}s}$ | $1.17\mathrm{~{}s}$ | $2.66\mathrm{~{}s}$

###  A.3 Parameters

Table 4: Parameters in optic nerve model

Parameters | Value | Parameters | Value
---|---|---|---
$R_{a}$ |  $4.8\times 10^{-5}\mathrm{~{}m}$ (Ref.[37, 6]) | $\mu$ |  $7\times 10^{-4}\mathrm{~{}Pa}\cdot\mathrm{s}$ (Ref.[41])
$R_{b}$ |  $6\times 10^{-5}\mathrm{m}$ (Ref.[72]) | $c_{csf,IOP}^{Na}$ |  $111\ \mathrm{mM}$ (Ref.[37])
$L$ |  $1.5\times 10^{-2}\mathrm{~{}m}$ (Ref.[37]) | $c_{\text{csf,IOP }}^{\text{K }}$ |  $3\ \mathrm{mM}$ (Ref.[37])
$e$ | $1.69\times 10^{-19}\mathrm{~{}A}\cdot\mathrm{s}$ | $c_{gl}^{\text{Na,re }}$ |  $7.57\ \mathrm{mM}$ (*)
$k_{B}$ | $1.38\times 10^{-23}\mathrm{~{}J}/\mathrm{K}$ | $c_{gl}^{K,re}$ |  $100.84\ \mathrm{mM}$ (*,Ref.[37])
$T$ |  $296.15\mathrm{~{}K}$ (Ref.[37]) | $c_{ax}^{\text{Na,re}}$ |  $10.17\ \mathrm{mM}$ (*)
$\eta_{ax}^{re}$ |  $5\times 10^{-1}$ (Ref.[37]) | $c_{ax}^{K,re}$ |  $100.04\ \mathrm{mM}$ (*)
$\eta_{gl}^{re}$ |  $4\times 10^{-1}$ (Ref.[37]) | $A^{re}_{ax,gl}$ |  $105\ \mathrm{mM}$ (*)
$\eta_{ex}^{re}$ |  $1\times 10^{-1}$ (Ref.[37]) | $\tau_{ex}^{OP}$ |  $0.16$ (Ref.[41, 40])
$\mathcal{M}_{ax}$ |  $5.9\times 10^{6}\mathrm{~{}m}^{-1}$ (Ref.[57]) | $\tau_{ex}^{SAS}$ |  $1$ (*)
$\mathcal{M}_{gl}$ |  $1.25\times 10^{7}\mathrm{~{}m}^{-1}$ (Ref.[57]) | $\tau_{gl}$ |  $0.5$ (*)
$z^{Na,K}$ | $1$ | $p_{CSF}$ |  $1.3\times 10^{3}\mathrm{~{}Pa}$ (Ref.[3])
$z^{Cl}$ | $-1$ | $p_{IOP}$ |  $4\times 10^{3}\mathrm{~{}Pa}$ (Ref.[3])
$z^{ax,gl}$ |  $-1$ (*) | $p_{OBP}$ |  $0\mathrm{~{}Pa}$ (Ref.[3])
$\gamma_{\text{ax,gl}}$ |  $1$ (Ref.[41, 40]) | $D_{ex,ax}^{Na}$ |  $1.39\times 10^{-9}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$\gamma_{pia}$ |  $1$ (Ref.[41, 40]) | $D_{ex,ax}^{K}$ |  $2.04\times 10^{-9}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$K_{\text{Na1,Na2}}$ |  $2.3393\mathrm{mM}$ (Ref.[76]) | $D_{ex,ax}^{Cl}$ |  $2.12\times 10^{-9}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$K_{K1}$ |  $1.6154\mathrm{mM}$ (Ref.[76]) | $D_{gl}^{Na}$ |  $1.39\times 10^{-11}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$K_{K2}$ |  $0.1657\mathrm{mM}$ (Ref.[76]) | $D_{gl}^{K}$ |  $2.04\times 10^{-11}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$I_{gl,1}$ |  $4.78\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ (**,Ref.[76]) | $D_{gl}^{Cl}$ |  $2.12\times 10^{-11}\mathrm{~{}m}^{2}/\mathrm{s}$ (Ref.[41])
$I_{gl,2}$ |  $6.5\times 10^{-5}\mathrm{~{}A}/\mathrm{m}^{2}$ (**,Ref.[76]) | $k_{ex}^{OP}$ |  $1.3729\times 10^{-8}\mathrm{~{}m}^{2}/\cdot\mathrm{s}$ (Ref.[40])
$I_{ax,1}$ |  $9.56\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ (**,Ref.[76]) | $k_{ex}^{SAS}$ |  $0\mathrm{~{}m}^{2}/\mathrm{V}\cdot\mathrm{s}$ (*)
$I_{ax,2}$ |  $1.3\times 10^{-4}\mathrm{~{}A}/\mathrm{m}^{2}$ (**,Ref.[76]) | $K_{ax}$ |  $1.67\times 10^{6}\mathrm{~{}Pa}$ (Ref.[30, 39])
$g_{gl}^{Na}$ |  $2.2\times 10^{-3}\mathrm{~{}S}/\mathrm{m}^{2}$ (Ref.[41]) | $K_{gl}$ |  $8.33\times 10^{5}\mathrm{~{}Pa}$ (Ref.[30, 39])
$g_{gl}^{K}$ |  $2.1\mathrm{~{}S}/\mathrm{m}^{2}$ (Ref.[41]) | $L_{dr}^{m}$ |  $8.89\times 10^{-13}\mathrm{~{}m}/\mathrm{Pa}\cdot\mathrm{s}$ (Ref.[40, 76])
$g_{gl}^{Cl}$ |  $2.2\times 10^{-3}\mathrm{~{}S}/\mathrm{m}^{2}$ (Ref.[41]) | $L_{pia}^{m}$ |  $8.89\times 10^{-13}\mathrm{~{}m}/\mathrm{Pa}\cdot\mathrm{s}$ (Ref.[40, 76])
$g_{leak}^{Na}$ |  $4.8\times 10^{-3}\mathrm{~{}S}/\mathrm{m}^{2}$ (**,Ref.[64]) | $L_{gl}^{m}$ |  $1.34\times 10^{-13}\mathrm{~{}m}/\mathrm{Pa}\cdot\mathrm{s}$ (Ref.[40, 76])
$g_{leak}^{K}$ |  $2.2\times 10^{-2}\mathrm{~{}S}/\mathrm{m}^{2}$ (**,Ref.[64]) | $L_{ax}^{m}$ |  $7.954\times 10^{-14}\mathrm{~{}m}/\mathrm{Pa}\cdot\mathrm{s}$ (Ref.[70])
$\bar{g}^{Na}$ |  $1.357\times 10^{1}\mathrm{~{}S}/\mathrm{m}^{2}$ (**,Ref.[64]) | $\kappa_{gl}$ |  $9.366\times 10^{-19}\mathrm{~{}m}^{2}$ (Ref.[40, 76])
$\bar{g}^{K}$ |  $2.945\mathrm{~{}S}/\mathrm{m}^{2}$ (**,Ref.[64]) | $\kappa_{ax}$ |  $1.33\times 10^{-16}\mathrm{~{}m}^{2}$ (Ref.[40, 76])
$g_{ax}^{Cl}$ |  $1.5\times 10^{-1}\mathrm{~{}S}/\mathrm{m}^{2}$ (*) | $\kappa_{ex}^{OP}$ |  $3.99\times 10^{-16}\mathrm{~{}m}^{2}$ (**,Ref.[40, 76])
$G_{pia}^{Na,K,Cl}$ |  $3\mathrm{~{}S}/\mathrm{m}^{2}$ (*) | $\kappa_{ex}^{SAS}$ |  $1.33\times 10^{-14}\mathrm{~{}m}^{2}$ (**,Ref.[40, 76])

  * a

Note: the ‘*’ estimated or induced from the concentration balance.

  * b

Note: the ‘**’ deduct proportional from reference.