Spatio-temporal modeling of saltatory conduction in neurons using Poisson-Nernst-Planck treatment and estimation of conduction velocity

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Spatio-temporal modeling of saltatory conduction in neurons using

Poisson-Nernst-Planck treatment and estimation of conduction velocity

Rahul Gulatia, Shiva Rudrarajua

aDepartment of Mechanical Engineering, University of Wisconsin-Madison, WI, USA

Abstract

Action potential propagation along the axons and across the dendrites is the foundation of the electrical activity ob-

served in the brain and the rest of the central nervous system. Theoretical and numerical modeling of this action

potential activity has long been a key focus area of electro-chemical neuronal modeling, and over the years, electrical

network models of varying complexity have been proposed. Speciﬁcally, considering the presence of nodes of Ranvier

along the myelinated axon, single-cable models of the propagation of action potential have been popular. Building

on these models, and considering a secondary electrical conduction pathway below the myelin sheath, the double-

cable model has been proposed. Such cable theory based treatments, including the classical Hodgkin-Huxley model,

single-cable model, and double-cable model have been extensively studied in the literature. But these have inherent

limitations in their lack of a representation of the spatio-temporal evolution of the neuronal electro-chemistry. In con-

trast, a Poisson-Nernst-Planck (PNP) based electro-diﬀusive framework accounts for the underlying spatio-temporal

ionic concentration dynamics and is a more general and comprehensive treatment. In this work, a high-ﬁdelity im-

plementation of the PNP model is demonstrated. This electro-diﬀusive model is shown to produce results similar to

the cable theory based electrical network models, and in addition, the rich spatio-temporal evolution of the underlying

ionic transport is captured. Novel to this work is the extension of PNP model to axonal geometries with multiple

nodes of Ranvier, its correlation with cable theory based models, and multiple variants of the electro-diﬀusive model

- PNP without myelin, PNP with myelin, and PNP with the myelin sheath and peri-axonal space. Further, we apply

this spatio-temporal model to numerically estimate conduction velocity in a rat axon using the three model variants.

Speciﬁcally, spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is investigated.

Keywords: Action potential, saltatory conduction, signal velocity, Poisson-Nernst-Planck, electro-diﬀusion,

neuronal electrophysiology, myelin, peri-axonal space, cable theory

1. Introduction

Electrical activity in nerve cells, enabled through the propagation of action potentials, is critical to the entire sig-

naling and communication cascade of the nervous system. Disruption of this requisite signaling can lead to a number

of neurological disorders such as the motor neuron diseases and is often linked with traumatic brain injury (TBI),

Alzheimers, cognitive impairment, depression etc [1, 2, 3, 4, 5]. To gain insight into the neuronal electrophysiology,

varied experimental investigations using the patch-clamp technique, electroencephalograms (EEG), electrocardiogram

(ECG), MRI, calcium imaging, voltage imaging etc have been reported in literature. Despite the wealth of information

achieved by these investigations, there is a need to supplement these studies with a robust numerical implementation

which has the potential to represent the electrophysiology to a far greater resolution as recorded by the experiments.

Based on numerous voltage clamp experiments on the giant squid, Hodgkin-Huxley came up with a ﬁrst mathe-

matical model in the form of an electrical circuit to describe the current through the neuronal membrane [6]. They

quantitatively detailed the respective ionic conductance in respect to the membrane voltage. Huxley delineated that

the action potential propagation along the axon closely follows Ohm’s law [7]. Using cable theory, they arrived at a

Email address: shiva.rudraraju@wisc.edu (Shiva Rudraraju)

Preprint submitted to Brain Multiphysics

arXiv:2210.14870v2 [q-bio.QM] 30 Nov 2022

one dimensional model of the action potential propagation. Based on the physiology of the neuron, the importance

of myelin sheath that surrounds the axon, has been emphasized. Degradation of this protective covering for example

with age can lead to slowdown of the signal or even signal disruption resulting in various diseases [8, 9]. To be able

to study the eﬀect of myelin on action potential propagation, the Hodgkin-Huxley model was modiﬁed to incorporate

the myelin sheath and to obtain the single-cable model [10, 11]. Experimental investigation of the action potential by

Barrett and Blight in the 1980’s lead to the discovery of an after-potential [12, 13, 14]. With the advent of advanced

microscopy techniques, the existence of a secondary electrical conduction pathway in the peri-axonal space under the

myelin sheath has been uncovered [10].

The cable theory based models namely Hodgkin-Huxley, single-cable, double-cable, etc., have greatly contributed

to our understanding of the neuronal electrophysiology. They provide us an excellent insight into the membrane

potential, action potential propagation along the neuron, the electric current propagating along the axon, the eﬀect of

saltatory conduction due to the presence of myelin sheath, the eﬀect of the submyelin peri-axonal space dictated by

the double-cable model, the conduction velocity implied by each of these models etc. The cable theory based models,

however, have some limitations. First, these are a one dimensional reduction of the complex heterogeneous spatial

propagation of the action potential. Secondly, the cable theory based models fail to describe the underlying spatial

ionic diﬀusion and the generated electric ﬁeld during the electrical conduction propagation. Therefore, these models

are not able to accurately describe the dynamics after a prolonged electrical activity or when the diameter is relatively

smaller, as in the case of dendrites. Further, the cable theory based models cannot be easily extended to account for

the membrane microenvironment, such as incorporating the membrane-glia interaction.

The Poisson-Nernst-Planck based electro-diﬀusive model has the capability to overcome the limitations of the

cable theory based models and provide a spatio-temporal representation of the electrical potential along with the ionic

distributions [15]. The PNP model is a more generalized model which can be reduced to the electroneutral model and

can subsequently be reduced to the one dimensional cable theory based model [16]. Qian and Sejnowski modelled one

of the ﬁrst intracellular dynamics incorporating one dimensional PNP theory [17]. PNP model has also been applied to

neuron-ECM-astrocyte interations [18]. Assuming electroneutrality, ionic dynamics have also been represented using

Kirchoﬀ-Nernst-Planck [19, 20]. However, the assumption of electroneutrality for nonuniform geometries is invalid

[21]. Using PNP model, the neuronal intracellular-extracellular dynamics have been represented for a single node of

Ranvier [21, 22]. It has been demonstrated that the dynamics of the PNP model resemble to that of the cable theory

at higher ion channel density [21]. The PNP model can also be coupled with mechanics to represent the complex

neuronal mechano-electrophysiology interactions [23, 24, 25, 26].

In this work, we extend the electro-diﬀusive PNP model to multiple nodes of Ranvier, enhancing our ability to

study the electrophysiology in a full length neuronal axon. We present novel variants of the PNP model based on the

discrete cable theory based models, i.e. PNP model, PNP model with myelination, and PNP model with myelin and

peri-axonal space. As an example, we demonstrate these models by simulating action potential conduction in a rat

neuron. Spatial saltatory conduction due to the presence of myelin sheath and the peri-axonal space is demonstrated.

Finally, we provide a detailed insight into the numerically estimated conduction velocity for a rat and squid neuron

using various representative PNP electro-diﬀusive models. The Finite element (FE) method is used to discretize the set

of PDEs underlying the PNP model. As suggested by an earlier work, non-homogeneous adaptive mesh is employed

[22]. Results indicate that the conduction velocity (CV) of the action potential increases with the presence of myelin

sheath and the peri-axonal space. The CV for the PNP with myelin is comparable to the single cable network but the

CV for the PNP with myelin and peri-axonal space model does not increase drastically as compared to the double

cable model. We observe that the action potential amplitude is lower for the PNP model when the myelin sheath is

present.

In section 2, we brieﬂy review and illustrate the well known models based on the one dimensional cable theory.

The electro-diﬀusive PNP model is presented in section 3. The mathematical formulation of the numerical framework

is elaborated in section 4. The simulation results of the various models of the PNP are detailed in section 5. Finally, a

discussion of the conduction velocity for a rat and a squid neuron is in section 6, followed by conclusion in section 7.

Hodgkin-Huxley

Single-cable

Double-cable

(HH)

(SC)

(DC)

Voltage

Length

Axon

Myelin

Peri-axonal space

ECM

Rmy

Cmy

Rpn

Rpa

gNa gKgL

ENa EKEL

30 -70

Vmy •

Vmy

•

Length

Vm-Vmy

-70

Voltage

1000 µm

100 µm

-70

Vm-Vmy

Figure 1: Electrical circuit of cable theory based models. (A) Schematics of cable theory based Hodgkin-Huxley circuit consists of membrane

capacitance, resistance oﬀered by the ion channels. Action potential propagates like a soliton through the axon and is depicted on the right.

Extracellular voltage is taken as 0 mV. (B) Electrical circuit of the single-cable model considers the presence of myelin sheath. The capacitance

and resistance oﬀered by the myelin are explicitly modeled. (C) Electrical network of the double-cable representation incorporates an additional

cable pathway for the submyelin peri-axonal space. Action potential jumps from one node of Ranvier to the next, resulting in a higher conduction

velocity.

2. Review of electrical network models of action potential propagation

2.1. Hodgkin-Huxley model

The classical work of Hodgkin and Huxley was a landmark model in terms of providing deep insights into the

ionic basis of action potential propagation in nerve cells. Based on the voltage clamp experiments on the Giant

Squid, the physiology of the initiation and propagation of action potential in a neuron was posed as a coupled set of

ordinary diﬀerential equations. This electrical network model takes into account the membrane capacitance and the

ionic currents due to the sodium ions, potassium ions and some leak current through the respective ion channels in the

neuronal membrane. The inﬂuence of the conductance of the respective ion channels, or conversely the resistance, on

the action potential was quantitatively estimated using experimental data. The resulting electrical circuit is depicted

in Figure 1(A), and the corresponding governing equation linking the evolution of the membrane potential with the

underlying ionic transport is the following:

∂Vm

∂t+¯

GNam3h(Vm−ENa)+¯

GKn4(Vm−EK)+GL(Vm−Vrest)=Iin j (1)

where Cmis the membrane capacitance of the lipid bilayer, Vmis the membrane potential, Vrest is the resting potential

of the nerve cells and Iin j is the injected current through the voltage clamp experiments to initiate the action potential

in the cell. ¯

Giand Eiare the peak conductance and the Nernst potential of the sodium or potassium ions, and GLis

the conductance of the membrane leak channel.

Numerous experiments have pointed towards the eﬀective behaviour of the intra-cellular region to that of a resistor

along the axon [27]. Utilizing cable theory, one can then arrive at:

∂2Vm

∂x2=Iin j (2)

where Riis the resistance per unit length of the intra-cellular region along the axon. The partial diﬀerential equation

(PDE) form of the Hodgkin-Huxley equation can then be written as:

∂Vm

∂t+¯

GNam3h(Vm−ENa)+¯

GKn4(Vm−EK)+Gm(Vm−Vrest)=1

∂2Vm

∂x2(3)

Using the above one-dimensional PDE, one can model the propagation of action potential along the length of the

axon.

2.2. Single-cable model

From the physiology of nerve cells, it is well known that the glial cells provide a protective covering to the axonal

membrane. This protective covering consists of multiple layers of myelin sheath along the axon, with gaps in between.

These axonal gaps, void of any myelin covering, are identiﬁed as the nodes of Ranvier. These myelin lamellae play an

indispensable role in the rapid movement of the action potential, since the direct ionic exchange with the extra-cellular

medium occurs only at the nodes of Ranvier. This leads to a local current and the action potential jumping from one

node of Ranvier to the next, more commonly referred to as saltatory conduction. The degradation of the myelin layers

is known to lead to a decline in the conduction velocity and is also linked with various neuronal disease conditions

[8, 9].

To accommodate this spatial heterogeneity of the myelin sheath, the standard Hodgkin-Huxley treatment needs

to be altered to model myelinated-axons. A simple electrical circuit to realize this, known as the single-cable model,

has been proposed [11, 28]. The ionic exchange between the intra-cellular regions and the extra-cellular regions takes

place at the membrane, but only at the location of the nodes of Ranvier. As an example, for a rat axon, the span of the

nodes of Ranvier is around 2.3µmand the nodes are separated by distance of 70µm−100µm[10]. The single-cable

model with the myelin sheaths having their respective capacitance and resistance is shown in Figure 1(B) [10].

2.3. Double-cable model

The presence of the submyelin peri-axonal region has been proposed as a potential pathway for rapid electrical

conduction along the axon [10, 29]. Due to the presence of two conduction pathways, each modeled using cable

theory, this treatment is referred to as the double-cable model. An electrical circuit representing such a double-cable

model can be seen in Figure 1(C) . The basic governing equations for the electrical circuit at the node of Ranvier are

similar to the set of equations from the single cable model/Hodgkin-Huxley model. In addition, using cable theory,

the partial diﬀerential equation modeling the peri-axonal space takes the form:

∂2Vm

∂x2+1

Rpa

∂2Vmy

∂x2=Cmy

∂Vmy

∂t+Vmy

Rmy

(4)

where Rpa is the resistance per unit length in the peri-axonal space, Cmy is the cumulative capacitance of the myelin

sheath, Rmy is the myelin resistance and Vmy is the potential in the peri-axonal region.

2.4. Comparison of action potential proﬁles for electrical network models

The main results of this manuscript will be discussed in Section 5, but it is worthwhile to present here a brief

comparison of the relative diﬀerences between the action potential (voltage) proﬁles predicted by modeling the elec-

trical network models described above. Figure 1 presents such a comparison of the voltage proﬁles. All these plots

were generated by solving the governing equations listed earlier in this section using an in-house 1D Finite Element

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Spatio-temporalmodelingofsaltatoryconductioninneuronsusingPoisson-Nernst-PlancktreatmentandestimationofconductionvelocityRahulGulatia,ShivaRudrarajuaaDepartmentofMechanicalEngineering,UniversityofWisconsin-Madison,WI,USAAbstractActionpotentialpropagationalongtheaxonsandacrossthedendritesisthefoundat...

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Spatio-temporal modeling of saltatory conduction in neurons using Poisson-Nernst-Planck treatment and estimation of conduction velocity

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