Study of Action Potential Propagation in Cardiac Tissues Using Cable

Theory

NATELA ZIRAKASHVILI1*, TEONA ZIRAKASHVILI2,3

1I.Vekua Institute of Applied Mathematics,

Iv. Javakhishvili Tbilisi State University,

11 University St., 0179, Tbilisi,

GEORGIA

2Ilia State University,

Tbilisi,

GEORGIA

3Tbilisi Heart and Vascular Clinic LTD,

Tbilisi,

GEORGIA

*Corresponding Author

Abstract: - The goal of the present paper is to study the propagation of action potential in cardiac tissue using

the cable equation. The paper discusses one-dimensional models of continuously coupled myocytes. Electrical

behavior in cardiac tissue is averaged over many cells. Therefore, the transmembrane potential behavior for a

single cell is studied. Using the monodomain model, in the absence of current at the beginning and end of the

cable (cell), the initial boundary problem is posed and solved analytically. The paper also discusses a one-

dimensional mathematical model of conduction in discretely coupled myocytes. The electrical behavior in the

tissue is studied in individual myocytes, each of which is modeled as a continuum connected through

conditions at the cell boundaries, which represent gap junctions. A stationary passive problem with Dirichlet

boundary conditions is stated and solved analytically using the bidomain model. The problems are solved by

the method of separation of variables. Numerical modeling of transmembrane potential propagation is

performed using MATLAB software. Transmembrane isopotential contours, and 2D and 3D graphs

corresponding to the obtained numerical results are presented.

Key-Words: - Cardiomyocytes; transmembrane potentials ; passive 1D cable equation; monodomain model;

bidomain model

Received: July 12, 2022. Revised: September 28, 2023. Accepted: October 8, 2023. Published: October 16, 2023.

1 Introduction

The article studies the propagation of action

potential in cardiac tissue using the cable equation.

Cable theory, one of the main problems of which is

the calculation of the membrane potential and which

has been developed in recent decades, is older than

the cable equation itself. It is a variation of the

equations developed by Lord Kelvin to model the

propagation of electrical signals in underwater

telegraphs. The cable theory was originally applied

to conducting potentials in the axon, for example,

by, [1]. Cardiomyocytes (heart muscle cells) differ

from nerve axons in their shape and size - roughly

they are very small cylinders. Variations of the

cable equation led to passive one-dimensional (1D)

cable equations, which are monodomain and

bidomain models and describe the electrical

behavior of cardiac tissue cell membranes and

propagation of action potentials.

Heart diseases are one of the leading causes of

death in the world, and there are many scientific

papers devoted to the study of the causes and

mechanisms of heart problems. The study, [2],

determined the distribution of intracellular,

extracellular, and transmembrane potentials induced

by current injection in the tissue in question. The

study, [3], describes the simulation of excitation

propagation in cardiac tissues based on nonlinear

reaction-diffusion type models taking into account

the monodomain model. The study, [4], investigates

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the mathematical model of heart tissue based on the

explicit representation of individual cells. A detailed

mathematical model is used in, [5], to study the

conductivity properties in small collections of

cardiomyocytes.

Although the monodomain and bidomain models

describe only the macroscopic behavior of syncytial

(cellular) tissue, they are used to explain passive

current measurement results in the lens, [6],

measurements of cable constants, [7], and

measurements of intracellular resistances in cardiac

vessels in cardiac tissue filaments, [8],

electrocardiogram, [9], [10], magnetocardiogram,

[11], four-electrode impedance measurements, [12],

and extracellular measurements of electrical

potentials generated in atrial or ventricular muscles,

[13], [14], [15], [16].

The current work discusses a one-dimensional

model of continuously coupled myocytes. In this

case, the electrical behavior in cardiac tissue is

averaged for many cells. So, the distribution of the

transmembrane potential in a single cell is studied.

Using a monodomain model, the propagation of the

transmembrane potential in a thin cylindrical

excitable myocyte is studied in the absence of

current at the beginning and end of the myocyte. A

1D mathematical model of the conductivity of

discretely coupled myocytes is also discussed.

Electrical behavior in the tissue is considered in

individual myocytes, each of which is modeled as a

continuum bound through conditions at cell

boundaries that represent gap junctions. A stationary

passive problem with Dirichlet boundary conditions

is posed and solved analytically using the bidomain

model. These problems are solved by the method of

separation of variables. Numerical results of

transmembrane potential propagation in

cardiomyocytes are obtained using MATLAB

software, and transmembrane isopotential contours,

and 2D and 3D graphs of the obtained numerical

results are presented.

2 Theoretical Aspects

The heart consists of transversely striated muscle

tissue, which ensures the rapid spread of the wave of

fiber contraction. As a result, all sections of the

heart contract as a single entity. The homogeneous

representation of cardiac tissue involves a large

number of identical myocytes, which can be thought

of as two interconnected spaces - intracellular and

extracellular. The cells are connected by gap

junctions (Figure 1).

Fig. 1: Schematic drawing of cardiac tissue

The cardiac muscle action potential (membrane

potential) is a brief change in voltage on the cell

membrane in heart cells caused by the movement of

charged atoms (ions) into and out of the cell via

proteins called ion channels. The cell membrane

separates extracellular and intracellular spaces with

potentials

e



and

i



, while

ei

V





is the

transmembrane potential.

An action potential is an excitation wave, which

as a brief change of the membrane potential in the

membrane of a living cell moves to a small area of

an excited cell (neuron or cardiomyocyte), as a

result of which the outer surface of this area

becomes negatively charged compared to the inner

surface of the membrane, while it is positively

charged in a non-excitation state. Sometimes the

action potential is called a propagating potential

because the excitation wave is actively transmitted

along the fiber of a neuron or muscle cell.

Cardiomyocytes are approximately cylindrical (very

small, measured in microns) whose length (e.g., x in

the direction of the cylinder’s long axis) is

sufficiently greater than its diameter. So we can

assume that the action potential of the cell depends

only on the length variable, and the problem can be

reduced to a single measurement. Thus, the article

discusses both continuously and discretely coupled

1D myocyte models. Intracellular, extracellular, and

transmembrane potentials are vector fields in space

and time,

i.e.

 

tx

ii ,





,

 

tx

ee ,





,

 

txVV ,

.

The electrical behavior of the cell membrane of

cardiac tissue and the propagation of the action

potential are described by monodomain and

bidomain models (equations). Both models use the

representation of cardiac muscle as two

interconnected spaces, intracellular and

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extracellular. In contrast to the monodomain model,

the bidomain model does not ignore the

extracellular space; rather, it takes into account

conductivity and the current flowing through it. The

state of the bidomain system is described by

intracellular

i



and extracellular

e



potentials. For

these models, the search variable is the

transmembrane potential

ei

V





, and

i

V





for monodomain.

Using the passive 1D cable equation derived in

Appendix A, the standard formulations for

monodomain and bidomain models are presented

below as extensions or variations of the cable

equation.

2.1 Monodomain Model

The monodomain model (equation) for the

transmembrane potential

 

txV ,

is an extension of

the cable equation presented in Appendix A. The

equation relates the spatial distribution of the

transmembrane potential to reaction conditions that

locally control (determine) the transmembrane

potential. 1D monodomain equation (A3) related to

the kinetics of the time-dependent active ion

channel

 

tIion

and external stimulus

 

tIstim

, can

be written as:

   

,

2



















tItI

t

V

c

x

Vstimionmi



(1)

where

 

txV ,

is the transmembrane potential,

i



is

the effective intracellular conductivity,



is the

tissue surface-to-volume ratio, and

2

1cmFcm





is the membrane capacity. Let us divide equation (1)

by



and

m

c

and we will gain:

    

tItI

ct

V

x

V

Dstimion

m









1

2

,

where

m

i

c

D







.

The monodomain equation is often presented in this

form and

D

is described as a diffusion coefficient.

2.2 Bidomain Model

The bidomain model, [17], [18], is a

phenomenological model that aims at encapsulating

the action of single-cell ion channels in a

homogenized representation of cardiac tissue

consisting of a large number of identical myocytes.

The model assumes the coexistence of two

continuous domains (intracellular and extracellular)

at all points in space. A rigorous mathematical

representation of the bidomain model was realized

by, [19]. There are alternative mathematical

formulations of the bidomain equations; the

standard version presented here belongs to, [17].

The bidomain equations are derived from

Maxwell’s equations of electromagnetism with

certain assumptions: 1) the first (quasi-static)

assumption is that intracellular current can only

flow between the intracellular and extracellular

regions, and the intracellular and extra-myocardial

regions can communicate with each other. So, the

current flows into and out of extra-myocardial

regions, but only in the extracellular area; 2) the

second assumption is that the heart is isolated. So,

the current leaving one domain must enter another.

In addition, the current density in each intracellular

and extracellular domain must be the same in

magnitude but opposite in sign and can be

determined as the product of the surface-to-volume

ratio of the cell membrane and the transmembrane

ionic current density per unit area.

According to Ohm’s Law:

EJ





,

where

J

is the electrical current,



is the

conductivity of space, and

Ε

is the electric field.

Using the quasi-static assumption, the electric

field

Ε

is defined as the gradient of the scalar

potential



:

.



Ε

As a result, we will gain:

.



J

Assume,

i



and

e



are the conductivity of

intracellular and extracellular spaces, respectively.

According to Ohm’s Law and the quasi-static

assumption, we will gain the following expressions

for

i

J

(intracellular) and

e

J

(extracellular) current

densities in each domain:

.

iii



J

.

eee



J

The change in current density in each domain is

equal to the current flowing through the membrane;

under the second assumption, we will gain:

,

mmei IA JJ

where



and



are the gradient and divergence

operators, respectively,

m

A

is the ratio of the cell

membrane surface to volume, and

m

I

is the

transmembrane current density per unit area. By

combining the two equations above, we gain

 

,

mmii IA



(2)

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 

,

mmee IA



(3)

By summing up these equations, we will gain:

   

.

eeii





Let us subtract

 

ei





from both sides. We

will gain:

       

.

eieeeiii





By using denotation

ei

V





we will gain:

    

.

eeii V





(4)

By inserting equation (2) in monodomain equation

(1), we will gain:

 

.















 ionmei I

t

V

c



By subtracting and adding

 

ei





and

considering

ei

V





notation, we will gain:

 

.















 ionmi I

t

V

cV



(5)

By combining equations (4) and (5) and adding

time- and space-dependent stimulus

 

),txIs

, a

bidomain model is obtained:

    

,

1seeii IV 



(6)

 

,

2sionmi II

t

V

cV 



















where



is the surface-to-volume ratio of the cell

membrane,

i



and

e



are intracellular and

extracellular conductivity,

m

c

is the membrane

capacity per unit area,

ion

I

is the sum of all ionic

currents, and

i

s

I

is the external stimulus.

It should be noted that the bidomain model can

be reduced to a monodomain model in two

particular cases: when the extracellular potential can

be neglected because of the extracellular

conductivity, and when the anisotropy ratio between

the effective intracellular and extracellular

conductivities is the same, [20]. In the latter case,

the intracellular and extracellular conductivities are

proportional and can be related as follows:

ie





By inserting this expression in equation (6), we will

gain:

,

12sionmi II

t

V

cV 





































which is identical to the monodomain equation, if

we choose the effective conductivity

i









1

.

3 Problem Statement and Solution

3.1 Monodomain Model

In this section, we discuss the 1D model of

continuously coupled myocytes. Because of the

assumption of continuity, in this case, the electrical

behavior in the tissue is averaged over many cells.

So, we will study the propagation of the

transmembrane potential for one cell.

Consider a thin cylindrical excitable cell of

length

L

. Let us solve the passive cable equation

(see Appendix A) with the following boundary and

initial conditions: 1) there is no current flowing into

the cable (myocyte) (at the beginning of the cable)

or out of the cable (at the end of the cable), and 2) at

the beginning of time interval, the current is a

function only of the spatial coordinate

x

of the

cable. 1) is a problem analogous to the insulated-end

beam thermal conductivity problem. Thus, the

boundary and initial conditions will be written as

follows:

 

,0, :For

,0, :0For

,



txVLx

txVx

x

(7)

   

.0, :0For xfxVt 

(8)

Using the method of separation of variables (see

Appendix B: after calculations of the integrals

included in (B12) and elementary algebraic

transformations) and considering the boundary (7)

and initial (8) conditions, we will obtain the

following expression:

   

 

     

 

.cos

1cos

1

sin

1

2

cos1cos

sin

2

,

2

1

2

1

2

1

22

m

tLn

n

tLn

n

e

L

nx

n

LL

e

L

nx

n

L

n

L

txV























































































3.2 Bidomain Model

The present section discusses the 1D mathematical

model of conductivity in discretely coupled

myocytes. Electrical behavior in the tissue is

considered in individual myocytes, each of which is

modeled as a continuum bound through conditions

at the cell boundaries, which are gap junctions.

Consider a chain consisting of cylindrical

myocytes of length L and radius a. The myocytes

are connected through gap junctions, as described

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by, [21]. The cylindrical coordinate system

 

zr ,,



is defined by

zx :

in the direction of the myocyte

length. The extracellular space has a cross-sectional

area

 

x

, which changes along

x

. The

intracellular potential

 

txVi,

is defined in

myocytes, and extracellular potential

 

txVe,

is

defined in the extracellular space. The 1D bidomain

model is constructed

x

by determining the average

intracellular

 

txVi,

and extracellular

 

txVe,

potentials as well as the transmembrane potential

 

eim VVtxV ,

.

The cable equation for each cell is as

follows:

.

11 















































x

V

rxx

V

rx

I

t

V

Cp e

e

i

ion

m

(9)

Here:

i

iA

R

r

and

e

eA

R

r

, where

i

R

,

e

R

are

resistances of intercellular and extracellular spaces,

respectively.

2

aAi





and

e

A

are the average

intracellular and extracellular cross-sectional areas

of the cell, respectively,

m

C

the membrane

capacity, and

ap



2

the cell circumference. The

intracellular current is given as

x

V

ri

i

1

and is

continuous in the cell.

ion

I

is the sum of all ionic

currents, which is taken as a constant current

applied to all points of the cell membrane, i.e.:

.

m

ei

m

ion R

VV

R

V

I



Assuming that gap junctions behave as ohmic

resistors, the potential drop at the junctions is

proportional to the current flowing through the

junctions.

 

,

1

x

V

rr

Vi

ij

i





where

 

i

V

is the leap in intracellular potential

along the gap junction, and

j

r

is the effective

resistance of the gap junction.

The steady-state passive problem with

Dirichlet boundary conditions is solved by the

method of separation of variables. For a constant

stimulus at one end of the cable (Dirichlet boundary

conditions), let us take the solution with a

geometrically decaying solution, [21],

   

,xVLxV ii





   

xVLxV ee





for the

decay constant

1



. The decay constant is related

to the space constant

g



by the following

expression:

.

g

L

e









For a given cell, the analytical solution of this

stationary problem can be found, which for the n-th

cell is proportional to:

.



























e

i

n

e

i

V





Thus, in the case of the stationary state, equation

(9) can be written as follows:

. ,0

1

eim

m

mm

ei R

p

xrrx

























The solution of this equation is given as follows:

   

,expexp 21 kxckxcmmm 



where

 

ei

m

rr

R

p

k

2

. So, the solutions are given

by the following formulas:

   

bx

rr

r

bx

rr

rm

ei

e

em

ei

i

i











,

Using the boundary conditions of current

continuity and continuity of the extracellular

potential and the leap in the intracellular potential at

the gap junction, the constants are determined by the

following formulas:

  

,

1

2

,, 21

















kLkL

ei

e

kL

m

kL

m

ee

rr

r

b

ecec

where



is the root of the following characteristic

equation:

  

 

.2 kLkL

kLkL

j

ei

j

ee

kL

R

rr

kr













m

j

jR

Lpr

R

is the effective dimensionless

resistance of the gap junction.

4 Numerical Realization

4.1 Realization of the Monodomain Model

The numerical simulations for the 1D model of

continuously coupled myocytes in the absence of

current at the beginning and end of the cable (cell),

were performed using MATLAB software for the

following data: time (



ranges from 2.5 to 4.5 ms

in humans) and length (



ranges from 1.3 to 2.2

mm

) for four pairs of constants:

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1)

mmms 3.1 ,5.2 



; 2)

mmms 6.1 ,8.2 



;

3)

mmms 8.1 ,3 



and

4)

mmms 2.2 ,5.4 



, myocyte (cell) length:

mmmL 135.0135 



, spatial (lengthy)

discretization

mmmx 006.06 



and temporal

discretization

mst02.0

were used. Numerical

values of the transmembrane potential

V

were

obtained during the variation of

x

(from

mm006.0

to

mm135.0

) and

t

(from

ms02.0

to

ms03.0

).

In practice, a system of contours (isolines) is

often used to analyze the measurement results. For

myocytes, some contours of transmembrane

potential

V

for the plane area in the vector field of

space (length)

x

and time

t

are presented. (It is

possible to find such a system of points, in which

numerical values of transmembrane potentials

V

are

equal. By connecting them, we get lines of equal

transmembrane potentials, so-called isopotential

contours). Figure 2 shows myocyte transmembrane

isopotential contours for four values of



and



.

As Figure 2 shows, the transmembrane isopotential

contours in all four cases look similar, in particular,

all of them are almost elliptic lines, and only their

numerical values differ slightly.

Figure 3 shows three-dimensional (3D) graphs

of the distribution of the transmembrane potential

V

in the vector field of space and time for the

myocyte for four values of



and



. As the Figure

shows, in all cases, the 3D graphs look almost the

same and only their numerical values differ slightly.

Fig. 2: Transmembrane isopotential contours for

different fixed values of time and space constants

(upper left figure:

mmms 3.1 ,5.2 



, upper

right figure:

mmms 6.1 ,8.2 



, lower left

figure:

mmms 8.1 ,3 



, lower right figure:

mmms 2.2 ,5.4 



)

Fig. 3: 3D graphs of the distribution of

V

transmembrane potential in the

x

t

vector field for

different fixed values of time and space constants

(upper left figure:

mmms 3.1 ,5.2 



, upper

right figure:

mmms 6.1 ,8.2 



, lower left

figure:

mmms 8.1 ,3 



, lower right figure:

mmms 2.2 ,5.4 



).

Figure 4 gives the graphs of variation of

V

transmembrane potential along

x

for four different

fixed constant values of



any length



and

different fixed values of

t

, in particular: upper left

figure:

mst01.0

, upper right figure:

mst02.0

, lower left figure:

mst05.0

, lower right figure:

mst09.0

. Figure 4 shows: 1) as the value of

t

increases different fixed constant values of



and

length



, the graph of variation of transmembrane

potentials get close to each other meaning that the

transmembrane potential will change after some

time only slightly when



and



change; 2) in all

four cases, as

075.0x

, for all four values of time

and length constants, the transmembrane potential

 

138.0, txV

.

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Volume 20, 2023

Fig. 4: Graphs of variation of transmembrane

potential

V

along

x

for different fixed values of

time



and space



constants and different fixed

values of

t

, in particular: upper left figure:

mst01.0

, upper right figure:

mst02.0

, lower

left figure:

mst05.0

, lower right figure:

mst09.0

.

Figure 5 shows the graphs of variation of

transmembrane potential

V

as

t

changes for

different fixed values of time



and space



constants and different fixed values of

x

in

particular upper left figure:

mmx006.0

, upper

right figure:

mmx012.0

, lower left figure:

mmx03.0

, lower right figure:

mmx054.0

.

In all four cases, 2D graphs look very similar and

only their numerical values differ slightly. Besides,

in all four cases

02.0t

, transmembrane potential

 

138.0, txV

.

Fig. 5. Graphs of variation of transmembrane

potential

V

, in case of

t

change, for different fixed

values of time



and space



constants and

different fixed values of

x

, in particular: upper left

figure:

mmx006.0

, upper right figure:

mmx012.0

, lower left figure:

mmx03.0

,

lower right figure:

mmx054.0

.

4.2 Realization of the Bidomain Model

The 1D mathematical model of conductivity in

discretely coupled myocytes discussed in section

3.2, was realized numerically, in particular, for 16

cells at fixed potentials at both ends of the myocyte

in the steady state. Numerical simulation was done

with MATLAB software for the following data

based on the obtained analytical solution:

cmL012.0

,

26

104cmAi



,

26

1065.1 cmAA ie 



,



62 104, 

 aaAi

cm

g09.0



,

2

7000 cmRm

,

cmRi150

,

cmRe 75

,

cmRj110

. The graphs corresponding to the

analytical solution are shown in Figure 6, in

particular, graphs of changes in the intracellular

potential

i

V

, extracellular potential

e

V

, and

transmembrane potential

V

for different fixed

values of space constant

g



: upper left figure:

mm

g09.0



, upper right figure:

mm

g05.0



,

lower left figure:

mm

g03.0



, lower right figure:

mm

g01.0



. As the figure shows, by reduction

g



, the geometric decay of intracellular,

intercellular, and extracellular transmembrane

potentials accelerates.

Fig. 6: Graphs of intercellular

i

V

, extracellular

e

V

,

and transmembrane

V

potentials, as the functions

of space for different fixed values of space constant

(upper left figure:

mm

g09.0



, upper right

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Volume 20, 2023

figure:

mm

g05.0



, lower left figure:

mm

g03.0



, lower right figure:

mm

g01.0



).

5 Conclusion

In this paper, the theoretical matters for models of

cardiac electrophysiology are considered, including

the introduction of single-cell action potential

models and the continuum tissue model.

Mathematically describing the local continuous or

discontinuous excitation of tissue is the focus of this

work.

The principal results presented in the paper can

be formulated as follows:

 Using the passive 1D cable equation, the

standard formulations for monodomain and

bidomain models are given as extensions of the

cable equation.

 The transmembrane potential propagation for a

single cell is studied; in particular, the

transmembrane potential propagation of a thin

cylindrical excitable cardiomyocyte in the

absence of current at the beginning and end of

the cardiomyocyte is studied.

 A 1D mathematical model of the conductivity of

discretely coupled cardiomyocytes is discussed.

The electrical behavior in cardiac tissue is

studied in individual cells, each of which is

modeled as a continuum bound through

conditions at cell boundaries that represent gap

junctions.

 Using MATLAB software for the above

problem, numerical results for the

transmembrane potential are obtained, from

which the contours of the transmembrane

isopotential for the plane area in the vector field

of length

x

and time

t

are construed. Some 2D

and 3D graphs are plotted.

 One of the most surprising (interesting) results

we obtained is that after a certain time, the

transmembrane potential almost stops changing

when



and



changes (Figure 4). It is

important to verify these theoretical predictions

experimentally.

Cardiovascular disease remains the leading

cause of death worldwide, most notably, heart

failure due to heart attack and fatal arrhythmias. The

immediate cause of fatal cardiac arrhythmias is still

not thoroughly understood, but in many cases, it

may be related to an improper spread of the cardiac

action potential. It should be noted that, despite

many years of research, the distribution of the action

potential of the heart muscle is still not fully

understood. So its study remains a pressing topic of

many modern scientific studies.

This research can be used in electrophysiology

to examine a wide range of arrhythmias to

understand the etiology of the disease and figure out

the solution. Also, the data can be used for some

electrophysiologic medical devices to perform a

comprehensive electrophysiologic study.

The work can be considered to be combined

with artificial intelligence in the future for more

efficacy of electrophysiologic diagnostic and

treatment devices.

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Volume 20, 2023

References:

[1] Hodgkin AL, Rushton WAH. The electrical

constants of a crustacean nerve fiber.

Proceedings of the Royal Society of London.

Series B, Biological Sciences 1946; 133: 444-

479.

[2] Sepulveda NG, Roth BJ, Wikswo JP. Current

injection into a two-dimensional anisotropic

bidomain. Biophysical Journal 1987; 55: 987-

999.

[3] Shuaiby SM, Hassan MA, et al. A finite

Element Model for the Electrical Activity in

Human Cardiac Tissues. Journal of Ecology

of Health & Environment 2013; 1(1): 25-33.

[4] Tveito A, Jæger KH, Kuchta M, Mardal K-A,

Rognes ME. A Cell-Based Framework for

Numerical Modeling of Electrical Conduction

in Cardiac Tissue. Frontiers in Physics 2017;

5: Article 48.

[5] Jæger KH, Edwards AG, McCulloch A,

Tveito A. Properties of cardiac conduction in

a cell-based computational model. PLoS

Computational Biology 2019; 15(5):

e1007042.

[6] Eisenberg RS, Barcilon V, and Mathias RT.

Electrical properties of spherical syncytia.

Biophysical Journal 1979; 25:151-180.

[7] Peskoff A. Electric potential in three-

dimensional electrically syncytial tissues.

Bulletin of Mathematical Biology 1979; 41(2):

163-181.

[8] Roth BJ. Longitudinal resistance in strands of

cardiac muscle. Ph.D. thesis, Vanderbilt

University, Nashville, TN, 1987.

[9] Tung L. A bi-domain model for describing

ischemic myocardial d-c potentials. Ph.D.

thesis. Massachusetts Institute of Technology,

Cambridge, MA, 1978.

[10] Miller WT, Geselowitz DB. Simulation

studies of the electrocardiogram, 1. The

normal heart. Circulation Research 1978; 43:

301-315.

[11] Geselowitz DB, Miller WT. A bidomain

model for anisotropic cardiac muscle. Annals

of Biomedical Engineering 1983; 11: 191-206.

[12] Plonsey R, Barr RC. The four-electrode

resistivity technique is applied to cardiac

muscle. Ieee Transactions on Bio Medical

Engineering 1982; 29(7): 541-546.

[13] Muller AL, Markin VS. Electrical properties

of anisotropic nerve-muscle syncytia. 1.

Distribution of the electric potential. Biofizika

1977; 22(2): 307-312.

[14] Barr RC, Plonsey R. Propagation of excitation

in idealized anisotropic two-dimensional

tissue. Biophysical Journal 1984; 45: 1191-

1202.

[15] Plonsey R, Barr RC. Current flow patterns in

two-dimensional anisotropic bi-syncytia with

normal and extreme conductivities.

Biophysical Journal 1984; 45:557-571.

[16] Sepulveda NG, Wikswo JP Jr. Electric and

magnetic fields from two-dimensional bi-

syncytia. Biophysical Journal 1987; 51(4):

557-568.

[17] Henriquez CS. Simulating the electrical

behavior of cardiac tissue using the bidomain

model. Critical Reviews in Biomedical

Engineering 1993; 21(1):1-77.

[18] Tung L. A Bidomain Model for Describing

Ischemic Myocardial D-C Potentials. Ph.D.

thesis, MIT, Boston, 1978.

[19] Neu JC, Krassowska W. Homogenization of

syncytial tissues. Critical Reviews in

Biomedical Engineering 1993; 21(2):137-199.

[20] Hunter P, Pullan A. Modeling total heart

function. Annual Review of Biomedical

Engineering 2003; 5: 147-177.

[21] Keener J, Sneyd J. Mathematical Physiology,

second edition. Springer, New York, NY,

2009.

[22] Plonsey R, Barr RC. Bioelectricity: a

quantitative approach. New York, NY, 2007.

[23] Akar FG, Roth BJ, Rosenbaum DS. Optical

measurement of cell-to-cell coupling in the

intact heart using subthreshold electrical

stimulation. American Journal of Physiology-

Heart and Circulatory Physiology 2001;

281(2): H533-42.

[24] Kamke E. Handbook of Ordinary Differential

Equations. Moscow, Nauka, 1971.

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Volume 20, 2023

APPENDICES

Appendix A. Derivating passive 1D cable

equation

Let us consider a long enough thin cylindrical

excitable cell. Let us denote the specific resistance

of the cell by , the membrane resistance by ,

and the membrane capacity by .

The variation of transmembrane potential

within a short interval is formulated as

follows: , where is the resistance of

the cell cytoplasm per unit length and is the

current along the membrane. Let us assume that

,

(A1)

The current along the membrane can be described

with the following formula: , where

is the transmembrane current per unit of length.

Again, letting ,

(A2)

Transmembrane current at every point x is the

combination of capacitance current and

membrane leakage current corresponding to the

membrane resistance . The capacitance current

within the membrane area of a given volume is

expressed with the formula:

The current due to membrane resistance is

expressed by Ohm’s Law:

By combining these two members, we gain:

Then, by inserting these two members in (A2) and

by considering (A1) (from (A1), ), the

following expression is obtained:

Therefore, the passive cable equation will be written

down as follows:

, (A3)

where denotes intercellular current.

Then, by an algebraic transformation of the

passive cable equation, namely by multiplying

equation (A3) by , the number of parameters is

reduced to two basic parameters. Thus, the passive

cable equation will be written as follows:

, (A4)

where is the constant of the space (length) of the

passive cable equation and is expressed by the

following formula:

which determines the distance along the cell at

which the injected potential decreases by factor e.

By defining the spatial constant as the distance at

which the potential at a cell point decreases by

factor e, it can be calculated that for healthy tissue,

the spatial constant is much greater than the length

of an individual cell, in particular for normal cardiac

excitation the spatial constant is much greater

than the length of one myocyte ( , [22], or

, [23], compared to the average

myocyte length of ). This means that the

change in potential along the length of an individual

cell is very small, and thus homogenizing the tissue

to represent the continuum without considering the

change in length of an individual cell is an

acceptable approximation.

is a time constant and is defined by the following

formula:

It determines the length of time over which the

injected potential decreases by factor e. Space and

time constants are useful parameters used to

measure and characterize specific properties of

excitable tissues.

Instead of passive membrane resistance, the

currents, due to active ion channels, may be

replaced by passive transmembrane currents. So:

,

where each ion is the transmembrane current

due to the motion of a particular type of charge-

i

r

m

r

m

c

V

x

xriV ll 

l

r

l

i

0x

.

l

ir

x

V



xii ml 

m

i

0x

.

m

li

x

i



c

i

r

i

.

t

V

ci mc 





.

m

rr

V

i

.

m

mm r

V

t

V

ci 





x

V

r

i

l

l



 1

.

11

2











































m

ll

lr

V

t

V

c

x

V

rx

V

rxx

i

m

mi r

V

t

V

c

x

V







2



l

ir

1





m

r

V

x

V

t

V

m







2





,

l

m

im r

r

r





mm1



mm5.1



mm1,0



,

mmcr





ionsr ii

ions

i

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Volume 20, 2023

bearing ion. Then the equation of the passive cable

can be written as follows:

Appendix B. Method of separation of

variables

After separating the variables, let us write down

as follows:

(B1)

After substituting (B1) into equation (A4), we gain:

Moving the members and equating them to the

negative characteristic yields:

As a result, the following equations are gained:

(B2)

(B3)

The solution of equation (B2) is [24]:

.)( )/( 2m

tk

cetT







(B4)

As for the solution of (B3), if we assume that

, then the roots of the equation are complex. If

, then the roots will be real and the solution

will be as follows, [24]:

This will not serve because in this case, the

boundary conditions do not allow for unique values

of a and b, and only trivial solutions are obtained.

So, if we consider that , we will get the

following expression, [24]:

(B5)

By substituting equalities (B4) and (B5) into

expression (B1), we will obtain:

.

1

sin

1

cos),(

2

)/( 2









































x

k

b

x

k

acetxV m

tk





(B6)

Boundary conditions and

mean the following:

Hence:

The following expression is obtained from (B1),

(B6) and :

 

.00

2

1

cos

0

2

1

sin

2

1

0'



























































k

b

k

a

k

X

The term containing sine drops out and will

remain. Thus, from (B5) we will gain:

(B7)

By applying the second boundary condition

, we will gain:

Generally, , when

, thus, we will gain:

(B8)

Hence:

(B9)

By substituting equality (B8) into (B7), we will

gain:

(B10)

By substituting equalities (B9) and (B10) into (B6),

we will gain:

.

Now, let us insert

.

Hence, we will have the Fourier decomposition with

cosines:

. (B11)

.

2

2









ionsmi i

t

V

c

x

V



 

txV ,

     

., tTxXtxV 

."' 2XTTXXT

m



 

2

k

.1

"' 22 k

X

T

m



,' 2TkT

m



 

.1" 22 XkX 



1

2k

1

2k

 

.

1

sinh

1

cosh 2

2



k

xb

k

xaxX 







1

2k

 

.

1

sin

1

cos 2

2



























x

k

bx

k

axX



 

0,0 tVx

 

0, tLVx

       

.0'0'  tTLXtTX

   

.0'0'  LXX

 

00' X

0b

 

.

2

1

cos 













x

k

axX



 

0' LX

   

.0

2

1

sin

2

1

'













 L

kk

LX



 

0sin 



,2,1 , nn



,2,1 ,

2

1

n

L

nk





.1

2

2











 L

n

k



 

,2,1 ,cos 













n

L

nx

axX



 

 

,2,1 ,cos, 2

1













nec

L

nx

atxV m

tLn

n





acdn

 

 

m

tLn

nn e

L

nx

dtxV





2

1

cos, 















 

 























1

0

2

cos

2

1

,

n

tLn

nm

e

L

nx

ddtxV





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Volume 20, 2023

By considering the initial condition ,

the exponential term drops out:

.

The Fourier decomposition with cosines will be

written down as follows:

.

By substituting this into (B11), the following

expression is obtained:

 

.cos

cos)(

2

)(

2

,

2

1

10

0

m

tLn

n

L

e

L

nx

d

L

n

f

L

df

L

txV































































(B12)

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed to the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

   

xfxV 0,

   















 xf

L

nx

ddxV n



cos

2

1

0, 0















L

nd

L

n

f

L

d

0

cos)(

2







WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE

DOI: 10.37394/23208.2023.20.24

Natela Zirakashvili, Teona Zirakashvili

E-ISSN: 2224-2902

239

Volume 20, 2023