Research ArticleSimulating Notch-Dome Morphology of Action Potential ofVentricular Cell: How the Speeds of Positive and NegativeFeedbacks on Transmembrane Voltage Can Influence theHealth of a Cell?

S. H. Sabzpoushan and A. Ghajarjazy

Department of Biomedical Engineering, Iran University of Science and Technology (IUST), Tehran 16846-13114, Iran

Correspondence should be addressed to S. H. Sabzpoushan; sabzposh@iust.ac.ir

Received 27 December 2019; Revised 23 March 2020; Accepted 18 July 2020; Published 4 September 2020

Academic Editor: Rumiana Koynova

Copyright © 2020 S. H. Sabzpoushan and A. Ghajarjazy. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

Ventricular action potential is well-known because of its plateau phase with a spike-notch-dome morphology. As such, themorphology of action potential is necessary for ensuring a correct heart functioning. Any distraction from normal notch-domemorphology may trigger a circus movement reentry in the form of lethal ventricular fibrillation. When the epicardial actionpotential dome propagates from a site where it is maintained to regions where it has been lost, it gives rise to the proposedmechanism for the Brugada syndrome. Despite the impact of notch-dome dynamics on the heart function, no independent andexplicit research has been performed on the simulation of notch-dome dynamics and morphology. In this paper, using a novelmathematical approach, a three-state variable model is proposed; we show that our proposed model not only can simulatemorphology of action potential of ventricular cells but also can propose a biological reasonable tool for controlling of themorphology of action potential spike-notch-dome. We show that the processes of activation and inactivation of ionic gatingvariables (as positive or negative feedbacks on the voltage of cell membrane) and the ratio of their speeds (time constants) canbe treated as a reasonable biological tool for simulating ventricular cell notch-dome. This finding may led to a new insight to thequantification of the health of a ventricular cell and may also propose a new drug therapy strategy for cardiac diseases.

1. Introduction

The so-called process, “excitation-contraction coupling” inthe cardiac myocyte, is unique in that it is responsible forthe coupling of electrical impulse to mechanical function.There are several ions, ion channels, and regulatory pathwaysparticipating in the generation of action potential (AP) ofcardiac cells. Ionic channels are large transmembrane proteinshaving aqueous pores through which ions can flow down theirelectrochemical gradients. The dynamics of electrical conduc-tance of an individual channel is controlled by its gate dynam-ics, i.e., gates’ speed. The gating dynamics influence the timecourse of the channel opening and closing [1].

The speed of gates, i.e., their time constants, ionic con-centrations, membrane voltage following time sequence,

and various regulatory pathways determine the morphologyof AP, which can be expressed as mathematical formalisms,making it feasible to use the computational approach to ana-lyze and elucidate the underlying mechanisms of the wholecardiac cell.

Ventricular AP is famous because of its plateau phasewith a spike-notch-dome (SND) morphology [3]. As such,the morphology of AP like Figure 1 is necessary for ensuringa correct heart functioning.

AS stated beforehand, APs are produced as a result of ioncurrents that cross the cell membrane via ionic channels andgates. Ion currents produce a net depolarization or repolari-zation of the membrane as different currents are invoked inresponse to the transmembrane voltage changes. Each typeof channel is highly selective for a specific type of ion. The

HindawiBioMed Research InternationalVolume 2020, Article ID 5169241, 16 pageshttps://doi.org/10.1155/2020/5169241

most common intracellular ion concentrations considered incardiac models are sodium, calcium and potassium.

Channelopathies due to mutations that modify the ionchannel function can perturb the form of the action potential,sometimes leading to cardiac dysfunctions or altered APmorphology and propagation [4, 5].

Following the pioneering mathematical description of theAP by Hodgkin and Huxley (HH) in neuronal cells [6], thefirst cardiac models considered in much the same way. Thechanges in the transmembrane potential, _v, produced by asum of ionic-gated currents, ∑ionIion, that is [2, 7],

c _v = Istim −〠ion

Iion, ð1Þ

where v and c are membrane’s voltage and capacitance,respectively, and Istim is stimulus current. Equation (1) thatis a general formalism for modeling cell’s membrane poten-tial is illustrated schematically in Figure 1(b).In this context,the inward ionic currents that increase cell’s voltage are con-sidered negative currents and outward ones as positivecurrents.

Today, as more developed experimental data of the dif-ferent currents and their dynamics became available, morecomplex models of the AP for specific cardiac cells have beenproposed [8].The mission of these models has been notice-able for several animal kinds, such as guinea pig [9], dog[10], rabbit [11], or rat [12], and also for humans [2, 13–15].

The general drawback of the above electrophysiologicalmodels is the increasing complexity of the models and conse-quently the very high computational costs associated with thesimulation of cardiac tissue as well as the difficulties andsometimes impracticality of model analyzing. Clearly, ahighly realistic ionic model is ideal as an in silico experimentaltool, allowing to safely examine multiple hypotheses and mak-ing predictions without experiencing any cardiac danger forthe patient or to check conditions not easily reproducible inclinical experiments. But, despite their more realistic descrip-tion, these models usually are often difficult to analyze mathe-matically and deducing biology principles from them andsometimes less compatible with the pathology of cardiac cells.

Simplified models have also been proposed for modelingAP dynamics. However, although they are simple, they are

not complete electrophysiological [16–18], so-called semielec-trophysiological models [19–23], i.e., somewhere between thesimple and the complex electrophysiological models. Thesesemiphysiological models preserve particular sets of propertiesof the action potential, like AP duration (APD); that is, thetime during which the voltage is above a certain threshold,characterizing the duration of the excited state, or the conduc-tion velocity (CV) [24]. In this respect, these can be viewed asmesoscopic models that bridge the gap between dynamics atthe molecular level (ion channel gating) and whole macro-scopic system, heart. These models are still manageable forboth computation and theoretical analysis.

An example of a semiphysiological model is the one byFenton and Karma [22], a three-state variables model of thecardiac action potential. Thismodel uses three transmembranecurrents, i.e., fast inward, slow inward, and slow outward,which resemble the set of physiological sodium, calcium, andpotassium currents, respectively. In the Fenton-Karma model,the state variables are the transmembrane potential and twogating variables that are used to regulate the inactivation ofthe fast inward and slow inward currents, respectively.

Despite the Fenton-Karma model simplicity, the model isless compatible with electrophysiology of cardiac cell in thesense that it uses discrete functions, Heaviside function.Although the Fenton-Karma model can reproduce fairly theaction potential of more complex models or experiments,on one hand, it fails to reproduce accurately the SND ornotch-dome (ND) pattern and on the other hand, it has noindependent and appropriate parameters for regulating ND.In fact, the Fenton-Karma model was originally created inorder to describe the propagation and properties of scrollwaves in cardiac tissue. Wave propagation behavior in thecardiac tissue is determined by the restitution and alternansproperties of cells. With the alternans, we mean a beat to beatmodification in the APD at fast pacing rates, where the onsetand subsequent evolution of alternans has been related to theshape of the restitution curves [25]. However, more recentresults have shown that models with the same restitutionproperties, but different AP morphologies, may give differentonsets for alternans [21]. The reason is that the dynamics ofalternans is also affected by ionic currents that depend signif-icantly on the form of the AP [26]. Furthermore, ionic cur-rents are also a basis factor for the occurrence of reexcitation

V (m

v)

Spike NotchDome

2

0 3

4

1

4

Time (ms)

(a)

Iion1

Iion2 …

C

–

V

+

(b)

Figure 1: (a) A typical morphology of ventricular AP with five distinct phases and spike-notch-dome (SND) as a significant part of an AP [2];(b) membrane voltage of a cell can be modeled by a capacitor that is charged by ionic currents.

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and phase 2 reentry in cases where the duration of the actionpotential is nonhomogeneous in the heart tissue [27].

A modification of the three-variable Fenton-Karmamodel has been proposed in [19]; it fits better the AP mor-phology. The modified model includes an additional gate thatmodulates the slow inward current, in a fashion that resem-bles the effect of the fast outward potassium current in elec-trophysiological models. This allows to reproduce the spikein phase 1 of the AP typical of epicardial cells [28]. Buenoet al.’s model is still less compatible with the electrophysiol-ogy of ventricular cell because it uses the Heaviside function.

Peñaranda et al. [29] included the effect of Ito, fast out-ward potassium current; following the same idea as in theFenton-Karma model, they divided the currents not by theircarrier ion, but by their function: inward or outward cur-rents, and among these, slow and fast currents to a total offour. With this assumption, they showed that it is possibleto reproduce the important characteristics of AP propagationand morphology. They also used their model to study reexci-tations in tissue presenting large dispersion of repolarization.There, it was shown that reexcitation is due to a slow pulseinduced by the L-type calcium current and propagates fromthe region of long APDs to the region of short APDs untilit encounters newly excitable tissue.

Channelopathies are diseases that appear because ofdefects in ion channels; this defection causes by either geneticor acquired factors. Mutations in genes encoding ion chan-nels, which damage channel function, are the most commoncause of channelopathies. Because ion channels are greatlyinvolved in cells, ion channel defects have been involved ina wide variety of diseases, including epilepsy, migraine,blindness, deafness, diabetes, hypertension, cardiac arrhyth-mia, asthma, irritable bowel syndrome, and cancer [30].

Cardiac channelopathies are responsible for many sud-den arrhythmic death syndrome cases. Mutations in calcium,sodium, potassium, and TRP channel genes have been iden-tified to cause a variety of cardiac arrhythmic disorders likeBrugada syndrome types, dilated cardiomyopathy, and longQT syndrome types [31].

Although a great number of researches have been con-ducted in ventricular cell AP modeling, approximately all ofthem have paid less independent and direct attention toND simulations and dynamics. In fact, there is nothing inelectrophysiology literature connecting ND-related diseasesto channelopathies.

Grandi et al. [13] developed a detailed mathematicalmodel for Ca handling and ionic currents in the human ven-tricular myocyte with the main aim to simulate basic excita-tion–contraction coupling phenomena; their model relies onthe framework of a rabbit myocyte model. They could simu-late AP morphology of epicardial cells with characteristicspike and dome (SD).

AP dynamics are influenced by its morphology [21, 32,33]. In a research by Bueno-Orovio et al. [19], they used pre-viously published data [34–42] and developed a minimalmodel of the APs of human ventricular myocytes; they mod-ified the three-variable model proposed by Fenton andKarma [22]. They argued that a three-variable formulationis not sufficient to reproduce SND morphologies. Therefore,

they add a fourth variable to obtain a more accurate AP mor-phology with SND. However, as we will see later, althoughour proposed model is a three-state variable one, it cansimulate SND morphology satisfactorily. It means that ourproposed model can be more computational cost-effectivethan Bueno-Orovio et al.’s model.

In recent years, a dynamic known as phase-2 reentry(P2R) has been recognized as a mechanism for triggeringlethal arrhythmias. P2R occurs when adjacent cells undergodramatic action-potential (AP) shortening from a normalND morphology to a loss-of-dome morphology. Local reex-citation ensues when ionic current during the AP dome stagepropagates from depolarized sites to hyperpolarized loss-of-dome sites. It is thought that P2R could degenerate into ven-tricular fibrillation and cause sudden cardiac death.

Human AP in the Brugada syndrome have been charac-terized by delayed or even complete loss of dome formation,especially in the right ventricular epicardial layers. It is shownthat under the condition of fast pacing rates, there is alterna-tion between “loss of dome” and “coved dome” in the APmorphology for the epicardial cells. Peñaranda et al. in theirpaper presented a study on Brugada syndrome, where theloss of the dome is achieved by changing the conductanceof a current in their model. Bueno et al. showed that, inone-dimensional cables, P2R can be induced by adjoininglost-dome and delayed-dome regions, as mediated by tissueexcitability and transmembrane voltage profiles, and reducedcoupling facilitates its induction. In two and three dimen-sions, sustained reentry can arise when three regions(delayed-dome, lost-dome, and normal epicardium) arepresent [19]. Ten et al. in their model for human ventricularcell demonstrated that the AP shows the characteristic SNDarchitecture found for epicardial cells [2]. Epicardial and Mcells, but not endocardial cells, display action potentials witha notched or SND morphology. They showed that a furthershift in the balance of currents leads to the loss of the actionpotential dome at some epicardial sites, which is manifestedin the ECG as a further ST segment elevation. Antzelevitchshowed that because the loss of the action potential domein the epicardium is generally not spatially uniform, a devel-opment of striking epicardial dispersion of repolarization inAP can be seen [27].

Considering above reviewed researches, it is clear that,although they name dome or ND or SND in their modelsby some means and for different proposes, no one has ana-lyzed or modeled dome, ND, or SND directly.

In this paper, we use a mathematical approach andpresent a new model for simulating biological phenomena,ND in ventricular AP. We consider agents involved inSND dynamics and propose simple yet sufficient principlesand hypothesis theories for ND generation. As we will see,our mathematical idea will provide a new insight to thegating mechanism of ionic channels.

2. Method and Mathematical Approach

As we stated earlier, AP is the product of interactions amongelectrophysiological agents, most importantly and generally,activation and inactivation gating variables [1]. AP is a

3BioMed Research International

voltage that is generated across the cell membrane. AP isproduced when interacting ionic currents flow throughtransmembrane ionic channels and charge membranecapacitance (see Figure 1(b)). Ionic channels can be openedand closed via gating process; gating process can be eitheractivation (shown by n) or inactivation (shown by h). Withthe activation process, we mean a process that opens a chan-nel more when membrane voltage increases. In inactivationprocess, a channel is closed more when membrane voltageincreases.

A gating process can generate positive or negative feed-back on membrane voltage. A positive feedback is createdwhen activation variable, n, generates inward current or inac-tivation variable, h, generates outward current. In the sameway, a negative feedback is created when activation variable,n, generates outward or inactivation variable, h, generatesinward currents, respectively.

All of these mean that n and h can be generally consid-ered two major interacting agents and mechanisms for APgeneration. In this paper, we assume a general formalismfor each ionic current as

Iion = gion · gating process½ � · v − Eionð Þ, ð2Þ

where gion is the base conductance of ionic channel and½gating process� is a mathematical compound that is com-posed of gating processes; these processes may be voltage ortime dependent or both.

During the plateau phase of AP (phase 2 in Figure 1(a)),we assume (or approximate) a general exponential decreas-ing (increasing) dynamics for h and an increasing (decreas-ing) one for n over considered time interval, as follows:

h tð Þ = e−t/τh , ð3Þ

n tð Þ = 1 − e−t/τn : ð4ÞNow, if we assume a compound gating variable, nh, then

the dynamics of nh can be written as:

nh tð Þ = e−t/τh 1 − e−t/τn� �

: ð5Þ

And for the speed (time derivative) of nh, we have:

_nh = −1τh

e−t/τh 1 − e−t/τn� �

+ 1τn

e−t/τhe−t/τn

= −1τh

e−t/τh + 1τh

+ 1τn

� �e−t/τhe−t/τn :

ð6Þ

Taking τh/τn = α or τh = ατn, one has:

_nh = −1τh

e−t/τh + 1τh

1 + αð Þe−t/τh e−t/τn

= 1τh

e−t/τh 1 + αð Þe−t/τn − 1� �

:

ð7Þ

Considering Equation (7) that is an amazing one in ourapproach, the sign of the speed of nh is determined by the

expression in bracket. Note that the term e−t/τn in the bracketis ≤1; then, the sign of the speed of nh is influenced by α, theratio of the speeds (time constants) of the gating processes hand n. If we name the time point at which the sign of thespeed changes as ts, then we have:

1 + αð Þe−ts/τn = 1→ e−ts/τn = 11 + α

→ tsτn

= ln 1 + αð Þ→ ts = τn ln 1 + αð Þ:ð8Þ

Equation (8) that is a key equation in our research givesthe time at which the sign of the speed of nh changes; i.e.,an increasing (decreasing) dynamics evolves to a decreasing(increasing) one. It is clear from Equation (7) that, for t > ts,nh has an increasing dynamics, where for t < ts has a decreas-ing one.

Now let us have a look at acceleration (second derivative)of nh. Using some differential mathematics manipulations, itcan be shown that:

_nh = _nh + n _h,€nh = €nh + _n _h + _n _h + n€h

ð9Þ

that leads to:

€nh = −1τh

2 e−t/τh 1 + αð Þe−t/τn − 1

� �−

1τh

e−t/τh 1 + αð Þ 1τn

e−t/τn

= −1τh

e−t/τh1τh

1 + αð Þe−t/τn − 1τh

+ 1τn

1 + αð Þe−t/τn�

= −1τh

2 e−t/τh 1 + αð Þe−t/τn − 1 + α 1 + αð Þe−t/τn� �

= −1τh

2 e−t/τh 1 + αð Þ2e−t/τn − 1

� �:

ð10Þ

Considering Equation (10) that seems similar to Equa-tion (7), it is interesting that the sign of the acceleration ofnh is determined by the expression in the bracket. If wedenote the time point at which the sign of accelerationchanges, by tin, so we can write:

1 + αð Þ2e−tin/τn − 1 = 0→ tin = τn ln 1 + αð Þ2� �: ð11Þ

In Equation (11), tin is the time at which the accelerationof nh becomes zero; i.e., tin is the inflection point of nhðtÞ. Inother words,tin is a point at which the graph of nhðtÞ changesits curvature.

In Figure 2, we have illuminated the above mathematicaldiscussions graphically. Figure 2(a) illustrates sample graphsof nðtÞ and hðtÞ, for three different values of α, where τn = 1 isfixed. It is clear that by increment of α, hðtÞ becomes slowerin comparison with nðtÞ. In Figure 2(b), we have sketchedcompound variable nhðtÞ against the time; it is seen that byincreasing α, while the amplitude value of nhðtÞ decreases,its skewness (peak value) shifts to the right. Another interest-ing finding in Figure 2(b) is that for early times, the dynamics

4 BioMed Research International

of nhðtÞ is like nðtÞ, where for later times, i.e., after the peakof nhðtÞ, the dynamics is similar to hðtÞ. In Figure 2(c), thetime point ts, i.e., the time at which the speed of nhðtÞ getsits maximum value, is plotted against α. In Figure 2(d), the

inflection point of nhðtÞ, tin, is plotted against α; it is clearthat by increment of α, the position of ts and tin can be shiftedto the right of the time axis; i.e., ts and tin happen at highervalues of time. Figure 2(e) depicts that tin is always greater

0 1 2

𝛼 = 0.1

3 4 5t

0

0.2

0.4

0.6

0.8

1

n(t)h(t)

𝛼 = 1.1

𝛼 = 2.1

(a)

0

0.2

0.4

0.6

0.8

nh

(t)

0 1 2

𝛼 = 0.1

3 4 5t

𝛼 = 1.1

𝛼 = 2.1

(b)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

t s

𝛼

(c)

0 5 10 15 20 25 300

1

2

3

4

5

6

7

t in

𝛼

(d)

0 5 10 15 20 25 300

2

4

6

8

𝛼

tstin

(e)

0 1 2 3 4 5t

0

0.2

0.4

0.6

0.8

1

nh

(t)

𝛼 = 0.1𝛼 = 1.1𝛼 = 2.1

(f)

Figure 2: Graphical illustrations of (a) nðtÞ and hðtÞ. (b) nhðtÞ for three different values of α. (c) ts against α. (d) tin against α. (e) ts and tinagainst α. (f) Scaling nhðtÞ so that it takes the maximum value of unit. In all panels, τn = 1 is fixed.

5BioMed Research International

than ts; it happens after ts. Inspection of the slope of curves inFigures 2(c) and 2(d) makes it clear that tin is more sensitiveto the variations of α in comparison with ts. In fact, it will beso interesting if we use Equations (8) and (11) and get:

∂ts∂α

= τn1

1 + α, ð12Þ

∂tin∂α

= 2τn1

1 + α: ð13Þ

Considering Equations (12) and (13), it is clear that ∂tin/∂α = 2ð∂ts/∂αÞ which means tin is twice more sensitive tothe variations of α than ts.

Substituting Equation (8) into Equation (5), it is easy toshow that:

nhmax = nh tsð Þ = α∙ 1 + αð Þ− 1+αð Þ/αð Þ: ð14Þ

So if we want to treat nh as an individual gating variablethat can take a maximum value of unit, we can scale Equation(5) using Equation (14). In Figure 2(f), we have done so.

In Figure 2(e), we have sketched ts and tin on the samescreen. It is seen that tin is always larger than ts; i.e., thechange in the curvature of nhðtÞ occurs after the maxi-mum (minimum) speed. It is also evident that by incre-ment of α, the distance between ts and tin in timedomain increases.

Figure 2 clarifies how the value of α influences thedynamics and morphology of nhðtÞ. As we will see later,the introduced novel parameter, α, plays a key role in ourproposed formalism and model; i.e., one may use the abovefindings and facts for regulation of the AP morphology viathe adjustment of the ratio of the speeds of activation andinactivation processes in a supposed system of ventricularcell.

Figure 3 depicts a typical simulation of AP, nhðtÞ, nðtÞ,and hðtÞ in a model of ventricular cell during the same timeinterval, where α is set to 0:65. You see how the evolutionsof nhðtÞ, nðtÞ, and hðtÞ are similar to the general forms ofFigure 2, particularly on the right side of the vertical dashedline where transmembrane potential, v, settles down towardrest potential; i.e., v has no variations. This figure will bemore discussed in Results.

20

–20–40–60–80

0

40

200 400 600 800 1000

(a)

(b)

(c)

(d)

0

0.25

0.150.1

0.05

0.2

0.3

200 400 600 800 10000

200 400 600 800 10000

0.4

0.20.1

0.3

0.5

200 400 600 800 10000

0.4

0.2

0.90.80.7

0.3

0.50.6

Figure 3: Typical realizations of (a) AP, (b) nhðtÞ, (c) nðtÞ, and (d) hðtÞ, in a ventricular cell model. Here, nðtÞ and hðtÞ are the gating process.Compare these general morphologies with those in Figure 2.

6 BioMed Research International

In this research, for explanation of our novel mathemat-ical approach, i.e., illumination of time constant ratiohypothesis, we propose a mathematical model that is com-patible with the biology of cardiac ventricular cell. Our modelis composed of three currents: IL, a leakage current (linearcomponent), I f , a dynamicless or fast current (nonlinearcomponent) and Id , a nonlinear dynamical (slow) current.Following HH general formalism in Equation (1), the pro-posed model is defined in Equations (15), (16), (17) and(18) as follows:

_v = − IL vð Þ + I f vð Þ + Id v, tð Þ� �, ð15Þ

where each current undergoes the general formalism ofEquation (2) as:

IL = gL · v − ELð Þ, ð16Þ

I f = gf ·m vð Þ · v − Ef

� �, ð17Þ

Id = gd · n v, tð Þ · h v, tð Þ · v − Edð Þ: ð18ÞWe define the dynamics of activation, n, and inactivation,

h, agents as follows:

_n = n v,∞ð Þ − n v, tð Þτn

, ð19Þ

_h = h v,∞ð Þ − h v, tð Þτh

, ð20Þ

n v,∞ð Þ = 11 + exp V12n − vð Þ/Knð Þ , ð21Þ

h v,∞ð Þ = 11 + exp V12n − vð Þ/Khð Þ : ð22Þ

The role of producing a regenerative process (positivefeedback) for making an upstroke phase of AP has beenassigned to I f [43]. Note that we say nothing about the

20

–20–40–60–80

0

40

200 400 600 800 10000

(a)

200 400 600 800 10000

0.25

0.150.1

0.05

0.2

0.3

(b)

200 400 600 800 10000

0.4

0.20.1

0.3

0.5

(c)

200 400 600 800 10000

0.4

0.2

0.90.80.7

0.3

0.50.6

(d)

Figure 4: Evaluation of our proposed model for simulating ventricular cell AP: (a) APmorphology generated by the model; (b) time course ofnhðtÞ; (c) time course of nðtÞ; (d) time course of hðtÞ.

7BioMed Research International

direction of I f ; i.e., it may be treated as an inward or outwardionic current arbitrary. Here, we may assume a fast activationgating process, mðvÞ, for I f as follows:

m vð Þ = 11 + exp V12m − vð Þ/Kmð Þ : ð23Þ

In the proposed model, Id delegates all dynamical activa-tion and inactivation agents; i.e., all voltage- and time-dependent positive and negative feedbacks on transmem-brane voltage, v.

3. Results

In this section, it is shown how AP morphology, particularlyND, that was obtained by complex electrophysiologicalmodels can be simulated by the presented model. Specially,it is shown how regulation of parameter α, i.e., adjustingthe ratio of the activation speed to the inactivation speed,can regulate AP morphology. In the following investigations,we have used OpenCOR [44] and Matlab as simulationmedium. Figure 4 illustrates a typical simulation with theuse of our model. It is seen that the proposed model canreflect the general morphology of ventricular APsatisfactorily.

The parameter values of the model for generatingFigure 4 are listed in Table 1. In this table, we see that Ef >0 means I f acts as an instantaneous inward current and isresponsible for generating the upstroke phase of AP (phase0 in Figure 1(a)). We also have Ed < 0 that means Id is an out-ward current and generally responsible for plateau phaseshaping and AP repolarization. Here, because Id is an out-ward current, activation and inactivation gating variablesact as negative and positive feedbacks on membrane voltage,respectively.

For more clarification of our basic idea behind the pro-posed formalism for illustration of the role of nh as a com-pound gating variable, we consider Figure 5 that depicts azoomed-in view of SND portion of a simulated AP and itscorresponding nh. Because Id is an outward current, its incre-ment decreases membrane potential. It means that when nhgets its peak, we expect a minimum in membrane potential,i.e., a notch or valley. Similarly, when nh gets its minimum(valley), we expect a peak (dome) in AP. Comparison of timeevolutions of AP and nh in Figure 5 justifies our approach forcontrolling notch and dome of an AP via regulating maxi-mum and minimum of nh.

As we stated earlier, our new idea in this research is thatthe ratio of the speeds of activation and inactivation gatingvariables in a ventricular cell model can determine the mor-phology of AP; particularly, we may regulate ND in AP via

regulation of α. Figure 6 proves this idea by a graphical illus-tration. In Figure 6(a), we have sketched AP generated by ourmodel for three different values of α. In Figure 6(b), thecorresponding graphs of nhðtÞ are depicted. It is seen thatthe alteration of the ratio of activation and inactivation pro-cess speeds can affect AP morphology via alteration of nhdynamics. Figure 6 also demonstrates how our approachis amenable to mathematical analysis and at the same timecompatible with the electrophysiology of action potential;i.e., we can adjust AP morphology with the use of a biolog-ically plausible parameter.

To show how our model and postulated activation toinactivation speed ratio approach can simulate well-knownventricular cell models, we fit it to two complex experimentaldata-based electrophysiological models: Luo and Rudy [9]and TNNP [2]. In Figures 7 and 8, simulation results areillustrated, respectively. In the left panel of each figure, simu-lated AP and gating variables, nhðtÞ, nðtÞ, and hðtÞ, aresketched against time, where in the right one, the AP of thecomplex models is drawn. It is seen that our model can reflect

Dome peakNotch valley

30

40

20

10

0

0 100 200 300 400

–10

–20

nh peak

nh valley

0.1

0.15

0.2

0.05

0

0 100 200 300

–0.05

–0.1

Figure 5: A zoomed-in view of SND of a simulated AP and itscorresponding nh. This figure justifies our approach for regulatingND in a ventricular cell model.

Table 1: Parameter value for generating AP in Figure 4.

Parameter gL EL gf Ef Km V12m gd Ed Kn V12n Kh V12h τn τh C α

Unit nS/cm2 mV nS/cm2 mV mV mV nS/cm2 mV mV mV mV mV ms ms pF/cm2 —

Value 0.01 -90 0.008 97.89 4.96 -38.1 0.072 -92.8 28.86 -29.4 -28.86 -29.4 300 65 0.01 0.22

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general dynamics of these two models reasonably. Theappended table in each figure lists the corresponding param-eter value for the respective case. The goodness-of-fit mea-sures are also reported in a separate table for each case. Thenearly unit R-square shows how well our mathematicalapproach can simulate the dynamics of AP.

4. Discussion and Conclusion

In this research, we showed that our simple three-state vari-able model can simulate AP of ventricular cells. In addition,we revealed that our model can be fitted to well-known com-plex electrophysiological models. Our model was based on anovel idea that the ratio of the speeds of activation and inac-tivation gating variables in a ventricular cell model can gov-ern the morphology of AP; particularly, we may regulateND in AP via regulation of speed ratio. The results showedthat our mathematical approach is useful to gain better bio-logical understanding or explain simpler a biological phe-nomenon; AP.

The proposed speed ratio approach suggests a new way oflooking at modeling and simulation of AP. The results dem-onstrated the effectiveness of our approach. We showedmathematically how the speed ratio coefficient, α, can influ-ence the morphology of compound gating variable, nh, andhow this gating variable which is used in an ionic currentcan affect membrane potential morphology by itself. In otherwords, speed ratio of activation and inactivation processescan be used as a tool for regulating AP morphology particu-larly ND shape. We see that α not only has an exact mathe-matical meaning but also has an exact and reasonablebiological interpretation. From an electrophysiology pointof view, while α actually delegates microscopic agents andplayers, it has great impacts on the macroscopic behavior,AP, of a cardiac cell.

As it was stated in Introduction, SND and particularlyND morphology have great impacts on cardiac arrhythmia

and diseases. Therefore, we may treat α as a quantitative fea-ture of the health of a ventricular cell. In this context, we mayextend our idea to a broader area and hypothesize a novelimpression about the quantification of the health of a neuronor cell in general. Following this idea, we may hypothesizethat the health of a cell is influenced by the ratio of the speedsof positive and negative feedbacks on transmembrane volt-age. We may also define a healthy range of α for any kindof cells, where with the healthy range, we mean an intervalof α values on which the normal AP morphology can be cre-ated. This new idea should be more investigated in futureresearches.

Here, we should emphasize that we are biomedical engi-neers, so we look at biological systems from an engineeringpoint of view. However, we have transactions with biologistsand physicians in explaining our ideas and hypothesis. Thereare a huge number of examples that translating an idea fromthe engineering point of view to a biological one or vice versahas led to answer medical problems or to find therapy strat-egies. It means that sometimes, an engineering hypothesiswhich is mathematically satisfactory should be validatedexperimentally by a biologist. Similarly, an observation orfinding by a biologist should be described mathematicallyby an engineer.

Experimental observations by physiologists have con-firmed the following electrophysiological phenomena in theprocess of action potential generation [1]:

(1) Inward ionic currents, depolarizing currents, forincreasing membrane potential

(2) Outward ionic currents, repolarizing currents, fordecreasing membrane potential

(3) Positive feedback on membrane potential for gener-ating fast upstroke

(4) Negative feedback on membrane potential for stabil-ity and rest potential generation

0 200 400 600 800 1000Time (ms)

–100

–50

0

50

Volta

ge (m

V)

𝛼 = 0.02

𝛼 = 0.5

𝛼 = 0.75

𝛼 = 0.75𝛼 = 0.02𝛼 = 0.5

(a)

0 200 400 600 800 1000Time (ms)

0

0.2

0.4

0.6

0.8

nh

(t)

𝛼 = 0.75𝛼 = 0.02𝛼 = 0.5

(b)

Figure 6: (a) Illustration of the effect of α on the general morphology of plateau phase of AP. We see how one can manage ND generation byregulating α. (b) Different morphologies of nhðtÞ generated by regulation of α. This figure validates graphically our hypothesis.

9BioMed Research International

(5) Activation and inactivation mechanisms of ionicchannel gating process

Therefore, if the formalism of a mathematical modeldeals with the above biological phenomena, then it can beconsidered as a realistic biological model. In this circum-stance, a finding or hypostasis that may be deduced fromthe model can be extended to the underlying biological sys-tem. However, this extension should be carried out by biolo-gists and can be the subject of a separate biological research.

We have involved all above sensations in our modelstructure, so we can consider it an experimental-basedmodel. Moreover, as we have shown in our paper, the pro-posed model can reproduce the AP morphology and dynam-ics of both Luo-Rudy and TNNP models, where Luo-Rudyand TNNP are based on experimental data [2, 9]. It means

that our model which fits these models can be regarded asan experimental-based model as well. This look is alsoenhanced by the fact that our model has Hodgkin-Huxley-(HH-) based formalism.

Complex electrophysiological models of cardiac cell aresloppy [45]. With the sloppy, we mean models in whichmany parameters are loosely constrained and only dependon a few “stiff” parameter combinations. In a sloppy complexelectrophysiological model, we expect that model behaviorscan be controlled by a relatively small number of parametersand agent combinations. Among parameters and agent com-binations, we should look for sets which deal with substantialconcepts that influence complex system behaviors. In a ven-tricular cell, substantial agents are related to the abovemen-tioned essential phenomena, i.e., ionic currents (inputs,outputs, and fast and slow with different underlying mecha-nisms) and gating process (activation and inactivation with

Measure R-square RMSE

Value 0.9710 0.0437

Parameter

Unit

LRd 1.26

gL

nS/cm2 pF/cm2

C

mV

EL

–85

nS/cm2

gf

3.01

mV

Ef

154

mV

Km

22.68

mV

V12m

–40.13

nS/cm2

gs

26.4

mV

Es

–106

mV

Kn

100

mV

V12n

48.68

mV

Kh

–70

mV

V12h

–3.83 0.01

–

𝛼

0.2

ms

145

𝜏n

ms

30

𝜏h

0.90.80.7

0.50.6

100 200 300 400 5000

h

0.25

0.150.1

0.05

0.2

100 200 300 400 5000

n

0.180.16

0.120.1

0.080.060.04

0.14

100 200 300 400 5000

nh

4020

–20–40–60–80

0

100 200 300 400 5000

V

(a)

4020

–20–40–60–80

0

100 200 300 400 5000

V

(b)

Figure 7: Fitting our model to the Luo-Rudy complex model: (a) simulation results: AP, nh, n, h; (b) AP of the Luo-Rudy complex model.Parameter value and goodness-of-fit measures are also reported in separate tables. The almost unit R-square shows how good our modelcan capture the morphology of AP.

10 BioMed Research International

different underlying mechanisms). These agents may inducepositive or negative feedbacks on membrane voltage, wherehaving at least one negative and one positive feedback onmembrane voltage is necessary for AP generation [1]. Wesee that our model has representatives of all necessary elec-trophysiological agents in a ventricular cell: ionic currents,gating process, and positive and negative feedbacks.

Our presented and verified hypothesis states that com-plex morphology and dynamics of ND can be explained bya simple theory of the speed ratio ðαÞ of activation and inac-tivation gating processes (h and n.). This statement suggeststhat the simple theories of macroscopic behaviors are hiddeninside complicated microscopic processes which makes thecomplex world understandable [46].

Mathematical models of biological systems, which aremainly proposed by mathematicians or engineers, especially

biomedical engineers, can play essential roles for clinicalinvestigations, prediction of the disease behavior, and sugges-tion of suitable therapy protocols. In these circumstances,experiments and mathematical models are complementary.Biomedical engineers use experimental observations to pro-pose appropriate framework for mathematical models, thenbenefit from the results of these models to improve futureexperiments. In this way and by using mathematical models,it will be possible for biologists to make predictions of situa-tions that are difficult to implement in the lab. For example,electrophysiological mechanism such as gating speed cannotbe easily measured experimentally. In this condition, oneusually extrapolates experimental measurements. Consider-ing today’s experimental tools and methods, basic electro-physiological mechanisms cannot be easily measured byexperiments. For example, patch clamp method which is

100 200 300 400 5000

4020

–20–40–60–80

0

V

Parameter

UnitTNNP 0.17

gL

nS/cm2 mV

EL

–121.94nS/cm2

gf

0.35mV

Ef

130mV

Km

15.81mV

V12m

–46.05nS/cm2

gs

1.31mV

Es

–82.69mV

Kn

30mV

V12n

–31.48mV

Kh

–17.9mV

V12h

–15.16pF/cm2

C

0.01–

𝛼

0.5ms

44

𝜏n

ms

22

𝜏h

Measure R-square RMSE

Value 0.9875 0.0444

0.40.35

0.250.2

0.150.1

0.05

0.3

100 200 300 400 5000

nh

0.7

0.50.40.30.20.1

0.6

100 200 300 400 5000

n

0.91

0.80.7

0.50.40.30.2

0.6

100 200 300

(a)

400 5000

h

(b)

4020

–20–40–60–80

0

100 200 300 400 5000

V

Figure 8: Fitting our model to Ten Tusscher et al. [2]: (a) simulation results: AP, nh, n, h; (b) AP of the Ten Tusscher et al. complex model.Parameter value and goodness-of-fit measures are also reported in separate tables. The almost unit R-square shows how good our model cancapture the morphology of AP.

11BioMed Research International

one of the most common experiments usually suffers frominterventions and effects of coupled cells. Also, no experi-mental techniques exist for measuring or controlling a sin-gle activation or inactivation gate speed. However, to showhow our theory, the speed ratio ðαÞ of positive and nega-tive feedbacks through activation and inactivation effects,can describe the complex phenomenon, ND, we use widelyused human ventricular cell by Priebe and Beuckelman [15]as a cell in laboratory and set up the following in silicoexperiment.

Evoking the sloppiness of biological systems and in orderto find the currents which are not explicitly involved in NDdynamics generation, we inhibit some currents in a cell.Results are summarized in Figures 9(a)–9(f). Each panel inFigure 9 depicts the general morphology of AP where therespective currents are inhibited. Figure 9(f) illustrates themorphology of AP in the ventricular cell where a minimumnumber of currents are involved in its generation. In otherwords, these currents are necessary to reproduce AP andND morphology qualitatively.

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(a)

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(b)

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(c)

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(d)

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(e)

0 500 1000Time (ms)

–100

–50

0

50

Volta

ge (m

v)

(f)

Figure 9: AP and ND morphology of PB cell where related ionic currents are present in a cell: (a) INa + ICa + Ito + IKr + IKs + IK1 + INaCa +INaK + IbNa + IbCa + Istim; (b) INa + ICa + Ito + IKr + IKs + IK1 + INaCa + INaK + IbNa + Istim (IbCa is blocked);(c) INa + ICa + Ito + IKr + IKs + IK1 + INaCa + INaK + Istim (IbCa + IbNa are blocked). (d) INa + ICa + Ito + IKr + IKs + IK1 + INaCa + Istim(IbCa + IbNa + INaK are blocked); (e) INa + ICa + Ito + IKr + IKs + IK1 + Istim (IbCa + IbNa + INaK + INaCa are blocked); (f) INa + ICa + Ito + IKr +IKs + Istim (IbCa + IbNa + INaK + INaCa + IK1 are blocked).

12 BioMed Research International

0 500 1000–100

–50

0

50

Time (ms)

Volta

ge (m

v)

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000–100

–50

0

50

Time (ms)

Volta

ge (m

v)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

–100

–50

0

50

0 500 1000Time (ms)

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

Figure 10: AP and ND morphology of PB cell where α is adjusted on a range of 1-10. Considering regulation of ND morphology withregulation of α, our hypothesis is validated experimentally.

(a)

(b)

(c)

20

–20

–40

–60

–80

0

40

0.35

–0.35

0.25

–0.25

0.2

–0.2

0.15

–0.15

0.1

–0.1

0.3

–0.3

0.4100 200 300 400 5000

–0.4

100 200 300 400 5000

100 200 300 400 5000

Figure 11: A typical simulation with the use of our model: (a) AP, (b) nhðtÞ, and (c) reciprocal of nhðtÞ against time axis. We see how thedynamics of nhðtÞ reciprocal during the plateau phase is similar to AP.

13BioMed Research International

In Figure 9(f), we note that ICa is the only current that cancreate positive and negative feedbacks on membrane voltagesimultaneously. This conception can be cleared mathemati-cally if we have a look at ICa equation in PB:

ICa = gCa · d · f · f Ca v − ECað Þ, ð24Þ

where d and f (and f Ca) are activation and inactivation gatingvariables, respectively.

In Figures 10(a)–10(j), we have validated our hypothesisby variation of α in a range of 1-10. You see how ND mor-phology can be biased by the speed ratio adjustment. Thereader can also compare Figure 10 with Figure 6(a).

It is interesting that ICa is an inward current in PB cell. Soits activation gate, d, creates a positive feedback and its inac-tivation gate, f , creates a negative feedback on membranevoltage. In a similar way, Id is an outward current in our pro-posed model in which activation and inactivation gates gen-erate negative and positive feedbacks, respectively. This factshows a beautiful perspective of the generality of our hypoth-esis that speed ratio of positive and negative feedbacks caninfluence ND dynamics.

Another amazing picture of our new idea is illustrated inFigure 11, where the two upper panels AP and nh are thesame as in Figure 5 and the lower one is the flip of nh overthe time axis; you see how the morphology of the plateauphase (the time evolution of AP after upstroke and beforedownstroke) is similar to the morphology of reciprocal of nh. The generalization of this interesting finding should bemore investigated in future researches.

In the light of the speed ratio approach, some drug ther-apy strategies may be invented in the future, i.e., drug dosagesthat regulate the speed of activation and inactivation processof ionic channels.

The amenability of our approach to mathematical analy-sis makes it very suitable and flexible for applications in insilico drug therapy experiments and also introduction of awide range of applications for it in cardiac arrhythmiasimulations.

Although the impacts of speed regulation on AP mor-phology were investigated in this research, its impacts onother properties of AP dynamics, like AP propagation in tis-sue and restitution [47, 48], should be more investigated infuture works. As a start point, we used our model with theparameter values in Table 1 where it generates an AP withND. Then, we applied the dynamic restitution protocol [36]for driving action potential duration (APD) and conductionvelocity (CV) restitution curves and the results are depictedin Figure 12. We see that our model can reflect generallythe restitution property in ventricular cells [49].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

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