The Action Potential in the Heart

Home , Cardiac action potential, Resting potential, Threshold potential, Ventricle (heart)

Zahra Dhanerawala MATH 390 FALL 2015

December 17, 2015

Figure 1: (A) Schematic of action potentials, recorded in different regions of the human heart. (B) Schematic of a ventricular action potential.20

1 Contents

1 Introduction 3

2 Ionic models for the cardiac cell 4 2.1 Luo-Rudy model: C a 2+ channels ...... 6

3 The FitzHugh-Nagumo (FHN)-type models 9 3.1 Aliev-Panﬁlov modiﬁcations to the FitzHugh-Nagumo equations ...... 9

4 Contemporary Clinical Implications 12

5 Appendix 13 5.1 The Hodgins-Huxley Equations ...... 13 5.2 The Generalised Fitzhugh-Nagumo Equations ...... 14 5.3 Four Phases of the Solution ...... 15

6 References 18

2 Abstract

This paper discusses the characteristics of the cardiac AP and the evolution in its understanding in the ﬁeld of systems biology. Several examples of mathematical models dis- cussed are attempts to explain a particular biological phenomenon (arrhythmias, resting potential, intracellular ionic regulation, etc.) in a larger context by examining how several variables interact in a particular model.

1 Introduction

In analyzing the Hodgkin-Huxley model for action potentials (APs) in the neurons, it was shown in 1952 that the amplitude and rate of rise of an action potential of the squid giant axon is dependent on the extracellular sodium concentration, which leads to a large and specific increase in the permeability of the cellular membrane to sodium ions3. Hodgkin and Huxley used the voltage-clamp technique to separate the membrane current into sodium and potassium com- ponents to compose a model in which these currents vary with the membrane electrical potential and time. These equations are experimentally determined to reproduce many of the electrical properties of the squid giant axon, including the shape and size of the action potential and the velocity of conduction. The range of biological phenomena that these equations describe have greatly increased since the initial experiments4. Many of these new mathematical models owe to the fact that the amount of biological data generated in the late 20t h and early 21s t century has overwhelmed the scientific community and its attempts to understand it. Genomics, proteomics, and other fields of ‘big data’ describe specific physiological parts of living processes, but there is no definite guide to understand these phenomena cohesively. Understanding physiological functions at a specific micro- molecular level, such as at those of proteins, has led to successful intervention in the practice of medicine. For example, many applications in drug therapy depend on knowing how a pro- tein behaves in context. This integrative knowledge of how proteins interact at a higher cellular level is crucial to examining in which disease states a receptor, enzyme, or transporter is rel- evant. An ignorance of these fundamental processes can lead to unpredicted side effects that may not be realized at the molecular level5. It has been reported widely that cardiac APs are shortened by the increase in external calcium and sodium concentration due to a slowly activated channel. This proposes a few explanations between the differing “fast" and “slow" response cells in the heart (outlined below). Calcium-rich solutions generally raise the resting potential which accelerates time- and voltage-dependent current changes and trigger re-polarization. This, imposed by the depolar- 2+ izing current, shortens the AP.Additionally, raising the concentration of C a enhances IK , the current produced by the background K + ions, which favours faster re-polarization. It is for this reason that the C a-sensitive current was identified as an increase in IK due to its dependence + 2 on membrane potential and K0 (the initial concentration of K ) .

3 There are two principal types of action potentials observed in the heart. One is that of the “fast" responding cells observed in conductive cardiac tissue: the myocardial fibers in the atrium, ventricles, His bundle, and the Purkinje fiber network. The other is that of the “slow" responding cells observed which have a prolonged upstroke in AP.There are found in the pacemaker fibers in the sinoatrial and atrioventricular nodes. The slow response is distinguished by a smaller amplitude and a slower upstroke, which accounts for a slow electrical conduction velocity relative to the fast response fibers. The action potential in the heart is initiated by the permeability of the cell membrane to sodium, potassium, and calcium ions. This is ac- complished by the variations in configurations of specific transmembrane proteins known as ion channels. These channels are selective to specific ions and alternate between the states of “open" and “closed" gates. Thus, the current is defined by the proportion of time spent in the open versus closed states. Like most cells in the body, the cells in the heart highly permeable to K + ions in their resting state. The N a +/K + ATPase in the cell is responsible for pumping N a + out of the cells and K + into the cells. This accumulation of K + into the cells creates a strong chemical gradient which allows for the flow of K + out of the cell. The inside of the cell becomes more and more electronegative and provides a negative feedback loop which slows down the efflux of K +. The electrostatic forces rises until they equal the force of the chemical gradient. The chemical and electrostatic forces are in equilibrium; thus, the electronegative potential inside the cell is known as the equilibrium potential. This state in which cells have a negative overall potential inside the cell with high permeability to K + ions is what cardiac cells are in when the action potential reaches them through the conduction pathway. The negative ion flux inside the cell is said to hyper-polarize it because it amplifies the potential difference already present between the cell and the extracellular environment in the resting state. When the ion flux renders the inside of the cell more positive, it is depolarized, as it reduces the charge difference prevalent in the resting state1.

2 Ionic models for the cardiac cell

It has proven difficult to come up with accurately representative models of the action potential specific to the cardiac tissue. One of the first developed cardiac cell models was that of FitzHugh (1960) and Noble (1960, 1962). This model explained the ‘plateau’ region found in cardiac cells as a consequence of reducing the amplitude and speed of activation of the delayed potassium current, IK .

4 Figure 2: The cardiac action potential ’plateau’ region

However, these early models did not take into account the voltage gated inward current of C a 2+ ions since these currents had not been discovered. Noble (1962) reduced the voltage dependence of the sodium activation process, but it wasn’t until voltage clamp experiments in the heart were carried out in the mid- to late-1960s that the calcium current was discovered6. Sub- sequent models were constructed quantitatively on the basis of experimental data and measured parameter values.

Figure 3: Schematic diagram of the cell model. The abbreviations representing ionic currents, pumps, and exchangers are deﬁned in Luo and Rudy (1996). The intracellular compartment is the sarcoplasmic reticulum (SR), which is divided into two subcompartments, the network SR (NSR) and the junctional SR (JSR). Dotted areas (in the myoplasm and the JSR) indicate presence of C a 2+ buffers.

5 Voltage-clamp experiments in the cardiac cell formed the basis for studying the AP through ionic interaction.

2.1 Luo-Rudy model: C a 2+ channels

One such model proposed by Luo and Rudy (1994) for ventricular action potential as the basis of study of arrhythmogenic activity of a single myocyte takes into account the C a 2+ current + + through the L-type (long-lasting) channel (IC a ), the N a /K pump, and the involvement of the sarcoplasmic reticulum, which regulates the calcium ion concentration in the cell. The objective of this model was to introduce the most important depolarizing and repolarizing currents based on data available at the time from single-cell and single-channel record- ings. These were not as morphologically complex as previous multicellular preparations and overcame many of the limitations of the earlier measurements. IN a , the fast sodium current;

IK , the delayed rectiﬁer potassium current; and IK 1, the inward rectiﬁer potassium current were formulated. A plateau potassium current, IK p was also added to the model. The following differential equation represents their numerical description of the rate of change in the ventricular action potential7,9:

d V 1 = (Ii + Is t ) (1) d t −Cm Also written simply as:

d V I C (2) = m d t

Where Cm is the membrane capacitance, Is t is the external current or stimulus, and Ii takes into account the various ionic currents through the voltage-gate channels or through the N a +/K + and N a +/C a 2+ pumps. The gated channels are described by the Hodgkin-Huxley deﬁnitions as solutions to a coupled system of differential equations. In the absence of an external current, equation (2) can be solved for I = 0. It is also necessary to specify the particular currents resulting from intercellular coupling:

d V I = (D V ) = Cm + Im (3) 5 · 5 d t D is a tensor (or multivariable gradient) of conductivities, and is the one-, two-, or three- 5 + dimensional gradient operator. With appropriate descriptions of Im , the ionic currents of N a , C a 2+, and K +, Eq. (3) constitutes a model for the electrical properties of cardiac tissue. The following equation also takes into account the dynamic changes of ionic concentrations in the action potential duration (APD):

d [B] (Ib AC ap ) = − · (4) d t VC zB F · · in which [B] is the concentration of some ion B, Ib is the sum of the ionic currents carrying

6 B, AC ap is the area of the capacitive membrane, VC is the volume of the compartment in which 2+ B has entered, zB is the valence number of ion B (ex. +2 for C a ), and F is Faraday’s constant. The change in V (electrical potential) due to ionic currents represents a negative feedback loop and regulates the ionic gates and current ﬂow. The equations that describe this phenomenon are13:

d y = (y y )/τy (5) d t ∞ −

1 τy = (6) αy + βy

αy yx = (7) αy + βy

where y represents any gating variable, τy , is its time constant, and yx is the steady-state value of y . αy and βy are rate constants as functions of v , the voltage. These systems of differential equations were solved using the integration algorithm outlined by Rush and Larsen14 and Victorri et al15. This algorithm used an adaptive time step that is always smaller than tma x = 1 millisecond. When V changes relatively slowly ( V 4 4 ≤ Vmi n = 0.2), t is Vma x /V and Vma x = 0.8 mV. When V changes rapidly ( V Vma x ), 4 4 4 4 4 ≥ 4 t is set to Vmi n /V . If t takes the values V Vma x , t is reduced until V < Vma x . 4The time step4 is ﬁxed in4 APD to 0.05 or 0.01 msec4 ≥ to 4 minimize4 variability caused4 by the4 time discretization. Ionic models describing the changes between concentrations of major ions inside cardiac cells (N a +, C a 2+, and K +) generally vary from the range of 10 to 60 ODEs. This ionic model for cardiac cells is particularly adapted to the guinea pig ventricular cell. The processes modeled are important to the understanding of the electrical activity and the AP of the cell, but they also aid in our understanding of the mechanical excitation-contraction coupling process7. This is a particularly interesting case in which the strength of contraction in heart muscle is dependent C a 2+ on the ratio N a + in the extracellular ﬂuid. Thus the prolonged repolarization is an essential feature of contraction in heart muscle.

7 Figure 4: Tracings showing major ionic currents that determine the shape of the AP. The fast kinetics and large amplitude of the fast Na+ current (IN a , B) result in the upstroke of the AP 2+ (A). The L-type C a current (IC a , C) is also activated quickly to support the action potential + plateau against the repolarizing K current (IK , D) and the total time-independent current (Iv , E). Finally, the large increase of IK and the late peak of Iv during its negative slope phase repo- larize the membrane to the resting potential. During the late repolarization and early postrepo- + 2+ 2+ larization phases, the N a -C a exchange current (IN aC a , F) is activated to extrude C a ions out of the cell and contributes an additional component of inward current. Currents shown are for 1 µF of membrane capacitance.

Observations of the various ionic currents are largely based on experimental data. The currents incorporated are carried across the sarcolemma by processes other than the gated ionic channels such as pumps and exchangers like cellular processes responsible for C a 2+ ion concentration inside the cell such as the C a 2+ uptake and release by the sarcoplasmic reticulum and buffering of C a 2+ in the myoplasm and the sarcoplasmic reticulum. Figure 4 shows the behaviour of various ionic currents during AP. Detailed explanations for the behaviour of the

8 currents can be found in Luo and Rudy (1994)7. The quantitative models of these ionic currents are constantly evolving, and there is no agreed upon ‘best’ model for cardiac tissues in all biological models. Luo and Rudy’s focus lies primarily in ionic currents and processes that regulate and determine the concentration of C a 2+ in the cell during AP.An important feature of the regulation of C a 2+ in the junctional sarcoplasmic reticulum in particular is the ability to simulate spontaneous release of C a 2+ from the organelle, which generates a non-speciﬁc current (Ins C a ) studied in arrhythmogenic activity. The variables are adapted to various models based on the importance of ionic and other transmembrane currents.

3 The FitzHugh-Nagumo (FHN)-type models

Other widely recognized models which use a phenomenological approach consist of the FitzHugh- Nagumo (FHN)-type models8. Initially published as a simplified, two-variable approach to the multi-system Hodgkins-Huxley equations, these models explain the basic properties of cardiac tissue and quantitatively reproduce important characteristics of AP propagation of cardiac tissue such as the duration and velocity of each action potential, the recovery period, and the shape of the cardiac action potential. The following is a FHN-type model for AP propagation that has been customized using experimental data from a canine myocardium. The benefits of these mathematical models been confirmed using biophysical models of cardiac tissue10,11.

3.1 Aliev-Panﬁlov modiﬁcations to the FitzHugh-Nagumo equations

Aliev and Panﬁlov’s (1996) approach to the FitzHugh-Nagumo model was used to study the canine myocardium in particular using a system of non-linear partial differential equations.

∂ u ∂ ∂ u = di j k u(u a)(u 1) uv (8) ∂ t ∂ xi ∂ x j − − − −

∂ v µ1v = (" + )( v k u(u a 1), (9) ∂ t µ2 + u − − − −

where k = 8, a = 0.15, "0 = 0.002. µ1 and µ2 are ﬁtted to experimental data. v (electrical potential) from the original FHN model (see Appendix) was changed to uv to improve the description of the shape of the action potential (see nullclines below). di j , the conductivity tensor, is determined on the basis of the biological model in use, i.e. that of experiments obtained by Nielson et al.12. Note that this is also the case in Luo and Rudy’s initial observations in Eq.([2.1.3). However, ionic models incorporate a larger number of currents and conductivities based on detailed, single-cell experimental data. All the variables in (8) and (9) are non- dimensional, including the temporal and spatial derivatives. The transmembrane potential E and time t are obtained by:

9 E [mV ] = 100u 80 (10) −

t [ms ] = 12.9t [t .u.]. (11)

in their respective dimensions.

For simpliﬁcation purpose, Er e s t is -80mV, and the amplitude of the pulse is 100mV. The time variable is scaled according to the assumption that the APD is measured at 90% of re- 12 polarization. APD0 = 330ms, according to experimental observations . k u(u a)(u 1) in (8) are the ‘fast’ processes: the initiation of an action potential and the ‘upstroke’− (see Appendix− for the FitzHugh-Nagumo phase plane solutions). Recovery is de- v termined by the time taken by v , or the term " µ1 . a is the threshold parameter. The ( + µ2+V u ) remaining parameters do not have clear physiological meaning but are reproduced to ﬁt their experimental data.

The left branch of the nullcline in the u vs. v graph (Figure 5) ut = 0 does not enter the region in which u < 0, preventing hyper-polarization, which is typical to the FHN model but has not been reported to exist in the myocardium. Another difference that was adapted to the heart model is the exchange of the linear nullcline of the slow variable vt = 0 to a quadratic term. Thus, the fast and slow variable nullclines are parallel to each other in a large region. The dependence of " on u and v also adapts to experimentally observed data by adjusting the parameters µ1 and µ2.

Figure 5: Nullcines of the Aliev Panﬁlov model

10 Figure 6: Nullcines of the FitzHugh-Nagumo model, Britton, N. (2012)

Figure 7: Pulse proﬁle in the duration of the action potential (APD)

The numerical simulations to the non-linear system of differential equations-particularly in the ionic models-run in the late 20t h century were mostly written in FORTRAN. The Runge-

11 Kutta-Merson algorithm and the Rush-Stanley approach mentioned earlier were used to solve the system using discretized time steps14,16. Aliev and Panfilov’s model thus provides a more realistic shape of the cardiac action potential compared to the original FitzHugh Nagumo equations, since the hyperpolarisation over- shoot is eliminated and an evident ‘plateau’ region is obtained. However, these equations do not explain dynamics of the action potential as thoroughly as the aforementioned ionic model. They are mainly used to model specific, multicellular cardiac cell experiments con- ducted. Changes in the extracellular ion concentrations or drug therapy that affects these concentrations will significantly alter any variables in such models. Gathering experimental data and using ionic models to ’fit’ the data and compute desired properties of the AP has been reported to be a better approach11.

4 Contemporary Clinical Implications

Clinical and physiological applications to mathematical models require a deeper mechanistic understanding of the cardiac excitatory process. One of the major shortcomings of mathematical models in cardiology is that explanations are isolated and piece-wise for a integrated, cohesive organ system and does not replicate the physiological environment (e.g. ion channels in the cellular membrane in the organ itself). Both of the models outlined are reductionist models used to elucidate the role of an individual component (i.e. ion channels) in the ar- rhythmogenicity and the action potential of the cell. Over the past decade, optical mapping studies are used to provide experimental evidence of these theoretical approaches. Addition- ally, biochemical models that explain electrophysiological processes as a coupled system have also developed21.

2+ Figure 8: Mapping simultaneous Vm and C a in the zebrafish heart in vivo. (A) Fluorescence of GCaMP5G in a fish expressing CaViar under the cmlc2 promoter. (B) Single optical section of a zebrafish heart expressing CaViar at 4 dpf. Cell membranes fluoresced in both the voltage channel (top) and the C a 2+ channel (bottom) as the AP propagated from the atrium to the ventricle. (C) Map of AP isochrones overlaid on a three-dimensional reconstruction of the heart. AP onsets were measured as time to reach 5% of maximum on the rising edge, with sub- frame timing achieved through spline interpolation. (D) Three-dimensional reconstruction of the electrical AP.Scale bars in (A-D) 50 µm.

12 The larval zebrafish has gained popularity as an efficient biological model in the past decade. It has also been used for optogenetics and optical mapping widely. Jennifer H. Hou et al. (2014) established through imaging that an inward C a 2+ flux is required to trigger electrical action potentials in the immature zebrafish heart which contrasts the adult zebrafish (and human) heart, where a N a + current regulates the upstroke of the action potential22. This opened up the possibility for a transition state between larval and adult zebrafish ionic current configurations. Detection of this transition required simultaneous observation of the voltage and C a 2+ dynamics in the intact heart. C a 2+ ion flow and voltage is measured simultaneously in vivo in the developing zebrafish heart. This also allows us to study the effects of genetic and phar- macological perturbations on development, on excitation-contraction coupling, and on C a 2+ handling in the future. The use of biological models like the zebrafish, optical mapping, and simulations combined can also be used in the future to screen drugs for their effects on cardiac repolarization prior to human use. It is important to keep in mind that all models are only partial representations of reality. However, through this iterative interaction between experiment and simulation that we will gain a deeper understanding of the systems we want to study.

5 Appendix

5.1 The Hodgins-Huxley Equations

The Hodgkins-Huxley equations are given by:

d V 3 4 Cm = = gN¯ a m h(V VN a ) g¯K n (V VN a ) gL (V VL ) (12) d t − − − − − −

d m τm (V ) = m (V ) m (13) d t ∞ −

d n τn (V ) = n (V ) n (14) d t ∞ −

d h τh (V ) = h (V ) h (15) d t ∞ − They involve four variables: the potential difference V , the sodium activation and inacti- vation variable m and h, and the potassium activation variable n. The functions m and n ∞ ∞ increase with V since they are activation variables, while decreases with V since it is an in- ∞ activation variable. gN a and gK approach their equilibrium values|which depend on voltage|at a ﬁnite rate if the voltage is held constant. It is observed that time dependent variables V and m are about a magnitude faster than h and n (time constants: 1ms vs. 10ms). Simpler models of nerve impulse use two variables, v and w , where v is the ‘fast’ excitation variable, similar

13 to the potential difference, and w is the ‘slow’ recovery variable, similar to the potassium con- ductance.

5.2 The Generalised Fitzhugh-Nagumo Equations

The advantage of this variation of the Hodgkin-Huxley equations is that it summarizes its information in terms of one-fast and one-slow variable. After its introduction, it also opened up possibilities to analyze the reduced two variable, fast-slow system in the phase plane, and thereby the qualitative interpretation of the underlying characteristics of the model and its ex- tension to more general models. The ‘simpler’ reduction of the Hodgkins-Huxley equations are obtained by the following ‘space-clamped’ equations:

d v d w ε f v, w , g v, w (16) d t = ( ) d t = ( ) which are also known as the generalised FitzHugh-Nagumo equations. Richard FitzHugh ﬁrst reduced the Hodgkins-Huxley model to two variables. J. Nagumo et. al built an electrical circuit that mimics one of the FitzHugh models using a a capacitor, a tunnel diode, a resistor, an inductor and a battery. Diodes are semiconductor devices that come in several varieties with nonlinear I V relations. The diode in the Fitzhugh-Nagumo circuit has an I V relation (f ) in the form of− a cubic, as depicted in Figure 9. From Kirchoff’s laws, the current− through the diode is zero for voltages across the diode less than the threshold voltage indicated by the position of the vertical axis in the ﬁgure. If the tunnel diode is operated in a circuit with voltage

V0 V V0 and load resistor R, the current is given by I = R− .

Figure 9: The Fitzhugh-Nagumo circuit

In the FHN model, g is traditionally a straight line where g (v, w ) = v c b w . The f − − function is cubic, f (v, w ) = v (v a)(1 v ) w or a piecewise linear function f (v, w ) = H (v a) − − − − −

14 v w , where H is the Heavyside function. H is unit discontinuous step function, whose value is zero− for a negative argument (i.e when a channel is ‘closed’) and one for positive argument (i.e when a channel is ‘open’). The cubic nature of f (v, w ) is due to fast positive feedback observed in APD. The following is a singular perturbation problem, which can be analysed using the method of matched asymptotic expansions.

d v d w ε = f (v, w ) = v (v a)(1 v ) w, = g (v, w ) = v b w (17) d t − − − d t − Where v (0) = v0, w (0) = w0.

5.3 Four Phases of the Solution

The system of linear differential equations is solved using a phase plane. Similar to a direction ﬁeld, a phase plane is helps us visualize how the solutions of this system would behave over extended period of time. We can do this by observing the behaviour of the nullclines, the set of points in the phase plane for which f (v, w ) = 0 and g (v, w ) = 0.

Figure 10: Nullcines of the FitzHugh-Nagumo model, Britton, N. (2012)

The nullcline f (v, w ) = 0 is in cubic form. In phase 1, the upstoke phase, v , the excitation variable, changes very quickly to attain t f = 0. We rescale the time variable for analysis by deﬁning a short time scale by T = " ,V (T ) = v (t ),W (T ) = w (t ). Thus:

d V = f (V,W ) = V (V a)(1 V ) W (18) d T − − −

d W = εg (V,W ) = ε(V b W ) (19) d T − d V Where W = w0 andV (0) = v0 and d T = f (V, w0). However, we are only interested in the

15 matching condition V (T ) h3(w0), where h is function deﬁned in the original Hodgkins- Huxley equation as T →. The time for this phase is negligible in terms of the t variable. In phase 2, the excited→ ∞ phase, we consider the longer t time scale and Eq. (18, 19). For the d w leading order, f (v, w ) = 0, continuing v = h3(w ) and w satisﬁes d t = g (h3(w ), w ) = G3(w ), which indicates that w increases until it reaches w , beyond which h w does not exist. The ∗ 3( ) time taken for this phase is:

w Z ∗ 1 t2 = dw (20) G3(W ) w0 In phase 3, the downstroke phase, v changes rapidly as the solution jumps from the right to left hand branch of the nullcline, the set of points in the phase plane for which f (v, w ) = 0. ˆ t t0 ˆ ˆ ˆ ˆ Thus, we redeﬁne the time scale as in phase 1, where T = −ε , V (T ) = v (t ), W (T ) = w (t ). Returning to the Eq. (18, 19) as in phase 1, we are again interested in the matching condition to phase 4, where lim Vˆ Tˆ h w T ( ) = 1( ∗) →∞ The time taken is negligible compared to the t time variable, as in phase 1. In phase 4, the recovery phase, the long t time scale is required as in phase 2. Since we have v h w and d w g h w , w G w , with w t w . Thus v and w decrease back = 1( ) d t = ( 1( ) ) = 1( ) ( 2) = ∗ to steady state. The above four phases describe the behaviour if the steady state lies in the left hand branch of f = 0, as illustrated in Figure 6. If the linear nullcline, g = 0, where shifted to the left so that the steady state would be established in the area of f = 0 as illustrated on the right, in the middle branch of f 0. Thus, in phase 4, w would drop until it reaches w . The system would = ∗ shift to the right-branch of f = 0.

Figure 11: Oscillatory phase plane of the FitzHugh-Nagumo model, Britton, N. (2012)

The periodic oscillation is described to the ﬁrst order by:

16 w ∗ Z 1 1 t2 = dw (21) G3(W ) − G1(W ) w ∗ The numerical solutions in the ﬁgure on the right are valid with ε = 0.01, a = 0.1, b = 0.5, c = 0.1. The steady state is established at (0.1, 0), surrounded by stable periodic relaxation oscillation. From experimental observations, the most important prediction of this model is the exis- tence of a “threshold" potential that generates voltage and recovery traveling waves away from the external stimulus. This mechanism is what is responsible for carrying information along the axon of the neuron. As spatial dimensions are included it resembles the ﬁring of action potentials in the neuron!17

Acknowledgements

Special thanks to Dr. Victor Grigoryan for his help with the concept and revision of this article. I would also like to thank Aiza Kabeer for helpful discussions.

17 6 References

1. Seifter, Julian, David Sloane, and Austin Ratner. Concepts in medical physiology. Lippin- cott Williams & Wilkins, 2005.

2. Kass, R. S., and R. We Tsien. “Control of action potential duration by calcium ions in cardiac Purkinje ﬁbers." The Journal of general physiology 67, no. 5 (1976): 599-617.

3. Hodgkin, Alan L., and Andrew F.Huxley. “Aquantitative description of membrane current and its application to conduction and excitation in nerve." The Journal of physiology 117, no. 4 (1952): 500-544.

4. Noble, Denis. “A modiﬁcation of the Hodgkin-Huxley equations applicable to Purkinje ﬁbre action and pacemaker potentials." The Journal of Physiology 160, no. 2 (1962): 317- 352.

5. Noble, Denis. “Modeling the heart–from genes to cells to the whole organ." Science 295, no. 5560 (2002): 1678-1682.

6. Reuter, Harald. “The dependence of slow inward current in Purkinje ﬁbres on the extracellular calcium-concentration." The Journal of Physiology 192, no. 2 (1967): 479-492.

7. Luo, Ching-hsing, and Yoram Rudy. “A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes."Circulation research 74, no. 6 (1994): 1071-1096.

8. Fitzhugh, Richard. “Thresholds and plateaus in the Hodgkin-Huxley nerve equations." The Journal of general physiology 43, no. 5 (1960): 867-896.

9. Keener, James, and James Sneyd. “Mathematical Physiology, Interdisciplinary Applied Mathematics 8." (1998).

10. Winfree, Arthur T. “Electrical instability in cardiac muscle: phase singularities and ro- tors." Journal of theoretical Biology 138, no. 3 (1989): 353-405.

11. Aliev, Rubin R., and Alexander V. Panﬁlov. “A simple two-variable model of cardiac excitation." Chaos, Solitons & Fractals 7, no. 3 (1996): 293-301.

12. Elharrar, V. I. C. T. O. R., and B. O. R. Y. S. Surawicz. “Cycle length effect on restitution of action potential duration in dog cardiac ﬁbers." American Journal of Physiology-Heart and Circulatory Physiology 244, no. 6 (1983): H782-H792.

13. Luo, Ching-hsing, and Yoram Rudy. “A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction." Circulation research 68, no. 6 (1991): 1501-1526.

18 14. Rush, Stanley, and Hugh Larsen. “A practical algorithm for solving dynamic membrane equations." Biomedical Engineering, IEEE Transactions on 4 (1978): 389-392.

15. Victorri, Bernard, Alain Vinet, Fernand A. Roberge, and Jean-Pierre Drouhard. “Numeri- cal integration in the reconstruction of cardiac action potentials using Hodgkin-Huxley- type models." Computers and Biomedical Research 18, no. 1 (1985): 10-23.

16. Beeler, Go W., and H. Reuter. “Reconstruction of the action potential of ventricular myocardial ﬁbres." The Journal of physiology 268, no. 1 (1977): 177-210.

17. Britton, Nicholas. Essential mathematical biology. Springer Science & Business Media, 2012.

18. Heinrich, Martin, Thomas Dahms, Valentin Flunkert, Stephen W. Teitsworth, and Ecke- hard SchÃull.˝ “Symmetry-breaking transitions in networks of nonlinear circuit elements." New Journal of Physics 12, no. 11 (2010): 113030.

19. Goktepe, S., and E. Kuhl. “Computational modeling of cardiac electrophysiology: a novel ﬁnite element approach." International journal for numerical methods in engineering 79, no. 2 (2009): 156-178.

20. Nerbonne, Jeanne M. “Molecular basis of functional voltage-gated K+ channel diversity in the mammalian myocardium." The Journal of Physiology 525, no. 2 (2000): 285-298.

21. Ch’en, FF-T., R. D. Vaughan-Jones, K. Clarke, and D. Noble. "Modelling myocardial is- chaemia and reperfusion." Progress in biophysics and molecular biology 69, no. 2 (1998): 515-538.

22. Hou, Jennifer H., Joel M. Kralj, Adam D. Douglass, Florian Engert, and Adam E. Cohen. "Simultaneous mapping of membrane voltage and calcium in zebraﬁsh heart in vivo re- veals chamber-speciﬁc developmental transitions in ionic currents." Frontiers in physiology 5 (2014).