The role of short term memory and conduction velocity restitutionin alternans formation

Ning Wei a, Yoichiro Mori a, Elena G. Tolkacheva b,n

a School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USAb Department of Biomedical Engineering, University of Minnesota, 312 Church St. SE, 6-128 Nils Hasselmo Hall, Minneapolis, MN 55455, USA

a r t i c l e i n f o

Article history:Received 29 April 2014Received in revised form12 November 2014Accepted 17 November 2014Available online 27 November 2014

Keywords:Action potential durationMapping modelSingle cellOne dimensional cableNumerical simulation

a b s t r a c t

Alternans is the periodic beat-to-beat short–long alternation in action potential duration (APD), which isconsidered to be a precursor of ventricular arrhythmias and sudden cardiac death. In extended cardiac tissue,electrical alternans can be either spatially concordant (SCA, all cells oscillate in phase) or spatially discordant(SDA, cells in different regions oscillate out of phase). SDA gives rise to an increase in the spatial dispersion ofrepolarization, which is thought to be proarrhythmic. In this paper, we investigated the effect of two aspectsof short term memory (STM) (α, τ) and their interplay with conduction velocity (CV) restitution on alternansformation using numerical simulations of a mapping model with two beats of memory. Here, α quantifiesthe dependence of APD restitution on pacing history and τ characterizes APD accommodation, which is anexponential change of APD over time once basic cycle length (BCL) changes. Our main findings are asfollows: In both single cell and spatially coupled homogeneous cable, the interplay between α and τ affectsthe dynamical behaviors of the system. For the case of large APD accommodation ðτZ290 msÞ, increase inα leads to suppression of alternans. However, if APD accommodation is small ðτr250 msÞ, increase in αleads to appearance of additional alternans region. On the other hand, the slope of CV restitution does notchange the regions of alternans in the cable. However, steep CV restitution leads to more complicateddynamical behaviors of the system. Specifically, SDA instead of SCA are observed. In addition, for steep CVrestitution and sufficiently large τ, we observed formations of type II conduction block ðCB2Þ, transition fromtype I conduction block ðCB1Þ to CB2, and unstable nodes.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Alternans (or the so-called 2:2 response), a condition in whichthere is a beat-to-beat alternation in the action potential duration(APD) of a periodically stimulated cardiac cell, has been linked to thegenesis of life-threatening ventricular arrhythmias (Pastore et al., 1999;Zipes and Wellens, 1998; Franz, 2003; Myerburg and Spooner, 2001).In the heart, alternans can be spatially concordant (SCA) or spatiallydiscordant (SDA). SCA is a phenomenon in which all cells oscillate inphase. SDA is the phenomenon of two spatially distinct regions dis-playing APD alternans of opposite phases. These regions are separatedby nodal lines in which no alternans is present. SDA has recently beenrelated to T wave alternans, a periodic beat-to-beat variation in theamplitude or shape of the T wave in an electrocardiogram (Pastoreet al., 1999; Zipes and Wellens, 1998). T wave alternans is associatedwith the vulnerability to ventricular arrhythmias and sudden cardiac

death. Therefore, it is accepted that SDA can promote the occurrence ofwavebreak and subsequently, reentry (Pastore et al., 1999; Pastore andRosenbaum, 2000; Pruvot and Rosenbaum, 2003) in the heart, andthus it is proarrhythmic. However, the mechanisms of SDA are notcompletely understood.

Several mechanisms of SDA formation have been proposed. Thefirst mechanism requires preexisting tissue heterogeneities. SDA can beformed via a timed stimulus or faster pacing rate around regions ofheterogeneities (Pastore et al., 1999, 2006; Chinushi et al., 2003).However, as indicated in several numerical studies, tissue heterogene-ity is not essential for SDA formation (Watanabe et al., 2001; Qu et al.,2000). For example, it has been revealed that SDA can form in aperfectly homogeneous tissue via a pure dynamic mechanism(Watanabe et al., 2001). The second proposed mechanism for SDA isa steep dependence of conduction velocity (CV) on the precedingdiastolic interval (DI) (Franz, 2003; Watanabe et al., 2001; Fenton et al.,2002; Fox et al., 2002b; Taggart et al., 2003). A third possiblemechanism is unstable intracellular calcium cycling, which refers tothe steep relation between calcium release versus sarcoplasmic reti-culum calcium load (Sato et al., 2013). The last possible mechanism

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Journal of Theoretical Biology

http://dx.doi.org/10.1016/j.jtbi.2014.11.0140022-5193/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ1 612 626 2719.E-mail address: talkacal@umn.edu (E.G. Tolkacheva).

Journal of Theoretical Biology 367 (2015) 21–28

(Mironov et al., 2008) is short term memory (STM). The originalassumption was that APD depends only on the preceding DI. However,STM implies the dependence of APD on the entire pacing history, notjust preceding DI.

The above four mechanisms of SDA formation in cardiac tissue canbe distinguished from the behaviors of nodal lines. Nodal lines formedby tissue heterogeneities are displayed by one of the following twoscenarios (Hayashi et al., 2007). First, once a nodal line forms, it candrift away from the pacing site on a beat-to-beat basis without reachinga steady state. Second, the nodal line may reach a steady state butremain pinned as pacing rate increases. In contrast, nodal lines formedby steep CV restitution reach steady state andmove towards the pacingsite at faster pacing rates (Hayashi et al., 2007). However, nodal linesformed by unstable calcium cycling are not oriented radially withrespect to the pacing site, and also does not move in response tochanges in pacing rate (Gizzi et al., 2013). On the other hand, it hasbeen suggested that STM is related to unstable behaviors of nodal lines,which means that the nodal lines undergo drifting when pacing rateincreases (Mironov et al., 2008). However, in experiments, it is difficultto identify the role of each individual mechanism (for example, CVrestitution and STM) in alternans formation. In addition, the interplaybetween these two mechanisms is still unclear.

Therefore, in this paper, we aimed to uncover the mechanisms ofSDA formation using numerical simulations of a mapping model withtwo beats of memory that was previously introduced in Tolkachevaet al. (2004). This is a rather simple yet powerful tool to determinethe contributions of STM to alternans formation in single cell. Inaddition, to identify the individual contributions of STM and CVrestitution to SDA formation, we will perform numerical simulationsof the mapping model in a homogeneous cable.

We adopted the mapping model in the following form (Tolkachevaet al., 2004):

Anþ1 ¼ FðDn; An; Dn�1Þ; ð1:1Þwhere An and Dn are APD and DI at the nth stimulus. Note that, in Eq.(1.1), APD depends not only on the preceding DI, but also on theprevious APD and earlier DI, which indicates the presence of twobeats of STM in the model. We considered the case of periodicpacing of constant basic cycle length (BCL), for which the relationAnþDn ¼ BCL holds.

2. Materials and methods

2.1. Mapping model with two beats of memory

For exact form of Eq. (1.1), we used a mapping model that wasintroduced in Fox et al. (2002a):

Anþ1 ¼ ð1�αMnþ1ÞGðDnÞ; ð2:1aÞ

Mnþ1 ¼ ½1�ð1�MnÞ expð�An=τÞ� expð�Dn=τÞ; ð2:1bÞwhere

GðDnÞ ¼ Qþ R1þexpð�ðDn�SÞ=TÞ

and α and τ are two aspects of STM, which represent rate-dependentrestitution and APD accommodation, respectively. Rate-dependentrestitution is measured as the difference between slopes of differentrestitution curves calculated at the same BCL (Tolkacheva and Zhao,2012). APD accommodation refers to the slow change of APD overtime after an abrupt change in BCL. Note that, large α refers to moredisparity between APD restitution curves, and large τ refers to moreAPD accommodations. We chose the following parameters in Eq. (2.1),

Q ¼ 115 ms; R¼ 121 ms; S¼ 42:5 ms; T ¼ 20:2 ms;

since these values allowed the mapping model to fit with the steadystate solutions from a physiological model of action potential.

In order to investigate the effect of STM on alternans formationusing mapping model (2.1), we varied α from 0 to 1 and chosethree different values of τ's: 11.4 s (large APD accommodation),260 ms and 200 ms (small APD accommodation).

2.2. Spatially coupled mapping model

In order to model a one dimensional cable, we adopted aspatially coupled mapping model (2.2) by adding diffusion andadvection terms to Eq. (1.1), as suggested in Fox et al. (2003):

Anþ1 ¼ FðDn; An; Dn�1Þþξ2∇2Anþ1�ω∇Anþ1; ð2:2Þwhere ξ¼1, ω¼0.35. The choices of ξ and ω were based on Foxet al. (2003) and Echebarria and Karma (2007), and detailedjustification can be seen in Appendix. Here, the diffusion termξ2∇2Anþ1 represents the coupling between cells, and the advec-tion term �ω∇Anþ1 depicts the asymmetry of influence by its leftand right neighbors, considering that cells are activated at differ-ent times by the propagating wave front (Echebarria and Karma,2002, 2007). The distribution of Dn(x), the nth DI along the cablehas been described in Watanabe et al. (2001):

dDnðxÞdx

¼ 1VðDnðxÞÞ

� 1VðDn�1ðxÞÞ

�dAnðxÞdx

; ð2:3Þ

where x is the position along the cable and V represents a CVrestitution curve. Note that, Anð0ÞþDnð0Þ ¼ BCL. We combined Eq.(2.2) with (2.3) to solve Anþ1 along a cable with length L¼4 cm,Fox et al. (2003). Due to the finite length of the cable, we imposedthe Neumann boundary condition, ð∂Anþ1=∂xÞjx ¼ 0;L ¼ 0, on thetwo ends of the cable to minimize the boundary effect. Toinvestigate the effect of CV restitution on SDA formation, we usedsteep (V1) and shallow (V2) restitution curves, as shown in Fig. 1:

V1ðDnÞ ¼ 0:4–0:3 expð�Dn=10Þ;V2ðDnÞ ¼ 0:4–0:03 expð�Dn=10Þ: ð2:4Þ

We discretized Eqs. (2.2) with (2.3) along a cable with length Lby the following scheme at a spatial step size ðΔx¼ L=NÞ of0.01 cm.

� ξ2

Δx2Anþ1ðxi�1Þþ 1þ2

ξ2

Δx2� ωΔx

!Anþ1ðxiÞ�

ξ2

Δx2� ωΔx

!Anþ1ðxiþ1Þ

¼ FðDnðxiÞ; AnðxiÞ; Dn�1ðxiÞÞ; i¼ 1⋯N;

Anþ1ðx0Þ ¼ Anþ1ðx1Þ; Anþ1ðxNÞ ¼ Anþ1ðxNþ1Þ:

DnðxiÞ ¼ BCLþ ∑i�1

j ¼ 0

ΔxVðDnðxjÞÞ

� ∑i�1

j ¼ 0

ΔxV ðDn�1ðxjÞÞ

�AnðxiÞ;

where xi is the ith site along the cable, Δx¼ xi�xi�1.

Fig. 1. CV as a function of Dn, for steep restitution curves V1 (solid) and shallow V2

(dashed).

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–2822

2.3. Pacing protocols

In order to study the mechanism of alternans formation afteran abrupt change in pacing rate, BCL was decreased from

BCL1 ¼ 500 ms to a different BCL2, ranging from 100 ms to300 ms. We applied 300 stimuli at BCL1 and sufficient numberof stimuli to reach steady state at BCL2.

In the cable, pacing was applied at the left end x¼0, and thefollowing parameters were recorded: steady state positions of nodesðXssÞ and time to reach steady state of the nodes ðtssÞ at BCL2. Here, Xss

was defined as a point for which APD node drifts less than 0.1 mm forthe 10 consecutive stimuli. tss was recorded from the beginning ofpacing at BCL2 until steady state is reached. (See Table 1 for details.)

2.4. Overpacing and conduction block

One of the general limitations of the mapping model is that in asingle cell, APD computed using Eq. (2.1) may be larger than

Table 1Parameters abbreviations.

Parameter Meaning Value

Dmin Minimum DI for which action potential can be initiated. 2 msBCL1 Initial BCL in the pacing protocol. 500 msBCL2 Second BCL in the jump pacing protocol. 100–300 msXss Steady state positions of nodes in the cable.tss Time to steady state of the nodes in the cable.

Athr1

Threshold for APD without overpacing. Athr1 ¼ BCL2�Dmin

Athr2

Threshold for APD including overpacing. Athr2 ¼ 2BCL2�Dmin

CB Conduction block in the single cell or cable.CB1 Type I conduction block in the cable (block occurs at x¼0).CB2 Type II conduction block in the cable (block occurs at xa0).V1 Steep CV restitution curve. V1 ¼ 0:4�0:3 expð�DI=10ÞV2 Shallow CV restitution curve. V2 ¼ 0:4�0:03 expð�DI=10Þx Position in the cable.L Length of the cable. 4 cm

Fig. 2. Periodic pacing of single cell leading to 1:1 responses (A). Overpacing ofsingle cell (B) and cable (C). In (A), action potentials occur as a result of periodicstimulation, i.e., AnþDn ¼ BCL. In (B), action potentials are shown for pacingprotocol in which BCL was changed from BCL1 to BCL2. Note that, ðn�1Þth andnth stimuli produce action potentials, but the ðnþ1Þth stimulus (dashed verticalline) fails to produce an action potential. However, if the ðnþ2Þth beat produces anaction potential, then Dnþ1 and Anþ2 can be obtained by Eq. (2.5). In (C), solid anddashed lines represent the wavefront and waveback of an action potential,respectively. The ðn�1Þth and nth stimuli produce action potentials that propagateto the end of the cable. But the ðnþ1Þth action potential can only be delivered forxoxn , i.e., CB occurs at x¼ xn . Note that, the ðnþ2Þth action potential can propagateto the end of the cable.

Fig. 3. CB (A) and overpacing leading to transition from 2:1 to 1:1 in single cell (B).APD (open circle) is displayed versus stimulus number at BCL2 when BCL waschanged from BCL1 ¼ 500 ms to BCL2 ¼ 170 ms. Here, α¼0.1 (A), α¼0.6 (B) andτ¼11.4 s. Dashed lines represent Athr

1 ¼ BCL2�Dmin and Athr2 ¼ 2BCL2�Dmin.

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–28 23

BCL�Dmin, where Dmin ¼ 2 ms is the minimum DI for which anaction potential can be initiated (Fox et al., 2002b, 2003; Otani,2007). Slight changes in Dmin did not affect our results significantly.If DIoDmin, then the cell does not produce an action potential,which will lead to a 2:1 response provided APDo2BCL�Dmin.However, if the cell is further stimulated, it may recover from 2:1back to 1:1 or 2:2 responses. This phenomenon is known asoverpacing, and it is shown in Fig. 2B.

Fig. 2B illustrates responses of mapping model (2.1) when BCLwas decreased from BCL1 to BCL2. The ðn�1Þth and nth stimuligive rise to action potentials, while the ðnþ1Þth stimulus fails toproduce an action potential, since An4Athr

1 ¼ BCL2�Dmin. How-ever, if AnoAthr

2 ¼ 2BCL2�Dmin, then the ðnþ2Þth stimulus will beable to produce an action potential, and

Dnþ1 ¼ 2BCL2�An;

Anþ2 ¼ FðDnþ1; An; Dn�1Þ: ð2:5ÞIf a cell cannot recover from 2:1 response as a result of overpacing,we defined it as conduction block (CB) for single cell. In general, ak : 1ðk42Þ response may be possible if APD4Athr

2 .The effect of overpacing on single cell dynamics for the mapping

model (2.1) when BCL was changed from BCL1 to BCL2 ¼ 170 ms isdemonstrated in Fig. 3. It shows an example of CB (A) and transition

from 2:1 to 1:1 responses (B) as a result of overpacing. Here, α¼0.1(A), α¼0.6 (B) and τ¼11.4 s. As observed in Fig. 3A, APDs (opencircles) are always greater than Athr

1 (dashed line) but less than Athr2

(dashed line), indicating a sustained 2:1 response.In contrast, in Fig. 3B, APDs are greater than Athr

1 and less thanAthr2 at the beginning of pacing at BCL2. But with continued

pacing, APD becomes less than Athr1 , thus indicating the transition

from 2:1 response to 1:1 behavior can only be observed in thepresence of overpacing.

In a cable, one may have DnðxÞZDmin for certain x, whereasDnðxÞoDmin for other x. Several scenarios of dealing with CB andoverpacing in the cable have been discussed in the literature. Forexample, in Otani (2007), the author developed a method thatuses the APD and CV restitution relations to generate predictionsregarding which sequences of premature stimuli are most likely toinduce type II CB and reentry. Specifically, block occurs when thetrailing edge of the preceding wave travels slower than the leadingedge of the current wave, allowing the latter to encroach andterminate on the former. In Fox et al. (2003), simulation wasterminated once DnðxÞoDmin occurred for any x in the cable. In Foxet al. (2002b), overpacing was considered for a cable consisting ofa simple mapping model without memory, Anþ1 ¼ f ðDnÞ.

In this paper, we implemented overpacing in the cable for themapping model with two beats of STM by using the approachshown in Fig. 2C, which illustrates responses of spatially coupledmapping model (2.2) when the cable was paced at x¼0 and BCLwas decreased from BCL1 to BCL2. As observed, solid and dashedlines represent the wavefront and waveback of an action potentialalong the cable, respectively. Let xn be the maximal value such that

D̂nðxÞZDmin for xoxn; ð2:6Þwhere D̂nðxÞ satisfies the following:

dD̂nðxÞdx

¼ 1

VD̂nðxÞ� 1VDn�1x

�dAnxdx

;

An0þD̂nð0Þ ¼ BCL:

Then we let Dnx¼ D̂nðxÞ for xoxn, where Dn(x) is the nth DI. Thatis, the ðnþ1Þth action potential can only propagate to xn. There-fore, Anþ1ðxÞ also exists for xoxn (we let Anþ1 ¼ 0 for xZxn toemphasize).

Let us consider the situation when the ðnþ2Þth action potentialcan propagate to the end. For xoxn, Dnþ1ðxÞ and Anþ2ðxÞ can beobtained by Eqs. (2.2) and (2.3). For xZxn, Dnþ1ðxÞ and Anþ2ðxÞ canbe obtained using the following equation:

dDnþ1ðxÞdx

¼ 1VðDnþ1ðxÞÞ

� 1VðDn�1ðxÞÞ

�dAnðxÞdx

;

Anþ2 ¼ FðDnþ1; An; Dn�1Þþξ2∇2Anþ2�ω∇Anþ2; ð2:7Þwhere Dnþ1ðxÞ is the DI that follows either Anþ1ðxÞ (for xoxn) orAn(x) (for xZxn). If the action potential cannot propagate to theend of the cable, CB occurs. Moreover, if CB occurs at x¼0, wedefined it as type I CB ðCB1Þ. If CB occurs at xa0, we defined it astype II CB ðCB2Þ.

3. Results

3.1. Dynamical responses of single cell

We first aimed to classify different dynamical behaviors of themapping model (2.1) at steady state (i.e., 1:1, 2:2, or CB) when BCLwas changed from BCL1 to various BCL2. Fig. 4 illustrates steadystate responses at different BCL2 as a function of α for τ¼11.4 s (A),τ¼260 ms (B), and τ¼200 ms (C). Black dashed lines outline theregions for 1:1 responses (filled circles or upper right corner

Fig. 4. Dynamic responses of single cell at steady state as a function of α when BCLwas decreased from BCL1 to different BCL2 for τ¼11.4 s (A), τ¼260 ms (B), andτ¼200 ms (C). Behaviors are classified as 1:1 (filled circles or upper right cornerregions labeled as 1:1), 2:2 (or alternans) (crosses) and CB.

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–2824

regions labeled as 1:1) and CB. For τ¼11.4 s and 260 ms (Fig. 4Aand B), we can see that when BCL2 is large, only 1:1 responses areobserved. As BCL2 decreases, CB or 2:2 (or alternans) (crosses)behaviors occur, which transition to 1:1 responses as α increases.Once BCL2 becomes sufficiently small, only CB is observed.

Fig. 4C shows different dynamical responses of the mappingmodel (2.1) for τ¼200 ms. Here, we can see a larger regionoccupied by alternans (crosses), especially for large α. Theseresults suggest that the effect of α is diminished as τ decreases.

The dynamical behaviors of a single cell are different for highand low values of τ. The cases τ¼11.4 s and τ¼200 ms are tworepresentatives of the respective regimes. The high τ regimeextends to about 290 ms and the low τ regime extends to around250 ms. There is a narrow transition regime between τ¼250 msand τ¼290 ms, for which we showed the case τ¼260 ms.

The individual roles of α and τ in alternans formation can beunderstood by considering the limit of Mnþ1 as τ approaches zero.Eq. (2.1) yields limτ-0Mnþ1 ¼ limτ-0½1�ð1�MnÞexpð�An=τÞ�expð�Dn=τÞ ¼ 0, since An, Dn40. Then Eq. (2.1) reduces toAnþ1 ¼ GðDnÞ, i.e., no memory. Thus, effect of α is diminished as τdecreases.

3.2. SDA in spatially coupled mapping model

3.2.1. Effect of overpacing on CBWe found that the effect of overpacing has more important

consequences on dynamics in the cable, in comparison to the singlecell. To illustrate it, we performed numerical simulations of spatially

coupled mapping model (2.2) when BCL was decreased from BCL1 toBCL2. Fig. 5A–E shows results for different stimulus number atBCL2 ¼ 220 ms, α¼0.175, τ¼11.4 s and CV¼ V1. As observed, Fig. 5Ademonstrates that CB1 occurred at the first beat as BCL wasdecreased to BCL2, after which SCA (Fig. 5B) and SDA (Fig. 5C andD) occurred sequentially. SDA persists for about 40 beats, and thenCB2 (Fig. 5E) is observed at the 59th beat, at which action potential isblocked in the middle of the cable.

In Fig. 5F, we set BCL2 ¼ 190 ms, α¼0.425, τ¼11.4 s andCV¼ V1. As stimuli number increases, we observed a transitionfrom CB1 (dashed line) to SDA (open and filled circles) and then to1:1 (gray line) behavior under overpacing. Note that, neither thetransition from CB1 to CB2 nor the transition from CB1 to SDA andthen to 1:1 response can be observed without overpacing. Instead,simulations would have terminated at the first beat.

3.2.2. SDA node behaviors while approaching steady stateWe observed two different behaviors of SDA nodes while approach-

ing steady state: stable and unstable node formation. Fig. 6A showsstable behavior of nodes for the spatially coupled mapping model (2.2)when BCL was decreased from BCL1 to BCL2 ¼ 240 ms, α¼0.175,τ¼11.4 s and CV¼ V1. Note that the node formed by the 188th and189th beats (black dashed) appears at the distal end of the cable andthen moves towards (gray and black nodes) the pacing site x¼0 as itapproaches steady state Xss.

Fig. 6B shows unstable behavior of the node for the spatiallycoupled mapping model (2.2) when BCL was decreased from BCL1 to

Fig. 5. Effect of overpacing in cable. Transition from CB1 (A) to SCA (B) and SDA (C, D) and then to CB2 (E) as BCL was decreased from BCL1 to BCL2 ¼ 220 ms. Here, α¼0.175,τ¼11.4 s and CV¼ V1. Transition (F) from CB1 to SDA and then to 1:1 response as BCL was decreased from BCL1 to BCL2 ¼ 190 ms. Here α¼0.425, τ¼11.4 s and CV¼ V1.

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–28 25

BCL2 ¼ 210 ms, α¼0.3, τ¼11.4 s and CV¼ V1. The node formed bythe 6th and 7th beats (black dashed) appears at the end of the cableand migrates towards (gray solid) the pacing site x¼0 first, and thenaway from (black solid) the pacing site x¼0 as it approaches steadystate Xss.

For the stable nodes, we calculated the steady state positionsðXssÞ and time to reach steady state ðtssÞ. Fig. 7A and B showsrepresentative examples displaying Xss and tss as a function of αrespectively. Both figures were obtained for the spatially coupledmapping model (2.2) when BCL was changed from BCL1 toBCL2 ¼ 130 ms, where τ¼200 ms and CV¼ V1. As depicted, Xss

increases as α increases and tss first decreases and then increases.Our results show that increase of α could result in steady statenodes being further away from pacing site. Nevertheless, thereappears to be a minimum tss as α is varied in Fig. 7B.

3.2.3. Interplay between CV restitution and STMFinally, we aimed to characterize different dynamical behaviors

of the spatially coupled mapping model (2.2) at steady state whenBCL was changed from BCL1 to various BCL2. Fig. 8 illustratessteady state responses at different BCL2 as a function of α for V2

(shallow CV restitution), τ¼11.4 s (A), τ¼260 ms (B), τ¼200 ms(C) and for V1 (steep CV restitution), τ¼11.4 s (D), τ¼260 ms (E),τ¼200 ms (F). As observed in Fig. 8A–C, at CV¼ V2, steady statebehaviors in the cable looked very similar to those of single cell ifSDA (cross) and SCA (square) are considered as alternans, sincesingle cells cannot exhibit spatial phenomena.

Fig. 8A and D illustrates that for τ¼11.4 s, steep CV restitution(V1, Fig. 8D) leads to appearance of more complicated dynamicalresponses, such as CB2 (diamond) (Fig. 5E), transition from CB1 toCB2 (star, CB1-CB2) (Fig. 5A–E) and unstable nodes (plus, unstable)(Fig. 6B).

However, Fig. 8B, C, E, and F illustrates that for τ¼260 ms and200 ms, the effect of steep CV restitution curves is negligible,except more formation of SDA, including the region of large α.

Furthermore, Fig. 8D and F illustrate that for V1, smaller τ(200 ms) leads to appearance of simpler dynamical responses,such as the inhibition of CB2 (diamond), transition from CB1 to CB2

(star, CB1-CB2), and unstable nodes (plus, unstable).Similar to the single cell case, the dynamical behaviors of the

cable are different for high and low values of τ. The cases τ¼11.4 sand τ¼200 ms are two representatives of the respective regimes.We also showed results for τ¼260 ms, at which the behavior isintermediate between the two regimes.

Our results indicate that shallow CV restitution together withlarge τ can help suppress SDA. Steep CV restitution coupled withlarge τ gives rise to CB2, the transition from CB1 to CB2 andunstable nodes, which are inhibited by either shallow CV or smallτ. Steep CV restitution together with small τ can lead to moreformation of SDA, including the region of large α. Our result alsoindicates that STM plays a dominant role in SDA formation.

4. Conclusion and discussion

In this paper, we investigated the effect of two aspects of STM (α, τ)and their interplay with CV restitution on alternans formation usingnumerical simulations of a mappingmodel with two beats of memory.In both single cell and spatially coupled homogeneous cable, theinterplay between α and τ affects the dynamical behaviors of thesystem. For the case of large APD accommodation ðτZ290 msÞ, anincrease in α leads to suppression of alternans. However, if APDaccommodation is small ðτr250 msÞ, an increase in α leads toappearance of additional alternans region. On the other hand, theslope of CV restitution does not change the regions of alternans in thecable. However, steep CV restitution leads to more complicateddynamical behaviors of the system. Specifically, SDA instead of SCAare observed. In addition, for steep CV restitution and sufficiently largeτ, we observed formations of CB2, transition from CB1 to CB2, andunstable nodes.

Fig. 6. Stable (A) and unstable (B) SDA nodal behaviors. APD of selective beats aredisplayed versus cable length when BCL was decreased from BCL1 to BCL2 ¼ 240 ms(A) and BCL2 ¼ 210 ms (B), α¼0.175 (A), α¼0.3 (B), τ¼11.4 s and CV¼ V1 (steep CVrestitution).

Fig. 7. Xss and tss as a function of α for stable nodes when BCL was decreased fromBCL1 to BCL2 ¼ 130 ms, τ¼200 ms and CV¼ V1 (steep CV restitution).

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–2826

To our knowledge, this is the first use of overpacing in a mappingmodel with two beats of memory. At the single cell level, the cell mayrecover from 2:1 rhythm back to 1:1 or 2:2 rhythm if the cell isstimulated beyond the beat at which DIoDmin. The same can be saidof the cable. This indicates the importance of pacing beyond conduc-tion block. In Fox et al. (2003), stimulation was terminated onceDIðxÞoDmin for any x in the cable. In Fox et al. (2002b), overpacingwas adopted using a simple mapping model without memory; APDonly depended on the preceding DI, making it relatively simple toimplement overpacing. Here, we overpaced the cable in a mappingmodel with two beats of memory. The dependence of APD on thepreceding APD and two preceding DIs gives rise to new difficulties inthe continuation of pacing after DIoDmin for any x in the cable. Theresolution of this difficulty is a technical contribution of our paper.Our use of continued pacing has allowed us to observe recoveriesfrom CB1 to 1:1 responses or SDA and transitions from CB1 to CB2

that we would not have been able to see otherwise.This is the first study in which a mapping model with two beats

of memory has been used to investigate SDA formation. Over the lastdecades, the study of SDA has become a major focus of researchbecause of its potentially crucial link to cardiac instability. Ournumerical results are in agreement with previous experimental andtheoretical studies of SDA.We confirmed that SDA can be observed ina homogeneous cable (Watanabe et al., 2001; Qu et al., 2000), andthat steep CV restitution facilitates SDA (Franz, 2003; Watanabe et al.,2001; Fenton et al., 2002; Fox et al., 2002b; Taggart et al., 2003). Wealso found that small τ, together with steep CV restitution, furtherfacilitates SDA formation. This agrees with the experimental resultsin Mironov et al. (2008) which states that stable SDA nodal lineswere associated with steep CV restitution and small τ. Finally, we

demonstrated that unstable behaviors of nodes are related to steepCV restitution and large τ. This is consistent with experimentalresults in Mironov et al. (2008), wherein the unstable driftingbehaviors of nodal lines during adaptation to change in BCL are seenfor large values of τ.

One limitation of our study is that we chose Dmin ¼ 2 ms. However,our simulation results demonstrate that sufficiently large increases inDmin might affect the dynamic behaviors of the cable model. Specifi-cally, for Dmin ¼ 20 ms, τ¼11.4 s and CV¼ V1 (steep CV restitution),we observed more incidences of CB1 as well as transitions from SCA orSDA to CB2, the latter of which were not present for Dmin ¼ 2 ms.

Another limitation of our study is that we chose the cablelength L to be 4 cm with 401 cells. It is clear that we will see moreincidences of SDA and nodes in a larger domain. However, ifsimulations are performed on a larger domain with lengthL44 cm, but the dynamical behaviors are only tracked on thefirst 4 cm segment of the simulation domain, we have checkedthat the simulation results are similar to the case when L¼ 4 cm.This indicates that the boundary conditions at x¼L do notappreciably affect the results of the simulation.

Acknowledgment

This work was supported by National Science Foundation GrantsCMMI-1233951 and NSF CAREER PHY-125541 to E.G.T. Y.M. wassupported by NSF Grant DMS 0914963, the Alfred P. Sloan Founda-tion and the McKnight Foundation. In addition, we would like tothank all the referees for careful reading of the paper and valuablesuggestions.

Fig. 8. Dynamical responses of the cable at steady state as a function of α when BCL was changed from BCL1 to different BCL2 for CV¼ V2, τ¼11.4 s (A), τ¼260 ms (B) andτ¼200 ms (C); CV¼ V1, τ¼11.4 s (D), τ¼260 ms (E), τ¼200 ms (F). Behaviors were classified as 1:1 response (filled circle or upper right corner regions labeled as 1:1),SDA (cross), SCA (square), CB1, CB2 (diamond), transition from CB1 to CB2 (star, CB1-CB2) and unstable nodes (plus, unstable).

N. Wei et al. / Journal of Theoretical Biology 367 (2015) 21–28 27

Appendix A. Derivation of ξ and ω

Numerical solutions to Eq. (2.1) together with Eq. (2.3) developunphysical spatial discontinuities along the cable, as observed inEchebarria and Karma (2002). This can be avoided by consideringthe cell coupling effect, which contributes diffusion and advectionterms, as has been suggested in Echebarria and Karma (2002, 2007).Therefore, in order to model a one dimensional cable, we adopted aspatially coupled mapping model (2.2) by adding diffusion andadvection terms to Eq. (1.1), as suggested in Fox et al. (2003).

Here, we discuss the values for ξ and ω that were used inEq. (2.2) following Fox et al. (2003). An interpretation of ξ and ωcan be found in Echebarria and Karma (2007). The starting point isthe following equation:

Anþ1ðxÞ ¼ ðGnf ÞðxÞ ¼Z þ1

�1GðyÞf ½Dnðx�yÞ� dy; ðA:1Þ

where G(y) is some normalized asymmetrical kernel that expressesthe diffusive coupling between neighboring cells. Let us define theFourier transform ðF ð�ÞÞ of the kernel to be ðF ðGÞÞðkÞ ¼ R þ1

�1 GðyÞe� iky dy. The exponential function in the above equation can beexpanded in the form e� iky ¼ 1� iky�ðkyÞ2=2þ⋯. It then foll-ows that

ðF ðGÞÞðkÞ ¼ 1� ibωk�bξ2k2þ⋯; ðA:2Þ

where the coefficients are defined as

bω ¼Z þ1

�1GðyÞy dy; bξ2

¼ 12

Z þ1

�1GðyÞy2 dy:

Now, consider the equation

ðI�ξ2∇2þω∇ÞAnþ1 ¼ f ðDnÞ: ðA:3ÞThis is the same as Eq. (2.2) except that the mapping model f onlydepends on Dn. Let us relate ξ and ω above with bξ and bω of Eq.(A.2). By taking Fourier transform of both sides, we have

F ðAnþ1Þ ¼1

1þ iωkþξ2k2F ðf Þ: ðA:4Þ

Therefore,

Anþ1ðxÞ ¼ ðGnf ÞðxÞ; where ðF ðGÞÞðkÞ ¼ 1

1þ iωkþξ2k2: ðA:5Þ

By Taylor expansion, we see that

ðF ðGÞÞðkÞ ¼ 1� iωk�ðξ2þω2Þk2þ⋯:

Comparing the above with Eq. (A.2), we have

bω ¼ω; bξ2¼ ξ2þω2:

If ω2{ξ2, then bξ2¼ ξ2. Therefore,

G¼F �1 1

1þ iωkþξ2k2

!:

We point out that Eq. (A.1) (or a discretized version thereof) has beenused in several other publications to describe electronic couplingsbetween cells. For example, the papers by Fox et al. (2002b) and Vinet(2000) used the Gaussian kernel GðyÞ ¼ expð�μy2Þ.

Note that Eq. (A.1) is an approximation. It is more realistic touse a biophysical ionic model, and it is thus of interest to relateEq. (A.1) to ionic models. For a simple two variable ionic model,Echebarria and Karma (2007) demonstrated that bω and bξ can becalculated explicitly in certain limiting cases, giving

bω ¼ 2Dc; bξ ¼ ðD� APDcÞ1=2;

where D denotes the diffusion coefficient in the two variable ionicmodel, c denotes CV of a propagating wave and APDc denotes APDat the bifurcation point. Using typical values of D, c and APDc , thelength bξ is on the order of 1 mm, while bω is on the order 0.1 mm.

The authors of Fox et al. (2003) adjusted the value of ωð ¼ bωÞ to0.35 mm in order to produce spatial patterns that were similar towhat they observed experimentally. We also investigated theeffect of changing ω on dynamic behaviors of spatially coupledmapping model, and determined that it has a minor effect. Forinstance, ω¼0.1 mm leads to simpler dynamic behaviors. There-fore, we used ξ¼1 mm andω¼0.35 mm in our paper, without lossof generality.

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