Ionic and Electrotonic Contributions to Short-Term Ventricular Action Potential Memory: An In Silico Study

Abstract

Electrical restitution (ER) is a determinant of cardiac repolarization stability and can be measured as steady action potential (AP) duration (APD) at different pacing rates—the so-called dynamic restitution (ER_dyn) curve—or as APD changes after pre- or post-mature stimulations—the so-called standard restitution (ER_s1s2) curve. Short-term AP memory (M_s) has been described as the slope difference between the ER_dyn and ER_s1s2 curves, and represents the information stored in repolarization dynamics due to previous pacing conditions. Although previous studies have shown its dependence on ion currents and calcium cycling, a systematic picture of these features is lacking. By means of simulations with a human ventricular AP model, I show that APD restitution can be described under randomly changing pacing conditions (ER_rand) and M_s derived as the slope difference between ER_dyn and ER_rand. Thus measured, M_s values correlate with those measured using ER_s1s2. I investigate the effect on M_s of modulating the conductance of ion channels involved in AP repolarization, and of abolishing intracellular calcium transient. I show that M_s is chiefly determined by ER_dyn rather than ER_rand, and that interventions that shorten/prolong APD tend to decrease/increase M_s.

1. Introduction

Electrical restitution, i.e., how action potential (AP) duration (APD) changes according to pacing cycle length (CL), is a fundamental property of cardiac muscle, as it describes AP stability under perturbation of the pacing rate [1]. The so-called restitution hypothesis predicts, in fact, how repolarization changes due to pre- or post-mature stimuli are quenched in a few beats or amplified toward repolarization alternans depending on features of the electrical restitution (ER) curves [2,3]. However, the use of ER properties to predict the onset of arrhythmias has often led to inconsistent results [4], mostly due to the dependence of ER on the protocol used to measure it [5,6,7,8]. In particular, the steady-state APD as a function of the constant-pacing CL is called the dynamic restitution (ER_dyn) curve, whereas the changes in APD in response to pre/post-mature stimuli (S2) delivered at a constant basic CL (BCL or S1) are called standard S1S2 restitution (ER_s1s2) and represented by a curve reporting APD vs. pre/post-mature CLs. The former is a global feature of repolarization at all CLs; the latter indicates a local property at one given BCL. The restitution hypothesis only refers to ER_s1s2, though there are cases in which the slope of ER_dyn—and not that of ER_s1s2—is determinant in preventing arrhythmias [9,10,11]. Also, global ER_dyn and local ER_s1s2 curves are differently affected by a partial block of ion channels, by ion concentrations, and by pharmacological treatments like antiarrhythmic drugs [1]. A unifying picture of restitution properties comes with the concept of short-term AP memory. As opposed to long-term AP memory, which refers to pacing-induced remodeling in protein modification and/or gene expression after hours or days of pacing [12], short-term AP memory (M_s) reflects the influence of pacing history on APD [13,14]. In the absence of short-term AP memory, the local response to a change in CL is equal to the global steady-state response, or, in other words, ER_dyn = ER_s1s2 at any BCL; thus, APD depends only on the preceding CL. On the contrary, in the presence of short-term AP memory, ERdyn ≠ ER_s1s2, i.e., APD depends not only on the preceding CL but also on the pacing history [3,6,15]. After previous attempts to quantify M_s by introducing parameters representing the influence of pacing history [5,16,17,18,19,20,21], the measure of M_s as the difference between the slopes of the ER_s1s2 and ER_dyn curves at any given BCL value has been introduced by Tolkacheva [14], who validated the procedure on isolated rabbit ventricular myocytes. Thus conceived, M_s is mainly determined by ion currents with slow recovery kinetics and by intracellular calcium cycling [22], though a systematic characterization of this dependence is still lacking. Also, since M_s is BCL-dependent, the concept of M_s restitution has been introduced [23]. In fact, the M_s value is small at longer BCLs, because at low pacing rates, the ion currents recover almost completely from inactivation; it increases up to a maximum at intermediate BCLs, where ion currents only partially recover from inactivation, and goes to zero at even shorter BCLs, at which ER_dyn and ER_s1s2 tend to become equal [23].

In the present study, computer simulations with an established human ventricular AP model allow the proposition of a novel method to measure M_s based on beat-to-beat restitution properties measured under randomly varying pacing CL. Numerical simulations also allow insight into the ion channels and intracellular calcium contribution to M_s, and study of the relative contribution of global ER_dyn and local ER_rand to it.

2. Materials and Methods

The simulations presented in this study have been performed with the O’Hara et al. endocardial AP model ([24], ORd model, for short), which is one of the most adopted to reproduce human ventricular electrical activity. The stiff ‘ode15s’ solver built into the R2024a version of Matlab (The Math-Works, Inc., Natick, MA, USA) was used to integrate the model’s equations. All simulations were run on a PC with an Intel(R) Core (TM) i7, 2.8GHz CPU. APs were elicited by simulating 0.5 ms long current injections with an amplitude 50% above the current threshold. APD was measured as the time between the maximum first derivative of membrane potential (V_m) during the initial fast depolarization phase and the time during repolarization, when V_m reached the value of −60 mV. Initial conditions at different BCLs or for any modulation of maximum conductance of ion channels were measured after conditioning pacing of 1000 beats.

Measure of standard electrical restitution: A conditioning pacing train of 150 beats was simulated at a given BCL. After the last conditioning beat, a stimulus was delivered at a CL (coupling interval CI) 100 ms shorter than BCL and the following APD measured. The conditioning train was then repeated and CI progressively prolonged, in steps of 5 ms, up to BCL + 100 ms. The collection of recorded APDs versus the preceding coupling interval CL is called the electrical restitution curve and denoted as ER_s1s2 in the rest of the paper.

Measure of random electrical restitution: A conditioning pacing train of 150 beats was simulated at a given BCL. After the last conditioning beat, CL was made, varying for 50 beats according to a uniform-random sequence of CLs, varying within a BCL ± 100 ms interval. The collection of APDs versus the preceding randomly varying CL is called the random restitution curve and denoted as ER_rand in the rest of the paper. Notably, random sequences were newly generated each time by the random generator function built into Matlab; thus, there are no two identical random sequences in any of the protocols performed in the present study. Also, BCL intervals of random variability greater and smaller than ±100 ms were systematically tried, finding the one in use the better compromise.

Measure of dynamic electrical restitution: The pacing rate was initially performed at BCL = 2000 ms for 1000 beats, and the last 25 APDs of the sequence were measured and averaged. The procedure was repeated with BCL = 1950 ms, and then, repeatedly, for BCL—50 ms, down to BCL = 350 ms. The averaged APDs thus measured are reported versus the corresponding BCLs, and describe the so-called dynamic restitution curve, indicated in this paper as ER_dyn.

3. Results

Measuring memory with standard and random restitution.Figure 1 shows a single run of the stimulation protocol described in Materials and Methods for measuring random restitution. The initial 150 beats at constant pacing rate (BCL = 450 ms) is followed by 50 beats with CL changing randomly from beat to beat (Figure 1B) within BCL ± 100 ms. The corresponding sequence of APDs is shown in panel C. APD variability is also evident in the superimposed AP waveforms of panel A measured during random pacing. The protocol was repeated for different BCL values from 350 up to 2000 ms, in steps of 50 ms. The APDs of the last 25 beats of each conditioning BCL sequence preceding random pacing were taken and averaged to construct the dynamic restitution curve ER_dyn reported in Figure 2A.

The scatter plot in Figure 2B represents ER_rand, where each APD of the random sequence of Figure 1C is reported as a function of the preceding CL; data points were fitted with a parabolic curve (red in the figure; R² = 0.94). Superimposed on the scatter plot, the corresponding segment of ER_dyn is reported as a green dotted line, also fitted with a quadratic curve. The values of the slopes of ER_dyn (S_dyn) and of ER_rand (S_rand) were measured at BCL = 450 ms and are reported in the figure together with their difference (ΔS). The black dots in Figure 2C represent the standard ER_s1s2 curve (see Section 2) measured at the same BCL as in panel B, with its parabolic fitting (R² = 0.98). Superimposed, the corresponding segment of ER_dyn is in green. Slope values of ER_dyn and ER_s1s2 are also reported, together with their difference.

The protocol of Figure 1 was then repeatedly applied, taking each time as initial conditions those at the end of the preceding run, within a BCL range from 350 up to 2000 ms. A summary of all the ER_rand (empty dots), their parabolic fittings (red lines), and the corresponding ER_dyn (green), is reported in Figure 3A. Similarly, Figure 3B reports ER_s1s2 curves, their fittings, and the corresponding ER_dyn. Panel C shows the slope of ER_dyn (S_dyn in green), and that of each fitting of ER_rand (S_rand, dotted line) and of ER_s1s2 (S_s1s2, continuous line), as a function of the corresponding BCL. Finally, panel D shows ΔS_rand,dyn (S_dyn−S_rand) and ΔS_s1s2,dyn (S_dyn−S_s1s2), as a function of the corresponding BCL; the two parameters tightly correlate (linear correlation shown in the inset in red, R² = 0.99). ΔS_s1s2,dyn has been used in the literature [14] to designate short-term AP memory and called M_s. In this study, I will use the same term to indicate ΔS_rand,dyn and ΔS_s1s2,dyn.

To compare the measure of M_s with ER_rand and with ER_s1s2 (ΔS_rand,dyn and ΔS_s1s2,dyn), the effect of reduction of two ion currents mainly involved in AP repolarization was simulated in the Ord model, and M_s measured with the two protocols (Figure 4). The peak value of M_s, as measured with ER_rand (panel A), increased under 50% reduction of the maximum conductance of repolarizing potassium current I_Kr (G_Kr, red in figure) and decreased under 50% reduction of the maximum conductance of L-type calcium current I_CaL (G_caL, blue in figure), remaining centered at BCL = 600 ms. The same behavior was shown with ER_s1s2 measurement (panel B). Panel C shows the percentage changes of the peak value of M_s under the two ion current modulations, as revealed through ER_rand (first and third columns) and ER_s1s2 measurement (second and fourth columns). Despite the larger absolute value of M_s found with ER_rand (Figure 3D), the percentage changes under G_CaL and G_Kr reduction were very similar with the two methods (Figure 4C).

Short-term AP memory will be measured in the rest of this paper as M_s = S_dyn−S_rand. Most of the simulations presented in this study have been performed with both procedures (ΔS_rand,dyn and ΔS_s1s2,dyn), with qualitatively similar results.

Pharmacological modulation of memory. The effect on M_s of progressively increasing (up to 75%, step 25%) or decreasing (down to −75%, step 25%) the maximum conductance of all the ion currents included in the ORd AP model was studied. The top panels of Figure 5A show the effect of ±50% changes of G_CaL, which affected the peak value of M_s in the case of decrease and modified its shape, but left the peak value unchanged in the case of increase. Notably, M_s changes were mostly due to changes in S_dyn rather than in S_rand (lower panels of Figure 5A). This appears more clearly in Figure 5B, where a detail of ER_dyn (black) and ER_rand (red) curves at BCL = 1000 ms (red star in Figure 5A) is reported in control conditions and under G_CaL increase, and the angle between the two increases mainly due to the increase in the slope of the ER_dyn curve (S_dyn). Figure 5C shows the effect of ±50% increasing/decreasing G_Kr on M_s, which led to decrease/increase the peak value of the M_s curve, leaving its shape largely unaltered. Notably, M_s reduction under G_Kr increase was mainly due to a decrease in S_dyn, whereas M_s increase under G_Kr decrease (see detail as blue star at BCL = 600 ms) was due to changes in both parameters (lower C panels and panel D).

The M_s curve, as defined above, represents the slope difference between ER_dyn and ER_rand at any given pacing BCL; thus, each point of the curve corresponds to a different APD, due to both intrinsic rate dependence and to the specific treatment (partial increase/decrease in G_max). The APD value corresponding to each point of the M_s curve can be evaluated by averaging the last 25 APDs of the constant-pacing BCL train preceding the random sequence (double arrow in Figure 1C). Thus, each M_s vs. BCL curve can be translated into an M_s vs. APD curve, which shows, all at once, the effect of each treatment on M_s and on APD. This transformation was done for the increase/decrease in G_CaL and G_Kr, and is reported in Figure 6, where the effects of ±50 % of G_max are shown. Figure 6B,D show that changes that shorten APD, either due to decreased depolarizing or increased repolarizing current, also decrease peak value and area underlying the M_s curve, whereas those prolonging APD have opposite effect. The area under the M_s vs. BCL curve will be used in this study as an estimate of total short-term memory in any given condition.

The effect of modulating maximum conductance of the late sodium current I_NaL was also explored, given the significant contribution of this current to AP ventricular repolarization [25]. The results are reported in Figure 7, which shows that 75% increase/decrease in G_NaL led to increase/decrease the peak (same BCL) value of M_s (Figure 7A), paralleled by an increase/decrease in APD (Figure 7B). Also, changes in M_s, particularly the peak value, were mainly due to changes in S_dyn rather that in S_rand (Figure 7C,D).

Given its electrogenic contribution played during both systole and diastole, the role of modulating the current I_NaK associated with the activity of a sodium–potassium pump was explored as well (Figure 8). Whereas a 50% decrease in G_NaK led to a significant decrease in the peak value of M_s (Figure 8A, left), mostly due to a significant decrease in S_dyn (Figure 8B, left), an increase of the same percentage amount did not lead to significant changes in M_s (right panels of the same figure).

To summarize the above results and take overall M_s changes into consideration, the area underlying the M_s curve was measured for each treatment. Figure 9A shows the almost symmetrical effect that changes in G_CaL and G_Kr had on this parameter. The insets show the effect that ±75% changes in the maximum conductance of these two currents had on the AP waveform measured at pacing BCL = 1000 ms. Changes in the area of M_s are also reported under G_NaL, G_Kto, G_K1, G_Ks, and G_NaK modulation in Figure 9B. They had smaller and, except for G_NaK, linear effects on M_s area and on APD.

The complete suppression of sarcoplasmic reticulum (SR) calcium cycling was then simulated by blocking SR ryanodine receptor calcium release (J_rel = 0 in the model) and SR calcium pump uptake (J_up = 0 in the model) (Figure 10), which led to prolongation of APD and flattening of intracellular calcium transient (Figure 10D), both measured at BCL = 1000 ms. Abolishing calcium cycling also led to a large increase in peak value and area of M_s (Figure 10A,B), mostly due to a dramatic change in ER_dyn and, consequently, of its slope (Figure 10C) and with a minimal effect on S_rand.

Electrotonic modulation of memory. A pattern seems to emerge from the above simulations, suggesting that interventions that decrease/increase APD also decrease/increase M_s (peak and area). To induce an APD shortening and study its effect on M_s without affecting specific ion currents or transporters, the electrical coupling between two ORd model cells via a coupling resistance R_j was simulated in a R_j range that prevented AP propagation. This consists in solving simultaneously the equations for the ORd model of cell 1 (membrane potential V₁ with its set of n ion currents I_ion1 and the stimulus current I_stim,1) and cell 2 (membrane potential V₂ with its set of n ion currents I_ion2 and the stimulus current I_stim,2):

$$\left\{\begin{array}{c}{\displaystyle \frac{V_1-V_2}{R_j}}=C_m{\displaystyle \frac{d{V}_1}{dt}}+{\displaystyle \sum _{i=1}^n}{I}_{ion1,i}+I_{stim,1}\\ {\displaystyle \frac{V_2-V_1}{R_j}}=C_m{\displaystyle \frac{d{V}_2}{dt}}+{\displaystyle \sum _{i=1}^n}{I}_{ion2,i}+I_{stim,2}\end{array}\right.$$

where the two equations are coupled by means of the electrotonic current, the term on the left-hand side of both equations (see for example [26]). In other words, only one of the two cells was electrically paced (I_stim,1 ≠ 0 and I_stim,2 = 0), and the second was electrically coupled via R_j to the source cell, only depolarized below-threshold, thus inducing an electrotonic polarization (shortening) of APD in the source cell. As R_j was made to decrease from infinite (control uncoupled condition) down to 1.5 GΩ, both the peak and area of M_s decreased (Figure 11A, numbers on figure are R_j values in GΩ). Further decrease in R_j led to near-threshold depolarizations in the sink cell, causing non-linear interactions in the cell pair, and were not considered. The overall changes of M_s area are shown as a function of G_j (1/R_j) in Figure 11E. When M_s was plotted against APD, the electrotonic shortening of APD was paralleled by a decrease in M_s peak and area (Figure 11B). Notably, the electrotonically induced decrease in M_s (red star in Figure 11A) is mainly due to a decrease in S_dyn rather than a change in S_rand, which indeed remained almost constant (Figure 11C). In fact, the electrotonic load of the resting cell not only shortened the APD of the source cell, but dramatically modified the shape of ER_dyn and, in turn, its slope (Figure 11D), leaving S_rand nearly unchanged.

A more straightforward way to modify APD and see how it affects M_s without acting on specific ion channels or transporters is simulating a constant current injection across the membrane. The result of injecting progressively increasing positive, i.e., polarizing, current is shown in Figure 12A (numbers refer to current amplitude in μA/μF), where the APD shortening and the decrease in peak and area of M_s can be seen. In contrast, when negative depolarizing constant current was injected, APD prolonged, the peak and area of M_s increased, and an M_s component at higher BCLs developed. Constant current injections higher than −0.1 μA/μF led to early after-depolarizations and alternans and were not considered. The overall M_s modification as a function of injected current is reported in Figure 12F. The role of S_dyn, rather than S_rand, in modifying M_s shape (Figure 12C,D) is clearer when looking at the dramatic modification of ER_dyn under current injection (Figure 12E). Once again, constant current-induced modifications of the AP also revealed the tendency of APD shortening/prolonging maneuvers to decrease/increase M_s (Figure 12B), respectively.

4. Discussion

Ventricular fibrillation (VF) underlies most sudden cardiac deaths and is due to electrical instability that can be described by electrical restitution properties of the heart [27,28]. However, electrical restitution is often insufficient to predict the electrical alternans that precede VF, due to short-term AP memory [13,21,29], which plays a key role in cardiac dynamics and, particularly, in the onset of irregular cardiac rhythms [23]. Despite the relevance of M_s for arrhythmogenesis, many aspects of this property remain unclear, particularly at the cellular level, and its dependence on ion currents is largely unexplored. This is partially due to the difficulty and the duration of the stimulation protocols used to measure it in vivo.

The aim of the present computational study is to show that (1) M_s can be more efficiently measured under random pacing conditions as the slope difference between local ER_rand and global ER_dyn curves. A further scope is to (2) show that M_s not only depends on the pacing CL but on APD as well. Also, (3) M_s changes under ion channels or electrotonic modulation are more frequently determined by changes in the global steady-state restitution ER_dyn curve, rather than in the local ER_rand or ER_s1s2 curves. Notably, the ORd model used to generate all the simulations of the present study is by far the most widely adopted numerical reconstruction of the human ventricular AP, which successfully reproduces repolarization dynamics, including rate-dependence and restitution properties, under physiological and pathological conditions (for a comparative study, see [30]). The model is used in its endocardial version, and differences in the ion currents modulation of M_s must be expected if simulations would have been conducted on mid-miocardial or epicardial AP types.

Measuring M_s with random restitution. Since short-term AP memory measures the difference between steady-state (ER_dyn) and instantaneous (ER_s1s2) sensitivity of APD to BCL changes, when ER_s1s2 and ER_dyn are identical, no memory exists [6,15,31]. Whereas ER_dyn is unique for a given cell, ER_s1s2 is locally defined for any given BCL where it is centered, thus making the slope difference also BCL-dependent (Figure 3C). The rate-dependence of M_s has been designed as “short-term memory restitution” [14] and identified by the M_s vs. BCL curve.

ER_rand also represents local restitution properties, and its slope resembles that of ER_s1s2, though slightly lower at all BCL values [32] (figures 2 and 3). ER_rand has previously been used to evaluate restitution properties both in silico [33,34,35,36] and in vivo. In vivo measurements of ER_rand have been reported by Zaniboni in patch-clamped rat ventricular cardiomyocytes [37], and by Wu and Patwardhan on tissue samples of dog left ventricular endocardial wall impaled with standard floating electrodes [38]. In fact, the slope difference between ER_rand and ER_dyn (ΔS_rand,dyn) tightly correlates with that between ER_s1s2 and ER_dyn (ΔS_rs1s2,dyn) in the ORd model (inset of Figure 3D). The same percentage dependence of ΔS_rand,dyn and ΔS_rs1s2,dyn from the maximum conductance of two of the main ion currents involved in AP repolarization points to the fact that both measurements of M_s describe the same phenomenon. Also, the M_s measurements reported in Figure 3 in the form ΔS_rand,dyn and ΔS_rs1s2,dyn reproduce both in amplitude and in BCL-dependence what was found by Tolkacheva in patch-clamped guinea pig and rabbit cardiomyocytes [14], strengthening the fact that the repolarization dynamics of the ORd model respond, in terms of short-term AP memory, coherently with other mammalian species.

Indeed, most of the simulations of the present study were carried out in both ways with qualitatively similar results, but only those measured with ER_rand are reported for brevity and denoted as M_s in the rest of this study. Moreover, the use of ER_rand can be advantageous in vivo since a single run of the protocol reported in Figure 1 provides the entire restitution plot (Figure 2B), whereas the measure of the standard restitution ER_s1s2 curve (Figure 2C) requires, for each point, a corresponding conditioning stimulation at the constant BCL, resulting in extremely long pacing times. To save time, frequently, only two-point ER_s1s2 curves are measured [14], which, though, adds approximation to the local ER slope thus derived. A further reason for choosing ER_rand is that, during the ER_s1s2 protocol, particularly at high pacing rate, the introduction of a single premature stimulation (S2) can frequently trigger APD alternans, whereas a randomly varying pacing rate has been shown to prevent this phenomenon [32,35], thus allowing ER measurements also at very short BCLs.

Ion current contribution to M_s. Although the role of specific ion currents in modulating short-term AP memory has been previously investigated both in vivo and in silico [14,39], a more systematic picture of how the different currents contribute to M_s is lacking.

In a study on patch-clamped rabbit and guinea pig ventricular myocytes, Tolkacheva and coworkers found that I_CaL plays a major role in determining the slopes of ER_dyn and ER_s1s2 [40]. They showed that I_CaL significantly affects S_dyn at different values of BCL, and S_s1s2 only at small BCL values, which agrees with the simulations I am presenting here on S_dyn and S_rand (see lower panels of Figure 5A). Also, the importance of I_Kr to the rate-dependent dynamics of ventricular AP and to the onset of APD alternans was suggested previously [40,41]. The simulations presented in this study show that M_s, particularly its peak value at 650 ms, is strongly dependent on G_Kr, which significantly affects both S_dyn and S_rand (Figure 5C, lower panels). Also, the late sodium current I_NaL affects M_s (Figure 7A), though to a smaller extent, whereas other ion currents have no or only minimal effect (Figure 9B). I_K1 for instance, which has been suggested to be linked to short-term memory [42], has only a negligible effect on M_s as defined in this study (Figure 9B), and I_TO, similarly, has only a small effect on M_s (see Figure 9B), most likely due to the fact that all simulations were performed on the endocardial version of the ORd model, where G_TO is four times less than in the epicardial one [24].

The small role I found for I_NaK in modulating M_s (Figure 8 and Figure 9B), compared, for example, with the large changes observed by Romero and co-authors [43] on restitution properties measured under the same modulation, was somewhat unexpected. It can be explained when considering (1) that the ORd model includes a novel formulation of I_NaK which, with respect with previous ones (including the Ten Tusscher and Panfilov model [44] used by Romero) also takes into account intracellular potassium, ATP, and pH levels; and (2) that Romero and co-authors estimate changes on the maximal value of both ER_dyn and ER_s1s2 curves, whereas I report differences between the two, actually between ER_dyn and ER_rand.

Of particular interest is the partial block of I_Kr (Figure 5C), where the expected steepening of ER_dyn [40] is accompanied by a divergent increase in S_rand and a consequent drop in the M_s value for very high pacing rates (BCL < 450 ms). This abrupt transition of M_s from low to high values at high pacing rates should be considered when studying the effects of I_Kr-blocking agents, like class III antiarrhythmics, on ER properties and APD stability, and their paradoxical and rate-dependent pro-arrhythmic action [1].

Despite differences, a unifying feature of ion channel modulation is that maneuvers that shorten APD lead to an overall decrease in M_s and, vice versa, maneuvers that prolong APD lead to M_s increase (Figure 6B,D and Figure 7B). Short-term AP memory is, in other words, not only rate-dependent but also APD-dependent. This is confirmed by the simulations of APD modulation by electrotonic interaction (Figure 11B) and by current injection (Figure 12B), where, interestingly, ER_dyn seems to be uniquely responsible for M_s changes. It is known from previous studies that the electrotonic coupling of a paced single-ventricular cell with a quiescent one with similar passive electrical properties leads to a decrease in the APD of the paced cell, which becomes able to respond successfully to more rapid stimulation [45], and is tempting to attribute this modification to the decrease in M_s that accompanies such electrotonic effects (Figure 11). Electrotonic coupling also suppresses intrinsic beat-to-beat variability of APD and contraction [46,47], which, though, were not included in the simulations of the present study.

Negative M_s. Some of the results presented in this study show negative values of M_s at high pacing rate, which has not been noted previously. Since M_s = S_dyn–S_rand, negative M_s only means that the slope of ER_rand is larger than that of ER_dyn or, in other words, that APD displacement after a sudden change in CL is larger than that at the steady state. Small or negative values of M_s are a constant feature of high pacing conditions and are exacerbated either by the reduction in S_dyn and/or by the increase in ER_rand.

The role of intracellular calcium. The role of calcium cycling in affecting the slope of global and local restitution properties is controversial, though there is evidence supporting its primary involvement in APD alternans [22,23,48]. In a study on patch-clamped guinea pig and rabbit myocytes, Tolkacheva has shown that intracellular calcium did not affect S_dyn nor S_s1s2 [39], whereas Goldhaber has shown that the complete block of SR calcium release by simultaneous application of thapsigargin and ryanodine did reduce the slope of both ER_dyn and ER_s1s2 [22]. Here, I present evidence that the complete block of SR function mainly affects the slope of ER_dyn, which undergoes a non-linear increase (Figure 10C), which is, in turn, responsible for the large increase in M_s (Figure 10A). This finding might underlie the fact that a high pacing rate fails to produce APD alternans in patch-clamped rabbit ventricular myocytes where calcium cycling was abolished by the usage of thapsigargin and ryanodine [22].

During cardiac pacing, even on a beat-to-beat basis, a shortening of pacing CL leads to a shortening of ventricular APD, which, in turn, leads to a decrease in the amplitude of calcium transient, which means, in other words, that, under a variable pacing rate, the time course of CL, APD, and amplitude of calcium transient should correlate in phase [49]. On the other hand, both positive and negative calcium–voltage coupling has been documented, i.e., conditions in which the increase in calcium transients prolongs or shortens the APD, due to opposing effects on APD of the various constituents of calcium homeostasis [50]. Indeed, under sudden changes in the pacing rate, both APD and the calcium transient respond with their restitution dynamics, though it is still not clear which one leads the other under increased variability of pacing rate [51,52,53,54]. Several hypotheses have been formulated in order to explain these facts [51,55], also by showing that intracellular calcium cycling can per se constitute a source of alternans [51,53,54,55,56]. Thus, by removing this source of alternance via the complete block of SR function (Figure 10), we are left with a higher S_dyn and, in turn, a higher M_s. This dramatic change in the dynamics of AP repolarization after removing the calcium transient reinforces the hypothesis that it is the calcium cycling that leads APD oscillations in their mutual relationship.

To summarize, the results of the present computational study provide further evidence that short-term AP memory is affected by ion currents involved in AP repolarization, and particularly by I_Kr and I_CaL, and by calcium cycling, and that M_s is not only BCL-dependent but also APD-dependent. The knowledge of M_s changes under ion channel modulation and under interventions that modify AP repolarization may increase our understanding of the genesis of arrhythmias and may help to unravel paradoxical effects of the presently adopted antiarrhythmic agents and to develop new ones.

Although short-term memory M_s is defined and measured in the present study at the cellular (zero-dimensional) level, the results can be relevant at higher levels of structural complexity. The occurrence of rotors for the initiation of reentry and, in turn, of VF, is known, for instance, to be due to tissue heterogeneity following remodeling after cardiac disease but also to dynamic factors which include short-term AP memory [57]. M_s has, in fact, been studied by means of numerical simulations on two-dimensional and three-dimensional cardiac tissue, where it can spontaneously convert multiple wavelet fibrillation to mother rotor fibrillation or to a mixture of both fibrillation types [58]. This was confirmed by optical mapping M_s in perfused whole rabbit hearts [59]. Also, in a three-dimensional numerical model of ventricular tissue, Cherry and Fenton have shown that short-term memory effects can suppress alternans, even when a steep APD restitution curve would predict it according to the restitution hypothesis [6]. The heterogeneity of short-term AP memory found between the epicardium, mid-myocardium, apex, and base of the heart has been attributed to the heterogeneous distribution of ion channel densities within these regions of the rabbit heart [60], which have been confirmed by optical mapping of the canine transmural ventricle [23,61]. The zero-dimensional results reported in the present study represent the cellular counterpart of these phenomena, though it must be kept in mind that in two- or three-dimensional measurements, more variables are in play, like the fiber orientation, gradients of APD, and electrotonic interactions.

Important to be mentioned is the role of M_s in the clinical setting, where short-term cardiac memory is characterized by persistent changes in the T wave on electrocardiograms (ECGs), which follow the resumption of sinus rhythm after a period of an altered ventricular activation sequence and which has been widely studied in animal models and in humans [62], together with the roles that specific ion currents play in its modulation [63,64]. On the other hand, further issues need to be faced in order to compare clinical data with single cell results like those presented in this study. In the clinical setting, for example, the pacing rates to induce short-term memory are limited by the basic sinus rate of the patient, e.g., using 150% of it [65], and the method adopted to estimate memory is often based on the estimation of the maximal slope of ER curves measured by monophasic AP recording catheters, rather than on the slope difference between ER_dyn and ER_s1s2, or between ER_dyn and ER_rand. Also in this regard, the choice I am proposing here of measuring ER_rand seems to be promising in the clinical setting, where it can save much recording time when compared to ER_s1s2 recordings. In summary, more work is required in order to fill the gap between studies on M_s at the single cell, tissue, and whole organ levels, which is perhaps the reason why some authors still find M_s involvement into arrhtythmia development controversial or even not significant [62].