Stochastic pacing reveals the propensity to cardiac action potential alternans and uncovers its underlying dynamics

Abstract

Key points

• Beat‐to‐beat alternation (alternans) of the cardiac action potential duration is known to precipitate life‐threatening arrhythmias and can be driven by the kinetics of voltage‐gated membrane currents or by instabilities in intracellular calcium fluxes.

• To prevent alternans and associated arrhythmias, suitable markers must be developed to quantify the susceptibility to alternans; previous theoretical studies showed that the eigenvalue of the alternating eigenmode represents an ideal marker of alternans.

• Using rabbit ventricular myocytes, we show that this eigenvalue can be estimated in practice by pacing these cells at intervals varying stochastically.

• We also show that stochastic pacing permits the estimation of further markers distinguishing between voltage‐driven and calcium‐driven alternans.

• Our study opens the perspective to use stochastic pacing during clinical investigations and in patients with implanted pacing devices to determine the susceptibility to, and the type of alternans, which are both important to guide preventive or therapeutic measures.

Abstract

Alternans of the cardiac action potential (AP) duration (APD) is a well‐known arrhythmogenic mechanism. APD depends on several preceding diastolic intervals (DIs) and APDs, which complicates the prediction of alternans. Previous theoretical studies pinpointed a marker called λ_alt that directly quantifies how an alternating perturbation persists over successive APs. When the propensity to alternans increases, λ_alt decreases from 0 to –1. Our aim was to quantify λ_alt experimentally using stochastic pacing and to examine whether stochastic pacing allows discriminating between voltage‐driven and Ca²⁺‐driven alternans. APs were recorded in rabbit ventricular myocytes paced at cycle lengths (CLs) decreasing progressively and incorporating stochastic variations. Fitting APD with a function of two previous APDs and CLs permitted us to estimate λ_alt along with additional markers characterizing whether the dependence of APD on previous DIs or CLs is strong (typical for voltage‐driven alternans) or weak (Ca²⁺‐driven alternans). During the recordings, λ_alt gradually decreased from around 0 towards –1. Intermittent alternans appeared when λ_alt reached –0.8 and was followed by sustained alternans. The additional markers detected that alternans was Ca²⁺ driven in control experiments and voltage driven in the presence of ryanodine. This distinction could be made even before alternans was manifest (specificity/sensitivity >80% for –0.4 > λ_alt > –0.5). These observations were confirmed in a mathematical model of a rabbit ventricular myocyte. In conclusion, stochastic pacing allows the practical estimation of λ_alt to reveal the onset of alternans and distinguishes between voltage‐driven and Ca²⁺‐driven mechanisms, which is important since these two mechanisms may precipitate arrhythmias in different manners.

Key points

• Beat‐to‐beat alternation (alternans) of the cardiac action potential duration is known to precipitate life‐threatening arrhythmias and can be driven by the kinetics of voltage‐gated membrane currents or by instabilities in intracellular calcium fluxes.

• To prevent alternans and associated arrhythmias, suitable markers must be developed to quantify the susceptibility to alternans; previous theoretical studies showed that the eigenvalue of the alternating eigenmode represents an ideal marker of alternans.

• Using rabbit ventricular myocytes, we show that this eigenvalue can be estimated in practice by pacing these cells at intervals varying stochastically.

• We also show that stochastic pacing permits the estimation of further markers distinguishing between voltage‐driven and calcium‐driven alternans.

• Our study opens the perspective to use stochastic pacing during clinical investigations and in patients with implanted pacing devices to determine the susceptibility to, and the type of alternans, which are both important to guide preventive or therapeutic measures.

Abbreviations

AP

action potential

APD

action potential duration

AR model

autoregressive model

ARMA model

autoregressive‐moving‐average model

CL

cycle length

DI

diastolic interval

ORd

O'Hara‐Virag‐Varró‐Rudy human ventricular cell model

SR

sarcoplasmic reticulum

λ_alt

alternans eigenvalue

Introduction

Alternation of the cardiac action potential duration (APD) from beat to beat is a well‐known arrhythmogenic mechanism. Alternans can lead to spatial dispersion of refractoriness potentiating conduction block, reentry and life‐threatening arrhythmias such as atrial or ventricular fibrillation (Pastore et al. 1999; Weiss et al. 2006; Franz et al. 2012; Karagueuzian et al. 2013; Taggart et al. 2014; Verrier & Malik 2015; Wagner et al. 2015). Alternans can occur in a variety of forms and mechanisms (Qu et al. 2010; Edwards & Blatter 2014; Wagner et al. 2015). At the cellular level, alternans can result from the dependence of APD on the previous diastolic interval (DI), due to the amplitude and kinetics of voltage‐gated ion currents (voltage‐driven alternans) (Koller et al. 1998; Pastore et al. 1999). Alternans can also result from a dynamic instability in intracellular Ca²⁺ cycling, in particular due to the non‐linear behaviour of Ca²⁺‐induced Ca²⁺ release and the bidirectional interaction between APD and the Ca²⁺ transient (Ca²⁺‐driven alternans) (Qu et al. 2010; Edwards & Blatter 2014; Wagner et al. 2015). At the tissue and organ levels, alternans is modulated by intercellular interactions, conduction velocity restitution and pacing rate (Watanabe et al. 2001; Echebarria & Karma 2002). These multiscale mechanisms interact dynamically and give rise to complex spatiotemporal alternans patterns (Sato et al. 2006; Mironov et al. 2008; Gizzi et al. 2013).

In the initial theory of alternans (Nolasco & Dahlen 1968; Guevara et al. 1981; Chialvo et al. 1990), APD is described as a non‐linear function of the previous DI (restitution function). Alternans occurs as a bifurcation when the slope of this function becomes ≥1. Hence, the propensity to alternans is classically evaluated by determining APD restitution slopes using downsweep or S1S2 pacing protocols (Kalb et al. 2004; Tolkacheva et al. 2006).

Several studies suggest that this approach is not always appropriate and some provide arguments against it. For example, none of the conventional restitution slopes predicted alternans in bullfrog myocardium (Kalb et al. 2004), and the authors suggested that the memory of previous pacing cycles must in fact be taken into account. In rabbit hearts, it was shown that these slopes were statistically different from 1 at the onset of alternans (Cram et al. 2011). Another study showed that during hypokalaemia, the predictive power of APD restitution in promoting alternans fails (Osadchii et al. 2010). Using a system permitting the real time control of the DI, further investigators (Wu & Patwardhan 2006) showed that in canine ventricle, alternans can occur even if the DI is kept constant from beat to beat. This demonstrates that APD does not depend only on the previous DI and that memory of previous pacing cycles must be considered.

Because APDs, DIs and cycle lengths (CLs) are intricately linked, a perturbation of one parameter will propagate from one beat to the next and influence the sequence of the other parameters over several cycles. To untangle these interactions, modern paradigms to evaluate alternans should therefore rely on multivariate analyses taking entire sequences of APDs, DIs and CLs into account. Intuitively, it would be a great advantage to have a marker that quantifies how a perturbation of one of these parameters is transmitted over subsequent cycles and how fast or how slow it decays. Theoretical studies (Li & Otani 2003; Otani et al. 2005) have shown that the activity of a cardiac cell can be decomposed into distinct eigenmodes, i.e. patterns that recur every beat after being scaled by specific numbers called eigenvalues (λ), producing the series 1, λ, λ², λ³, … over successive beats, as illustrated in Fig. 1 A. For any |λ| < 1, the series converges to 0, and negative eigenvalues reflect a change of sign every beat. Thus, the most negative eigenvalue (λ_alt) is precisely the factor by which the most prominent alternating perturbation is multiplied after every beat. λ_alt represents an ideal marker of alternans, because it quantifies directly how an alternant perturbation decays over successive cycles. When the propensity to alternans increases, λ_alt decreases from 0 towards –1, which is the limit of dynamic stability. When λ_alt = –1, a perturbation no longer dissipates but keeps alternating.

Figure 1. Signification of λ_alt and bidirectional interactions between the AP and the Ca²⁺ transient .

A, the eigenvalue λ_alt represents the factor by which an alternating perturbation decays with every beat. For λ_alt close to 0, the alternating eigenmode dissipates rapidly and will not be manifest in a real system subject to continuous perturbations. With λ_alt getting closer to –1, alternating sequences are more likely to appear and to endure because the alternans eigenmode dissipates more slowly. The susceptibility to alternate becomes apparent for λ_alt around –0.5 and is quite manifest for λ_alt < –0.8. If λ_alt < –1, the system is unstable and the smallest perturbation is amplified. B, theoretical model of the interactions between CL, APD and the Ca²⁺ transient (CT). The interactions (details in the text) are marked with dotted arrows labelled with Greek letters.

We previously presented a theoretical framework forming the basis for an experimental estimation of λ_alt (Lemay et al. 2012). Central to this approach are pacing protocols comprising stochastic beat‐to‐beat variations of pacing cycle length. These stochastic perturbations continuously excite all the eigenmodes, but those with eigenvalues close to 0 dissipate rapidly while those with eigenvalues close to –1 tend to persist in an alternating manner, revealing the susceptibility to alternans.

Our aim in the present study was to demonstrate the practical usefulness of this approach in isolated cardiac myocytes. We show that the experimentally estimated value of λ_alt is an appropriate marker to quantify the propensity to alternans. In addition, we derived further markers to discriminate between voltage‐driven and Ca²⁺‐driven alternans, and tested them in experiments and computer simulations using models of rabbit and human ventricular myocytes. Our results show that it is possible to discriminate between voltage‐driven and Ca²⁺‐driven alternans with a sensitivity and specificity >80%, even when the propensity to alternans is moderate, thus permitting information to be gained regarding the ionic mechanism of alternans before alternans fully develops. This opens the perspective of implementing new strategies to evaluate the propensity to alternans, which, in a translational setting, would be of assistance for the prevention and the therapy of associated arrhythmias.

Methods

Ethical approval

All experiments (see protocols below) were conducted at the University of California, Los Angeles (UCLA). Animals were handled in accordance with the ethical principles and guidelines of UCLA Institutional Animal Care and Use Committee (ARC no. 2003‐063‐33) and conformed to the Guide for the Care and Use of Laboratory Animals published by the US National Institutes of Health. The authors understand the ethical principles under which the journal operates. The present work complies with the animal ethics checklist of the journal.

Isolation of rabbit ventricular myocytes

Rabbit cardiac myocytes were isolated as previously (Madhvani et al. 2011). Briefly, hearts were rapidly excised from New Zealand White (NZW) male rabbits (age: 3–4 months, weight: 1.9–2.1 kg, n = 21 rabbits) under deep anaesthesia induced by an intravenous injection of sodium pentobarbital (100 mg kg⁻¹) and heparin sulphate (1000 U). The adequacy of anaesthesia was confirmed by the lack of pedal withdrawal reflex, corneal reflex, and motor response to pain stimuli by scalpel tip. The hearts were placed in Ca²⁺‐free Tyrode solution containing (in mmol l⁻¹) 140 NaCl, 5.4 KCl, 1 MgCl₂, 0.33 NaH₂PO₄, 10 glucose and 10 Hepes, adjusted to pH 7.4. The aorta was cannulated and the heart perfused retrogradely at 37°C on a Langendorff apparatus with Ca²⁺‐free Tyrode buffer containing 1.65 mg ml⁻¹ collagenase. All solutions were continuously bubbled with 95% O₂–5% CO₂. After enzymatic digestion, the hearts were swirled in a beaker to dissociate cells. The Ca²⁺ concentration was gradually increased to 1.8 mmol l⁻¹, and the cells were stored at room temperature. The animals were supplied by Charles River Laboratories (Wilmington, MA,USA) and maintained in cages with standard lab chow and water available ad libitum.

Patch clamp experiments

The myocytes were bathed in a solution containing (in mmol l⁻¹): 136 NaCl, 5.4 KCl, 1.8 CaCl₂, 1 MgCl₂, 0.33 NaH₂PO₄, 10 glucose and 10 Hepes, adjusted to pH 7.4. Membrane potential was recorded (sampling rate: 10 kHz) at 34–36°C using the whole‐cell patch clamp technique in the current clamp mode using AxoPatch200B (Axon Instruments, Molecular Devices, Sunnyvale, CA, USA). The pipettes (resistance 1–2 MΩ) were filled with (in mmol l⁻¹): 110 K‐aspartate, 30 KCl, 5 NaCl, 10 Hepes, 0.1 EGTA, 5 MgATP, 5 creatine phosphate, 0.05 cAMP adjusted to pH 7.2. In some experiments, ryanodine was added to the bath solution at a concentration of 100 μmol l⁻¹ to completely suppress Ca²⁺ release from the sarcoplasmic reticulum. Additional experiments were performed using the perforated patch technique using amphotericin B (240 μg ml⁻¹) in the pipette solution.

Pacing protocols

Ramp protocol without stochastic variations

The cells were first paced with 2 ms current pulses at approximately twice the threshold of the resting cell for 3 min at a starting basic cycle length (CL) of 250–400 ms to allow for accommodation. Subsequently, CL was progressively decreased on a beat‐to‐beat basis over 10–25 min. To minimize transient alternans due to abrupt CL changes, the descending CL ramps were designed such that instantaneous pacing rate (1/CL) increases linearly with time.

Ramp protocol with stochastic variations

The series of pacing CLs were designed as described above and stochastic variations of CL (Gaussian distribution with zero mean and a standard deviation (SD) of 10 ms if not specified otherwise) were implemented to the entire CL series.

Computer simulations of action potentials

Computer simulations were conducted with the Mahajan et al. rabbit ventricular cell model (Mahajan et al. 2008), which corresponds directly to our experimental preparation, and the O'Hara‐Virag‐Varró‐Rudy (ORd) human ventricular cell model (O'Hara et al. 2011). The models were subjected to similar pacing protocols.

The Mahajan et al. model was run using a constant time step of 0.005 ms. Gating variables were integrated using the method of Rush and Larsen, the state occupancies of the Markovian model of the L‐type Ca²⁺ channel were computed as described by Milescu et al. (2005) and membrane potential and ion concentrations were integrated using the forward Euler algorithm. The ORd model (MATLAB source code downloaded from Dr Rudy's website: www.rudylab.edu) was run using the ‘ode15s’ MATLAB ordinary differential equation solver.

Analysis of APD, DI and CL series and markers of alternans

In experiments and simulations, activation was defined for every action potential (AP) at –35 mV during depolarization and APD was determined at a potential 10 mV above the minimal diastolic potential preceding the AP. Series of consecutive APDs, DIs and CLs (APD_n, DI_n, CL_n) were derived from these fiducial time points with the convention CL_{n  =} APD_n + DI_n (Lemay et al. 2012). In this convention, DI_n–1 is the DI immediately before APD_n (which occurs during CL_n), and DI_n is the DI immediately after APD_n.

Markers to characterize alternans were derived starting from the notion that APD depends on several previous DIs and APDs (‘memory’; Kalb et al. 2004, 2005; Mironov et al. 2008). This dependence can be formalized as a generalized restitution function g (Kalb et al. 2005):

$$\mathrm{AP}{\mathrm{D}}_n=g(\mathrm{D}{\mathrm{I}}_{n-1},\mathrm{AP}{\mathrm{D}}_{n-1},\mathrm{D}{\mathrm{I}}_{n-2},\mathrm{AP}{\mathrm{D}}_{n-2},...).$$

Because CL was explicitly controlled in our study, we expressed the generalized restitution function in terms of APD and CL as:

$$\mathrm{AP}{\mathrm{D}}_n=f(\mathrm{AP}{\mathrm{D}}_{n-1},\mathrm{C}{\mathrm{L}}_{n-1},\mathrm{AP}{\mathrm{D}}_{n-2},\mathrm{C}{\mathrm{L}}_{n-2},...).$$

Assuming that beat‐to‐beat variations of APD and CL are small enough such that non‐linearities of f are not involved, one can linearize f around mean APD and CL values as (Kalb et al. 2005; Lemay et al. 2012):

$$\begin{array}{ccc}\hfill \delta \mathrm{AP}{\mathrm{D}}_n& =& {\alpha }_1\delta \mathrm{AP}{\mathrm{D}}_{n-1}+{\alpha }_2\delta \mathrm{AP}{\mathrm{D}}_{n-2}+...+{\beta }_1\delta \mathrm{C}{\mathrm{L}}_{n-1}\hfill \\ & & +{\beta }_2\delta \mathrm{C}{\mathrm{L}}_{n-2}+...+{\epsilon }_n,\hfill \end{array}$$

where δAPD_n and δCL_n are the respective deviations of APD_n and CL_n from their local mean values and ε_n is a noise term representing intrinsic fluctuations of APD arising, e.g. from stochastic channel gating (Zaniboni et al. 2000; Lemay et al. 2011), stochastic Ca²⁺‐induced Ca²⁺ release (Cannell et al. 2013) or noise in the measuring apparatus. The coefficients α_i and β_i quantify the response of APD to a perturbation of the ith previous APD or CL and correspond respectively to the partial derivatives ∂f/∂APD_n−i and ∂f/∂CL_n−i, respectively.

The above equation represents a discrete‐time autoregressive‐moving‐average (ARMA) model (Ljung 1999) taking the series of (controlled) CLs as input and producing the series of (measured) APDs as output. The coefficients α_i and β_i characterize this input–output behaviour of the investigated cardiac cell. If CL variations are absent or negligible (e.g. CL is kept constant), the system reduces to an autoregressive (AR) model, in which the contributions of the coefficients β_i are lost:

$$\delta \mathrm{AP}{\mathrm{D}}_n={\alpha }_1\delta \mathrm{AP}{\mathrm{D}}_{n-1}+{\alpha }_2\delta \mathrm{AP}{\mathrm{D}}_{n-2}+...+{\epsilon }_n.$$

To illustrate this with a few examples, we first consider the immediate response of APD to an isolated perturbation of CL applied during steady state pacing (e.g. S1S2 protocol). In this situation, for the APD immediately following the perturbation and neglecting the noise term, the equation becomes δAPD_n = β₁δCL_{n −1}. Therefore, the first moving average coefficient β₁ directly reflects the conventional instantaneous S1S2 restitution slope. Second, we consider a setting in which alternans is principally determined by the kinetics of membrane currents during the previous DI (voltage‐driven alternans). In such a situation, APD variations are strongly influenced by β₁, in agreement with classical restitution theory. In contrast, if alternans is caused predominantly by fluctuations of the Ca²⁺ transient (due, e.g., to variations of sarcoplasmic Ca²⁺ loading and release) which are indirectly transmitted to APD, the latter will be predominantly governed by the autoregressive coefficients α_i while the β_i values will remain relatively small and exert only a weak effect. However, the α_i and β_i values cannot be ascribed directly to either the voltage or the Ca²⁺ driving mechanism (a detailed analysis is presented below in the section ‘Insights from an iterated map model’).

Given a series of CLs and APDs, the coefficients α_i and β_i can be estimated using dedicated algorithms (Ljung 1999). However, further analysis is required to extract information about alternans and stability. As we showed previously (Lemay et al. 2012), by applying a Z‐transform to the equation of the ARMA model, so‐called transfer functions can be derived to characterize the response of the cell in the frequency domain as well the stability of this response:

$$H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)=\frac{{\beta }_1{z}^{-1}+{\beta }_2{z}^{-2}+...}{1-{\alpha }_1{z}^{-1}-{\alpha }_2{z}^{-2}-...},$$

$$\begin{array}{ccc}\hfill H_{\mathrm{CL}\to \mathrm{DI}}\left(z\right)& =& 1-H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)\hfill \\ & =& \frac{1-({\alpha }_1+{\beta }_1)z^{-1}-({\alpha }_2+{\beta }_2)z^{-2}-...}{1-{\alpha }_1{z}^{-1}-{\alpha }_2{z}^{-2}-...}.\hfill \end{array}$$

The roots of the denominator, called ‘poles’ or ‘eigenvalues’ (λ), determine the stability of the system. The system is stable only if |λ| < 1 for all λ values (Ljung 1999). For a cell prone to alternans, one of the eigenvalues will dominate and approach –1 when alternans is imminent. This eigenvalue (λ_alt) thus characterizes the propensity to alternans. It can be understood as the number by which a perturbation of APD is scaled before being transmitted to the next beat (Fig. 1 A).

The roots of the numerators, called ‘zeros’ (ζ), determine the phase relationships between CL and APD variations. While the zeros do not provide information on stability, they nevertheless allow untangling the relative contributions of the coefficients α_i and β_i to the alternating behaviour of APD. Therefore, the major advantage of introducing controlled stochastic variations of pacing CL is to permit the estimation of the coefficients β_i, which are necessary to provide a full picture of the dynamic response of APD, not only in terms of the susceptibility to alternans but also in terms of the underlying dynamics.

We analysed the series of APDs, DIs and CLs using a sliding window (unless specified otherwise: 100 beats slid in steps of 25 beats). For each window, the series were first detrended using a 2nd degree polynomial. Then, a 2nd order AR model (protocols without stochastic CL variations) or ARMA model (protocols with stochastic CL variations) was used to fit the data (Ljung 1999). λ_alt was defined as the eigenvalue closest to –1. While λ_alt  = –1 represents full persistent alternans, values close to but still >–1 already indicate the susceptibility to alternate (see Fig. 1 A). We therefore defined thresholds of –0.5 and –0.8 for a moderate and a high susceptibility to alternans, respectively. We showed in previous computational work (Lemay et al. 2012) that the dominant zero of H _CL→DI (ζ_alt) is near 0 if alternans is voltage driven, whereas it is close to λ_alt if alternans is Ca²⁺ driven. The zero ζ_alt and the difference ζ_alt – λ_alt thus represent markers of the primary mechanism of alternans. Our analysis thus reports λ_alt (susceptibility to alternans), and, for protocols with stochastic CL variations, ζ_alt, ζ_alt – λ_alt and β₁ (further markers characterizing the dynamics of alternans). In addition, mean APD was plotted vs. mean DI in each window to generate a dynamic restitution curve. The slope of the plot provides an estimate of the dynamic restitution slope (Kalb et al. 2004; Tolkacheva et al. 2006).

The ARMA model is a good approximation of the generic APD restitution function under the assumption that APD and CL variations around their mean values remain small enough such that non‐linearities remain weak. Hence, for the linear analysis using the ARMA model to remain applicable, the stochastic variations of pacing CL must be kept small enough such that non‐linearities remain weak. If non‐linearities need to be accounted for, the analysis can be extended with non‐linear terms (Armoundas et al. 2002; Dai & Keener 2012). However, this approach would have the disadvantage that more data (longer series of APDs/CLs) would be required to obtain reliable non‐linear model identification.

All analyses and simulations were conducted using MATLAB (The MathWorks, Natick, MA, USA). AR and ARMA models were identified using a least squares algorithm (function ‘armax’ from the MATLAB System Identification Toolbox).

Statistics

The ability of ζ_alt, and β₁ to discriminate between control experiments and experiments with ryanodine was evaluated in terms of specificity and sensitivity by computing receiver operating characteristic curves.

Insights from an iterated map model

It must be noted that the coefficients α_i and β_i and the markers λ_alt and ζ_alt are interrelated and cannot be ascribed either to voltage‐driven or to Ca²⁺‐driven alternans. Furthermore, in a physiological setting, alternans is never either fully voltage driven or fully Ca²⁺ driven. While one mechanism typically predominates, both mechanisms contribute jointly to the genesis of alternans (Edwards & Blatter 2014; Groenendaal et al. 2014). To gain deeper insights into the relationships between λ_alt, ζ_alt, and β₁ during Ca²⁺‐driven vs. voltage‐driven alternans, we investigated the following iterated map model (Fig. 1 B) implementing the interactions between the AP and the Ca²⁺ transient (δ denotes a small deviation from the mean):

$$\begin{array}{ccc}\delta \mathrm{AP}{\mathrm{D}}_n\hfill & =\hfill & \mu ·\delta \mathrm{AP}{\mathrm{D}}_{n-1}+\rho ·\delta \mathrm{C}{\mathrm{T}}_n\hfill \\ & & +\nu ·\delta \mathrm{C}{\mathrm{L}}_{n-1}+{\epsilon }_{\mathrm{APD},n},\hfill \\ \delta \mathrm{C}{\mathrm{T}}_n\hfill & =\hfill & \gamma ·\delta \mathrm{C}{\mathrm{T}}_{n-1}+\varphi ·\delta \mathrm{AP}{\mathrm{D}}_{n-1}\hfill \\ & & +\kappa ·\delta \mathrm{C}{\mathrm{L}}_{n-1}+{\epsilon }_{\mathrm{CT},n},\hfill \end{array}$$

where CT_n is the peak of the Ca²⁺ transient during the nth action potential. In the first equation and as illustrated in Fig. 1 B, μ and ν describe the dependence of APD on the previous APD and CL independently of the Ca²⁺ transient and ρ describes the specific contribution of the Ca²⁺ transient in determining APD, which reflects Ca²⁺‐to‐APD coupling. For ρ > 0 (positive Ca²⁺‐to‐APD coupling), a larger Ca²⁺ transient leads to a longer APD, whereas for ρ < 0 (negative Ca²⁺‐to‐APD coupling), a larger Ca²⁺ transient leads to a shorter APD. In the second equation, γ represents the feedback of CT on itself, φ represents the influence of APD on the next Ca²⁺ transient (APD to Ca²⁺ coupling), and κ is a parameter describing the direct effect of a change of CL on the next Ca²⁺ transient (CT restitution). This latter parameter is expected to be positive, because shortening CL results in a smaller Ca²⁺ transient during the next AP since the sarcoplasmic reticulum has less time to refill and since the recovery of the L‐type Ca²⁺ current from inactivation may still be incomplete. Most cardiac cell models reproduce this behavior. ε_APD,n and ε_CT,n are corresponding error terms.

Although this map model is a simplification of the detailed ionic mechanisms shaping APD and the Ca²⁺ transient, it offers the advantage of being tractable analytically.

Case of principally Ca²⁺‐driven alternans

If alternans is essentially driven by unstable Ca²⁺ cycling rather than by the voltage‐dependent kinetics of membrane currents, μ and ν are small and APD variations reflect the variations of the Ca²⁺ transient via the parameter ρ. If the contributions from μ and ν are neglected, the system reduces to:

$$\begin{array}{ccc}\delta \mathrm{AP}{\mathrm{D}}_n\hfill & =\hfill & \rho ·\delta \mathrm{C}{\mathrm{T}}_n+{\epsilon }_{\mathrm{APD},n},\hfill \\ \delta \mathrm{C}{\mathrm{T}}_n\hfill & =\hfill & \gamma ·\delta \mathrm{C}{\mathrm{T}}_{n-1}+\varphi ·\delta \mathrm{AP}{\mathrm{D}}_{n-1}\hfill \\ & & +\kappa ·\delta \mathrm{C}{\mathrm{L}}_{n-1}+{\epsilon }_{\mathrm{CT},n}.\hfill \end{array}$$

Combining both equations (neglecting the error terms) and solving for δAPD_n, we obtain:

$$\delta \mathrm{AP}{\mathrm{D}}_n=(\gamma +\varphi ·\rho )·\delta \mathrm{AP}{\mathrm{D}}_{n-1}+\rho ·\kappa ·\delta \mathrm{C}{\mathrm{L}}_{n-1},$$

which corresponds to an ARMA model with α₁ = γ + ϕρ and β₁  = ρκ.

Using Z‐transformation, the transfer functions H _CL→APD(z) and H _CL→DI(z) are obtained as:

$$\begin{array}{ccc}\hfill H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)& =& \frac{\rho \kappa z^{-1}}{1-(\gamma +\rho \varphi )z^{-1}}\mathrm{and}{H}_{\mathrm{CL}\to \mathrm{DI}}\left(z\right)\hfill \\ & =& 1-H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)\hfill \\ & =& \frac{1-(\gamma +\rho \varphi +\rho \kappa )z^{-1}}{1-(\gamma +\rho \varphi )z^{-1}}.\hfill \end{array}$$

The pole λ_alt and the zero ζ_alt of H _CL→DI(z) are:

$${\lambda }_{\mathrm{alt}}=\gamma +\rho \varphi ,$$

$${\zeta }_{\mathrm{alt}}=\gamma +\rho \varphi +\rho \kappa ,$$

and the difference ζ_alt – λ_alt is:

$${\zeta }_{\mathrm{alt}}-{\lambda }_{\mathrm{alt}}=\rho \kappa .$$

This result indicates that Ca²⁺‐driven alternans is caused by λ_alt  = γ + ρϕ approaching –1, and because κ > 0, the sign of ζ_alt – λ_alt permits to infer the sign of ρ, i.e. the sign of Ca²⁺‐to‐APD coupling.

Case of principally voltage‐driven alternans

Conversely, if alternans is essentially voltage driven, the parameter ρ becomes negligible and the iterative map model for alternans reduces to:

$$\delta \mathrm{AP}{\mathrm{D}}_n=\mu ·\delta \mathrm{AP}{\mathrm{D}}_{n-1}+\nu ·\delta \mathrm{C}{\mathrm{L}}_{n-1}+{\epsilon }_{\mathrm{APD},n},$$

i.e. to an ARMA model with α₁ = μ and β₁  = ν, with corresponding transfer functions:

$$\begin{array}{ccc}\hfill H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)& =& \frac{\nu z^{-1}}{1-\mu z^{-1}}\mathrm{and}{H}_{\mathrm{CL}\to \mathrm{DI}}\left(z\right)\hfill \\ & =& 1-H_{\mathrm{CL}\to \mathrm{APD}}\left(z\right)=\frac{1-(\mu +\nu )z^{-1}}{1-\mu z^{-1}}.\hfill \end{array}$$

The pole λ_alt and the zero ζ_alt of H _CL→DI(z) are:

$${\lambda }_{\mathrm{alt}}=\mu \mathrm{and}{\zeta }_{\mathrm{alt}}=\mu +\nu .$$

Thus, the difference ζ_alt – λ_alt is ν, which corresponds to β₁. Thus, in the classical alternans model (Nolasco & Dahlen 1968), ν corresponds to the S1S2 restitution slope and is expected to be close to 1 at the onset of alternans. Since λ_alt is then expected to be –1, we expect that ζ_alt will be around 0 in the case of essentially voltage‐driven alternans.

Mixed case

If both voltage and Ca²⁺ dynamics contribute to alternans to a similar extent, ζ_alt, λ_alt and β₁ become more complicated functions which are less straightforward to link to the parameters of the iterated map model. However, we expect a continuum of possible values for ζ_alt, λ_alt and β₁.

Results

λ_alt quantifies the susceptibility to alternans even without stochastic cycle length variations

The APD series of a ventricular myocyte subject to a ramp pacing protocol without stochastic variations is shown in Fig. 2 A. The progressive decrease of CL led to an overall APD decrease and eventually resulted in manifest alternans. The insets in Fig. 2 A and the corresponding APs shown in Fig. 2 B illustrate that alternans is difficult to identify by visual inspection before its full development. These insets also depict the intrinsic beat‐to‐beat variability of APD. In Fig. 2 A, λ_alt is shown with different background colours based on the thresholds defined in the Methods section (green: λ_alt > –0.5, low susceptibility; orange: –0.5 > λ_alt > –0.8, moderate susceptibility; red: λ_alt < –0.8, high susceptibility). λ_alt decreased from around –0.2 to –1. Intermittent alternans appeared in the form of bursts of alternation when λ_alt reached –0.8 (inset), followed by full sustained alternans, hence motivating the selection of a threshold of –0.8 as a marker of the imminence of alternans. Figure 2 C shows the dynamic restitution curve for this experiment. Globally, the dynamic restitution slope remained <1.

Figure 2. Ramp protocol without stochastic variations in a ventricular myocyte .

A, pacing CL, APD and λ_alt (sliding window of 100 beats) vs. beat number. Odd and even APDs are shown in red and cyan, respectively. λ_alt is shown with distinct background colours (green: 0 > λ_alt > –0.5; orange: –0.5 ≥ λ_alt ≥ –0.8; red: λ_alt < –0.8). Data points in magenta indicate that λ_alt was complex (the real part is shown). B, APs corresponding to the first 11–13 beats of the insets labelled a–d in A. C, plot of mean APD vs. mean DI in the sliding window, using the colour defined above for λ_alt. The onset of sustained alternans is denoted by the arrow.

The ramp protocol was conducted with seven cells (from three animals). In six out of these seven cells, λ_alt decreased progressively during the ramp protocol and alternans appeared when λ_alt reached –0.8, although the APD and CL at which alternans appeared was different between the individual cells. This indicates that λ_alt reflects the propensity to alternans, and is not linked to a given APD or CL. In one cell, alternans did not develop although CL was decreased to 150 ms at the end of the protocol, and λ_alt did not decrease below –0.5.

Stochastic cycle length variations provide insight into the mechanisms governing alternans

Figure 3 illustrates an experiment in which a myocyte was subjected to a ramp protocol with stochastic variations. With this protocol, λ_alt progressively decreased as well. Sustained alternans developed around beat number 2600 when λ_alt reached –0.8. In contrast to cells paced without stochastic variations (e.g. Fig. 2), sustained alternans was then characterized by occasional phase reversals, visible as transitions between periods in which even APDs were longer than odd APDs (marked in different colours) and periods in which odd APDs were longer than even APDs. These phase reversals occurred irregularly. During the descending CL ramp, APD exhibited a progressive trend to alternate (Fig. 3 A, insets), with bursts of alternans that lasted progressively longer in response to the continuous perturbations caused by stochastic pacing.

Figure 3. Ramp protocol with stochastic variations in a ventricular myocyte .

A, pacing CL, APD, λ_alt, ζ_alt (cyan), β₁ (S1S2 restitution slope, grey) and difference ζ_alt – λ_alt (purple) vs. beat number, computed in the sliding window. Odd and even APDs are shown in red and cyan, respectively. λ_alt is shown with distinct background colours, as in Fig. 2. Data points in magenta indicate that λ_alt was complex (the real part is shown). B, plot of ζ_alt vs. λ_alt during the experiment. C, plot of β₁ vs. λ_alt during the experiment. D, plot of mean APD vs. mean DI in the sliding window, using the colour defined in A for λ_alt. The onset of full alternans is denoted by the arrow.

Importantly, incorporating stochastic CL variations permitted the estimation of two additional markers, β₁ and ζ_alt. β₁ (corresponding to the conventional S1S2 restitution slope), slightly increased during the experiment and reached 0.2 at the onset of alternans, i.e. ≪1. Interestingly, ζ_alt decreased gradually in parallel with λ_alt, but it always remained slightly more positive. Consequently, the difference ζ_alt – λ_alt was positive but remained close to 0. These relations between β₁, ζ_alt and λ_alt are further illustrated in Fig. 3 B (ζ_alt vs. λ_alt) and Fig. 3 C (β₁ vs. λ_alt). Figure 3 B shows that ζ_alt evolved along the diagonal defined by ζ_alt  = λ_alt but remained above it, and Fig. 3 C shows that β₁ remained near 0. The dynamic restitution slope (Fig. 3 D) was approximately 0.5 at the onset of alternans, i.e. ≪1.

Similar results were observed in 8 out of 10 cells (from six animals), in which λ_alt reached –0.8 when alternans developed. In these experiments, ζ_alt followed a course similar to λ_alt but always remained >λ_alt (ζ_alt – λ_alt always positive), and β₁ followed a course near 0 (β₁ < 0.5). The observation that ζ_alt remained close to λ_alt suggests that alternans was Ca²⁺ driven in these experiments (Lemay et al. 2012), and the positive sign of ζ_alt – λ_alt indicates that Ca²⁺‐to‐APD coupling was positive. In one of the 10 cells, the seal was lost before the end of the pacing protocol; alternans had not developed and λ_alt remained >–0.8. Interestingly, in one other cell that exhibited alternans, ζ_alt also followed a course similar to λ_alt but ζ_alt – λ_alt was negative, suggesting according to theory (Lemay et al. 2012) that Ca²⁺‐to‐APD coupling was negative in this particular cell.

Ryanodine changes the dynamics governing alternans

It is established that Ca²⁺‐driven alternans results from unstable cycling of intracellular Ca²⁺, especially due to a steep relation linking Ca²⁺ influx and sarcoplasmic Ca²⁺ load to Ca²⁺ release (Qu et al. 2010). Because the ryanodine receptor is a central element in Ca²⁺‐induced Ca²⁺ release, we evaluated the behaviour of APD and the different markers during similar stochastic ramp experiments in the presence of ryanodine (100 μmol l⁻¹).

Ryanodine caused APD shortening and alternans occurred at shorter cycle lengths. Therefore, the standard deviation of the stochastic pacing interval variations was reduced to 3–5 ms. Figure 4 illustrates a representative experiment with ryanodine. In this example, λ_alt progressively decreased until sustained alternans developed (around beat 2350) when λ_alt reached –0.8. However, ζ_alt did not decrease conjointly with λ_alt but fluctuated around 0 during the entire experiment, whereas β₁ and λ_alt – ζ_alt increased towards +1. Figure 4 B shows that ζ_alt remained around 0 independently of λ_alt, and Fig. 4 C shows that β₁ evolved along the diagonal defined by β₁  = –λ_alt. As in experiments without ryanodine, alternans exhibited irregularly occurring phase reversals once it had fully developed. These phase reversals thus appeared as a consequence of the stochastic CL variations. The dynamic restitution slope at the onset of alternans (Fig. 4 D) was approx. 0.8, i.e. close to 1.

Figure 4. Ramp protocol with stochastic variations in a ventricular myocyte exposed to 100 μmol l⁻¹ ryanodine .

Same layout as in Fig. 3.

Out of 15 cells (from nine animals) exposed to ryanodine, manifest alternans with λ_alt < –0.8 was observed in four cells. In these experiments, ζ_alt always remained close to 0 and ζ_alt – λ_alt increased above 0.5. In 10 of the remaining 11 cells, λ_alt nevertheless decreased to values <–0.5, APD exhibited intermittent bursts of alternans, and the tendency of ζ_alt to remain near 0 and of β₁ to increase was also observed. Thus, ryanodine changed the dynamics governing alternans and the relationship between the stochastic CL series and the resulting APD series in a manner compatible with a switch from essentially Ca²⁺‐driven to essentially voltage‐driven alternans.

In a next step, we examined the capability of the markers ζ_alt and β₁ to discriminate between experiments with and without ryanodine. Because APD at the onset of alternans was different for every cell, we conducted this analysis independently of APD and CL by considering only the relationships between ζ_alt, β₁ and λ_alt. For this purpose, we pooled the results of all the estimations of these three markers in all experiments (including experiments in which sustained alternans was not observed), as shown in Fig. 5 A. In the plot of ζ_alt vs. λ_alt, the distribution of the points and the regression lines for ryanodine vs. control experiments confirm that ζ_alt followed λ_alt in control experiments whereas it remained near 0 in the presence of ryanodine. The plot of β₁ vs. λ_alt confirms that β₁ changed only moderately in control experiments whereas it followed the diagonal β₁  = –λ_alt in the presence of ryanodine. To analyse the discriminating capability of ζ_alt and β₁ in terms of specificity and sensitivity, the data were collected in bins of λ_alt with a width of 0.1, and a receiver operating characteristic curve was constructed for each bin, as illustrated in Fig. 5 B. This analysis shows that the specificity and sensitivity of ζ_alt and β₁ (calculated from a single series of 100 consecutive CLs and APDs) to discriminate between the presence and the absence of ryanodine reaches >80% when λ_alt < –0.4 and increases with λ_alt approaching –1. Since the discrimination threshold cannot be determined a priori, we also determined in Fig. 5 C the specificity and sensitivity of the criteria ζ_alt > 0.5·λ_alt and β₁ > –0.6·λ_alt (illustrated by the green dotted lines in Fig. 5 A). Overall, ζ_alt was more specific and β₁ more sensitive, and the sensitivity/specificity was >80% even for λ_alt > –0.5, i.e. for a low susceptibility to alternans without any manifest alternation in APD time series.

Figure 5. Discriminative capability of ζ_alt and β₁ .

A, plot of ζ_alt (top) and β₁ (bottom) vs. λ_alt for all experiments pooled together (blue: control without ryanodine (10 cells from 6 animals); magenta: 100 μmol l⁻¹ ryanodine (15 cells from 9 animals)). Blue and red lines are corresponding regression lines. B, receiver operating characteristic curves for ζ_alt (top) and β₁ (bottom) for data in different bins of λ_alt. C, specificity and sensitivity of the criteria ζ_alt>0.5·λ_alt (top) and β₁ > –0.6·λ_alt (bottom) to detect the use of ryanodine (these criteria are shown as green dotted lines in A).

To ascertain whether excessive cell dialysis could affect the results, additional experiments (two under control conditions, two with 100 μmol l⁻¹ ryanodine) were conducted using the perforated patch technique. Similar behaviours of the different markers were observed, suggesting that using the ruptured patch technique did not alter the main outcomes of the analyses.

Computer simulations confirm experimental observations

To gain deeper insights into the progression to alternans during the ramp protocols and to support the notion of a switch from principally Ca²⁺‐driven to principally voltage‐driven alternans when the ryanodine receptor is blocked, we ran computer simulations using the Mahajan et al. rabbit ventricular myocyte model (Mahajan et al. 2008), which corresponds to our experimental preparation and which was especially developed to replicate the APD restitution curves observed experimentally as well as the alternans of APD and the Ca²⁺ transient at rapid pacing rates.

Figure 6 illustrates a simulation of a ramp protocol with stochastic variations. Alternans of APD appeared in bursts before becoming persistent at the end of the simulation. However, the peak of the Ca²⁺ transient already exhibited bursts of alternans early in the simulation, before APD alternans became manifest (insets in Fig. 6 A), indicating that alternans originated initially from unstable Ca²⁺ cycling and resulted later in visible APD alternans. The progression of λ_alt and ζ_alt during acceleration of pacing was similar to that in the experiments without ryanodine, with ζ_alt following λ_alt but being always less negative. The difference ζ_alt – λ_alt always remained in the interval between 0 and 0.2. The marker β₁ remained low as well. At the onset of APD alternans, when λ_alt reached –0.8, ζ_alt – λ_alt was 0.06 and β₁ amounted to 0.07. The slope of the dynamic restitution curve (Fig. 6 D) remained in the range 0.4–0.6 during the entire protocol.

Figure 6. Ramp protocol with stochastic variations in the Mahajan et al. model .

A, pacing CL, APD, λ_alt, ζ_alt (cyan), β₁ (S1S2 restitution slope, grey), difference ζ_alt – λ_alt (purple) computed in a sliding window (100 beats), and peaks of the Ca²⁺ transients. Odd and even APDs and corresponding Ca²⁺ transient peaks are shown in red and cyan, respectively. λ_alt is shown in distinct bands using different colours (green: 0 > λ_alt > –0.5; orange: –0.5 ≥ λ_alt > –0.8; red: λ_alt < –0.8). B, plot of ζ_alt vs. λ_alt during the simulation. C, plot of β₁ vs. λ_alt during the simulation. D, plot of mean APD vs. mean DI in the sliding window, using the colour defined above for λ_alt.

To test whether Ca²⁺ cycling could be stabilized and the regime of alternans switched from essentially Ca²⁺ driven to essentially voltage driven, we simulated the application of ryanodine by decreasing the strength of the Ca²⁺ release current through ryanodine receptors (g _RyR) by 90%, as illustrated in Fig. 7. As in the experiments, APD was decreased by this intervention and a ramp protocol descending to CLs < 100 ms was used, with stochastic variations having a SD of 2 ms. Similar to the control model, λ_alt gradually decreased towards –1 until full alternans developed. However, the peaks of the Ca²⁺ transients exhibited manifest alternans only in conjunction with APD alternans (insets). Furthermore, as in the experiments with ryanodine, ζ_alt clearly remained close to 0 and, consequently, ζ_alt – λ_alt increased to 1. β₁ increased to 1 as well. The dynamic restitution slope was steeper than in the control model (maximal slope: 1.1; slope at the onset of alternans: 0.7; Fig. 7 D). This behaviour of β₁, ζ_alt and the dynamic restitution slope (close to 1, 0 and 1, respectively) is in agreement with voltage‐driven alternans, i.e. with a steep dependence of APD on the previous DI as proposed in classical restitution theories.

Figure 7. Ramp protocol with stochastic variations in the Mahajan et al. model with Ca²⁺ release strength reduced by 90% .

Same layout as in Fig. 6.

Thus, in terms of λ_alt, ζ_alt, β₁ and dynamic restitution slope, the behaviour of the Mahajan et al. model was very consistent with that observed in the experiments, and decreasing g _RyR by 90% clearly changed the dynamics of alternans and switched its mechanism from principally Ca²⁺ driven to principally voltage driven. In the experiments, we used a high ryanodine concentration that blocks Ca²⁺ release and leads to Ca²⁺ accumulation in the sarcoplasmic reticulum (SR). This accumulation was confirmed in the Mahajan et al. model. This resulted in relatively smaller variations of [Ca²⁺] in the SR during the AP cycle, which, in turn, decreased Ca²⁺ release fluctuations. The feedback mechanism that causes Ca²⁺‐driven alternans was therefore depressed.

To evaluate the robustness of our approach, we ran the simulations shown in Figs 6 and 7 ten times with the same descending ramps but with different realizations of the stochastic variations of CL and analysed λ_alt, ζ_alt, β₁ and ζ_alt – λ_alt in terms of mean and SD over the 10 different runs. There were subtle differences from run to run, but the mean behaviour of the alternans markers was similar and consistent with the individual simulations shown in Figs 6 and 7. The SD of the markers was typically in the range of 0.1 far from the alternans regime and the SD of λ_alt strongly decreased when λ_alt approached –1. We also examined the effect of changing the SD of the stochastic CL variations. When these variations were decreased by half, the average behaviour of the markers was similar, and their SD was lower. However, when the SD of CL variations was doubled, some stimuli applied after particularly short stochastic CLs failed to be captured because they fell within the refractory period. These observations suggest that in an experimental setting, the SD of CL variations should be optimized to provide a sufficient signal‐to‐noise ratio in the measured APD series without causing stimulation failure.

Ca²⁺‐driven alternans is also identified in a human ventricular cell model

To examine whether the stochastic pacing approach could also be applicable in the context of the human heart, we conducted simulations using the O'Hara et al. (ORd) human ventricular model (O'Hara et al. 2011). Figure 8 A illustrates the behaviour of APD, Ca²⁺ transient amplitude and alternans markers in the original control model (ramp descending from 1000 ms; SD of stochastic variations: 3 ms). To demonstrate the influence of Ca²⁺ dynamics on alternans generation, Fig. 8 B shows the behaviour of the ORd model when the Ca²⁺ concentrations in all subcellular compartments were clamped to their initial values.

Figure 8. Ramp protocol with stochastic variations in the ORd human ventricular cell model under control conditions and under conditions of Ca²⁺ clamp .

A, control ORd model. Pacing CL, APD, markers λ_alt, ζ_alt, β₁ and ζ_alt – λ_alt (sliding window of 16 beats) as well as Ca²⁺ transient amplitude are shown vs. beat number. Layout similar to Figs 6 A and 7 A. B, ORd model with all Ca²⁺ concentrations clamped to their initial values. Same layout as in A. C, superimposed plots of mean APD vs. mean DI (left) in the sliding window, ζ_alt vs. λ_alt (middle), and β₁ vs. λ_alt (right) for the two situations investigated in A and B.

In the control model, alternant fluctuations of Ca²⁺ transient amplitude appeared before those of APD, and they were also more prominent. The decrease of λ_alt, towards –1 was accompanied by a parallel decrease of ζ_alt whereas β₁ and λ_alt – ζ_alt, remained <0.3, similar to the behaviour of the rabbit myocytes. This is in line with the notion that alternans is Ca²⁺ driven in the control model, as reported (O'Hara et al. 2011). Under conditions of Ca²⁺ clamp, APD was prolonged due to the missing Ca²⁺‐based inactivation of the L‐type Ca²⁺ current. But importantly, the behaviour of ζ_alt (close to 0), of β₁ (close to 1) and λ_alt – ζ_alt (close to 1) during the decrease of λ_alt reflected the disappearance of Ca²⁺‐driven alternans and the appearance of voltage‐driven alternans. The dynamic restitution slope (Fig. 8 C) was larger in the Ca²⁺‐clamped model than in the control model. The trajectories in the plots of ζ_alt vs. λ_alt and β₁ vs. λ_alt (Fig. 8 C) were characteristic of Ca²⁺‐driven alternans in the control model and of voltage‐driven alternans in the Ca²⁺‐clamped model. These results therefore suggest that Ca²⁺‐driven alternans can also be correctly identified in human ventricular myocytes using stochastic pacing protocols.

Discussion

We investigated the susceptibility to alternans of isolated rabbit ventricular myocytes by using a pacing protocol combining a progressive decrease of CL with stochastic CL variations. The analysis of the resulting sequences of APDs using an autoregressive‐moving‐average model permitted the experimental estimation of the eigenvalue λ_alt, which was shown in theoretical studies to be the ultimate marker of alternans (Li & Otani 2003; Groenendaal et al. 2014). Another innovative aspect of our analysis and the principal advantage of stochastic pacing is that it allows the extraction of further markers to characterize alternans (ζ_alt, ζ_alt – λ_alt, β₁).

In a previous computational study (Lemay et al. 2012), we showed that these markers exhibit distinctive signatures in a mathematical model which can be parameterized to exhibit voltage‐driven or Ca²⁺‐driven alternans (Sato et al. 2006). To ascertain these distinct signatures experimentally, we used ryanodine to depress Ca²⁺‐induced Ca²⁺ release, a key player in Ca²⁺‐driven alternans (Qu et al. 2010; Edwards & Blatter 2014; Wagner et al. 2015). Our experiments demonstrated the theoretically anticipated change of ζ_alt and β₁ suggestive of a shift from Ca²⁺‐driven to voltage‐driven alternans. Our findings are consistent with the notion that rabbit ventricular myocytes are prone to Ca²⁺‐driven alternans, as shown recently using dual voltage and Ca²⁺ recordings (Kanaporis & Blatter 2015).

We also found that ζ_alt and β₁ discriminated between the presence and absence of ryanodine with a specificity and sensibility reaching already ∼80% when the propensity to alternans was low (λ_alt > –0.5). Therefore, our approach can not only serve to detect the imminence of alternans by using λ_alt, but also to identify whether it is prone to occur via a Ca²⁺‐driven or voltage‐driven mechanism.

Further support to our approach was provided by simulations with the Mahajan et al. model of the rabbit ventricular myocyte (Mahajan et al. 2008), which was specifically developed to replicate APD dynamics and the Ca²⁺ cycling behaviour during rapid pacing. The behaviour of λ_alt, ζ_alt and β₁ in the model was consistent with our experimental observations. We then conducted simulations using a recent model of the human ventricular myocyte. The simulations suggest that it should also be possible to identify Ca²⁺‐driven alternans in human ventricular myocytes using stochastic pacing protocols.

Our work finally shows that conventional restitution slopes are poor markers to identify the occurrence of alternans (except in the case of a purely voltage‐driven mechanism without memory of previous pacing cycles as in classical restitution theory (Nolasco & Dahlen 1968; Guevara et al. 1981; Chialvo et al. 1990)). This further stresses the importance of revisiting the classical restitution concept in the genesis of alternans.

It must be underlined that both voltage and Ca²⁺ driving mechanisms contribute jointly to varying degrees to the generation of alternans rather than only one or the other (Edwards & Blatter 2014; Groenendaal et al. 2014). In the experiments, the mechanisms of unstable voltage dynamics that were revealed by the application of ryanodine were presumably present before, and they were certainly present in the simulations since the formulation of membrane currents was not changed. It is possible that both our experiments and simulations represented situations in which one mechanism clearly outweighed the other, and, in future studies, it would therefore be important to also investigate situations in which both mechanisms contribute to alternans to a similar extent and how our alternans markers evolve during a gradual transition from one to the other mechanism.

Extension of the analytical framework from cell to organ

In this study, we conducted experiments with single cells. In intact tissue, electrotonic interactions and conduction velocity restitution may further influence the mechanisms of alternans and give rise, e.g., to spatially discordant patterns (Qu et al. 2010). While this adds a level of complexity to the mechanisms underlying alternans, it does not preclude the validity of our analytical method. Indeed, dominant eigenvalue analysis based on principal components analysis or maximum likelihood estimation has recently been proposed (Petrie & Zhao 2012) and successfully applied (Kakade et al. 2013) on optical recordings of membrane potential in whole hearts during a perturbed downsweep protocol. Moreover, in clinical studies, QT interval stability analysis based on the magnitude of eigenvalues obtained via autoregressive modelling of the QT interval was shown to correlate with the propensity to ventricular tachyarrhythmias (Chen et al. 2011). These studies therefore suggest that our method is also applicable at the multicellular tissue and organ levels. We postulate that stochastic pacing may actually render it more reliable by continuously exciting the dominant eigenmodes that need to be identified. A recent study (Dvir & Zlochiver 2013) suggests that stochastic pacing in itself reduces the propensity to alternans and even prevents the transition to spatially discordant alternans by shifting the APD restitution slope, an effect explained on the basis of non‐linear switched system theory. In a translational context, stochastic pacing could therefore be used both as a diagnostic and preventive measure.

Nevertheless, these different aspects will require to be carefully evaluated in further studies. Such investigations are needed to clarify whether our approach can be used gainfully in the setting of the human heart. In this situation, it should be underlined that the recorded signal (electrogram or ECG) should be of sufficient duration and sampled at a sufficiently high rate to capture small APD or QT interval variations in the millisecond range. To deploy the maximal efficiency of ARMA model identification, cardiac CL series should also be fully uncorrelated (which is not the case during sinus rhythm or conventional pacing protocols). When full alternans appears during stochastic pacing, non‐linear model identification can then be applied to provide further insights into the characteristics of the period‐doubling bifurcation (Armoundas et al. 2002; Dai & Keener 2012).

Limitations

It was recently suggested that at the subcellular level, Ca²⁺‐driven alternans results from the interplay of the randomness of Ca²⁺ spark activation, the refractoriness of Ca²⁺ release units and spark recruitment (Qu et al. 2010). Mechanistically, Ca²⁺‐driven alternans can also involve early afterdepolarizations (Qu et al. 2010). We have not tested whether our approach can discriminate between these different factors.

Our study also has the limitation that we recorded APs but not Ca²⁺ transients. For this reason, we could not quantify the exact relative contribution of Ca²⁺‐driven and voltage‐driven alternans. A recent study (Groenendaal et al. 2014) conducted using simultaneous voltage and Ca²⁺ recordings showed that both unstable Ca²⁺ cycling and unstable ion channel dynamics contribute together to alternans; however, at the limit determined by λ_alt  = –1, one mechanism typically predominates. Simultaneous recording of voltage and Ca²⁺ thus offers an important tool for future studies. Nevertheless, our approach has the advantage that it can discriminate between Ca²⁺‐ and voltage‐driven alternans without the need to measure Ca²⁺ transients, which would hardly be feasible in a clinical setting.

Conclusion

Eigenmode analysis is emerging as a highly valuable tool to probe the dynamics of cardiac electrophysiological systems. The potential of this approach deserves to be explored further and opens prospects for future basic and translational research.

Additional information

Competing interests

The authors have no competing interests to disclose.

Author contributions

Patch clamp experiments were conducted at the Department of Anesthesiology and Perioperative Medicine, Division of Molecular Medicine, University of California, Los Angeles. Computer simulations were conducted at the Department of Physiology, University of Bern. J.P.K. and E.D.L. conceived and designed the study. R.V.M., M.A., N.P.B., R.O. and Y.P. acquired the data. Y.P. and J.P.K. analysed the data. J.P.K. and Y.P. drafted the manuscript. All authors contributed to the interpretation of the data and to a critical review of the manuscript for important intellectual content. All authors approved the final version of the manuscript, agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved and all persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

Funding

This work was supported by the Swiss National Science Foundation (grant number 31003A‐135016/1 to J.P.K.) and the National Heart, Lung and Blood Institute at the National Institutes of Health (grant number P01HL078931 to J.N.W.).

Translational perspective

The alternation (alternans) of cardiac action potential duration is an intricate dynamical phenomenon leading to life‐threatening heart rhythm disorders. Alternans can result from instabilities in ion current dynamics (voltage‐driven alternans) or in intracellular calcium cycling (calcium‐driven alternans). We examined the susceptibility to alternans of rabbit ventricular myocytes using pacing protocols comprising stochastic (random) beat‐to‐beat variations of pacing cycle length and tested the hypothesis that such an approach can provide more information about the dynamics of alternans compared to a pacing protocol without variations. From the series of resulting action potential durations, we derived novel markers that not only signal the imminence of alternans, but also discriminate between voltage‐driven and calcium‐driven alternans. We demonstrated that our approach can identify the change from calcium‐driven to voltage‐driven alternans dynamics induced by blocking the calcium‐induced calcium release from the sarcoplasmic reticulum. Our findings lead to the prospect of implementing stochastic pacing protocols for diagnostic purposes, e.g. in patients undergoing cardiac catheterization investigations or having an implanted pacemaker or defibrillator. This approach may assist to determine the patients’ susceptibility to alternans and associated arrhythmias and to identify to what type of alternans the patients are predisposed. Making this distinction is relevant for pharmacotherapy, because the propensity to calcium‐driven vs. voltage‐driven alternans may require different preventive or therapeutic measures (e.g. drugs acting on potassium currents vs. calcium fluxes) and be associated with different contraindications for certain drugs. Thus, the potential of stochastic pacing deserves further investigations and opens prospects for future basic and translational research.