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Conceived and designed the experiments: EdL JPK. Performed the experiments: JPK. Analyzed the data: ML JPK. Contributed reagents/materials/analysis tools: ML EdL JPK. Wrote the paper: ML JPK.

The authors have declared that no competing interests exist.

Alternans of cardiac action potential duration (APD) is a well-known arrhythmogenic mechanism which results from dynamical instabilities. The propensity to alternans is classically investigated by examining APD restitution and by deriving APD restitution slopes as predictive markers. However, experiments have shown that such markers are not always accurate for the prediction of alternans. Using a mathematical ventricular cell model known to exhibit unstable dynamics of both membrane potential and Ca^{2+} cycling, we demonstrate that an accurate marker can be obtained by pacing at cycle lengths (CLs) varying randomly around a basic CL (BCL) and by evaluating the transfer function between the time series of CLs and APDs using an autoregressive-moving-average (ARMA) model. The first pole of this transfer function corresponds to the eigenvalue (λ_{alt}) of the dominant eigenmode of the cardiac system, which predicts that alternans occurs when λ_{alt}≤−1. For different BCLs, control values of λ_{alt} were obtained using eigenmode analysis and compared to the first pole of the transfer function estimated using ARMA model fitting in simulations of random pacing protocols. In all versions of the cell model, this pole provided an accurate estimation of λ_{alt}. Furthermore, during slow ramp decreases of BCL or simulated drug application, this approach predicted the onset of alternans by extrapolating the time course of the estimated λ_{alt}. In conclusion, stochastic pacing and ARMA model identification represents a novel approach to predict alternans without making any assumptions about its ionic mechanisms. It should therefore be applicable experimentally for any type of myocardial cell.

Cardiac arrhythmias are frequent complications of heart disease and an important cause of morbidity and mortality. The rhythmic activity of the heart relies on the action potential, a bioelectrical signal characterized by complex dynamics involving ion currents and intracellular calcium cycling. When these dynamics become unstable, arrhythmogenic patterns can emerge. A typical example is the beat-to-beat alternation of action potential parameters, a phenomenon called alternans, which represents a well known mechanism precipitating severe arrhythmias. Alternans results from the interaction of action potentials during consecutive beats. Classically, this interaction is investigated by describing the dependence of action potential parameters on previous diastolic intervals and action potential durations. However, experiments have shown that quantitative markers derived in this way are only approximate or even inappropriate to predict alternans. Here, we devised a novel procedure for the reliable prediction of alternans, based on introducing small random variations of pacing intervals followed by signal processing in the frequency domain. Using a biophysical model of the cardiac cell, we demonstrate that our algorithm accurately predicts the onset of alternans during pacing at an accelerating rate or during the application of a drug. Our approach may thus open new perspectives for the clinical evaluation of arrhythmias.

In cardiac physiology, alternans designates the alternation of action potential (AP) parameters (e.g., AP duration (APD), calcium transient) from beat to beat _{m}), ion currents and intracellular calcium cycling, which can together lead to different types of dynamical instabilities

The classical understanding of the genesis of alternans is based on the concepts of restitution functions and iterated map models

However, it has been shown that the criterion α = 1 for the onset of alternans is only approximate or even inappropriate. Indeed, alternans can be present even if α<1, or conversely, APD may not alternate although α>1

In experiments, restitution is conventionally investigated by pacing at steady-state and by introducing premature or delayed stimuli (S1S2 protocol, S1S2 restitution curves), or by decreasing BCL stepwise and examining steady-state APD vs. DI at the end of each step (dynamic restitution curves). Results at odds with the classical theory have then motivated researchers to develop refined pacing protocols and analyses incorporating the notion of memory to investigate alternans, such as the “perturbed downsweep protocol”

In mathematical cardiac cell models, seminal insights have been obtained using eigenmode analysis _{alt} = min{Re(λ)} = −1. However, eigenmode analysis requires accessing the internal model variables and, therefore, it is not feasible experimentally. Thus, applications of eigenmode analysis have so far been limited to computer simulation studies

In previous work _{t→a}) and between the series of pacing intervals and the series of DIs (H_{t→d}), respectively. In the present study, we developed a generalized framework for a straightforward and accurate prediction of alternans. We devised an approach permitting to quantify the eigenvalue λ_{alt} and these transfer functions by using only experimentally measurable quantities (APD, DI) without the requirement to access internal model variables. The first step of this approach consists of using pacing intervals varying stochastically around a mean BCL. In the next step, the poles (including λ_{alt}) and zeros of the transfer functions H_{t→a} and H_{t→d} are identified by fitting an autoregressive-moving-average (ARMA) model to the recorded values of APD and DI

The power of this approach was evaluated in the cardiac cell model of Sato et al. _{m}-driven (alternans attributable to the gating kinetics of membrane ion channels, with a steep APD restitution curve), 2) Ca^{2+}-driven with positive Ca^{2+} to APD coupling (large Ca^{2+} transients generating longer APDs), and 3) Ca^{2+}-driven with negative Ca^{2+} to APD coupling (large Ca^{2+} transients generating shorter APDs). In the latter two versions, alternans originates from an instability of Ca^{2+} cycling and occurs even in the presence of shallow APD restitution curves, thus reproducing recent experimental and theoretical findings _{alt} obtained using ARMA model identification during stochastic pacing with the exact control value of λ_{alt} derived using eigenmode analysis. Our results provide the proof of principle that the eigenvalue λ_{alt} can be estimated from the time series of pacing cycle lengths, APDs and DIs, and thus that the criterion λ_{alt} = −1 could be utilizable experimentally.

We used the model of Sato et al. ^{2+} cycling.

The three different versions of the model were stimulated with 1-ms current pulses of 50 µA/µF as in the original study of Sato et al.

In all ionic cardiac cell models, including the Sato et al. model, the state of the cell at any time

At a given basic cycle length (BCL), there exists a unique function that maps _{i}^{th} stimulus) to _{i}_{+1} (at the onset of the ^{th} stimulus). At steady state, _{BCL}. As shown by Li and Otani _{BCL} is a small perturbation of _{BCL} and _{i}_{+1,r} is the ^{th} element of _{i}_{+1} and _{i}_{,c} is the ^{th} element of _{i}_{BCL}, applying the modified

As shown previously _{alt}) being equal to −1.

To derive the transfer function between a time series of pacing cycles fluctuating around BCL and APD, we first need to define and compute a vector ^{th} APD (_{i}_{BCL}) in response to a perturbation δ_{i}^{T} denotes transposition. Similar to the computation of _{BCL}, by applying the modified _{i}^{th} pacing interval (preceding the ^{th} AP) from BCL as_{BCL} = d_{i} of the pacing cycle length from BCL as^{T}

The transfer function H_{t→a}(^{−1}, where ^{−1} is a diagonal matrix with elements of the form 1/(_{t→a}(_{t→a}(_{n}(^{th} degree, and the poles of H_{t→a}(_{d}(

The transfer function H_{t→d}(_{i}_{i}_{i}

The frequency response in terms of gain and phase shift, corresponding to the ratio of the discrete-time Fourier transforms of the output (APDs or DIs) and the input (cycle lengths) is obtained by substituting ^{−1}^{−1} corresponding to the frequency of alternans (once every 2 beats).

The S1S2 restitution slope S_{S1S2} simply corresponds to_{dyn} can be inferred from the steady state response to a constant and continuous change δ^{2πif} = 1, and S_{dyn} corresponds to

The derivation presented above, based on eigenmode analysis, requires the knowledge of the internal variables of the cellular model. However, in a practical setting, only a restricted set of quantities are measurable, such as APD and DI. As proposed previously, the notion of memory in cardiac tissue can be approached by considering that APD depends not only on the previous DI but on several previous APDs and DIs _{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{n}_{t→a}(_{n}(_{d}(_{n}(_{d}(

Comparing the result of the previous section (Eq. 11) with Eq. 19, we first conclude that the transfer functions H_{t→a}(_{t→d}(

It must however be noted that modern cardiac cell models always exhibit features with very short time scales (e.g., the gates of the Na^{+} current) as well as features with considerably longer time scales (e.g., the filling of the sarcoplasmic reticulum with Ca^{2+}). Thus, mathematically speaking, cellular models of cardiac electrical activity belong to the category of stiff systems, for which it is known that some components dissipate considerably faster than others. These rapidly dissipating components correspond to eigenmodes which have eigenvalues close to 0. It then appears likely that all these models will have only few significant eigenmodes. Indeed, in the Beeler-Reuter ventricular cell model

The S1S2 restitution slope S_{S1S2} corresponds to_{dyn} is related to the coefficients α_{i}_{i}_{n}(_{d}(_{BCL}/_{BCL} represents a particular case with α_{1} = −α, β_{1} = α, and α_{2} = α_{3} = … = β_{2} = β_{3} = … = 0, for which the alternans criteria S_{dyn} = 1, S_{S1S2} = 1 and λ_{alt} = −1 are all equivalent.

For each BCL tested (in decremental steps of 5 ms from 1000 ms to 500 ms, and then in steps of 1 ms), the 3 versions of the Sato et al. model were paced at this BCL until a steady state 1∶1 response was obtained or until sustained alternans was documented. Because our goal was to be as close as possible (within a reasonable computational limit) to the true steady state when considering all model variables, steady state was considered to be attained when the relative beat to beat variation of all model variables was <10^{−7}. Steady state defined according to this criterion was obtained after 150–1000 beats. In presence of a stable 1∶1 response, the following protocols and analyses were conducted:

_{t→a} and H_{t→d} were computed as described above (Eqs. 1–12). The eigenvalue closest to −1 was defined as λ_{alt}, and S_{S1S2} and S_{dyn} were derived according to Eqs. 13 and 14. The eigenvalue closest to +1 was defined as the principal memory eigenvalue, λ_{mem}.

_{dyn}. Classical S1S2 stimulation protocols were used to construct S1S2 restitution curves for various BCLs (i.e., S1S1 intervals) and to measure S_{S1S2}. For every BCL at which the S1S2 protocol was conducted, steady state conditions were first obtained as described above.

_{t→a} and H_{t→d} together with their poles and zeros. S_{S1S2} and S_{dyn} were estimated according to Eqs. 20 and 21. H_{t→a} and H_{t→d} were also computed directly from the discrete Fourier transforms (DFT) of APDs, DIs and pacing cycle lengths (CL) as H_{t→a}(_{t→d}(

The results of these three analyses were compared and evaluated in terms of the ability of S_{S1S2}, S_{dyn} and λ_{alt} to predict the onset of alternans in the different versions of the Sato et al. model. In some simulations, a random error was added to APD to mimic experimental measurement error.

_{m}-driven model, the Ca^{2+}-driven model with positive Ca^{2+} to APD coupling and the Ca^{2+}-driven model with negative Ca^{2+} to APD coupling, respectively. Shortening these BCLs by 1 ms resulted in sustained alternans.

^{−7}). The vertical dotted lines denote the bifurcations to alternans. _{alt}. The eigenvalue closest to +1 is the principal memory eigenvalue λ_{mem}.

In _{alt}. reached −1 in all three model versions. The principal memory eigenvalue λ_{mem} remained close but always less than +1 at all BCLs tested. All the other eigenvalues were close to 0. Eigenvalues with an absolute value <0.1 (not shown) correspond to eigenmodes which dissipate by >99% after 2 beats, and which therefore have only very small influences on the dynamics of the model.

The time course with which the model stabilizes towards steady state can be inferred from λ_{mem}. Because λ_{mem} is the eigenvalue closest to 1 (in absolute value), it determines the slowest time scale in the model. During pacing at a given BCL, the corresponding eigenmode (E_{mem}) decays as

Each panel represents the dynamic restitution curve, i.e., steady-state APD vs. DI (blue) and several S1S2 restitution curves (labels indicate BCL, i.e., the S1S1 pacing cycle length). For each S1S2 restitution curve, a broad range of S1S2 intervals were explored, yielding DIs from 0 to 1000 ms. The intersections of the S1S2 restitution curves with the dynamic restitution curve are marked with red diamonds. The vertical gray bars denote memory amplitude at DI = 800 ms.

In all versions of the model, S1S2 and dynamic restitution functions never overlapped. As shown previously _{m}-driven model, the S1S2 restitution curves formed the closest pattern gathered around the dynamic restitution curve, and the S1S2 restitution curves were always monotonically increasing. In contrast, in the two Ca^{2+}-driven versions of the model, the S1S2 restitution curves deviated substantially from the dynamic restitution curve, especially at larger DIs. In the Ca^{2+}-driven model with positive Ca^{2+} to APD coupling, the prominent increase of APD at long DIs was the consequence of larger Ca^{2+} transients, resulting in AP prolongation. In the Ca^{2+}-driven model with negative Ca^{2+} to APD coupling, the S1S2 curves were non monotonic and each curve exhibited a segment with a negative slope. This phase of decreasing APD with increasing DI was the consequence of larger Ca^{2+} transients, resulting in this case in AP shortening.

Cherry and Fenton _{m}-driven and the Ca^{2+}-driven model versions with positive and negative Ca^{2+} to APD coupling, respectively. According to this criterion, the Ca^{2+}-driven models exhibit a larger amount of short term memory compared to the V_{m}-driven model. However, memory amplitude and λ_{mem} cannot be compared directly, because the former reflects APD changes over 2 (or a very few) beats, whereas the latter reflects the longest time scale of the model dynamics.

In the example illustrated in ^{2+}-driven cell model with positive Ca^{2+} to APD coupling was first paced at a constant BCL of 400 ms and exhibited a stable 1∶1 response at steady state. Subsequently, the cell was paced at CLs varying randomly with a SD of 5 ms around 400 ms. ^{rd} order ARMA model, accounting for >99% of APD variance with a residual variance <1%. The pole of the ARMA model closest to −1 was −0.780, very near to λ_{alt} = −0.790 computed using eigenmode analysis.

The model was paced at cycle lengths (CLs) varying randomly around 400 ms with a SD of 5 ms. ^{rd} order ARMA model (Eq. 18), and residual. The identified coefficients were: α_{1} = −0.2329, α_{2} = −0.7401, α_{3} = 0.0395, and β_{1} = 0.1351, β_{2} = −0.1058, β_{3} = −0.0293. The poles of the ARMA model were _{1} = −0.780, _{2} = 0.053, _{3} = 0.961. _{t→a} and H_{t→d} (gain and phase shift) derived using eigenmode analysis, computed from the ARMA model coefficients and obtained directly from ratios of discrete Fourier transforms (DFTs, on 512 consecutive cycles), respectively.

_{t→a} and H_{t→d} of the ARMA model with those derived using eigenmode analysis and those calculated directly from the ratios of the Fourier transforms of the APD, DI and CL time series. The three computations were all in agreement. Furthermore, the transfer functions obtained with the ARMA model matched almost exactly those predicted using eigenmode analysis, except at low frequencies <0.05 beat^{−1}.

These transfer functions represent the model behavior in terms of gain and phase shift in the frequency domain. The negative gain of H_{t→a} indicates that at a mean CL of 400 ms, variations of APD are small relative to variations of CL, and thus that the effects of APD restitution are moderate. Conversely, the gain of H_{t→d} around 0 shows that CL variations translate primarily to DI variations. However, the increase of H_{t→d} to +3.5 dB at f = 0.5 beat^{−1} indicates that DI variations at higher frequencies are actually amplified, revealing the propensity of the model to generate alternans.

_{S1S2} and S_{dyn}, respectively) and the markers λ_{alt} (alternans eigenvalue) and z_{td1} (first zero of the transfer function H_{t→d}) as a function of BCL. The restitution portraits (

_{S1S2} and S_{dyn}) and transfer function markers (alternans eigenvalue λ_{alt} and first zero of the transfer function H_{t→d} z_{td1}) as a function of BCL, identified using eigenmode analysis (EA), conventional restitution protocols (RP), and ARMA model identification during stochastic pacing (SD of CL: 5 ms), respectively. ARMA model identification was conducted on series of 30 successive cycles using a 3^{rd} order model; data points with error bars represent mean±SD for

_{S1S2} and S_{dyn} derived using eigenmode analysis were indistinguishable from those obtained using conventional restitution protocols. However, both S_{S1S2} and S_{dyn} were poor predictors of alternans as they always were <1 at the critical BCL at which alternans appeared. In contrast, λ_{alt} was always exactly −1 at the onset of alternans as predicted by theory. The behavior of z_{td1} further reflects the different dynamic mechanisms governing restitution. While z_{td1} remains near 0 in the V_{m}-driven model, it follows a nearly parallel course to λ_{alt} in the Ca^{2+}-driven models, but with z_{td1} > λ_{alt} for positive Ca^{2+} to APD coupling and z_{td1} < λ_{alt} for negative Ca^{2+} to APD coupling.

_{alt} in all three versions of the ventricular cell model, and this estimation was excellent at regimes when λ_{alt} was close to −1 (a feature which is essential for the practical prediction of alternans). Similarly, z_{td1} could be accurately estimated when it was larger than >0.5 (in absolute value). When these markers were close to 0 (e.g., in the V_{m}-driven model), the estimation became less reliable, in agreement with the notion that poles and zeros near 0 exert only a small influence on the dynamics of a time series, which renders their identification difficult

The combination of stochastic pacing and ARMA model fitting also permitted the reliable estimation of S_{S1S2} according to Eq. 20 without actually conducting an S1S2 protocol. However, the estimates of S_{dyn} with the ARMA model according to Eq. 21 were prone to a large variability (not shown). This is explained by the fact that H_{t→a} and H_{t→d} obtained with the ARMA model do not capture the transfer functions with a sufficient reliability at very low frequencies (see _{dyn} is given by H_{t→a} and H_{t→d} at _{dyn} with ARMA model fitting is thus prone to be less robust.

Representations of frequency response spectra are intuitively easier to interpret than corresponding sets of poles and zeros. Therefore, we investigated how the aspect of the transfer functions H_{t→a} and H_{t→d} behaves at regimes closer and closer to the bifurcation to alternans. As a reference, we first computed these transfer functions for the classical first-order memoryless restitution function APD_{n} = _{n−1}) _{t→a} = α/(_{t→d} = z/(^{2πif}, as can be deduced from Eqs. 12 and 15–19 and as we showed previously

_{t→a} and H_{t→d} in a theoretical first-order memoryless model given by APD_{n}_{n}_{−1}), with α being the slope of _{t→a} and H_{t→d} in the three versions of the Sato et al. model, computed by eigenmode analysis at various BCLs.

From _{t→d} and a negative gain (attenuation) for H_{t→a}. However, for all models and at regimes progressively closer and closer to the bifurcation to alternans (lighter redder curves), CL variations resulted in a more and more positive gain for both H_{t→d} and H_{t→a} at frequencies >0.4 beat^{−1}, with a peak at 0.5 beat^{−1}. This observation can be interpreted as an increasing propensity to alternans. This gain reached values up to 20 dB, which corresponds to amplification by a factor of 10. Thus, in regimes close to the development of alternans, variations of APD and DI may reach a level which is comparatively one order of magnitude higher compared to variations of CL.

The behavior of the transfer functions in the V_{m}-driven model (_{m}-driven Sato et al. model. In contrast, the behavior in the Ca^{2+}-driven models was clearly different. With positive Ca^{2+} to APD coupling (^{2+} to APD coupling (^{−1}, the phase shift of H_{t→a} is −2π instead of −π, a difference explained by the presence of a zero (_{td1}) more negative than λ_{alt}. Second, a singularity appears at ^{−1} in H_{t→d} (abrupt change in polarity) when this zero leaves the unit circle at −1 (see

Similar to ^{−1}. This range corresponds to time scales of >20 beats and is thus determined by long-lasting effects of memory (poles and zeros near +1), which cannot be captured accurately by the ARMA model. These effects manifest themselves in ^{−1}.

The analyses presented above were conducted in stationary regimes, for which mean BCL and cellular properties did not evolve with time. However, in electrophysiological experiments, the propensity to alternans is typically assessed by decreasing BCL (either stepwise or progressively) until alternans appears. Therefore, we examined whether determination of λ_{alt} using ARMA model identification would permit to anticipate the onset of alternans during a slow decrease of BCL. Results with the Ca^{2+}-driven model with positive Ca^{2+} to APD coupling are shown in

_{alt} and S_{S1S2} during the ramp decrease of CL with superimposed variations (SD = 1 ms). λ_{alt} and S_{S1S2} were estimated every 15 beats using a 2^{nd} order ARMA model from APDs and CLs in a window spanning the 30 preceding cycles. _{i}_{i}_{−1}) during the ramp decrease of CL without stochastic variations. _{alt} and S_{S1S2} in _{alt} and S_{S1S2} were estimated every 75 beats using a 2^{nd} order ARMA model from APDs and CLs in a window spanning the 150 preceding cycles. Data are presented as mean (bold lines) ± SD (shaded areas). In all panels, vertical dotted lines indicate the onset of macroscopic alternans (in _{alt} vs. mean BCL in the windows in which λ_{alt} was evaluated. _{alt}>−0.85, and the dotted orange arrows represent the extrapolation of these regression lines towards λ_{alt} = −1. The occurrence of alternans is marked with vertical dotted lines.

In _{alt} as well as S_{S1S2} were estimated from the data in each window using a 2^{nd} order ARMA model. In the illustrated example, λ_{alt} progressively approached −1 and the onset of alternans coincided with the moment when λ_{alt} reached −1 (vertical dotted line). Thus, observing the course of λ_{alt} as it gets closer to −1 allows anticipating alternans.

_{S1S2} is a poor predictive marker, as its value was only 0.4 at the onset of alternans. In _{i}_{i}_{−1}) reveals that microscopic alternans (micro-alternans) actually appeared at the same moment as anticipated in _{alt}. Similar results were obtained with the V_{m}-driven model and the Ca^{2+}-driven model with negative Ca^{2+} to APD coupling.

In an experimental setting, APD is always subject to measurement error. To evaluate how our analyses would be influenced by measurement error, we conducted 10 simulations as in _{alt} and S_{S1S2}. In these simulations, the SD of the random CL deviations was increased to 5 ms and the SD of the error added to APD was 1 ms. As illustrated in _{alt} (in absolute value) and a larger variability of the estimates, which necessitated increasing the number of cycles used for ARMA model fitting to 150. However, the estimation of λ_{alt} became progressively more accurate as it approached −1, and extrapolating the time course of λ_{alt} towards its intercept with the line λ_{alt} = −1 permitted to anticipate the onset of alternans as in the control situation without adding noise to APD. Interestingly, the estimation of S1S2 was not influenced by measurement noise.

It is also informative to analyze λ_{alt} as a function of BCL, as done in _{alt} vs. BCL was conducted for data points with λ_{alt}>−0.85. By extrapolating the regression line to λ_{alt} = −1, it was possible to anticipate the BCL at which alternans developed.

We then evaluated whether estimating λ_{alt} during a slow change of cellular properties would also allow anticipating the onset of alternans. The V_{m}-driven model was paced at a stationary rate (BCL = 320 ms) and the conductance of the slow delayed rectifier K^{+} current (I_{Ks}) was progressively reduced at a rate of 0.2% per second, starting from its nominal value of 100%, to mimic the slow application of an I_{Ks} channel blocker. _{alt} (estimated in windows of 30 cycles as in _{alt} progressively approached −1 and alternans appeared when λ_{alt} reached −1. This indicates that observing the course of λ_{alt} as it gets closer to −1 may also allow anticipating alternans during pharmacologic interventions. _{Ks} conductance. Manifest alternans appeared later than in _{alt}.

_{alt} during stochastic pacing (mean CL: 320 ms; SD of variations: 1 ms). λ_{alt} was estimated every 15 beats using a 2^{nd} order ARMA model from APDs and CLs in a window spanning the 30 preceding cycles. _{alt} in _{alt} was estimated every 75 beats using a 2^{nd} order ARMA model from APDs and CLs in a window spanning the 150 preceding cycles. Data are presented as mean (bold lines) ± SD (shaded areas). In all panels, vertical dotted lines indicate the onset of macroscopic alternans (in

The sensitivity of the estimation of λ_{alt} on noise added to the APD time series was investigated in _{alt} and the simulation was repeated 10 times. In these simulations, the SD of the random CL deviations was 2 ms and the SD of the error added to APD was 1 ms. The number of cycles used for ARMA model fitting was adjusted to 150. Adding noise to the APD time series resulted in a slight underestimation of λ_{alt} (in absolute value) but did not affect the time of its intercept with the line λ_{alt} = −1. Thus, predicting the onset of alternans was not precluded by the noise added to APD.

To investigate whether ARMA model identification during stochastic pacing offers a significant advantage over a simpler time domain analysis consisting of quantifying the decay of APD oscillations following a perturbation, we examined the response of the Sato et al. model to a step change of BCL. An example is illustrated in ^{2+}-driven model with positive Ca^{2+} to APD coupling after a step decrease of BCL from 400 to 390 ms. The step decrease of BCL caused transient decaying APD alternans, followed by an exponential convergence of APD to its new steady state at BCL = 390 ms. These two patterns reflect the alternans and memory eigenmodes, respectively. To quantify the decay of the alternans eigenmode, an exponential function was fitted to the absolute value of the APD difference series (|ΔAPD|). The time constant of this function provided an estimate of λ_{alt} of −0.800, which was close to the control value of −0.820 derived using eigenmode analysis. However, as shown in ^{rd} order ARMA model, identification over 30 cycles). In the presence of noise, the variability of λ_{alt} estimates was significantly smaller for ARMA model identification during stochastic pacing. Similar results were obtained at other BCLs and for the two other versions of the cell model. Thus, in the presence of noise, ARMA model identification is more robust than quantification of the exponential decay of APD alternation following a step decrease of BCL.

^{2+}-driven model with positive Ca^{2+} to APD coupling to a step decrease of BCL from 400 to 390 ms. _{alt} = −0.800. _{alt} = 0.643. _{alt} obtained with different methods, with eigenmode analysis serving as control. In the presence of noise added to APD, stochastic pacing and ARMA model identification was superior to the time domain analysis with exponential fitting (n = 37; * p<0.05, two-tailed Fisher-Snedecor's

Alternans is a clinically relevant phenomenon leading to dispersion of refractoriness, which precipitates conduction block and reentrant arrhythmias ^{2+} handling at the cellular level, but also intercellular interactions at scales ranging from tissue to the whole organ

In the present study, we revisit restitution by examining it in a generalized framework based on eigenmode analysis, a sound mathematical approach for the characterization of dynamical systems. Previous studies based on eigenmode analysis ^{−1}. Based on engineering notions of signal processing, we then devised a practical method to determine these transfer functions together with their poles and zeros using only time series of CLs, APDs and DIs. Our key finding is that the propensity to alternans can be quantified and monitored and thus the onset of alternans can be anticipated using the eigenvalue (first pole) λ_{alt} obtained via ARMA model identification during pacing at intervals varying randomly. The results of the computer simulations, conducted with models exhibiting three fundamentally different ionic mechanisms of alternans, not only support the general validity of this approach but also suggest that it may also be applied in non stationary regimes such as during a slow acceleration of the (average) pacing rate or the progressive application of a drug. Our approach is general because it can be applied to any model based on a biophysical description of ion currents as well as any higher-dimensional iterated map model of cardiac dynamics, such as the models of Vinet et al.

Our approach develops its full strength when pacing at randomly varying intervals is considered. Our simulations provide the proof of principle that stochastic pacing permits the estimation of the transfer functions H_{t→a} and H_{t→d} and thus of their first pole as a marker for the propensity to alternans. It is worth to note that in our simulations, ARMA model identification was excellent in estimating S_{S1S2}, even in the presence of measurement noise. Thus, the S1S2 restitution slope can actually be determined by stochastic pacing without the need to conduct an S1S2 protocol. However, the performance of ARMA model identification to estimate S_{dyn} was low. This observation is explained by the fact that the determination of S_{dyn} (using Eq. 21) is exquisitely sensitive to errors in the determination of the higher order coefficients of the ARMA model and that these errors cannot be decreased by increasing the order of the ARMA model. Indeed, in our simulations, >99% of APD variance was already described with a model of order 2 or 3, and increasing this order provided neither a better description of the dynamics, neither a higher reliability in computing λ_{alt} or restitution slopes. This indicates that the dynamics of the Sato et al. model, which has 16 variables, can be well represented by a lower dimensional model of order 2 or 3 in near stationary regimes. This also explains that the estimation of S_{dyn} using ARMA model identification is prone to a large variability, and thus that it would not be superior to a conventional pacing protocol in a practical experimental setting.

In this study, we used APD and DI as the system's output time series. It must be noted that our approach can also be used to derive the transfer function between pacing cycle lengths and any other output parameter such as the peak Ca^{2+} transient or the peak of a given ion current. Therefore, in an experimental setting, ARMA model identification during stochastic pacing could also be applied on series of peak Ca^{2+} transients, or, a fortiori, on local conduction velocities or mechanical parameters (e.g., peak force or shortening). Our approach also offers the advantage to be versatile. For example, exploring the frequency response of the Ca^{2+} transient in addition to the response of APD may uncover additional insights regarding the primary cause of alternans (voltage vs. Ca^{2+} driven), which may be pertinent in determining appropriate clinical therapeutic strategies (e.g. conventional pharmacotherapy targeting ion channels vs. new agents that may target the cellular Ca^{2+} handling machinery). This analysis was however beyond the scope of this work.

As demonstrated in the _{dyn} = 1, S_{S1S2} = 1 and λ_{alt} = −1 are equivalent. However, the equivalence of these criteria breaks down as soon as cardiac dynamics exhibit memory. Memory can thus be defined as any deviation from the first order map behavior. Thus, the notion of memory clearly explains why, in a more general setting, any prediction of alternans based on S_{dyn} or on S_{S1S2}

In a theoretical study, Tolkacheva et al. _{n}_{+1} = _{n}_{n}_{dyn}. However, our approach provides additional insights in the Z and frequency domains and links restitution to eigenmode analysis. As a principle, frequency domain analysis permits to understand cardiac dynamics in response to any arbitrary sequence of pacing intervals, including stochastic pacing and pacing at cycle lengths varying in an oscillatory manner.

In this latter context, the recent studies of Wu and Patwardhan deserve attention. To demonstrate memory effects, these investigators paced a ventricular cell ^{−1}), memory effects should become apparent at this frequency in graphical representations of transfer functions. Accordingly, at frequencies ≤0.01 beat^{−1}, the three versions of the Sato et al. model are characterized by manifest phase shifts (

Stochastic pacing and ARMA model identification would be straightforward to implement in any electrophysiological apparatus. Therefore, our approach could readily be translated to

Obviously, our theoretical framework must withstand the challenge of experimental validation. Because experimental data such as APD measurements are always subject to measurement error and because APD variability may also result from the stochastic gating of ion channels

From a theoretical point of view, it will also be necessary to extend the theory to multicellular systems in order to understand the influences of intercellular interactions, the effects of conduction velocity restitution

In conclusion, stochastic pacing combined with ARMA model identification represents a novel frequency domain approach to study cardiac dynamics. This approach should be applicable experimentally for the accurate evaluation of the propensity to alternans and the prediction of its onset. Because its mathematical foundation does not make any a priori assumptions about the ionic mechanisms of alternans, it pertains to any type of myocardial cell or tissue, irrespective of species, disease status or pharmacological interventions.