Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Abstract

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

Keywords: cardiac arrhythmias; dynamical systems; early warning signals; period-doubling bifurcations.

Fig. S1.

Spontaneously beating aggregates of 7-d-old embryonic chick cardiac cells following treatment with 1.7 μm E4031. The aggregates are the dark, ovoid-shaped regions located on the dish. The aggregates have an average size of 289 μm ± 29 μm. Imaging of several aggregates took place in parallel. The imaging takes place within a 1 cm² field of view, and the beat dynamics are recorded by monitoring the changes in light intensity from a pixel on the edge of each of the aggregates.

Fig. 1.

Period-doubling bifurcation in an aggregate of embryonic chick cardiac cells following treatment with a potassium channel blocker. (A) Interbeat intervals (IBI) through time following treatment with E4031, a potassium channel blocker, at t = 0. There are spaces between segments of the interbeat intervals (2–3 min in duration) for data storage reasons. A period-doubling bifurcation takes place at approximately t = 45 min. The traces below the interbeat intervals—i, ii, iii, and iv—represent time series corresponding with the interbeat intervals in the first, fourth, fifth, and tenth sets of interbeat intervals. Each trace is 37.5 s in duration. (B) A blown-up picture of the fifth set of interbeat intervals (from approximately t = 42 min to t = 48 min) corresponding with trace iii in A.

Fig. S2.

Interbeat intervals computed from spontaneously beating aggregates that undergo period-doubling bifurcations following treatment with E4031, a potassium channel blocker. Shown are the beat dynamics from eight spontaneously beating aggregates that undergo period-doubling bifurcations. Aggregates i–vi have stops in the recording (each stop is roughly 2–3 min in duration), and aggregates vii and viii are continuously recorded. The timing of the onset of the period-doubling bifurcation ranges from roughly 10 min to 55 min. The slope measure and the lag-1 autocorrelation coefficient in the neighborhood of the period-doubling bifurcation from these aggregates are analyzed in Fig. 4.

Fig. S3.

Interbeat intervals computed from spontaneously beating aggregates that do not undergo a transition in qualitative dynamics following treatment with E4031. Shown are the beat dynamics from eight spontaneously beating aggregates that do not undergo period-doubling bifurcations. These aggregates maintain their intrinsic beat frequency throughout the duration of the experiment. Analysis of these aggregates—as well as additional examples—are used to compute the false positive rate of the slope measure in Fig. S6.

Fig. 2.

Detection of noise amplification and oscillations in the ACF of the aggregate’s interbeat intervals in the neighborhood of the period-doubling bifurcation following treatment with potassium channel blocker. (A) Return maps of the interbeat intervals from the first (trace i), fourth (trace ii), and fifth (trace iii) sets of interbeat intervals from Fig. 1A. (B) Histograms of 250 detrended interbeat intervals for the three sets of interbeat intervals (traces i, ii, and iii) from Fig. 1A. Deviation represents the deviation of each interbeat interval from the mean value computed through the detrending process. The noise amplifies as the system approaches the period-doubling bifurcation. (C) Autocorrelation function (ACF) for a window of 20 detrended beats centered on the 150th beat for three sets of interbeat intervals. Damped oscillations emerge in the ACF in trace iii, consistent with the oscillations in the interbeat intervals observed in Fig. 1B.

Fig. S4.

Detection of noise amplification and oscillations in the ACF as the slope at the fixed point of a continuously perturbed linear map approaches −1. (A) Representative return maps for three values of slope A (the slope at the fixed point): −0.05 (Left), −0.65 (Middle), and −0.95 (Right). For these simulations, we applied normally distributed noise with an SD $\sigma =$ 0.01. (B) Histograms of the deviation from the mean for the last 4,000 values of x (as shown above in the return maps) for the three values of slope A. The red curves represent the PDFs computed using Eq. 4. (C) ACFs for the three above values of slope A. The blue dots represent the numerically computed ACF, and the red curve represents the analytical expression for the ACF calculated using Eq. 4.

Fig. 3.

Detection of noise amplification and oscillations in the ACF for a model of the data in the neighborhood of a period-doubling bifurcation. (A) Return maps computed from Eq. 1 for $\gamma =$ 3.0 (Left), 1.75 (Middle), and 1.5 (Right). (B) Histograms and PDFs corresponding to the data that make up the return maps. Deviation represents the deviation of each x value from the mean value of the sequence of x values. The red curve represents the analytic expression of the PDF computed from the linear approximation, Eq. 3. (C) The blue dots represent the ACF determined numerically, and the red curve represents the analytic expression computed from the linear approximation, Eq. 4, for all three values of γ.

Fig. 4.

Slope of a return map and the lag-1 autocorrelation coefficient represent quantitative measures that assess how far the aggregates’ dynamics are from a period-doubling bifurcation. (Traces i–viii) Each panel is a representative example of an aggregate for which we observed and captured a period-doubling bifurcation in the dynamics of the interbeat intervals. Each panel displaying the slope measure (Upper) is based on the interbeat interval trace given below it (Lower). The slope (in blue) represents the slope of a linear regression of a return map composed of a sliding window of the previous 20 detrended interbeat intervals. The lag-1 autocorrelation coefficient of the same sliding window of detrended interbeat intervals (in red) is consistent with the slope measure. The black hatched line represents the time at which the slope measure goes below −0.75 for five consecutive beats, which we consider the early warning signal threshold. The red hatched line represents the beat at which the slope first goes below −0.98 for at least five consecutive beats, which we consider the time at which the system goes through the period-doubling bifurcation.

Fig. S5.

Predictor quality depends on early warning threshold. Using 23 aggregates that undergo period-doubling bifurcations, we computed estimates of the fraction of aggregates that had gone through the early warning signal threshold as a function of beats before the bifurcation for three threshold values: −0.9 (blue), −0.75 (red), and −0.6 (green); 50% of the aggregates had gone through the threshold 11, 115, and 195 beats before the bifurcation for thresholds −0.9, −0.75, and −0.6, respectively.

Fig. S6.

False positive rate as a function of early warning signal threshold. Using data from 14 aggregates for which a transition in the qualitative dynamics did not take place following E4031 treatment, we computed the false positive rate as a function of early warning signal threshold. The red hatched line represents the early warning signal threshold equal to −0.75. See Fig. S6: False Positive Rate for a full description of the methods.

Fig. 5.

Slope of a return map and the lag-1 autocorrelation coefficient represent quantitative measures that assess how far the nonlinear map (Eq. 1) is from the period-doubling bifurcation. (A) The slope measure (in blue) represents the slope of a linear regression through a return map composed of a sliding window of the previous 20 detrended values of x (as given in B). The lag-1 autocorrelation coefficient of a sliding window composed of the previous 20 detrended values of x (in red) is consistent with the slope of the linear regression of the return map. Slope(γ) represents the slope of the fixed point as calculated using Eq. 1 and the current value of γ as given in C. The black dashed line represents the early warning, and the red hatched line represents the period-doubling bifurcation. (B) The value of x as numerically generated by Eq. 1 as a function of time. (C) The value of γ as a function of time.

Fig. S7.

Detection of noise amplification and oscillations in the ACF as a continuously perturbed 2D map simulating action potential duration in response to periodic stimulation approaches a period-doubling bifurcation. (A) Time series of action potential duration—200 values of the variable A from Eq. S1—for three values of stimulus period B: 300 ms (Left), 215 ms (Middle), and 202 ms (Right). The system approaches the period-doubling bifurcation as stimulus period B approaches 200 ms. (B) Histograms of the deviation from the mean computed from 801 values of the action potential duration variable A for three values of stimulus period B. Note that as the system approaches the period-doubling bifurcation, the noise amplifies. (C) ACFs computed from 801 values of the action potential duration variable A computed for the three above values of stimulus period B. Oscillations in the ACF emerge as the system approaches the period-doubling bifurcation. See Fig. S7: Analysis of a Continuously Perturbed 2D Model for a full description of the model and the methods of analysis.