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. 2010 Feb;57(2):316-24.
doi: 10.1109/TBME.2009.2028652. Epub 2009 Aug 18.

Detecting Space-Time Alternating Biological Signals Close to the Bifurcation Point

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Free PMC article

Detecting Space-Time Alternating Biological Signals Close to the Bifurcation Point

Zhiheng Jia et al. IEEE Trans Biomed Eng. .
Free PMC article

Abstract

Time-alternating biological signals, i.e., alternans, arise in variety of physiological states marked by dynamic instabilities, e.g., period doubling. Normally, a sequence of large-small-large transients, they can exhibit variable patterns over time and space, including spatial discordance. Capture of the early formation of such alternating regions is challenging because of the spatiotemporal similarities between noise and the small-amplitude alternating signals close to the bifurcation point. We present a new approach for automatic detection of alternating signals in large noisy spatiotemporal datasets by exploiting quantitative measures of alternans evolution, e.g., temporal persistence, and by preserving phase information. The technique specifically targets low amplitude, relatively short alternating sequences and is validated by combinatorics-derived probabilities and empirical datasets with white noise. Using high-resolution optical mapping in live cardiomyocyte networks, exhibiting calcium alternans, we reveal for the first time early fine-scale alternans, close to the noise level, which are linked to the later formation of larger regions and evolution of spatially discordant alternans. This robust method aims at quantification and better understanding of the onset of cardiac arrhythmias and can be applied to general analysis of space-time alternating signals, including the vicinity of the bifurcation point.

Figures

Fig. 1
Fig. 1
Uncertainty in detection of alternating signals. (a) Uncertainty in the bifurcation point: as a bifurcation parameter increases, a critical transition takes place, e.g., transition from a 1:1 response to 2:2 response to external perturbation. For a given system’s dynamics and in the presence of noise (including natural variability), there exists an uncertainty zone, for which it is challenging to determine the exact state of the system. (b) Uncertainty in transition zones between opposite phases of oscillation: over space, it is possible for neighboring regions to exhibit opposite phase of alternation. Between them a “nodal line” or no-alternation zone must exist. In real experiments, the width of this zone will depend on the system’s dynamics and the noise in the system. (c) TP of alternation in white noise: maximum-length alternating sequence, m, in a random binary signal of length n (left); TP of alternation in a random binary signal of length n (right). Presented empirical data (10 000 trials per point) show 95% confidence interval for m and TP. Based on these data, a threshold TP for alternans detection can safely be chosen above the curve (right).
Fig. 2
Fig. 2
Identification of time-alternating signals and derivation of (5). (a) Derivation of (5): the denominator is 2n all possible combinations of a binary string of length n. For the numerator: a given alternating sequence of length m can be positioned in (nm−1) positions with (nm−2) unassigned bits (central case), or positioned in two positions with (nm−1) unassigned bits (border case). The sum of both cases gives the total number of possibilities. (b) Time-alternating signals are identified based on their TP. From the original traces (synthetic signal is shown for simplicity), a representative signal parameter is extracted, e.g., peak height of calcium concentration. Then derivative and sign are taken along the time dimension to accentuate the alternating patterns and the signal is binarized. Interruptions in alternating patterns are found by locating the zeros after another derivative.
Fig. 3
Fig. 3
Specificity of alternans detection based on TP. (a) Specificity of alternans detection: the stars are the theoretical probabilities (5), while the solid lines represent empirical results, computed from white noise datasets. The theoretical and empirical data match perfectly for TP of 60%, 75%, and 90%. Signal length n ≥ 13 is needed for p < 0.05 (≥95% specificity of detection) at TP ≥ 60%, while shorter signals will suffice when enforcing stricter TP thresholds. (b) Spatial locations are said to exhibit alternans, if their TP exceeds some critical value, e.g., TP ≥ 60%. Two neighboring pixels A and B satisfying this criterion in the interval from T1 to T2, have their uninterrupted alternating sequences in the [t1, T2] and [T1, t2] intervals, respectively. There will be at least 20% temporal overlap for the different spatial locations, allowing the determination of RP, used for characterization of SDAs.
Fig. 4
Fig. 4
Sensitivity of alternans detection as a function of SNR and AR. (a) Variable SNR was generated by adding white noise to four perfect alternans sequences (TP = 100%) of length n = 30, having different AR = 6%, 10%, 20%, and 50%. Curves show 95% probability of detecting alternans of different AR at the corresponding SNRs and TP values. For TP ≥ 60%, all signals with SNR below the cross point with the dark horizontal line will be misclassified (false negative). Setting TP < 100% increases sensitivity, i.e., allows detection of alternans at lower SNR. (b) For low (left) and high (right) SNR, the 95% sensitivity limit of detection is shown at three different TP thresholds. Y-axis scale (ARs) is different for low and high SNR. Alternans in the area above the curves will be obscured by noise, while detecting alternans is safe with 95% sensitivity under the curves.
Fig. 5
Fig. 5
Frequency as control parameter for alternans development in space-time. (a) Experimental setup: cardiomyocytes are cultured on rectangular strips and paced during experiments with a line electrode on one side. A preprocessed Ca2 + fluorescence intensity signal from a single pixel is shown. (b) Spatial alternans patterns over different pacing frequencies. Gray scale represents AR with white areas not satisfying the TP criterion. Images on the left are for TP ≥ 60% (18/30 beats); images in the middle and to the right are the result of imposing TP = 100% (8/8 beats), with the eight beats selected at the beginning or the end of the 30 beat record. (c) Time series—the original intensity traces and derivatives of peak height in calcium are shown for two selected points A and B over different pacing frequencies. Blue and red in the left column represent the original data for points A and B, respectively. Blue and red in the middle and right column refer to derivative values during odd and even beats, respectively. For a perfectly alternating segment, all odd and even beats should be evenly distributed above and under the zero line, e.g., derivatives for A and B at 3.12 Hz (TP = 100% for both locations).
Fig. 6
Fig. 6
Identification of SDAs. (a) Phase-based identification of SDA patterns over different pacing frequencies. Color shows the magnitude of alternans at different spatial locations, quantified by RP × AR%. Red and blue identify opposite RP; green areas are regions with no detected alternans. Images on the left display all points with confirmed alternans; images in the middle show confirmed SDAs based on proximity criterion. (b) Enhancements of alternans by using the product of peak height (PH) and calcium transient duration (CTD) are due to changes in the same direction for both in response to frequency, and lead to improved noise resistance. (c) Large-scale alternans (AR ≥ 50) in solid contiguous in-phase regions are shown in contrast to previous examples with low AR and speckled spatial distribution. Two traces extracted from sites A and B illustrate spatial discordance.

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