The Dynamics of the Human Heartbeat

Logan GilbertPomona College Department of Mathematics

May 15, 2015

1 Introduction and Overview

Perhaps contrary to popular belief, one’s heart rate isn’t a constant lub-dub; it’s in con-stant flux. And different people’s heart rates follow different patterns over time. It could besinusoidal, seemingly random, have erratic bursts, or appear to have some structure withoutbeing entirely predictable. Most of these variants, however, are maladaptive; it turns outthat a very predictable or very random heart rate over time is often an indicator of some sortof heart malfunction. This flies in the face of the traditional biological theory that a healthybody always tends towards homeostasis. Instead, some level of variation is good, and thismakes sense from a naive evolutionary perspective: a successful organism is one which canadapt to (sometimes rapidly) changing conditions. In particular, we might imagine situa-tions in which we need to suddenly increase our heart rate substantially (e.g. to run awayfrom a predator) or situations in which decreasing our heart rate is more efficient (e.g. whileresting). Heart rate ought to be adaptive to one’s environment, neither fully predictable norentirely random.

Figure 1: Only the second sample shown is from a healthy patient. [3]

Indeed, this theory has some merit: it’s known that one’s heart rate variability canchange over short time intervals - it’s not just the result of chronic conditions like heartdisease. Your HRV will decline when you’re stressed and increase when you’re not; athletesuse heart rate variability monitors on a daily basis to decide how hard they should train on

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each day. There are even iPhone apps which measure your HRV by detecting changes in thedarkness of your fingertip when you place it over the camera. [2]

So without too much effort, we can develop a reasonable qualitative understanding of thehuman heart beat and see some important implications of this field. But if we want to saysomething more precise about heart rates and be able to compare samples from healthy andunhealthy samples in a quantitative way, we’ll need a mathematical model.

2 Modeling Considerations

The natural place to begin an analysis is the simplest: we will abstract the heart ratefrom its natural environment, forget entirely about the context from which the data came,and simply analyze it as a time series. It will turn out that there is substantial analysis tobe done in this way, but note that it ignores the larger dynamics of the biological system inwhich the heart is embedded.

To begin our analysis, we’ll need some context regarding time series. First, in practicewe will only gather a sample of finite size and resolution. Naturally our measurements willbe subject to some degree of noise. As in any other statistical application, our goal will beto work around these constraints to say something meaningful about the “population” ofhuman heartbeats. Because of these constraints, when we later discuss the possibility thatheart rate time series are fractal or multifractal, we will not mean that they are literallycompletely self-similar. Informally, “fractal” will simply mean that there are strong auto-correlations in the time series.

Another difficulty we’ll have to confront is that of non-stationarities. Stationary datacomes from a population in which the underlying probability distribution function is thesame across time intervals. So, for example, we’d be just as likely to see a large value inthe interval [0,10] as in the interval [100,110]. Since moments uniquely define a probabilitydistribution, this is equivalent to the data having the same (or “stationary”) moments acrosstime. In particular, the mean and variance of stationary data is constant.

Stationary data is nice to work with. Unfortunately for us, however, heart rate data isnon-stationary. This is intuitive: one doesn’t expect the mean heart rate of a person whileexercising to be anywhere near their mean heart rate while asleep. But the complexity iseven greater than that: heart rate data turns out to be non-stationary in socially stressfulsituations and retains a great deal of complexity even while at rest.[1, 4]

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3 Specific Techniques and Mathematical Formalism

We are now in a position to analyze a technique used to analyze time series similar tothe ones often encountered in heart rate data. It’s known as detrended fluctuation analysis(DFA) and the goal of the analysis is to estimate the correlation dimension of the time series.This is useful because the estimate can then be compared to the correlation dimension oftest data with known characteristics.

Before jumping into DFA, we discuss the meaning and calculation of the correlation di-mension. There are a host of methodologies one might use in various contexts to calculatedimension; in practice one is often looking for an algorithm which is computationally lightwhile correlating highly with other measures of dimension. The correlation dimension hap-pens to satisfy both of these properties.

To calculate it, we begin by computing a measure known as the correlation integral. Thecorrelation integral is a function of a chosen value epsilon - we calculate it as limN→∞

gN2 ,

where N is the number of data points considered and g is the number of pairs of points (notnecessarily successive) which lie within epsilon of each other. Thus the correlation integralis effectively measuring the probability that two randomly selected points will be near oneanother. We then plot the correlation integral against varying values of epsilon (in a log-log plot) and find the slope of the graph. The intuitive understanding of why this yields ameasure of dimension comes when you imagine increasing epsilon on, say, the real line vs.increasing it in Rn: a given increase in epsilon along the line will result in just a marginallygreater number of points within epsilon of each other. But in Rn one opens up an entiren-dimensional space around each point where other points might be.

As mentioned before, knowing the dimension of one’s data is useful because one can thencompare the data to test data for which one knows whatever mathematical properties areof interest. But it also allows one to calculate the Hurst exponent of the data (H = 2 − d,where H is the Hurst exponent and d is the dimension), which describes the rate at whichautocorrelations in the time series diminish as one increases the time between sample values.This is an excellent measure of long-range dependency in the time series. [9]

Now that we know what we’re looking for, we detail the technique itself. Here’s how itworks: first, we split the time series into a number of equally-sized intervals and integrateit. Integration involves calculating the local average and subtracting this from each point,thereby helping to eliminate local trends; this is the detrending part of DFA. Within eachinterval, a trend line is constructed (the least-squares line works, though one may also per-form the analysis with a higher-order polynomial trend line). Naturally our data will notfit the various trend lines exactly. Thus we measure the fluctuations about the trends byfinding the root-mean-square deviation between the (integrated) data and the trend in eachinterval. This process is repeated for many different bin sizes, and the average fluctuationsize is plotted against the bin size in each case. If the plot is linear, we have strong reason

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to believe that there is self-similarity. And in particular, the slope of the plot gives us thecorrelation dimension. [6, 8]

4 Experimental Results

DFA has been conducted on healthy human hearbeat data, and it originally appeared asthough this simple fractal analysis was sufficient to describe the time series. Later, however,it was speculated that different parts of the signal were governed by different fractal scalingexponents. Such a time series is known as multifractal, and it turns out that heart rate datais better modeled in a multifractal way. [7] The techniques necessary for conducting such ananalysis are outside the scope of this paper.

A significant portion of the research focuses on the differences between time series sampledfrom healthy patients and patients with various heart conditions, including atrial fibrillationand congestive heart failure. These studies typically result in the same conclusion: the pres-ence of a heart condition correlates strongly with a decline in the complexity of the heartrate time series. [6] It is unclear which direction the causation flows; probably the heartcondition and the associated loss of complexity are attributable to a common factor.

In the future, it would be interesting to model the physiological systems which controlheart rate so one could understand the biological underpinnings of these patterns. Theauthor of this paper would also be interested to know how the Hurst exponents describinghuman heart rate compare to those of other organisms.

5 Conclusion

Analysis of human heart rate time series reveals that the data have an interesting struc-ture characterized by long-range autocorrelations; it is neither random nor entirely pre-dictable. The tools used to conduct these analyses are fairly sophisticated; scientists andreaders wishing to stay abreast of the research are well-advised to brush up on their knowl-edge of time series.

References

[1] Amaral, Luis, et al. Behavioral-Independent Features of Complex Heartbeat Dynamics.Phys. Rev. Letters 86, 6026. 25 June 2001. Web. 30 April 2015.

[2] Caceres, J. Hernandez et al. Towards the Estimation of the Fractal Dimension of HeartRate Variability Data. The Internet Journal of Cardiovascular Research 2.1 (2004): np.Web. 18 April 2015.

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[3] Cipra, Barry A. A Healthy Heart is a Fractal Heart. SIAM News 36.7 (2003): np. Web.18 April 2015. https://www.siam.org/pdf/news/353.pdf.

[4] Gao, J., et al. Multiscale Analysis of Heart Rate Variability in Non-Stationary Environ-ments. Frontiers in Physiology v.4 (2013): 119. Web. 30 April 2015.

[5] Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, MarkRG, Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank, Phys-ioToolkit, and PhysioNet: Components of a New Research Resource for Com-plex Physiologic Signals. Circulation 101(23):e215-e220 [Circulation Electronic Pages;http://circ.ahajournals.org/cgi/content/full/101/23/e215]; 2000 (June 13).

[6] Goldberger, Ary L. et al. Fractal Dynamics in Physiology: Alterations with Disease andAging. PNAS Vol. 9 Suppl. 1 (February 19, 2002): 2466-2472. Web. 18 April 2015.

[7] Ivanov, Plamen et al. Multifractality in Human Heartbeat Dynamics. Nature 399(6735)(3 June 1999): 461-465. Web. 30 April 2015.

[8] Kantelhardt, Jan. Fractal and Multifractal Time Series. April 4, 2008. Web. 30 April2015. http://arxiv.org/pdf/0804.0747v1.pdf.

[9] Theiler, J. Estimating the Fractal Dimension of Chaotic Time Series. The Lincoln Lab-oratory Journal 3.1 (1990):63-86. Web. 30 April 2015.

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