Dataset:
Goldberger, A., Amaral, L., Glass, L., Hausdorff, J., Ivanov, P. C., Mark, R., ... & Stanley, H. E. (2000). PhysioBank, PhysioToolkit, and PhysioNet: Components of a new research resource for complex physiologic signals. Circulation [Online]. 101 (23), pp. e215–e220. RRID:SCR_007345.
A healthy heart is not a metronome. Its beat-to-beat intervals fluctuate in a structured, non-random way. The paper by Kiyono et al. argues this fluctuation has a fractal, scale-free structure - the same statistical signature we see in systems sitting at a critical point (like water at exactly 100°C, or a magnet at its Curie temperature). This is not trivial: it means the heart's regulatory system is continuously tuned to operate near criticality, which is associated with maximum adaptability and sensitivity.
We measure the time between consecutive R-peaks (the big spike in an ECG). That gives us a sequence of intervals - say 0.82s, 0.79s, 0.85s, ... We then normalise it:
This removes the mean heart rate and units, leaving only the fluctuation structure.
We sum b(i) to get:
This turns the heartbeat series into a random walk. Why? Because integration amplifies long-range correlations and makes them easier to detect. A purely random heartbeat would give a standard Brownian walk. A correlated one gives a different kind of walk - described by the Hurst exponent.
We chop B(i) into windows of size s and fit a polynomial to each window. The residuals are the fluctuations after removing local trends. This is the core of Detrended Fluctuation Analysis (DFA) - it lets us study correlations without being fooled by slow drifts (like the subject gradually relaxing).
The RMS of those residuals is F(s). If:
then H is the Hurst exponent. For healthy hearts, H ≈ 1.0, meaning the fluctuations are 1/f noise - long-range correlated, not random (H=0.5 would be white noise).
At each scale s, we compute lag increments:
and build a histogram. The key finding: this is not Gaussian. The tails are much fatter - large fluctuations happen far more often than a Gaussian predicts.
The right fit is a q-Gaussian (from Tsallis non-extensive statistics):
When q=1 this reduces to a Gaussian. When q>1 we get power-law tails. A healthy heart gives q ≈ 1.5, meaning extreme heart rate changes are significantly more probable than classical statistics would predict - the heart is built to handle (and produce) surprises.
This is the key test for scale invariance. We compute the PDF at many different scales s = 16, 32, 64, 128, 256 beats. Each PDF has a different width. We then rescale:
- x-axis: divide by
$s^H$ - y-axis: multiply by
$s^H$ (to conserve probability)
If the system is truly scale-invariant, all curves collapse onto one master curve. This is exactly what happens for a healthy heart - and it's what we'd expect from a system at a critical point, where there is no preferred length (or time) scale.
| Property | Random heart | Healthy heart |
|---|---|---|
| Hurst H | 0.5 (white noise) | ~1.0 (1/f, long-range correlated) |
| PDF shape | Gaussian | q-Gaussian, fat tails |
| Scale invariance | No collapse | Clean collapse |
The healthy heart sits in a critical state - not by accident, but because it's continuously regulated there. This gives it maximum dynamic range and the ability to respond rapidly to wildly different demands. A diseased heart (e.g. congestive heart failure) loses these correlations and becomes either more random or more periodic - both are worse.