Modeling the Not-So-Steady Heart

Physics 6, 77
Mimicking the stable but not perfectly periodic beating of the heart and similar biological systems requires a new kind of mathematical model.
Inconstant heart. Heart rates usually speed up and slow down in a cycle that is correlated with the breathing cycle. A new type of mathematical model can mimic this not-quite-periodic nature of the heart as well as its stability in the presence of environmental fluctuations.

At rest, your heart beats about 70 times per minute, but this rate can vary over time by as much as 20 percent. A research team has now formulated a new category of mathematical models that mimic this not-perfectly-periodic signal, a phenomenon that also occurs in many other biological systems. As the team describes in Physical Review Letters, the new models, which they call “chronotaxic,” can also reproduce the stability of these systems in the presence of the ubiquitous fluctuations, or noise, imposed by the environment. This framework could lead to new diagnostic procedures for abnormal heart function, as well as more general analysis techniques for oscillations in other biological systems.

Many systems in biology, such as heart beats, chemical production in the cell, and neural processes, exhibit nearly periodic behavior but with a frequency that oscillates in time. For example, the heart rate speeds up and slows down in a cycle that closely corresponds to the cycle of inhaling and exhaling. These variations may be important because a steady heart rate that deviates little from its average value is a predictor for heart disease, according to some studies [1]. So researchers have tried to develop techniques to analyze electrocardiograms (ECGs), the voltage signals produced by the heart, and also to develop mathematical models that reproduce their basic properties.

Previous models have involved the heart and lungs as interacting oscillators, but they tended to synchronize at a single frequency, unless there was a separate mechanism to vary the frequency. However, models with a variable frequency became erratic and unpredictable in the presence of environmental perturbations (random noise in the equations representing small body movements, temperature changes, and the like). “Stabilization [in the face of perturbations] is crucial for the operation of the heart,” says Aneta Stefanovska of Lancaster University in the UK.

To better model this stability, Stefanovska and her colleagues devised their new framework around “nonautonomous” systems, in which an external driver, or “master” oscillator (which could be the lungs), acts on a “slave” oscillator (such as the heart). The team set the master frequency with a specific periodic time dependence and showed that the slave would be drawn to follow this varying but predictable frequency, in a pattern that remains stable against perturbations.

To visualize this “chronotaxic” behavior, as the team calls it, they used a standard “phase space” plot, where a solution to the equations (the ECG, for example) is represented by a point that moves around on the plane. The point’s starting location depends on the details of a specific test, and if the system settles into a simple oscillation with a fixed frequency, the point permanently orbits the origin at a fixed speed on a circular path called a limit cycle.

Solutions to Stefanovska’s model ultimately orbit on a limit cycle but at a speed that varies with time. Including a moderate amount of noise in the model doesn’t alter this fate but simply causes some fluctuations in the point’s position. The graphical explanation is that the master in the chronotaxic model generates a moving equilibrium point, or “attractor,” that acts like a pace car setting the target motion as it orbits on the limit cycle [2]. All possible solutions are drawn toward the attractor, regardless of noise.

For a real-world test, Stefanovska and her colleagues measured the ECG of a healthy individual who was asked to breathe at a prescribed rate that varied in time, allowing the team to track the “master” contribution from the lungs. They found that the heart rate variations tracked the breathing rate, and further analysis showed that a chronotaxic attractor could account for the data. Stefanovska says future experiments may identify attractors coming from other drivers besides the lungs. Using such ECG analysis, she hopes in the future to distinguish “changes that are resulting from the heart’s own dysfunction from those resulting from changes that come from other systems influencing its function.”

Mathematician Peter Kloeden of Goethe University Frankfurt in Germany says the work suggests that, paradoxically, a person with smaller variations in their heart rate might have a weaker influence from the master and therefore a greater susceptibility to random perturbations. Federico Lombardi, a cardiologist from the University of Milan, believes the usefulness of the model is that it simplifies a complex signal like an ECG by isolating the inputs that are predictable from those that are random.

–Michael Schirber

Michael Schirber is a Corresponding Editor for Physics Magazine based in Lyon, France.


  1. M. M. Wolf, G. A. Varigos, D. Hunt, and J. G. Sloman, “Sinus Arrhythmia in Acute Myocardial Infarction,” Med. J. Aust. 2, 52 (1978), Pubmed:
  2. P. E. Kloeden, “Pullback Attractors in Nonautonomous Difference Equations,” J. Difference Eq. Appl. 6, 33 (2000)

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Nonlinear DynamicsMedical PhysicsBiological Physics

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