I have been measuring my heart rate recovery after exercise and I see that it can be fit reasonably well using a single exponential:

$HeartRate(t) = HR_{max} \times e^{-t/\tau} + HR_{resting}$

This observation is further supported by this paper: https://pubmed.ncbi.nlm.nih.gov/24494748/

From Wikipedia: "A quantity is subject to exponential decay if it decreases at a rate proportional to its current value"

So, the exponential decay suggests that the ability to slow down the heart depends on the heart rate itself. A faster beating heart somehow produces more of the signal to slow down the heart.

Is this the case? What is the mechanism that allows for such a feedback loop of the heart rate? Is the signal sent through the nerves as the heart muscles contract, or is the blood flow that allows a signal to travel?

Is there another way to explain the mono-exponential decay?

Here is an example from yesterday's run. I sprinted and then walked very slowly a few times. My heart recovery lifetime ($\tau$) including only the first three decays is 53 +/- 4 seconds. The last decay is after I ran up the stairs and then sat down, and its lifetime is of only 22 seconds.

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  • 1
    $\begingroup$ Exponential decay is characteristic of a first-order reaction, and where the exponential factor arises because the integral of 1/x (dx) is ln(x). Radioactive decay is a great example. For any first-order reaction, we can calculate the half-life, once the first-order rate constant is known. This is independent of concentration ... $\endgroup$
    – user338907
    Commented Nov 20, 2021 at 22:46
  • $\begingroup$ ... The half-life is usually expressed as ln(2)/k but in your case, it will be ln(2)*τ which I calculate to be about 36.7s. Thus after 36.7 seconds, HRmax will have decreased to half its original value (no matter what that initial value was), and after a further 36.7 seconds it will have decreased to half that value again, and so on $\endgroup$
    – user338907
    Commented Nov 20, 2021 at 22:48

2 Answers 2


I think most people would take exponential decay in a circumstance like this to be the null hypothesis. That is, if you had something not exponential decay, that would be curious and interesting and worth understanding better.

Exponential decay is a pretty fundamental concept in the universe in general and biology in particular.

So, the exponential decay suggests that the ability to slow down the heart depends on the heart rate itself. A faster beating heart somehow produces more of the signal to slow down the heart.

I would not think about it quite this way, and maybe that's why you're thinking there is something profound here about "ability to slow". Rather, I would think in homeostatic terms and consider the "set point". If your "set point" for heart rate is high during exercise and low during rest, you'd expect exponential decay if the physiological signal being acted on is a difference between the current rate and the set point. When the rate is much higher than the set point, you'd generate a bigger error signal and this would result in a faster decay.

More concretely, you could think in terms of regulators of heart rate like CO2 and catecholamines, as both of these are likely to follow an exponential pattern, CO2 because transfer out of tissues into the blood and out of the blood into the lungs are both going to be faster when CO2 concentrations are higher. Catecholamines are going to decrease by exponential decay through enzymatic degradation: when there are more molecules around to react, the rate of degradation will be faster.

  • $\begingroup$ Thank you! The precise nature of the "error signals" in homeostasis will keep me entertained for some time. I study the kinetics and dynamics of very simple chemical systems, and it is quite easy to take simple systems out of mono-exponential behavior. Even the systems that you mention - diffusion of CO2 and enzymatic degradation will often show non-exponential behavior unless special conditions are met. The emergence of mono-exponential behavior from a complex combination of dynamics over different time and size scales is quite fascinating, even if trivial! $\endgroup$
    – MaxParadiz
    Commented Nov 19, 2021 at 22:42

The answer by Bryan Krause is right on point from a system's perspective. From a cardiology point of view, heart rate recovery (HRR) post-exercise has been a significant consideration in cardiopulmonary exercise testing (CPET) in recent years.

Put succinctly, heart rate is influenced by the sympathetic and parasympathetic nervous systems. Increased sympathetic activity increases the HR and increased parasympathetic activity decreases it. The response of both systems to the cessation of exercise is a huge determinant of the behavior of HR post-exercise. So, abnormal HRR can potentially reflect imbalances between the sympathetic/parasympathetic systems.

In Chronic Heart Failure, the failing cardiac muscle prompts inappropriate neurohormonal activation of both systems, which is sometimes described as "flooring the gas pedal and the brake pedal at the same time". This can be directly observed (see Figure) in the HRR patterns of CHF patients, which is blunted because both the sympathetic and parasympathetic systems exhibit more inertia to change. Note that a lower HR is also achieved.

Adapted from Zweerink et al.

In fact, blunted heart rate recovery in CHF patients confers very poor prognosis.

Another interesting tidbit: In heart transplant recipients, one can essentially observe the behavior of a "denervated" heart, as all nerve connection have been severed*. In pediatric heart recipients, heart rate recovery (defined as peak HR - HR post 1-minute) was extremely slow compared to normal controls:

Adapted from Singh et al.

This finding highlights the significance of neural input to achieving a "normal", exponential HR decay.

*some innervation can be observed months-years post-transplantation


  • Zweerink et al. Chronotropic Incompetence in Chronic Heart Failure: A State-of-the-Art Review. Circulation: Heart Failure. 2018 Aug;11
  • Singh TP. Longitudinal changes in heart rate recovery after maximal exercise in pediatric heart transplant recipients: evidence of autonomic re-innervation? J Heart Lung Transplant. 2007 Dec;26
  • $\begingroup$ Welcome Anastasios. Please take our tour and refer to the help center as and when for guidance. Great first post. Enjoy Biology. $\endgroup$ Commented Nov 20, 2021 at 12:24

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