Why is Heart Rate Recovery after exercise reasonably well described by a mono-exponential decay?

Question

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.

Answer 1

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.

Answer 2

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.

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:

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

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

References

• 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