Loop Gain, Delay and Oscillatory Breathing
Why a working control system can swing, and what can be inferred from that on an exercise test
A real exercise test
A 63-year-old man on a cycle ergometer, breathing into a mouthpiece. The work rate climbs smoothly throughout, so nothing is being applied to him in bursts. These are the two channels a routine test provides.
His ventilation does not settle, and his CO₂ follows the same rhythm in the opposite direction. The equipment is working normally and nothing is being applied to him at that rhythm. The rest of this page explains where the rhythm comes from.
1. A shower with a long pipe
The recording above is returned to later. To explain it, it helps to start with a system that has the same structure but is easier to think about.
You get into a shower. The water is too cold, so you turn the tap towards hot. Nothing happens, so you turn it further. Then the water becomes too hot, so you turn it back, and after a delay it becomes too cold again.
Nothing in that system is broken. The tap works, the boiler works, and your skin reports the temperature accurately. The average temperature over five minutes would be close to correct. The problem is in the timing.
Before reading further, try it. The target is 38 °C and the water is currently too cold.
🚿 Setting the shower temperature
The target is 38 °C. The water is currently too cold. Turn the tap and keep the temperature at 38 °C for as much of the 60 seconds as you can.
2. Open loop and closed loop
Before going any further it is worth being precise about what kind of system this is, because it is the structural idea the rest of the page depends on.
An open loop runs in one direction:
tap position → water temperature
Set the tap and the temperature follows. Nothing the temperature does affects the tap.
A closed loop feeds the output back to the input:
tap position → water on your skin → your hand → tap position
In both cases the water takes time to travel down the pipe. That delay is a property of the plumbing and it is there whether or not anyone is listening.
The difference is that the output becomes part of the input. You are not setting the tap once. You are continuously setting it based on the result of your own previous settings.
Almost every physiological regulator is closed in this sense, and so is the respiratory control system. Once a loop is closed, its behaviour is no longer a property of any single component. It is a property of the whole loop, including how long the signal takes to travel around it.
3. Why oscillation happens
The obvious explanation for the shower is that you were too heavy-handed with the tap. That is part of the answer but not all of it, and the missing part matters.
A heavy hand on its own does not cause oscillation. Suppose the mixer were at the shower head, so the water responded instantly. You could turn the tap as hard as you liked, feel the result immediately and stop. The temperature would not overshoot, because the information you need arrives before you have time to make things worse.
A long delay on its own does not cause oscillation either. Suppose the pipe were twenty seconds long but you only ever moved the tap a fraction at a time. The temperature would drift slowly to where you wanted it and stay there. Late information is harmless if nobody acts on it forcefully.
Oscillation requires a strong correction applied to information that has gone out of date. Either ingredient on its own is stable. The combination is not.
This explains what happened in the shower. When the water at your skin is cold, the water you have already asked for is still in the pipe and you cannot feel it yet, so you keep turning. By the time it arrives you have asked for far more than you needed, and the same thing then happens in the other direction. Each correction is appropriate for a situation that has already passed.
The strength of the correction as it travels once around the loop is called the loop gain. If a disturbance comes back around the loop smaller than it started, the oscillation dies away. If it comes back the same size or larger, it does not.
The loop has four parts, and the same four appear for the rest of this page.
4. Splitting the loop gain in two
"How strongly the loop reacts" is not a single quantity. It is two quantities multiplied together, and they describe different parts of the loop.
| The shower | Breathing | |
|---|---|---|
| Plant gain Gp | °C the water changes per notch of tap | mmHg the CO₂ falls per L/min of ventilation |
| Controller gain Gc | notches you turn per °C you feel wrong | L/min of ventilation per mmHg of CO₂ |
| Loop gain LG | Gp × Gc | Gp × Gc |
Multiplying them cancels the units:
(°C per notch) × (notches per °C) leaves a pure number.
Neither factor on its own can easily be compared between two people, because one is measured in mmHg per litre and the other in litres per mmHg, and both depend on how hard the person is working. Their product is dimensionless, which makes it possible to compare loop strength without carrying the units of either individual gain. Whether two such numbers are genuinely comparable still depends on the operating point, the frequency and the measurement method, so this is a convenience rather than a guarantee.
What each factor is
In the shower, the plant is the plumbing. A sensitive tap has a high plant gain: a small movement changes the temperature a lot. A worn tap that has to be turned a long way has a low one. Neither is faulty. They are different taps.
In the shower, the controller is the person. It is how far you turn the tap per degree of error you feel.
In respiratory control, neither block corresponds to a single organ. The plant is the whole dynamical relationship between the controller's output and the variable the controller senses, which includes the lungs, the CO₂ stores of blood and tissue, and the circulation that carries the signal. The controller is chemoreflex dynamics rather than a person making decisions. "Plumbing" and "person" are useful labels for learning the structure, not a description of the anatomy.
In a person, the equivalent of the tap is breathing and the equivalent of the water temperature is CO₂. Plant gain is approximately the mean CO₂ divided by the mean ventilation, so it falls as ventilation rises. Someone breathing 8 L/min at rest has a sensitive plant, because each extra litre changes their CO₂ noticeably. The same person exercising at 40 L/min has a less sensitive one. In resting heart failure the measured plant gain is about 0.89 mmHg per L/min, and during exercise about 0.48. Alveolar ventilation roughly triples from rest to exercise, which accounts for most of the difference.
Try it
Three controls: how sensitive the plant is, how hard the controller corrects, and how long the delay is. The product of the two gains is shown in the middle. The controls are labelled for breathing, and the toggle at the top relabels the same model as the shower.
One loop, three controls
Moving either gain changes their product. In this model the delay sets the cycle length and the gains set the size of the swings.
A teaching model rather than a physiological simulator. It is built so that the delay and the gains act separately, which makes the two ideas easier to see. In a real delayed feedback system they interact.
A property of the body rather than of the controller, so nothing turns it. Sliding it changes how closely the graduations on the dial are spaced.
Grab a handle and turn it, or focus it and use the arrow keys.
A single disturbance at 60 s, such as a sigh, and nothing after it. A loop below 1 settles eventually; the loop gain decides how long that takes.
In this simulator the gains and the delay act separately: the gains set the size of the swings and the delay sets the cycle length. That separation is a property of the teaching model rather than of a real delayed loop, where the two interact. Section 5 says more about where the delay actually goes.
5. Stability does not mean silence
This is probably the most useful idea on the page, and it is the one most often lost when oscillatory breathing is described as either present or absent.
Three words tend to be used as though they were interchangeable. They are not, and almost everything in this section depends on keeping them apart.
| Term | What it means |
|---|---|
| Oscillation | The signal goes up and down. |
| Resonance | The system amplifies disturbances at some frequencies more than others. |
| Instability | A disturbance grows rather than decays. |
Stability tells you whether an oscillation grows or decays. It does not tell you whether an oscillation is present. A system can be stable, resonant and continuously oscillating at the same time.
A loop gain below 1 does not mean there is no oscillation. It means that a single disturbance dies away instead of growing. Whether anything is oscillating at any given moment depends on whether the system is being disturbed, and real systems are disturbed constantly.
A shower is never perfectly steady, and breathing is never perfectly regular, so the loop is disturbed all the time. It responds most to disturbances arriving near the frequency at which its own delay has brought the returning signal into step with the disturbance that produced it. Near that frequency the response is amplified, by roughly
1 / (1 − loop gain)
which is about 1.3 times at a loop gain of 0.22, 2.4 times at 0.58 and 10 times at 0.9.
That expression is the intuition rather than the complete description. Writing the loop gain as a single number hides the fact that it is a property of one frequency. The delay has not disappeared from the problem: it is what decides which frequency the loop responds to most strongly, while the gains decide how large that response is. A full treatment needs the magnitude and phase of the loop at each frequency, which is not necessary here.
Switching the simulator to continuous disturbance shows the consequence: a visible oscillation in a system that is formally stable. It wanders rather than keeping strict time, because it is driven by noise rather than sustaining itself. A truly self-sustaining oscillation is regular. Real recordings usually are not.
Note also how modest the effect is at a loop gain of 0.58. The response is concentrated in a narrow band of frequencies rather than dominating the recording. Raising the loop gain towards 1 makes it emerge from the background.
6. Same output, different mechanism
Consider two bathrooms that produce identical temperature recordings, with the same cycle length and the same size of swing.
- Shower A has a sensitive tap and a person who moves it gently.
- Shower B has an insensitive tap and a person who moves it a long way.
Their loop gains are equal, because a high plant gain times a low controller gain equals a low plant gain times a high controller gain. The "same swing, two causes" button in the simulator shows this. The two traces lie on top of each other while the underlying values are quite different.
Two systems can produce the same observed oscillation while having different plant and controller gains. A single output trace cannot tell you which of them produced it. Separating the two requires measuring one of the factors independently.
This matters clinically because the two cases would call for different treatment. One would need the plant changed, the other the controller. It also explains why a second measurement channel is valuable, which is the subject of section 9.
7. What would convince us this is resonance
Before returning to the patient, it is worth asking what evidence would actually distinguish a resonating feedback loop from ordinary wandering. A large oscillation on the time trace is not enough on its own, because slow drifts, changes in effort and measurement artefacts all produce something that looks like a swing.
Three things together are much more convincing:
- A narrow peak in the frequency content. The variation should be concentrated near one cycle length rather than spread across many.
- Coherence between the two channels. Ventilation and CO₂ should oscillate together at that same cycle length, rather than one of them wandering alone.
- The phase relationship a negative feedback loop would produce. Breathing more removes CO₂, so the two should be close to opposite, and the departure from exactly opposite is itself informative.
A resonance can satisfy all three criteria without looking like an obvious oscillation on the trace. Clinical definitions of oscillatory breathing are visual thresholds: a minimum number of cycles, a minimum amplitude, for a minimum fraction of the test. A loop with a moderate loop gain can fail all of them while still producing a measurable narrow-band response. "No oscillation seen" and "no resonance present" are different statements.
8. Back to the patient
The toggle on the simulator relabels the same model as a shower and back again. Nothing else changes: the three controls, the arithmetic and the curves stay the same, and even the numbers on the axes stay put, because 38 is a reasonable shower temperature in °C and a normal arterial CO₂ in mmHg.
Ventilation takes the place of the tap position. CO₂ takes the place of the water temperature, and moves in the opposite direction because breathing harder removes CO₂.
The same recording, with the measurements
A 63-year-old man on a cycle ergometer. The quantities below are measured directly from these two traces.
The CO₂ sits 153° away from his ventilation rather than the 180° of an exact mirror. That shortfall carries information about the effective delay around the closed loop. Turning it into a number of seconds is a model-based inference rather than a direct measurement, because lung mixing, the gas stores, chemoreceptor dynamics and the measurement itself all contribute to it.
This recording meets all three criteria from the previous section. The variation is concentrated near a cycle length of 36.5 s, the two channels vary together at that cycle length, and the CO₂ sits close to opposite the ventilation. Taken together those are consistent with a resonating feedback loop rather than with drift or artefact.
This pattern during exercise is called exercise oscillatory ventilation, and it is associated with a poor prognosis in heart failure. It has usually been described as a pattern that is either present or absent. The loop model suggests a different way of thinking about it: rather than classifying the pattern, loop gain provides a continuous measure of how strongly the system resonates. That is a proposal about how to describe the phenomenon rather than an established clinical standard.
9. What two channels can and cannot tell you
A routine exercise test records ventilation and end-tidal CO₂ at the mouth. These provide the two observable signals needed to characterise the plant within this model. The size of the resonance gives the loop gain. The controller gain is then whatever is left over. Within this simplified decomposition of the gain into two factors:
Gc = loop gain ÷ plant gain
The plant is estimated from the two measured signals. The loop gain is estimated from the resonance. The controller gain is then calculated from those two estimates, and inherits the assumptions of both. No instrument is placed on the respiratory centre. Published methods generally recover at most two of the three quantities, so the third has to be assumed, and it is worth knowing which one that is before quoting any number.
End-tidal CO₂ is also not the same thing as the arterial CO₂ the chemoreceptors actually sense. It is a usable stand-in for it, and treating it as one is another of the assumptions this framework rests on.
The phase between the two channels is where the most care is needed. In the recording above the CO₂ sits about 153° from the ventilation rather than at exactly 180°. That shortfall carries information about the effective delay around the closed loop. Turning it into a number of seconds is a model-based inference rather than a direct measurement, because lung mixing, the gas stores, the dynamics of the chemoreceptor response and the measurement itself all contribute to it. The phase is observed. The circulation time is inferred, under assumptions that should be stated whenever the number is quoted.
Most patients are not at a loop gain of 1
The textbook account of oscillation says it occurs when loop gain reaches 1, the point at which a disturbance exactly replaces itself. Measured values in patients are lower than that.
| Cohort | Loop gain |
|---|---|
| Cunha 2023, 250 exercising patients | 0.222 ± 0.193 (range 0 to 0.915) |
| Sands 2017, 25 heart-failure patients at rest | 0.43 ± 0.21 |
| An exercising cohort of ours (n = 26, unpublished) | 0.58 ± 0.15 |
None of these averages is close to 1, and the highest single value among 250 patients was 0.915. On these measurements, oscillatory breathing is usually a lightly damped resonance rather than a self-sustaining instability, which is the practical form of the point made in section 5.
This has a consequence for anyone calculating a controller gain. Assuming a loop gain of 1 in order to solve for it, which is what the shortcut Gc = 1 ÷ Gp does, means dividing by 1 where the divisor should be closer to 0.58. Controller gains obtained that way come out roughly 1.75 times too large. It also produces a recognisable symptom: a loop gain of exactly 1.000 in every patient measured, which restates the assumption rather than reporting a result.
10. A classification rule is not a measurement
The patient at the top of this page has a cycle length of 36.5 seconds. A widely used clinical definition of exercise oscillatory ventilation requires a cycle length between 40 and 140 seconds. He falls outside that window by three and a half seconds, so the rule applied as written returns no oscillatory ventilation.
This is not an argument that the classification rule is wrong. It illustrates the difference between a classification rule and a physiological measurement. The rule answers a yes-or-no question using thresholds chosen for reproducibility. The signal still contains a measurable resonant response, whichever side of the threshold it falls. Both can be true at once, and they answer different questions.
A high plant gain is sometimes read as a sign that the lungs are the problem. It is a physical property rather than a fault, and everybody has one. What determines the strength of the feedback is the product of the two gains. A sensitive plant with a low controller gain behaves perfectly well.
Relevance to this course
Most models in this course map an input to an output. This one connects the output back to the input, which produces behaviour an open-loop model cannot: a system that is accurate on average while being wrong at almost every individual moment.
It is also an example of the difference between what is observed and what is inferred. The traces are data. The loop, its gains and its delay are a model imposed on those data, and the model is useful precisely because it makes claims that could turn out to be wrong.
What comes next
The harder question is how these quantities are estimated from a real cardiopulmonary exercise test. That reduces to three questions about the recording:
- Where is the resonance?
- How strong is it?
- What does its phase tell us?
Each of those needs a method, and each method rests on assumptions worth setting out in full. That belongs in a separate methods document.
Sands SA, Mebrate Y, Edwards BA, et al. Resonance as the mechanism of daytime periodic breathing in patients with heart failure. Am J Respir Crit Care Med 2017;195(2):237–246. doi:10.1164/rccm.201604-0761OC. The resonance argument, and measurements of plant gain, controller gain, both delays and loop gain in 25 heart-failure patients using dynamic inspired CO₂. The loop gain framework used on this page is theirs.
Cunha GJL, Maltês S, Rocha BML, et al., Agostoni P. Beyond exercise oscillatory ventilations: the prognostic impact of loop gain in heart failure. Eur J Prev Cardiol 2023;30(6):498–505. doi:10.1093/eurjpc/zwad021. Loop gain measured from the resonance in 250 exercising patients, with the finding that a binary classification of oscillatory ventilation loses statistical significance once loop gain is included in the model.
Khoo MCK, Kronauer RE, Strohl KP, Slutsky AS. Factors inducing periodic breathing in humans: a general model. J Appl Physiol 1982;53(3):644–659. doi:10.1152/jappl.1982.53.3.644. The original multi-compartment model, which predicts oscillation from assumed parameters rather than measuring it. Its controller gain is a modelled steady-state slope rather than the frequency-specific gain discussed here, and the two are not interchangeable.
Hall MJ, Xie A, Rutherford R, Ando S, Floras JS, Bradley TD. Cycle length of periodic breathing in patients with and without heart failure. Am J Respir Crit Care Med 1996;154(2 Pt 1):376–381.
Francis DP, Willson K, Davies LC, Coats AJ, Piepoli M. Quantitative general theory for periodic breathing in chronic heart failure and its clinical implications. Circulation 2000;102(18):2214–2221. doi:10.1161/01.cir.102.18.2214
Sands SA, Edwards BA, Kee K, et al. Loop gain as a means to predict a positive airway pressure suppression of Cheyne-Stokes respiration in patients with heart failure. Am J Respir Crit Care Med 2011;184(9):1067–1075. doi:10.1164/rccm.201103-0577OC
Delos JB. A mathematical model of Cheyne-Stokes or periodic breathing. Math Biosci 2024;378:109318. doi:10.1016/j.mbs.2024.109318. Round-trip delays recovered from apnoea timing: 5 to 20 s in healthy adults, 20 to 31 s in heart failure.
Dhakal BP, Lewis GD. Exercise oscillatory ventilation: mechanisms and prognostic significance. World J Cardiol 2016;8(3):258–266. doi:10.4330/wjc.v8.i3.258. A review of the visual definitions referred to above, which originate with Kremser 1987 (amplitude above 15% of resting ventilation for at least 66% of the test), Ben-Dov 1992, Corrà 2002, Leite 2003 and Sun 2010 (at least 3 cycles, amplitude above 30% of mean ventilation, cycle length 40 to 140 s).
For the modelling background, see Module 4 on dynamic systems and first-order differential equations.