## Abstract

Transient responses of ventilation (V̇e) to limb motion can exhibit predictive characteristics. In response to a change in limb motion, a rapid change in V̇e is commonly observed with characteristics different than during a change in workload. This rapid change has been attributed to a feed-forward or adaptive response. Rate sensitivity was explored as a specific hypothesis to explain predictive V̇e responses to limb motion. A simple model assuming an additive feed-forward summation of V̇e proportional to the rate of change of limb motion was studied. This model was able to successfully account for the adaptive phase correction observed during human sinusoidal changes in limb motion. Adaptation of rate sensitivity might also explain the reduction of the fast component of V̇e responses previously reported following sudden exercise termination. Adaptation of the fast component of V̇e response could occur by reduction of rate sensitivity. Rate sensitivity of limb motion was predicted by the model to reduce the phase delay between limb motion and V̇e response without changing the steady-state response to exercise load. In this way, V̇e can respond more quickly to an exercise change without interfering with overall feedback control. The asymmetry between responses to an incremental and decremental ramp change in exercise can also be accounted for by the proposed model. Rate sensitivity leads to predicted behavior, which resembles responses observed in exercise tied to expiratory reserve volume.

- control of breathing
- exercise
- limb motion
- ventilation

explaining characteristics of the transient ventilation (V̇e) response to human exercise dates back to Krogh and Linhard (12), but apparently are still incompletely understood (8). The early rapid change in V̇e to a step change of exercise is grossly attributed to “neural” factors. More recently, sinusoidally changing workload (3) and pedaling speed (4) studies have revealed independent effects dealing with this rapid phase. Similar effects have been observed in treadmill experiments using platform tilt to separate load from limb motion (16). In response to load changes, a first-order or single-exponential dynamics model accounts for the main part of the response (9). The response to limb motion per se is not as simple to describe. Sinusoidal changes in pedaling frequency results in a reported phase lead response in several subjects (4). A phase lead corresponds to V̇e, which anticipates the change in pedaling speed and is inconsistent with a first-order single-exponential model. So a predictive response is indicated.

Human predictive responses to sinusoidal inputs have been extensively studied in eye movements (1). What has been found is that humans were able to track sinusoidally moving targets and correct phase shift to zero by predicting target motion. Note that the specific adaptation focused on was to phase shift, not to the magnitude of the response. Many types of models have been proposed to explain this observation. Most are difficult to correlate to specific physiological mechanisms. However, the common basis of these models is rate sensitivity. Rate sensitivity in physiological systems was initially proposed by Clynes (7) to explain transient responses of many physiological sensors. One common characteristic of rate sensitivity is unidirectionality or asymmetry between positive and negative rate responses. This is another characteristic reported for the fast component of ventilatory exercise transient responses especially for heavy exercise (16). Adaptation leading to unidirectionality has been estimated to take on the order of a few minutes following a step change treadmill exercise test.

## MODEL DERIVATION

Prior studies based on empirical modeling of exercise V̇e responses to changes in load have shown that the main component of the response can be described by a first-order model, as shown in Fig. 1, in terms of a LaPlace Transform transfer function, with exercise load considered as input and V̇e as output. A small delay in V̇e response has also been observed in some subjects, but will be ignored as a first approximation. The strategy followed here is to develop a minimal model that captures the main response features. To simplify model interpretation, a unit step input of load from rest is assumed with the transfer function gain set equal to the change in V̇e from resting level (ΔV̇e). The corresponding step response is also shown. ΔV̇e is assumed to vary as a function of exercise level. Missing from this transfer function is an initial rapid response term whose magnitude is expected to change with the level of training (10). For bicycle exercise, preceding the step load change by loadless pedaling reduces this rapid response term. The dynamic effect of pedaling rate changes per se has been studied using sinusoidal forcing (4). Shown in Fig. 1 are magnitude and phase data replotted from that reported as the mean data for all subjects. Also shown in the figure are predicted magnitude and phase responses based on estimating time constant τ = 0.575 min (34.5 s) by visual fit to the magnitude data (see appendix a and b for formulas). Note that the phase predictions depart significantly from the mean data, while the magnitude data can be reasonably predicted. Inspection of the individual phase data showed that phase lead or V̇e preceding pedaling rate was observed in several subjects. This first-order lag model is inconsistent with this phase data. This motivated a change in the model structure according to the block diagram shown in Fig. 2, which incorporates rate sensitivity to changes in pedaling rate. Since both input paths sum to a net ventilatory drive, a simpler model structure also shown in Fig. 2 was assumed as equivalent. Since the magnitude vs. frequency data are relative to the steady-state level, a single parameter τ_{r} can be used in interpreting normalized sinusoidal data. Rate-sensitive parameter τ_{r} was considered as adjustable by an adaptive feed-forward mechanism with ΔV̇e normalized to 1.0. For a step input, ΔV̇e = steady-state (low frequency) V̇e in l/min, and ΔV̇e τ_{r}/τ = transient change in V̇e at the time of step initiation, as shown in Fig. 3. For a step return to rest, the transient change = −ΔV̇e τ_{r}/τ.

## RESULTS

#### Sinusoidal responses.

The τ_{r} parameter value required to best fit the phase data was determined as a function of pedaling rate frequency, as shown in Fig. 3. Also shown in Fig. 3 are idealized lines fitted to these estimates of τ_{r} for model description at arbitrary frequencies. The resultant phase predictions of the model are shown in Fig. 3. There are two things to note: the resultant phase predictions fit the data well, and the data suggest rate parameter changes with values of 0.42 min (25.2 s) and 0.23 min (13.8 s), depending on the frequency of pedaling rate changes, with a value of 0.42 for low frequencies and 0.23 for higher frequencies. The implications of this change in parameter value will be discussed later. A step change should correspond to zero frequency and the higher τ_{r} value of 0.42 min. The model magnitude predictions are shown in Fig. 3 and compared with the same mean data of Fig. 1.

Further support for the existence of an adaptive mechanism for adjustment of rate sensitivity was provided by data reported by Casaburi et al. (4) for one subject. Figure 4 shows magnitude and phase responses for sinusoidal changes in pedaling rate for one subject compared with the model predictions, without including rate sensitivity (τ_{r} = 0). Note that model predictions are close, except for the phase at the highest frequency (0.5 counts/min), where there is a suggestion of the emergence of rate sensitivity to reduce the phase-tracking error only for this frequency.

Wells et al. (16) compared sinusoidal responses of treadmill speed and load and reported a 77° (1.34 radians) shorter phase lag for speed responses at 1 count/min sinusoidal frequency. This difference can be explained by a τ_{r} value of 0.66 min [39.6 s, where 1.34 = atan (ωτ_{r})] according to our rate sensitive model, where ω is angular velocity. Thus sinusoidal treadmill data appear to be consistent with rate sensitivity to treadmill speed.

#### Step responses.

The adequacy of the model shown in Fig. 1 to describe ventilatory responses to step changes in exercise load has been previously established. The current modified model incorporates the same model with τ_{r} = 0 for load changes, so it is also consistent with these previous results. The components of the predicted responses of the rate-sensitive model to a unit step change in pedaling rate (e.g., rest to pedaling with load) are shown in Fig. 2. Note that the initial fast ventilatory response is determined by the ratio τ_{r}/τ multiplied by the steady-state change in V̇e due to exercise. The qualitative shape of these responses agree with previous published responses for treadmill as well as bicycle studies (8, 10, 13, 17). A larger initial transient is typically measured for a rest-to-treadmill transition consistent with a larger τ_{r}/τ value. Rest to bicycle exercise has measureable but usually smaller initial transients, which also can be described by a different parameter value.

#### Adaptation.

As shown in the results section, the sinusoidal exercise data involving limb motion fits reasonably well with an adaptive adjustment of a feed-forward ventilator drive, as summarized by parameter τ_{r}. Parameter τ_{r}/τ determines the size of the step rapid initial ventilatory rise, as well as the rapid fall following the end of exercise. An unchanging value of τ_{r} is consistent with equal rapid initial rises and final decreases. However, an adaptive decrease in τ_{r} will decrease the final rapid fall magnitude. In this way, adaptation of the rapid decrease could be tied to an adaptive decrease in τ_{r}. τ_{r} = 0 results in zero magnitude of the rapid decrease in V̇e. Heavy exercise has been reported to lead to a decrease in the rapid change at the termination of exercise for a step decrease in exercise (13). This will now be addressed.

Mateika and Duffin (13) has reported that the rapid change in V̇e following step changes in limb motion decreases during heavy exercise. Their data for a single subject for treadmill exercise are replotted in Fig. 5 and are compared with model predictions. Two levels of exercise for two different durations are compared. For the lower level of exercise shown in Fig. 5, a value of the rate parameter τ_{r} = 1.57 min (94.2 s) led to a reasonable fit to the initial “on” transient for both exercise durations. A slow response τ = 2 min (120 s) fit both “on” and “off” transients. The τ_{r} parameter appeared to slightly decrease as a function of exercise duration. A value of 1.43 min (85.8 s) and 1.29 min (77.4 s) best visually fit the data for the shorter and longer duration “off” transients, respectively. At a higher level of exercise, a value τ_{r} = 1.09 min (65.4 s) was estimated for the “on” transient, and τ_{r} = 0 for the “off” transient for both durations. Again, an adaptive change of parameter τ_{r} as a function of exercise level and duration was suggested by these data. The ΔV̇e parameter accounted for the steady-state exercise load response and was increased from 28 to 70 l/min to account for the higher steady-state V̇e level. The τ was estimated as 2 min (120 s) for “on” and “off” slow responses for the lower exercise level. At the higher exercise level, “off” slow responses also had a τ of 2 min. The “on” τ for the higher level of exercise had to be increased to 3.8 min (228 s) to best visually fit the data. The rapid V̇e response appeared to be approximately constant for the transition from rest to both exercise levels. The level and duration of exercise decreases the value of τ_{r} and approached a zero value at higher exercise levels, leading to the absence of any rapid decrease in V̇e. The general features of the rapid exercise response have been previously reviewed by Duffin (8). The main difference here is the new interpretation based on a model incorporating rate sensitivity.

Figure 6 compares model predictions of rate-sensitive and single-exponential models with data reported by Whipp and Ward (17) for bicycle exercise of a trained subject. A similar “on” transient compared with treadmill exercise can be seen, with the main difference being the smaller value of the τ_{r} parameter of the rate-sensitive model in comparing Figs. 5 and 6. Slow response τ values are comparable (2 vs. 2.97 min). The main features of the rapid and slow exercise transients for bicycle exercise appear consistent with the present model predictions. All prior model fits were made by adjusting parameters for best visual fits. For Fig. 6, a different fitting procedure was used to compare residuals of the fit for both models considered in an unbiased way. A Nelder-Mead least squares method (Matlab function) was used to iteratively adjust parameters to minimize the summed squared difference between model predictions and V̇e measurements. The root mean square value of V̇e prediction residuals using the rate-sensitive model was 1.4 l/min and 4.6 l/min for the single-exponential model. Comparing both fits in Fig. 6 shows that the main difference occurs within the first minute following the step change. Only the rate-sensitive model can describe this fast component.

Hagberg et al. (10) found that the initial rapid change in V̇e at the start of bicycle exercise increased over a 9-wk training period of nonathletes, which led to a smaller oxygen (O_{2}) deficit. Interpreting the change in V̇e they reported in terms of rate parameter τ_{r}, a value of 0.31 min (18.6 s) corresponds to pretraining and 0.42 min (25.2 s) posttraining. Except for different parameter values, the model and data comparisons resembled Fig. 6, which also involved bicycle exercise, so they will not be shown. The other estimated parameters were as follows: pretraining τ = 1.4 min (84 s), ΔV̇e = 42.4 l/min and after training τ = 1 min (60 s), ΔV̇e = 34.9 l/min. Reduction of τ and increased τ_{r} after training both contribute to repayment of O_{2} debt by changing the initial V̇e change τ_{r}/τ ΔV̇e from 9.4 l/min to 14.8 l/min and reducing the time required to reach steady state. The proposed rate-sensitive model appears to account for the major features of step changes in exercise up to moderate levels.

In Fig. 5, the response to heavy exercise can be characterized as exhibiting unidirectional rate sensitivity. In other words, a rapid augmented ventilatory response results for a positive increase in the rate of limb motion, but no rapid change for a negative rate. One consequence of unidirectional rate sensitivity has been emphasized by Clynes (7) as providing a response proportional to the amount of a stimulus. Thus the effect of such a stimulus will persist long after the stimulus ends. In the case of moderate exercise, the absence of a rapid decrease in V̇e at the end of exercise will lead to a reduction of O_{2} debt, since V̇e will be maintained at a higher level immediately following exercise.

#### Ramp responses.

Miyamoto and Niizeki (14) found that human exercise responses to bicycle load ramp increases in load, followed by ramp decreases back to rest, were not symmetrical. Their results can be interpreted as an example of rate sensitivity leading to a decrease in phase shift between exercise level and V̇e. A slow ramp increase or decrease corresponds to a very-low-frequency sine wave. The time delay between the input load and V̇e can then be predicted as τ_{r} − τ min (see appendix b), where a positive value corresponds to a prediction. This simple prediction corresponds to a lateral shift of the ramp input, which can be used to fit their data. A value of τ_{r} = 0 and τ = 98 s for the ramp increase and τ_{r} = τ and τ = 98 s for the ramp decrease does provide a reasonable visual fit to their data. Instead, they fitted a single-exponential model for the ramp increase using a least squared method and reported a τ = 132 s. Applying this model prediction to data reported for a 7-min incremental ramp is shown in Fig. 7 for τ_{r} = 0. The decremental ramp shows approximately a net zero time delay, so τ_{r} can be estimated as τ_{r} = τ = 132 s. The predicted model response is identical to the input decremental ramp and is not shown in Fig. 7. The asymmetry between incremental and decremental ramp exercise responses can then be explained by an adaptive change in rate sensitivity. Rate sensitivity is negligible in response to an incremental ramp and increases for a decremental ramp. This observation implies that there is a threshold level of exercise level before rate sensitivity can have an effect. Since rate sensitivity seems to disappear at a higher moderate exercise level, as discussed earlier, an exercise range is suggested where rate sensitivity can influence V̇e. Since the ramp data were collected at a fixed pedaling rate, rate sensitivity may involve additional indexes of limb motion than pedaling rate alone.

## DISCUSSION

Rate sensitivity as modeled here only has an effect when there is a change in limb motion. Thus it does not affect the steady-state ventilatory response to exercise. In the present model, the steady-state response is set by parameter ΔV̇e, which will change with exercise load. Since the time of Krogh and Lindhard (12), one part of the rapid response in V̇e has been known to involve expiratory reserve volume (ERV). ERV decreases rapidly at the start of exercise (11) and is responsible for an increase in tidal volume (Vt) by as much as a liter. This increase in Vt when multiplied by an exercise level respiratory rate could easily account for the ∼20 l/min seen in Fig. 5 at the start of exercise. The ERV decrease does not appear to change much as a function of exercise level (6, 11). This matches the relatively constant value of the rapid V̇e change at the start of exercise, as seen in Fig. 5. During carbon dioxide (CO_{2}) inhalation, despite comparable increases in V̇e as exercise, no fast response component at the onset has ever been reported, which also correlates with a lack of ERV change (6, 11) and limb motion. A change in ERV during supine exercise is also small (11) and correlates with a smaller fast response component (15) compared with the upright response. Thus evidence is mounting that limb motion, rate sensitivity, and ERV changes are linked. Since ERV effects on Vt must be mediated by expiratory muscles, expiratory muscle control must play a prominent role.

Rate sensitivity in response to limb motion as proposed here offers an explanation for phase alignment of V̇e to pedaling rate similar to what has been found in eye movement tracking (4). Rate-sensitive adjustment appears to be triggered at a threshold level of exercise, since rate sensitivity appears to be absent during incremental ramp exercise, but present during the decremental ramp following reaching a target exercise level. Rate sensitivity may not be tied to pedaling rate alone, since it seems to occur even with constant pedaling rate ramp exercise. Exercising limb neural afferents are known to sense length, rate of change of length, force, and rate of change of force (7). So an additional limb motion rate sensitivity tied to a rate of change of force seems possible. The input to our proposed model as shown in Fig. 2 can be considered to be exercise power level P. The implications of the current model to O_{2} debt as discussed earlier in no way justifies an input connection to O_{2} production rate or even CO_{2} production rate. A purely mechanical input due to neural anticipation or feedback sensors was the intent. For rotary motion like bicycle pedaling: P = T (torque) × ω (angular velocity). Rate dP/d*t* = T dω/d*t* + ω dT/d*t*. Even at constant ω, there could be a limb motion rate sensitivity to dT/d*t*. Rate sensitivity is especially prominent during step initiation of exercise, but diminishes with prolonged duration and intensity of exercise, as suggested by step response data analysis.

From a control system standpoint, rate sensitivity compensates for the relatively long (1–2 min) response time of V̇e during exercise. This slow lag response can lead to a transient change in the arterial blood partial pressures of CO_{2} and O_{2} from resting levels. With rate-sensitive lead compensation this transient change can be minimized. Immediately following step initiation of exercise, the current rate-sensitive model predicts an instantaneous increase in V̇e, which can even lead to a drop in arterial partial pressure of CO_{2} if not matched with an instantaneous CO_{2} production rate. In fact, both types of responses have been observed (5). A perfect prediction matching lead and lag τ values was observed during decremental ramp exercise (14). This matching has also been observed for some subjects during sinusoidal exercise (4). Step “on” transient in exercise usually involves partial compensation, which varies with training level (10). Rate compensation is absent during the “off” transient following heavy exercise (13), which helps minimize O_{2} debt by maintaining high V̇e following exercise. The present model appears useful to unify interpretation of these seemingly disconnected observations.

Since feedback control may operate through the rate-sensitive path focused on in the present study, there are possible implications on control loop stability. Rate sensitivity in effect cancels out phase lag connected with the slow exercise response. This should then promote control loop stability, since excessive phase shift is a known cause of instability. The slow exercise response has previously not been looked at from a stability standpoint, but the current results suggest that it could play a role in pathological situations such as hypersensitivity of afferent neural sensors. Rate sensitivity may have functional importance beyond serving as the basis of another empirical model.

The model of the rapid ventilatory response proposed here predicts an instantaneous change in V̇e in response to a step of exercise. A more gradual change at the onset with a τ of ∼8 s is observed (9). This detail was left out to focus on the main contribution of rate sensitivity and to keep the model as simple as possible. There are also separate time delays to rapid and slower responses, which can be considered, which again provide a better model fit but do not significantly alter the main features of the response.

Finally, the possible range of contribution of rate sensitivity to the exercise V̇e response is shown in Fig. 8, which plots responses for different values of τ_{r}/τ. The τ was assumed to be 1.0 min (60 s) in Fig. 8. A τ_{r}/τ value of unity was the maximum observed for all responses considered in this study.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: S.M.Y. and T.K. conception and design of research; S.M.Y. performed derivations; S.M.Y. and T.K. analyzed data; S.M.Y. and T.K. interpreted results of experiments; S.M.Y. and T.K. prepared figures; S.M.Y. drafted manuscript; S.M.Y. and T.K. edited and revised manuscript; S.M.Y. and T.K. approved final version of manuscript.

## Appendix A

#### Derivation of Step Response

The LaPlace transform of the unit step response of the transfer function shown in Fig. 4 assuming input *x* and output *y* is for *X* (s) = 1/s:
(A1)
(A2)where s is the LaPlace transform operator.

#### Taking the Inverse Transform

(A3) (A4)## Appendix B

#### Derivation of Magnitude and Phase Frequency Response and Ramp Time Delay

A slow ramp can be considered a very-low-frequency sine wave; the phase of the rate-sensitive model can be derived from the transfer function:
(B1)
(B2)where *j* is the imaginary operator.
(B3)
(B4)
(B5)
(B6)
(B7)
(B8)
(B9)

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