## Abstract

The relationship between power output and the time that it can be sustained during exercise (i.e., endurance) at high intensities is curvilinear. Although fatigue is implicit in this relationship, there is little evidence pertaining to it. To address this, we developed a phenomenological model that predicts the temporal response of muscle power during submaximal and maximal exercise and which was based on the type, contractile properties (e.g., fatiguability), and recruitment of motor units (MUs) during exercise. The model was first used to predict power outputs during all-out exercise when fatigue is clearly manifest and for several distributions of MU type. The model was then used to predict times that different submaximal power outputs could be sustained for several MU distributions, from which several power-endurance curves were obtained. The model was simultaneously fitted to two sets of human data pertaining to all-out exercise (power-time profile) and submaximal exercise (power-endurance relationship), yielding a high goodness of fit (*R*^{2} = 0.96–0.97). This suggested that this simple model provides an accurate description of human power output during submaximal and maximal exercise and that fatigue-related processes inherent in it account for the curvilinearity of the power-endurance relationship.

- mathematical
- motor unit
- contractile properties
- muscle power
- human

the maximum period over which an exercise task can be sustained (i.e., endurance) at a constant power output is an inverse and curvilinear function of intensity (28). This power-endurance relationship provides a useful framework for understanding exercise tolerance, and parameters derived from it have been used to represent exercise-related constructs, such as “critical power” and “anaerobic work capacity” (23). With respect to these parameters, the research effort has tended to focus on their “energetic basis” (25), construct validity (33), and technical aspects associated with the accuracy of their estimation (5). By contrast, far less research has focused on the physiological mechanisms underlying these parameters and the power-endurance relationship from which they are derived.

Fatigue during exercise contributes to the failure to sustain exercise and limits endurance (12), particularly as it applies to high-intensity exercise above a critical threshold, such as the “critical power” or “maximal lactate steady state”. Given that the rate of fatigue during constant-load exercise is an inverse and curvilinear function of intensity (9), then fatigue is likely to contribute directly to the curvilinear power-endurance relationship. Although metabolic aspects of the power-endurance relationship have been explored (20, 25), until very recently little attempt has been made to directly address this link between muscle fatigue and the power-endurance relationship or the physiological processes that relate to them. Liu et al. (22) proposed a phenomenological model of muscle fatigue that was applied to all-out handgrip exercise, and Sih et al. (29) modified this model, extended its application to submaximal constant-load tasks (cycling and running), and compared the model outputs with power-endurance curves in the literature. The model proposed by Sih et al. (29) is based on the concept that motor units (MUs) that contribute to force production exist in one of four possible discrete states of activity and fatigue can cycle through these various states at different rates under the control of “brain command”. Although this model provides accurate predictions of human power-endurance curves, the authors acknowledge that the physiological basis of the “activation dynamics” of MUs in this model is not explicit and does not, for example, account for the influence of fiber type on any contractile properties. In fact, the contractile properties of all MUs are assumed to be the same, and, because MUs exist in a small number of discrete states, there is no allowance for the fact that contractile properties, such as fatigue, vary as a continuous function of time.

We developed a phenomenological model of the oxygen uptake response during exercise that was based on just a few contractile properties and, similar to the model of Sih et al. (29), recruitment of MUs (19). However, unlike the model proposed by Sih et al. (29), our model is based on the concept that contractile properties, including fatigue, vary as a continuous, rather than a discrete, function of time and MU type. Although the physiological basis of our model might be more realistic, its validity in predicting time-dependent changes in mechanical power output has not been established. Therefore, we aimed to explore this validity and the causal link between fatigue and the power-endurance relationship. In doing so, we also used the model to shed light on the effect of proportion and distribution of MU “type” on the power-endurance relationship, for which there is an absence of empirical data. We first explored the ability of the model to predict fatigue responses under conditions when they were manifest (i.e., “all-out” exercise) before examining power-endurance curves generated by the model. Then we used the model to test the effect of MU type on the power-endurance relationship and examined the extent to which model outputs under these exercise conditions conformed to published data collected from humans using a different approach from that used by Sih et al. (29).

## METHODS

#### Background to the model.

During exercise, work can be done on an external system at a given rate (i.e., mechanical power output) through the repeated actions of many muscles acting about several joints. The power output generated by each of these muscles is a function of the actions of several hundred MUs and the hundreds of muscle fibers that belong to each of these MUs. Although the muscle fiber contracts to generate muscle power, the MU is taken to be the basic physiological entity in our model.

The model predicts the total muscle power output (P_{out}) of a system of MUs distributed across the muscles involved in the exercise task (19). As described below, the equations of the model capture essential features of skeletal muscle and its MUs. All muscles that contribute to power output are viewed as a single population of MUs. These MUs vary continuously with respect to their “type” (e.g., slow vs. fast), and their recruitment is ordered from slowest to fastest as the power output is increased (7, 13, 17, 24). A small number of contractile properties of MUs are included in the model and include the maximum power output (P_{max}), onset to fatigue, rate of fatigue, and minimum power output (P_{min}) at complete fatigue. The first three of these properties are assumed to be distributed continuously across a continuum of MU type, whereas the last property is taken to be either the same for all MUs or vary as a function of MU type. A summary of the nomenclature used to describe the model equations is shown in Table 1.

#### MU type.

The type of MU is often related to the twitch speed of the muscle fibers that belong to it, and by the simplest convention is either “slow” or “fast”. However, within either category, there is considerable variation in muscle fiber contractile properties and overlap between categories (21). The proportion of MUs of a particular type can also vary considerably within a muscle, between muscles, and between individuals (10, 26). Therefore, we envisaged type as a continuous property that could be described conceptually by a type continuum. This continuum includes all MUs combined across all muscles that contribute to the total power output, and these MUs vary in their twitch speed from the slowest to the fastest. The position of a given MU on this type continuum is denoted by *x*. The slowest MU is located at the start of the continuum, *x* = 0, whereas the fastest MU is located at the end of the continuum, *x* = 1.

The distribution of MUs along the type continuum is quantified using a weighting, or density, function, *w*(*x*). This function is dimensionless and defined on the region *x* = 0 to 1. The proportion of MUs between *x* and *x* + δ*x* is defined to be
*w*(*x*) = 1, for which there are equal numbers of MUs at all points along this entire continuum. This uniform distribution has an equal proportion of MUs above and below the midpoint (0.5) and would represent a fiber-type proportion of 50% slow twitch and 50% fast twitch. As illustrated later in Fig. 3, both the uniformity of the distribution (linear or nonlinear) and the proportion of MU type can be altered by changing the density function. Figure 3*A*, *inset*, shows three linear distributions of MU type that yield proportions (%) of slow-fast of 50:50 [*w*(*x*) = 1], 75:25 [*w*(*x*) = 1 − *x*], and 25:75 [*w*(*x*) = *x*]. Figure 3*B*, *inset*, shows three nonlinear distributions of MU type that yield proportions of slow-fast of 67:33 [*w*(*x*) = 12*x*(1 − *x*)^{2}], 50:50 [*w*(*x*) = 6*x*(1 − *x*)], and 33:67 [*w*(*x*) = 12*x*^{2}(1 − *x*)].

#### MU recruitment.

Increasing P_{out} depends on increasing the recruitment of MUs and firing frequency of already active MUs (2, 24). To simplify our model, we ignored the influence of MU firing frequency on power output and focused on MU recruitment per se. Firing frequency tends to vary in proportion to force and MU recruitment across the full range of these responses (1, 11). On this basis, we assumed that our modeling of MU recruitment captures the essential behaviors of MU recruitment and firing frequency. MU recruitment is defined by movement along the type continuum from *x* = 0 toward *x* = 1. The fastest MU recruited at a particular time is denoted by *M*(*t*), where 0 ≤ *M*(*t*) ≤ 1, so that, at time *t*, all MUs from *x* = 0 to *x* = *M*(*t*) have been recruited.

#### MU power output: maximum, fatigue, and minimum.

The power output of a MU (P_{MU}) can be expressed per kilogram of muscle (W/kg). It is assumed that the recruitment of a MU evokes its P_{max} instantaneously, that this maximum power is maintained for a finite period before fatigue begins, and that the fatigue process continues until some minimum value (see below). There is approximately a threefold difference in the P_{max} of muscles composed of mainly slow or fast MUs (4), and so a very slow MU will have a much smaller initial power output than a very fast MU. On this basis, the model assumed that the initial P_{max} (P_{init}) varied as a positive and linear function of the MU's position on the type continuum [P_{init} = P_{init}(*x*)]. This initial power output, P_{init}(*x*), is assumed to be a decreasing linear function of MU type,
_{0} and P_{1} are the initial power outputs of the slowest MU (*x* = 0) and fastest MU (*x* = 1), respectively (Table 1).

A MU can sustain its P_{max} for a finite period before fatigue begins. A very slow MU can sustain its P_{max} considerably longer than a fast MU (32). Therefore, this period during which the P_{max} can be sustained, τ, was assumed to vary according to the MU's position on the type continuum [τ = τ(*x*)], so that τ(*x*) is a linearly decreasing function of *x*,
*T* is a constant representing the period over which the slowest MU can operate before the onset of fatigue. Since τ(1) = 0, the power output of the fastest MU at *x* = 1 will decline from its maximum value instantaneously with the onset of its recruitment.

It is assumed that fatigue (i.e., decline in power output) will eventually occur in all recruited MUs, and that fatigue is a simple exponential process occurring at a given rate (α). Since a very fast muscle fiber fatigues much more rapidly than a very slow muscle fiber (32), the model assumes that the rate of fatigue (W/s) of a MU varies as a linear function of its position on the type continuum (−α*x*), so that this rate increases continuously between the slowest and fastest MUs (Table 1). Note that the slowest MU at *x* = 0 will not fatigue under these assumptions.

For a contracting muscle, it is difficult to know if a completely fatigued MU still contributes to the P_{out} and whether or not its P_{min} varies as a function of MU type. To explore this, we examined the effect of three scenarios differing with respect to P_{min}: two scenarios where P_{min} was the same for all MUs [P_{min}(*x*) = 0 or 100 W/kg], and one scenario where P_{min} varied as a linear function of MU type [P_{min} (*x*) = 100(1 − *x*)], such that slower MUs have higher values of P_{min}.

#### Time-dependent changes in MU power output.

The model described above predicts the P_{MU} at point *x* on the continuum at time *t*, P_{MU}(*x*,*t*) to have a power output,
_{MU} during exercise. Although this function is not smooth (i.e., it has a discontinuous derivative), its primary use in the model is by integration, where smoothness is not a necessary condition. We acknowledge that the time-dependent recruitment of MUs during submaximal exercise in vivo could be a smoother process than the very small steps predicted by the model, but there is little physiological evidence pertaining to the smoothness of this process and the accuracy of the model predictions (see results) suggest that this is not a critical problem in the context of the present study. Figure 1 illustrates the three stages of power output, maximum (stage I), fatigue (stage II), and minimum (stage III), of three MUs that occupy different positions along the type continuum (0 < *x* < 1). Figure 1, *left*, illustrates a slow MU, which initially has a low power output, which is sustained for a long period (stage I) before fatigue sets in at a slow rate (stage II). Figure 1, *right*, shows a fast MU, which initially has a high power output (stage I), but enters stage II sooner. During stage II, the fatigue rate is much faster, and stage III is also entered sooner. In this example, all MUs have the same, nonzero, power output when fully fatigued. MUs occupying the end points on the type continuum will not show all three stages of power output. The power output of the slowest MUs (*x* ≅ 0) will show no or minimal decline between stages I and III, since the P_{max} is low and approximately equal to the P_{min} [P_{init}(0) = P_{0} = P_{min}]. The power output of the fastest MUs (*x* ≅ 1) will show no or little maintenance of power output (stage I) before fatigue occurs (stage II).

#### Power output and MU recruitment.

P_{out}(*t*) may vary with time during a normal exercise task and depends on the simultaneous recruitment of many MUs within and between muscles. P_{out} at the onset of exercise, P_{out}(0), is given by the extent of MU recruitment and the P_{max} of each MU (P_{MU}) recruited according to the equation,
*w*(*x*)]. Conversely, *Eq. 5* can also be used to find the initial MU recruitment required to generate a given power output at the onset of exercise. In this case, the left-hand side [P_{out}(0)] is known, and the equation can be solved numerically to find *M*(0), the fraction of MUs recruited at the onset of exercise.

#### Time-dependent changes in MU recruitment.

The time that a MU situated at point *x* is first activated is denoted *t*_{act}(*x*) [note: *t*_{act}(*x*) = *M*^{−1}(*x*)]. During all-out exercise when the maximum P_{out} is generated, it is assumed that all MUs are activated at the start of exercise, hence *t*_{act} is zero for all MUs. By contrast, in studies of the power-endurance relationship, subjects are required to sustain a submaximal power output (P_{out}) for as long as possible at high intensities at or above critical power (P_{crit}). During submaximal exercise, *t*_{act} is zero only for those MUs required to achieve the desired power output at the onset of exercise, but, as fatigue occurs, additional MUs will be recruited after exercise onset (i.e., each with *t*_{act} > 0 s) to maintain the required power output. After the onset of exercise, the power output at time *t* can be calculated as
_{MU}) on the continuum, *x*, multiplied by the weighting function (i.e., how many MUs there are of that type). By evaluating the power at *t* − *t*_{act}, the MUs are not activated until they are needed, and at the activation power they operate at their full, initial power. This equation can be solved numerically to find the fastest MU recruited, or fraction of the total MU pool that is recruited, to maintain the required power output at any time during exercise [i.e., *M*(*t*)].

#### P_{max} during all-out exercise.

The P_{max} predicted by the model will occur at *t* = 0 when all MUs have been recruited (*M* = 1) and none have fatigued. Consistent with *Eq. 5*, the P_{max} is given by

#### P_{crit}.

The model can also be used to calculate the maximum long-term or steady-state power output, P_{crit}
_{min} is a constant, then P_{crit} = P_{min}. However, if P_{min} varies across the MU continuum, these quantities will be different. This model output, P_{crit}, is conceptually identical to the concept of critical power articulated in the literature and that is estimated from the power-endurance relationship (23, 25).

#### Model outputs: fatigue, power-endurance relationship, and the effect of MU type.

The curvilinearity of the relationship between power output and endurance implies, but does not demonstrate explicitly, that fatigue contributes significantly to this relationship. By contrast, fatigue is manifest explicitly during all-out exercise. Moreover, there is experimental evidence pertaining to the effect of MU type on fatigue during all-out exercise, whereas there is none pertaining to the power-endurance relationship. Therefore, to explore the causal link between fatigue and the power-endurance relationship, we first tested the effect of MU type on power output, fatigue, and MU recruitment during all-out exercise to establish the extent to which model outputs corresponded with experimental observations on humans. Since the results supported the validity of the model, with more confidence in its accuracy, we then used the model to predict the times that a range of submaximal power outputs could be sustained to construct the power-endurance relationship. To address the absence of experimental data, the effect of MU type on the power-endurance relationship was then tested. To shed light on fatigue during exercise at these submaximal power outputs and its causal link to the power-endurance relationship, the time-dependent changes in MU recruitment predicted by the model (see *Power output and MU recruitment* above) were also explored.

#### Data analysis and curve-fitting.

Numerical analyses were performed using MATLAB (R2010a), and numerical integration was performed using basic quadrature methods. Where the functions were discontinuous (e.g., see lower *right* panels of Fig. 4), the quadrature was done over each continuous section separately and summed. We propose a model that predicts both human power output responses during all-out exercise and the time over which a submaximal power output can be sustained at different power outputs (i.e., power-endurance relationship). To test this and enable a unique fit of the model to be achieved, the model was simultaneously fitted to two sets of human data (5, 34) collected under these two conditions. Although these two data sets were collected on two groups of subjects by different investigators, they were collected on similar subjects (i.e., untrained, young, and healthy) under similar experimental conditions (i.e., cycle ergometry). All-out exercise data for eight subjects were provided by Vanhatalo et al. (34), and we analyzed the average power output values measured at 1-s intervals during a 3-min test performed under “control” conditions, but excluded the first 5 s of data during which power output was rising. The power-endurance data were taken directly from the publication by Busso et al. (5) (Fig. 2). Mean endurance times for six power outputs were used and then normalized to a 0–1 scale by setting P_{crit} at zero and maximum power at 1, based on estimates of P_{crit} (243 W) and maximum power (750 W) obtained from these studies.

Best fit was defined as minimizing the average least squares residual, lsq, error for both models simultaneously
*N*_{1} is the number of data points in *dataset 1* (*N*_{1} = 6), and *N*_{2} is the number of data points in *dataset 2* (*N*_{2} = 176). During model fitting, three different P_{min} functions [P_{min} = 0, P_{min} = P_{0}, P_{min} = P_{0}(1 − *x*)], a range of linear MU distributions to cover all possible fast-to-slow MU ratios and the remaining parameters, P_{0} and P_{1}, the initial power output of the slowest and fastest units, respectively, and α, the fatigue rate, were tested to identify the best fit model functions and parameters. Confidence intervals for the parameters were found using standard boot-strapping methods. However, in the case of *T* (the period over which the slowest MU can operate before the onset of fatigue), no confidence is given, as this parameter was always estimated to be zero.

## RESULTS

#### All-out exercise: model predictions.

Figure 2 shows the model prediction of power output over time during all-out exercise when all MUs are activated at *t* = 0 s and P_{min} is constant for all MUs (100 W/kg). The density function [*w*(*x*) = 1] corresponds to an equal proportion (50%) of slow and fast MUs. Figure 2, *insets*, show the power outputs of individual MUs along the type continuum (*x*) at *t* = 0, 20, and 60 s. At *t* = 0, all units were recruited and generated maximum power. At *t* = 20 s, the fastest 20% of units have started to fatigue, and the very fastest (i.e., those located at *x* = 1) are now operating close to their minimum output power. At *t* = 60 s, the fastest MUs have fatigued and are contributing minimal power (P_{min} = 100 W/kg), whereas the slowest 40% of MUs are still operating at their P_{max}. All other parameter values are as given in Table 1.

As with all models, it is important to perform basic sensitivity analysis to show the effect of the most important parameters and functions on the model output. The key inputs to the model are the MU distribution function, *w*(*x*), and the P_{min}. While the other parameters, e.g., fatigue rate, initial power outputs, etc., will have a quantitative effect on the model outputs, these are the parameters most likely to have a qualitative effect. Figure 3*A* shows the effect on power output of altering the distribution of MU type. Figure 3, *insets*, show the three linear MU distributions used: a uniform distribution (also shown in Fig. 2; dashed line in Fig. 3), a distribution skewed toward faster MUs (dotted line), and a distribution skewed toward slower MUs (solid line). The main figure shows the power output during all-out exercise, corresponding to these three distributions. Increasing the proportion of faster MUs increases the maximum power and rate of fatigue, and vice versa. The power outputs of the three MU distributions cross over at *t* ≅ 40 s and then converge at a later time to the same asymptotic power output, because power output in the completely fatigued state (i.e., P_{min}) is the same for all MUs.

MU type might not be distributed in a linear manner, as in Fig. 3*A*, and so we tested the effect of a nonlinear (unimodal) distribution (Fig. 3*B*). Although the effect is similar between linear (Fig. 3*A*) and nonlinear distributions (Fig. 3*B*), there is a subtle difference: the dashed line in both figures represents the same proportion of slow-fast MUs (i.e., 50:50), and the power output decreases relatively more rapidly at the start of exercise for the linear compared with nonlinear distribution and is due to the relatively larger number of very fast and fatiguable fibers (toward *x* = 1).

Power-time profiles in Figs. 2 and 3, *A* and *B*, employ a P_{min} after complete fatigue that is constant for all MUs (P_{min} = 100 W/kg). If a lower value is used for P_{min}, the asymptotic power output at large time will also be lower. Figure 3*C* uses a minimum power that varies between MUs and is proportional to MU type [P_{min} = 100(1 − *x*)]. In this case, the final fully fatigued power output is strongly dependent on the MU distribution, with distributions more skewed toward slow MUs having a much higher final power output.

#### Power-endurance relationship.

It is thought that endurance during submaximal exercise is linked to fatigue and the progressive recruitment of faster MUs until task failure occurs (see discussion). Figure 4, *left*, shows the fraction of the total MU pool required to sustain exercise at five submaximal power outputs and how it varies as a function of time [*M*(*t*)]. For all power outputs, *M*(*t*) increases over time as fatigue occurs in some MUs at a rate disproportional to the power output. This is illustrated in Fig. 4, *right*, which shows the output of MUs across the spectrum while maintaining a power output of P_{out} = 105 W/kg. At *t* = 0, approximately the slowest 65% of MUs are recruited to achieve the required power output. By *t* = 50 s, the fastest of these MUs have fatigued, and, to maintain the required total power output, additional faster MUs were recruited and generate a relatively higher power output. If the required power output is high enough, eventually all units will have been recruited, and failure occurs when *M*(*t*) = 1, as illustrated in Fig. 4, *left*, for *M*(*t*) responses to 105, 140, and 175 W/kg.

The time to failure illustrated in Fig. 4 for three out of the five power outputs is endurance, and the model can be used to calculate the endurance time for a range of submaximal power outputs and yield a power-endurance curve. Figure 5*A* shows examples of these curves for three linear MU distributions and a constant P_{min} (P_{min} = 100 W/kg). The curve has a vertical asymptote at the P_{crit} value; power outputs below this threshold can be sustained for an infinite length of time. As the required power output is increased, the endurance time decreases until the required power output is above P_{max} and can no longer be achieved. In this case P_{crit} is the same for all of the illustrated MU distributions, but P_{max} is higher for the distributions with more fast MUs.

The sensitivity of the model output (power-endurance) to variation in key model parameters was tested. Figure 5, *B* and *C*, shows power-endurance curves for different MU distributions and values of P_{min}. Provided *w*(*x*) has been chosen to sum to 1 (i.e., it is analogous to a probability distribution for the probability of a MU being located at that point on the continuum), then a constant value of P_{min} will result in the asymptote that represents P_{crit} being constant for different MU distributions (Fig. 5). If P_{min} can vary across the continuum, this will result in different values of P_{crit} for different MU distributions (Fig. 5*C*). The value of P_{max} will only depend on the proportions of fast and slow MUs.

#### Model fitting.

An important test of the model is the accuracy with which it predicts human responses during all-out exercise (i.e., power-time profiles) and submaximal exercise sustained to failure (i.e., power-endurance relationship). To do this and also enable a unique fit of the model to be achieved, the model was fitted simultaneously to two sets of human data collected by different investigators and under different conditions (see methods, *Data analysis and curve-fitting*). During this process of model fitting, different P_{min} functions [0 and 100 W/kg; P_{0}(1 − *x*)] and MU distributions (linear, nonlinear) were tested to identify which combination of P_{min} and MU distribution provided the best fit. Significantly better fits were obtained when P_{min} varied as a linear function of MU type [P_{min}(*x*) = P_{0}(1 − *x*)], and the MU distribution was linear. Using this optimal combination of P_{min} and MU distribution, the goodness of fit of the model to the two sets of human data is illustrated in Fig. 6 for all-out exercise (*top*, *R*^{2} = 0.97) and the power-endurance relationship (*bottom*, *R*^{2} = 0.96). The parameter estimates and confidence intervals related to this unique, best fit are given in Table 2.

These model parameter estimates associated with this unique fit to human data (Table 2) shed light on the muscle contractile properties of the participants involved. They suggest that, on average, the MU type of the muscles recruited during exercise (i.e., cycling) were skewed slightly toward a faster type, and that there was a very wide range of P_{max} between MUs (P_{0} to P_{1}). Fatigue is reflected in the fatigue constant (α) and the time delay associated with the onset of fatigue. It is of some interest that this latter parameter was not significantly different from zero, suggesting that the onset of fatigue of MUs is coincident with the onset of their activation during exercise, regardless of their type.

## DISCUSSION

The present study shows that a phenomenological model (19), based on just a few contractile properties and the recruitment of MUs during voluntary exercise, accurately captures the essential qualitative features of human P_{out} during intense exercise and highlights the underlying muscle physiology important to fatigue during high-intensity exercise and the power-endurance relationship. In this way, it supports the recent observations by Sih et al. (29), who also showed that a MU-based model can accurately predict human fatigue and power-endurance curves. However, the present model goes further to clarify some of the underlying muscle physiology involved and the important contribution of MU type and its influence on a small number of contractile properties to the relationship between power output and endurance.

The curvilinear relationship between power output and endurance provides a useful framework for understanding exercise tolerance. Ever since Hill (18) interpreted this curvilinearity as evidence of an aerobic and anaerobic contribution to the tolerance of exercise at higher intensities, metabolism has been the dominant framework for thought and experimenting about the mechanistic basis of the power-endurance relationship (20). Morton and colleagues (20, 25), in particular, have explored the mathematical basis of the power-endurance relationship and the link between “bioenergetic” models of exercise and parameters derived from this relationship. However, a more comprehensive understanding of the power-endurance relationship requires the development of physiological models to complement existing bioenergetic models. Although muscle fatigue at intensities above P_{crit} is thought to be essential to the power-endurance relationship (20), the mechanistic link between them has not been made clear. Since metabolic responses during exercise are largely attributed to contracting skeletal muscle, knowledge about the contractile behavior of muscle and its MUs is essential to a physiological understanding of these responses and their causal link to the power-endurance relationship.

Our phenomenological model predicts time-dependent changes in mechanical power output on the basis of differences in contractile properties of MUs and their recruitment during exercise (19). Physiological background to the model and its equations capture several features of skeletal muscle and its MUs (see methods). Despite the simplicity of this background and the delimiting of the model to only properties of the MU, it is important to our understanding of the physiological basis of the power-endurance relationship to see how well such a model can describe fatigue and the power-endurance relationship observed in humans.

Fatigue, a decline in the maximum voluntary force or power output (12), is manifest explicitly during all-out exercise; whereas it is only implicit in the power-endurance relationship. Therefore, in using the model to explore the causal link between fatigue and the power-endurance relationship, it was important to first verify the accuracy of the model in predicting fatigue during all-out exercise. The model estimates of power output during all-out exercise (Figs. 2 and 3) illustrate the fatigue typical of this exercise, with half-times (∼40–50 s) similar to those observed in human subjects performing running or cycling exercise (6, 8). The effect of the proportion of type of MUs on P_{max} and rate of fatigue during all-out exercise (Figs. 2 and 3) was also consistent with human studies (3, 27). These observations support the validity of the model to predict fatigue in humans performing all-out exercise and demonstrate the extent to which just a few basic features of contractile properties and recruitment of MUs can predict fatigue manifest explicitly under this condition.

For all-out exercise, it was assumed that all MUs were recruited at exercise onset, remained active throughout exercise, and fatigued in accordance with their times to onset of fatigue and rates of fatigue, as determined by their position on the continuum of MU type. During all-out exercise, there is minimal reserve for additional MU recruitment. By contrast, the power-endurance relationship is established from a small series of submaximal exercise bouts, so that the extent of MU recruitment at the onset of exercise is not maximized and is proportional to the power output. Electromyographic evidence suggests that, when the intensity of submaximal exercise is high enough (e.g., >P_{crit}), MUs fatigue, and, to sustain the required power output, additional MUs are recruited (35) at a rate that is a positive function of exercise intensity (14, 16). This intensity- and fatigue-dependent rise in MU recruitment was predicted by the model (Fig. 4). According to the model, the failure to sustain exercise occurred when MU recruitment was maximized, and the time at which this occurred (i.e., endurance) became disproportionately less as the initial submaximal power output increased (Fig. 4). Consequently, the relationship between power output and endurance was curvilinear and demonstrated that this curvilinearity was linked causally to the onsets and rates of fatigue of the MUs involved.

The present model is based on experimental evidence obtained using a variety of nonhuman preparations, and so it is of interest to test the extent to which the model output corresponds with real human data. In addition, since the model can be rearranged to estimate either the power output as a function of time (i.e., all-out exercise) or the time that a given power output can be sustained for (i.e., power-endurance relationship), it is important to test the accuracy of model estimates under these two scenarios. To address this, a unique fit of the model was obtained by fitting it simultaneously to all-out exercise data (34) and power-endurance data (5) obtained from young, healthy subjects performing cycle ergometer exercise. Despite the limitations inherent in this approach, there appears to be reasonable agreement between both sets of human data and the model estimates (Fig. 6), further supporting the conclusion that MU behavior and fatigue processes described by it constitute the essential physiological basis of the curvilinear relationship between power output and endurance.

There were outcomes of this fitting process that shed further light on the model and underlying physiology. The model assumes that there is a finite delay between the onset of recruitment of a MU (*Eq. 3*) and its onset of fatigue. However, the unique fit of the model to the two sets of human data suggested that this delay was not different from zero [i.e., τ(*x*) = 0; see *Eq. 3*] and thus, for the majority of MUs, there is no discernible delay between the onset of recruitment and beginning of fatigue. The distribution, which is distinct from the proportion, of type of MU affected how well the model fitted the human data, suggesting that the shape of the distribution of MU type and contractile properties related to it exerts an important influence on time-dependent changes in human power output.

Given that all contractile properties inherent in the model are influenced by the “type” of MU, then the model should provide insight into these effects. Human studies demonstrate that, during all-out exercise, P_{max} and rate of fatigue are proportional to the fraction of fast-twitch muscle fibers (3, 27), as predicted by the model (Figs. 3 and 4). By contrast, we were not aware of any experimental data on the effect of muscle fiber type on the power-endurance relationship. The asymptote of this relationship (“critical power”) has been linked to maximal aerobic power (33) and thus would be expected to be higher in slower, more fatigue-resistant MUs (31). This expectation was predicted by the model, but only when the P_{min} after complete fatigue varied as a function of the type of MU (*Eq. 3*; Fig. 5*D*). The degree of curvature of the power-endurance relationship, or its linear form, is positively correlated with the anaerobic ATP production during exercise (15) and thus should be proportional to the relative number of faster, more fatiguable MUs (30). This expectation was always predicted by the model (Fig. 5) and is not surprising, given the degree of curvature of the power-endurance relationship is determined by the P_{max}, which varied as a function of MU type and was greatest for the fastest MU, and the P_{crit} (i.e., *Eq. 8*), which was either independent of MU type or lowest for the fastest MU (*Eq. 3*). Importantly, the model outputs show that the fatigue implied in the extent of curvature of the power-endurance relationship is a function of the fatigability and maximum P_{MU} involved in the exercise.

The model also provides insight into an effect of fiber type not yet tested experimentally. Establishing the muscle fiber type using conventional techniques is based on calculating a proportion of one type (e.g., slow) relative to another type (e.g., fast). However, a contractile property such as the onset or rate of fatigue can vary considerably between muscle fibers of the same type (32). Consequently, the distribution of this property within a population of MUs of the same type might influence power output in ways that cannot be predicted from conventional measurements of proportion. We explored this by examining model predictions of power output during all-out exercise for linear and nonlinear distributions of MUs that had the same proportions of MU type, and a relatively faster rate of fatigue (all-out exercise) was observed for the linear distribution. A similar observation was made for the degree of curvature of the power-endurance relationship. These data further reveal that the distribution of MU type, independent of the proportion of MU type calculated using a conventional scheme, influences power output during all-out exercise and the curvature of the power-endurance relationship. This might help explain how two individuals with the same apparent “muscle fiber type” can have different power output responses during all-out exercise and different estimates of anaerobic capacity derived from the power-endurance relationship.

#### Limitations and perspective.

To our knowledge, the present model is the first to predict the power-endurance relationship on the basis of physiological properties of MUs and their recruitment during exercise. The model equations represent just a limited number of features of skeletal muscle and MU contractile behavior (e.g., MU firing rate has been ignored), do not accommodate potential differences in mass or volume of contracting muscle, and do not represent other physiological systems (e.g., cardio-respiratory). Moreover, like all models, the present model is limited by the experimental data on which the equations are based, and these data have not been collected under uniform experimental conditions. However, a positive feature of the model is that the equations have been “scaled” in such a way as to accommodate differences in power outputs caused by the mass and/or number of muscles involved and, therefore, can be applied to any species. The present model might also form the basis of a more complex model, which incorporates behavior of other physiological systems and, in so doing, can be used to study the contribution that variation in the behavior of these systems makes to the tolerance of high-intensity exercise at and above P_{crit}.

#### Conclusion.

A phenomenological model based on a limited number of contractile properties and simple recruitment behavior of MUs appears to adequately describe temporal features of human P_{out} during submaximal and maximal exercise. This suggests that these contractile properties, particularly those related to fatigue, and their dependence on MU type are the fundamental basis of the curvilinear relationship between power output and endurance. This knowledge expands the conventional “metabolic” view of this relationship to one that is more physiological and connected to the fatiguability and time-dependent recruitment of MUs.

## DISCLOSURES

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

## AUTHOR CONTRIBUTIONS

Author contributions: A.J. and S.G. conception and design of research; A.J. and S.G. performed experiments; A.J. and S.G. analyzed data; A.J. and S.G. interpreted results of experiments; A.J. and S.G. prepared figures; A.J. and S.G. drafted manuscript; A.J. and S.G. edited and revised manuscript; A.J. and S.G. approved final version of manuscript.

- Copyright © 2012 the American Physiological Society