This study introduces a multi-timescale mechanical model to quantify proximity to performance limits during endurance exercise. The model represents power output using a set of rolling averages, each associated with a characteristic time constant, and identifies the dominant component as the one approaching its historical maximum at any given time. To demonstrate this framework, real-world data were collected from 21 male professional cyclists during an 11-day training camp. Data from the first 10 days were used to construct individual maximal mean power (MMP) profiles across multiple time scales. On the final day, cyclists completed a fatiguing protocol (~2000 kJ of work) followed by 3-min and 12-min maximal time trials. During exercise, the ratio between each exponentially weighted component and its corresponding historical maximum was computed, and the maximum ratio was used to track proximity to performance limits. At the end of the time trials, this ratio reached 98.6% (94.3%–101%) and 101% (98.5%–103%) for the 3-min and 12-min efforts, respectively (median and interquartile range), indicating convergence toward maximal performance capacity. Notably, in both trials the dominant component corresponded to a slower time scale (~1 h), rather than to components matching the nominal duration of the efforts. These findings suggest that performance limits emerge from the interaction of multiple time scales and are not solely dictated by the duration or intensity of the task. This framework extends the traditional use of MMP from a post hoc descriptive tool to a real-time dynamical measure of performance capacity.

Zignoli, Giorgi, Kolodziej, Martinez-Gonzalez, Leo and Laursen (2026). Tracking Performance Limits Using Multi-Timescale Maximal Mean Power Ratios. European Journal of Sport Science. doi.org/10.1002/ejsc.70179

The model represents your power output using a set of rolling averages, each one associated with a characteristic time constant, and identifies the dominant component as the one approaching its historical maximum at any given time.

In practice, your power output is passed through twelve exponentially weighted moving averages. Each is a rolling average over a time constant τ, from 12 seconds up to 4 hours, updated each second with EWM += (P − EWM)·Δt/τ. Short time constants react quickly to surges; long ones track sustained effort.

For every time scale we keep the highest value you have ever reached, which is your historical maximum for that time scale. While you ride, we compare each live rolling average to its historical maximum. The largest of those ratios is the maximum ratio, your proximity to the limit, where 100% means you are at it. The time scale that produces the largest ratio is the dominant, or limiting, component. What is left over is your reserve, the margin to the limit, which is 100% minus the maximum ratio.

This is Figure 2 from the paper, drawn on your reference activity. The top panel shows your power output and the maximum ratio over the whole ride. The bottom panel shows the limiting time scale, the τ of whichever rolling average is closest to its historical maximum at each moment. Hover to read any point.

This is the main result of the paper. Cyclists rode a 3 minute and a 12 minute maximal effort after about 2000 kJ of fatiguing work. At the point of exhaustion, the limiting time scale was not the 3 or 12 minute component you would expect from how long the effort lasted. It was a much slower component, around 1 hour. In the paper's words, performance limits emerge from the interaction of multiple time scales and are not solely dictated by the duration or intensity of the task.

To show this on your own ride, the tool takes the hardest effort of the length you choose from the first half of the ride, when you were fresher, and the hardest from the second half, when you were more fatigued. It runs each against your profile and reports the limiting time scale for each. If the limit has moved to a slower time scale when you were more tired, you will see it below. This is only an approximation. Unlike the paper's protocol, the two efforts are not guaranteed to be equally hard.

Figure 3 in the paper is a conceptual diagram, adapted from Gastin (2001). It is not data from this study. It suggests which energy system is likely to dominate for a given limiting time scale: the alactic ATP-PCr system at the shortest scales, anaerobic glycolysis a little longer, and aerobic metabolism at the long scales. The paper offers it as a way to interpret the model and to generate hypotheses, not as a measurement. Read the system labels in the replay the same way.

Build a session out of blocks of constant power and run it against your profile, without needing a real ride. You can see when your reserve would reach zero and which time scale would be the first to limit you. A quick way to test a pacing idea.

Each rolling average is a first order filter, EWM += (P − EWM)·Δt/τ, reset to an initial value of zero at the start of every session (Equation 3 in the paper). Because it starts from zero, the long time scales need a long warm up. A one hour ride can never fully load the four hour average. This follows from the method, and it is why your long time scale ceilings sit below a classic mean-max, which is just the best average for each duration looked at after the fact.

Fixed settings, chosen to match the paper. The twelve time constants are τ = 12, 30, 60, 120, 180, 300, 600, 1200, 1800, 3600, 7200 and 14400 seconds. Missing or coasting power is treated as 0 W. No pre-smoothing is applied. The energy system labels in section 3 are a conceptual overlay adapted from Gastin (2001), not measured data.

After Zignoli, Giorgi, Kolodziej, Martinez-Gonzalez, Leo and Laursen (2026), Tracking Performance Limits Using Multi-Timescale Maximal Mean Power Ratios, European Journal of Sport Science.