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Re: Non-random fluctuations in power output: methodological limits and complexity of interpretation

Dear Editor

In line with the previous work of their laboratory, Tucker et al [1] recently proposed a new point of view on the dynamic control mechanisms of the athlete during self-paced exercise. The authors should be thanked for their innovative contribution to exercise physiology, and we think that the tools they used need some complements.

Firstly, some methodological caution should be taken into account for forthcoming studies. Indeed, the number of analyzed data points (100 for the entire trial and only 20 for each interval) seems insufficient for non-linear analyses. A recent study showed a dramatical decrease in performance of spectral analysis with short series. [2] So we suggest that a minimum of 512 points should be considered to improve the validity of the method. Furthermore, the authors also used the Higuchi method for the determination of a fractal dimension on the number of available points. However Higuchi's algorithm primarily assesses geometric fractal properties, and much more relevant methods for the assessment of statistical fractal properties have widely been developed. So we suggest that a temporal method as Detrended Fluctuation Analysis (DFA) should be associated to spectral analysis, and completed by a method like ARFIMA/ARMA modeling, which provides a statistical probability for series to be fractal.

Secondly, data showed the presence of an endspurt leading the non-stationary of the signal. The authors concluded themselves that "the dominant low frequency cycle is probably caused by, or related to, the decrease in power output evident in most athletes during the middle period of the event, and the endspurt which occurs at the end of the event". That is precisely the reason why fractal analysis have to be applied on stationary signals. So we suggest that the drift of the data across the events should be corrected. The absence of steady-state is confirmed by the facts that "the dominant frequency spikes found when analyzing shorter time epochs at the beginning, middle and end of the time trial were different to those found when analyzing the frequency spectrum of the entire time trial". This result seems in contradiction with the self-similarity properties of fractal series. [4-5] In the same line, data reporting multiple peaks frequency peaks seems fundamentally opposed to 1/f-like scaling properties. [4-5] From a general point of view, non random fluctuations are not automatically fractal [3], and even if series in this study were indeed fractal, fractal properties are conceived to be generated by the interaction of several subsystems [4-5] and do not allow to conclude to the presence of one central governor.

References

1 Tucker R, Bester A, Lambert EV, et al. Non-random fluctuations in power output during self-paced exercise. Br J Sports Med. 2006, 40: 912-917.

2 Delignieres D, Ramdani S, Lemoine L, et al. Fractal analyses for ‘short’ time series: A re-assessment of classical methods. J Math Psychol. In press

3 Torre K, Delignières D, Lemoine L. Detection of long-range dependence and estimation of fractal exponents through ARFIMA modeling. Brit J Math Stat Psy. In press

4 Beltz BC, Kello CT. On the intrinsic fluctuations of human behavior. In: Sohob SP, ed. Trends in Cognitive Psychology. New York: Nova Publishers 2006.

5 Goldberger AL. Nonlinear dynamics, fractals and chaos theory: Implications for neuroautonomic heart control in health and disease. In: Bolis CL, Licinio J, eds. The Autonomic Nervous System. Genova: World Health Organization 1999.