Mean Normalized Gain: A New Method for the Assessment of the Aerobic System Temporal Dynamics during Randomly Varying Exercise in Humans

Figures

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  Figure 1. (A) Illustration of the 4-stage shift register used to generate the pseudorandom binary sequence protocol (PRBS). The module addition feedback (∑) sums the first and fourth stage values and tests the “if” statement. The value is then inserted into the first stage and the entire system shifts to the right, and the sequence repeats after 15 values. Each unit is maintained for 30s to create the PRBS protocol in the time-domain (B). The system input stimulates the oxygen uptake (V˙O2) response (−), here represented by the mean signal of all participants (n = 8) during the first visit. The V˙O2 data regarding the first PRBS sequence (warm-up) were excluded a priori. (C) To illustrate the frequency domain analysis, Fourier transformations were used to decompose the V˙O2 time-domain response of the second PRBS sequence into amplitudes of sinusoidal functions at specific frequencies (i.e., harmonics). The amplitudes of the system input (i.e., work rate in Watts) and output (i.e., V˙O2) at different frequencies are displayed in (D).

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  Figure 2. (A) Mean ± SD of 10 simulations of the system gain (gAmp) calculated in the frequency domain from data generated using 10 different values of system time constants (τs), amplitudes and baselines (see Table 2). (B) linear (– –) relationship between τs and the average absolute oxygen uptake gain at each tested harmonic (h) for each of the 10 simulations. (C) mean ± SD of data displayed in A normalized by the gAmp at h1. The mean normalized gain (MNG) was calculated as the mean of the gAmp between h2, h3, and h4 (please see text and Equation 4). (D) relationship between τs and MNG. This relationship was fitted by a linear (– –) and sigmoid (–) function. Notice in (D) that the normalization procedure isolated the relationship between amplitude and τs from other sources of system distortion such as system gain and baseline. The correlation coefficient “r” was used to indicate the degree of correlation between τs and MNG. Please see text for further details regarding the sigmoid function.

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  Figure 3. Computer simulations were performed to generate different oxygen uptake (V˙O2) responses considering different values of time constant (τs) that defines the speed of the V˙O2 adjustment to random exercise. The V˙O2 data were transformed to frequency domain and the mean normalized gain amplitude (MNG) was obtained considering the normalized system gain obtained from different frequency ranges. The equations describe how the MNG was obtained from the V˙O2 data. The x-axis scale was converted to log10 for a better visualization of the sigmoidal characteristics of the relationship between MNG and τs. The symbols “•” represent the simulated data from Figure 2D. Please see Equation (4) and text for further details regarding the equation parameters and procedures. Notice that the linear portion of the sigmoid function is always located at the physiological portion of tau values (i.e., from 10 to 100 s).

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  Figure 4. Relationship between time constant (τ) calculated from low-pass filtered oxygen uptake data and mean normalized gain (MNG). Both indexes were obtained from eight repetitions for all participants (n = 8) of the pseudorandom binary sequence test. The individual values (black circles), the regression line (dashed) and 95% CI (solid thin lines) are shown in comparison to the simulated data (solid thick line for MNG vs. τs as in Figures 2C).

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  Figure 5. Correlation between the mean normalized gain amplitude (MNG) estimated under the influence of different time of the day (A, Morning vs. Afternoon) and different day of testing (B, Day 1 vs. Day 2). The agreement level between the factors plotted in (A,B) are displayed in (C,D), respectively. r, Pearson's correlation level; p, statistical significance level; and n, sample size.

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  Figure 6. Correlation between the time constant (τ) estimated under the influence of different time of the day (A, Morning vs. Afternoon) and different day of testing (B, Day 1 vs. Day 2). The agreement level between the factors plotted in (A,B) are displayed in (C,D), respectively. The arrow indicates an outlier. r, Pearson's correlation level; p, statistical significance level; n, sample size.

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  Figure 7. Relationship between effect size of the mean normalized gain amplitude (MNG) or time constant from low-pass filtered data (τ) with the sample size needed to find statistical significance by student t-test (A,B) or paired t-test (C,D) considering different number of exercise repetitions (symbols). The desired power and the significance level considered for the sample size calculations were 0.8 and 0.05, respectively.

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  Figure 8. Relationship between the confidence interval (CI95, reported as percentage of the mean) of the time constant (τ) and mean normalized gain (MNG) obtained from different numbers of exercice repetitions. The τ was obtained from raw and filtered (0.075 Hz) oxygen uptake data. The τ CI95 stabilized at ≈70% after four exercise repetitions while the MNG CI95 was constant at ≈30% across all exercise repetitions.

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  Figure 9. Relationship between mean response time (MRT) calculated from filtered oxygen uptake data and mean normalized gain (MNG). Both indexes were obtained from eight repetitions for all participants (n = 8) of the pseudorandom binary sequence test. As the individual values (black circles), the regression line (dashed) and 95% CI (solid thin lines) are shown in comparison to the simulated data (solid thick line for MNG vs. τs as in Figure 2C).