# Adiabatic Invariant of Center-of-Mass Motion during Walking as a Dynamical Stability Constraint on Stride Interval Variability and Predictability

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Protocol

#### 2.2. Adiabatic Invariant I (Global Stability)

#### 2.3. SI Variability, Predictability, and Complexity

#### 2.4. Statistical Analysis

## 3. Results

#### 3.1. Population

#### 3.2. SI Variability, Predictability, and Complexity

#### 3.3. Phase-Space Dynamics

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Adiabatic Invariants and Random Noise

## References

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**Figure 1.**Two typical plots of whole-body COM vertical trajectories in phase space $(Q,P)$ in the two studied conditions: CTRL (green) and METRO (blue). Attractors, computed as the mean cycle in phase space, are also displayed in the two studied conditions: CTRL (dark green line) and METRO (dark blue line). The same subject has been chosen in both conditions.

**Figure 2.**Typical plots of the SI time series obtained in the two studied conditions: CTRL (green) and METRO (blue). Parameters in the CTRL condition are: SI = 1.12 s, CV = 0.0282, H = 0.988, S = 1.59, and D = 1.35. Parameters in METRO condition are: SI = 1.12 s, CV = 0.0132, H = 0.383, S = 2.20, and D = 1.51.

**Figure 3.**(

**A**) Comparison of the mean values of CV in CTRL and METRO conditions. The error bar is equal to 1 SD. (

**B**) Boxplots comparing the distribution of H in CTRL and METRO conditions. (

**C**) Same graphical representation as in (

**A**) for $\pi \phantom{\rule{0.166667em}{0ex}}I$. The stars (*) denote significant differences between the means or medians between the two conditions.

**Figure 4.**(

**A**) Computed pairs $\left(\frac{f}{{f}_{m}},\frac{\overline{{E}_{k}}}{{E}_{km}}\right)$ (points) compared to a global regression of the form (Equation (3)) (black line and gray band indicating the 95% confidence interval). The coefficient of determination ${R}^{2}$ of the regression is indicated. The model (Equation (2)) is also shown (dashed red line). (

**B**) Computed pairs with the linear trend (2) removed $\left(\frac{f}{{f}_{m}}-1,\frac{\overline{{E}_{k}}}{{E}_{km}}-\frac{f}{{f}_{m}}\right)$. The densities of the points along the two axes are shown above (density of $\frac{f}{{f}_{m}}$) and to the right side of the plot (density of $\frac{\overline{{E}_{k}}}{{E}_{km}}-\frac{f}{{f}_{m}}$).

**Table 1.**General characteristics of our population. Results are reported in the form mean ± SD. The number of gait cycles performed by the participants in 10 min is reported in the form median [q1–q3], regardless of the condition.

N | 25 |
---|---|

Age (years) | 22.8 ± 5.2 |

Mass (kg) | 68.1 ± 13.6 |

Height (m) | 1.65 ± 0.32 |

Sex (M/F) | 9/16 |

Gait cycles | 532 [513–552] |

**Table 2.**Comparison between results in conditions CTRL and METRO for the SI analysis. Results are reported in the form mean±SD if a paired t-test was performed, or median and first-third quartiles [q1–q3] if a Wilcoxon signed rank test was performed. Significant p-values are in bold.

Condition | SI (s) | CV | H | D | S |
---|---|---|---|---|---|

CTRL | 1.184 [1.126–1.269] | 0.0261 ± 0.0071 | 0.848 [0.781–0.951] | 1.633 ± 0.116 | 1.759 ± 0.228 |

METRO | 1.187 [1.119–1.273] | 0.0193 ± 0.0060 | 0.373 [0.265–0.560] | 1.667 ± 0.129 | 1.891 ± 0.249 |

p | 0.258 | 0.001 | <0.001 | 0.282 | 0.063 |

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**MDPI and ACS Style**

Buisseret, F.; Dehouck, V.; Boulanger, N.; Henry, G.; Piccinin, F.; White, O.; Dierick, F.
Adiabatic Invariant of Center-of-Mass Motion during Walking as a Dynamical Stability Constraint on Stride Interval Variability and Predictability. *Biology* **2022**, *11*, 1334.
https://doi.org/10.3390/biology11091334

**AMA Style**

Buisseret F, Dehouck V, Boulanger N, Henry G, Piccinin F, White O, Dierick F.
Adiabatic Invariant of Center-of-Mass Motion during Walking as a Dynamical Stability Constraint on Stride Interval Variability and Predictability. *Biology*. 2022; 11(9):1334.
https://doi.org/10.3390/biology11091334

**Chicago/Turabian Style**

Buisseret, Fabien, Victor Dehouck, Nicolas Boulanger, Guillaume Henry, Florence Piccinin, Olivier White, and Frédéric Dierick.
2022. "Adiabatic Invariant of Center-of-Mass Motion during Walking as a Dynamical Stability Constraint on Stride Interval Variability and Predictability" *Biology* 11, no. 9: 1334.
https://doi.org/10.3390/biology11091334