Restoring Walking Complexity in Older Adults Through Arm-in-Arm Walking: Were Almurad et al.’s (2018) Results an Artifact?

in Motor Control
View More View Less
  • 1 University of Montpellier
  • 2 Union Nationale Sportive Léo Lagrange
  • 3 Montpellier University Hospital

The analysis of stride series revealed a loss of complexity in older people, which correlated with the falling propensity. A recent experiment evidenced an increase of walking complexity in older participants when they walked in close synchrony with a younger companion. Moreover, a prolonged experience of such synchronized walking yielded a persistent restoration of complexity. This result, however, was obtained with a unique healthy partner, and it could be related to a particular partner’s behavior. The authors’ aim was to replicate this important finding using a different healthy partner and to compare the results to those previously obtained. The authors successfully replicated the previous results: synchronization yielded an attraction of participants’ complexity toward that of their partner and a restoration of complexity that persisted in two posttests, 2 and 6 weeks after the end of the training sessions. This study shows that this complexity restoration protocol can be applied successfully with another partner, and allows us to conclude that it can be generalized.

Complexity is considered an essential feature for living systems, providing them with both robustness and adaptability (Whitacre, 2010). As such, complexity represents an important scope for research focusing on evolution (Whitacre & Bender, 2010) or health (Lipsitz & Goldberger, 1992). Goldberger, Peng, and Lipsitz (2002) developed the hypothesis of the loss of complexity with aging and disease (see also Harrison & Stergiou, 2015; Stergiou & Decker, 2011), and more specifically, Hausdorff, Edelberg, Mitchell, Goldberger, and Wei (1997) showed a typical decrease of complexity in stride duration dynamics in elderly people and showed that this decrease in complexity correlated with falling propensity (see also Buzzi et al., 2003; Haudorff, 2007; Herman, Giladi, Gurevich, & Hausdorff, 2005; Kurz, Markopoulou & Stergiou et al., 2010).

Almurad, Roume, Blain, and Delignières (2018) explored the hypothesis of a possible restoration of walking complexity in older people. This work was based on the framework of complexity matching, initially introduced by West, Geneston, and Grigolini (2008). The complexity matching effect states that information transfer is maximized when interacting systems share similar complexity levels. Marmelat and Delignières (2012) proposed an additional hypothesis, suggesting that interacting systems tend to harmonize their complexities in order to improve their synchronization. This effect has been evidenced in several experiments (Abney, Paxton, Dale, & Kello, 2014; Almurad, Roume, & Delignières, 2017; Coey, Washburn, Hassebrock, & Richardson, 2016; Delignières & Marmelat, 2014; Marmelat & Delignières, 2012; Stephen, Stepp, Dixon, & Turvey, 2008). In particular, Almurad et al. (2017) showed that walking in synchrony, side-by-side, was effectively governed by a complexity matching effect, revealed by a close attunement of the complexities of the two partners. They also evidenced that the complexity matching effect increased with coupling strength and was stronger in close arm-in-arm walking than in just side-by-side walking. Note that, in a similar protocol, Nessler et al. (2011) evidenced an alteration of walking complexity in both participants, which was interpreted as the result of the active control of synchronization. The authors, however, did not check for a possible convergence of complexity levels within the dyads, which represents the typical signature of complexity matching.

More interestingly for the present purpose, Mahmoodi, West, and Grigolini (2018, 2020) formally showed that, when two systems with different complexity levels interact, the least complex system is “attracted” by the most complex, yielding an increase of the complexity of the former. This theoretical hypothesis was experimentally tested by Almurad et al. (2018), who showed that a prolonged training of walking in synchrony with a young and healthy companion allowed for a restoration of walking complexity in older participants. In this experiment, elderly participants were invited to walk in close synchrony, arm-in-arm, with a young experimenter. The experiment lasted 4 weeks, with three sessions per week. The results confirmed that synchronization was achieved through complexity matching and that, during synchronization bouts, walking complexity in participants tended to match that of the experimenter. The evolution of the intrinsic complexity of walking was assessed during bouts performed in isolation at the beginning of each week: the results showed a significant restoration of complexity at the beginning of the fourth week, and this effect was persistent in a posttest performed 2 weeks after the end of the training sessions.

However, a potential bias of this experiment was related to the fact that a unique individual served as the companion for all participants, yielding a possible artifact in the obtained results. The main aim of the present work was to replicate this experiment in a protocol in which the participants were accompanied by another young guide. In addition, we completed the previous protocol by performing two additional posttests, 4 and 6 weeks after the end of the training sessions, in order to assess the medium-term persistence of the obtained effect. Finally, we compared the data obtained in the present experiment with those of Almurad et al. (2018) in order to check for possible differences in the evolution of the participants’ walking complexity in the two experiments.

The experimental hypotheses were the following:

  1. (a)If an older person is invited to walk in synchrony, arm-in-arm with a healthy companion, we should observe a complexity matching effect within the dyad.
  2. (b)Considering the asymmetry of complexities (older participants exhibiting lower levels of complexity than their companion), complexity matching should result in an increase of complexity in the older person.
  3. (c)A prolonged training of walking in synchrony with healthy partners should induce a perennial restoration of complexity in older participants.

Materials and Methods

Participants

Almurad et al. (2018), in a within-factor analysis of variance (ANOVA) with repeated measures, obtained an effect size of Cohen’s d = 0.48. A power analysis (Faul et al., 2017) showed that in order to replicate this effect size, with 95% power, 12 participants should be necessary. Twelve participants (four males and eight females, mean age = 72.0 years, SD = 8.13) were then involved in the study. They were recruited from within associations or via advertisements with health professionals. They presented no contraindication to the practice of autonomous walking (musculoskeletal, cardiovascular, respiratory, or neurological pathologies). They were randomly assigned to two groups, experimental and control. One of the participants of the control group, however, was unable, for health reasons, to complete the whole protocol and was excluded from statistical analyses. Finally, the experimental group included six participants (one male and five females, mean age = 69.83 years, SD = 7.2; mean weight = 78.33 kg, SD = 12.5; mean height = 166.67 cm, SD = 11.5), and the control group included five participants (two males and three females, mean age = 75 years, SD = 8.8; mean weight = 74 kg, SD = 10.4; mean height = 165.2 cm, SD = 10.3). The healthy guide was a female (28 years old, weight = 58 kg, height = 168 cm).

This study was conducted in accordance with the 1964 Helsinki Declaration and validated by the Euromov International Review Committee (no. 1711C). The participants signed an informed consent form and were not rewarded for their participation.

Experimental Procedure

For this study, we strictly followed the protocol of Almurad et al. (2018) in order to check its validity and reliability. The experiment was performed on a covered athletic track (circumference 200 m). The participants were asked to perform walking training for 4 consecutive weeks. Each week included three training sessions, on Monday, Wednesday, and Friday. During each session, the participants had to perform four 15-min walking sequences. On the Monday session, the participants started with a solo sequence in which they had to walk alone, in the most regular way, for 15 min, at their preferred speed. This solo sequence allowed us to evaluate the complexity of the series of stride durations produced by the participant at the beginning of each week.

Each participant was accompanied by the guide for all the other walking sequences (three sequences on Monday, four sequences on Wednesday, and four sequences on Friday). The participants in the experimental group walked arm-in-arm with the young guide (see Figure 1) with an explicit instruction to synchronize their steps with those of their guide. The participants in the control group walked next to an experimenter without physical contact and without any synchronization instruction. For both groups, the experimenter adapted her walking speed to that of the participant.

Figure 1
Figure 1

Illustration of the arm-in-arm walking condition. The participants are explicitly instructed to synchronize their steps with those of their guide.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

Between two successive walking sequences, the participants had a rest of about 10–15 min. All participants performed the same amount of training (44 sequences, 12 hr of walking). The difference between the experimental group and the control group was only in terms of physical contact and the synchronization of steps during walking.

The participants performed a posttest (solo sequence) 2 weeks after the end of the protocol (Week 7). For the experimental group, we added two supplementary posttests, the first one 4 weeks after the end of the protocol (Week 9) and the second one 6 weeks after the end of the protocol (Week 11).

Data Collection

We recorded data via two soles containing force sensitive resistors positioned at the heel. The soles were connected to a Schmitt trigger (LM 393AN), a device that makes the linear signal of the force-sensitive resistor sensors a (on/off) switch via a voltage threshold. The output of the Schmitt trigger was connected to the GPIO interface of a Raspberry Pi model A+. A Wi-Fi dongle (EDIMAX EW7811Un) connected to the USB port of the Raspberry was configured as a hotspot, which allowed us to launch and distance the two devices. We had developed a small box containing the Raspberry Pi, the Schmitt trigger, and a battery (2000 mAh). This device was put in a bag worn on the belt of the participants during training. The bag in its entirety weighted 0.4 kg.

Regarding the acquisition software, we powered the Raspberry Pi by the version of the Raspbian distribution of February 9, 2016. Then, to record the data, we wrote a script in Python 3, which uses the internal clock of the Raspberry to time each touch of the heel against the ground and then calculate the series of stride durations of the subject.

Statistical Analyses

All analyses were performed on the series of stride durations of the right leg. Each raw series contained between 700 and 1400 data points. When analyzing data, we observed local trends related to periods of increase or decrease in walking speed, especially at the beginning of the series, essentially due to the time needed for participants to reach the comfortable speed for performing the test. Therefore, since fractal analyses can be distorted due to local trends in the series, these initial acceleration/deceleration phases were removed for each series.

The resulting stride series had an average length of 647.58 points for solo sequences (SD = 191, max = 1,101, min = 257) and 633.78 points for duo sequences (SD = 140, max = 1,004, min = 222). The majority of the recorded series presented the minimum number of points required for a valid fractal analysis (Delignières et al., 2006).

We first applied the Windowed detrended cross-correlation (WDCC) analysis proposed by Roume et al. (2018) to assess the nature and the strength of synchronization between the dyads during the duo trials. WDCC computes the cross-correlation function within short windows of 15 points, for lags ranging from −10 to 10. The data are linearly detrended within each interval before the computation of cross-correlation coefficients. A sliding window procedure is used for obtaining multiple assessments of the cross-correlation function. WDCC functions were computed for each participant and each duo sequence and then point-by-point averaged, for each participant, within each week. Finally, we computed a weekly averaged WDCC function across participants for each group, and we tested the signs of the cross-correlation coefficients with two-tailed location t tests, comparing the obtained values to zero (Roume et al., 2018). In WDCC functions, the complexity matching effect is revealed by a significant positive peak at lag 0, indicating an immediate synchronization between systems. WDCC could also reveal discrete, step-by-step corrective processes: in that case, positive peaks could appear at lag −1 and/or lag 1, depending of the leader/follower statuses within the dyad (Roume et al., 2018). Note that WDCC correlation coefficients could take values in the interval [−1, 1].

We used the detrended fluctuation analysis (DFA, Peng et al., 1994) to estimate the complexity of each data series. For this analysis, we chose to start the intervals from 10 up to N/2 (N being the length of the series). We used the evenly spaced DFA algorithm proposed by Almurad and Delignières (2016), which was proven to provide a better estimate of the scaling exponent, especially for a short series. Note the α-DFA exponents obtained in the walking series are expected to vary between 0.5 and 1, the exponent approaching 1 in young and healthy participants, revealing optimal complexity. The loss of complexity is revealed by weaker exponents, 0.5 theoretically corresponding to a complete loss of complexity.

In order to assess the effects of the experimental protocol on the complexity of stride series in the solo trials, we applied a two-factor ANOVA 2 (group) × 5 (week), with repeated measurements on the second factor (including the 4 weeks of the training protocol and the posttest). We used the Bonferroni post hoc test for locating significant ANOVA effects. In a second step, we applied a one-way ANOVA 7 (week), with repeated measurements, for assessing the persistence of the effects of synchronized walking in the experimental group over the three posttests performed after the end of the rehabilitation protocol.

Finally, in order to compare our data with that obtained by Almurad et al. (2018), we applied a three-factor ANOVA 2 (group) × 5 (week) × 2 (guide), with repeated measurements on the second factor (including the 4 weeks of the protocol and the first posttest). We used the Bonferroni post hoc test for locating significant ANOVA effects.

Results

We confirmed that synchronized walking was dominated by a complexity matching effect. In all cases, the weakly averaged WDCC functions presented a positive peak at lag 0, revealing the immediate synchronization expected from a complexity matching effect (Figure 2). However, this peak had a higher average value for the experimental group (about 0.3) than the control group (about 0.1). The lower value obtained in the control group suggested a less marked, even intermittent synchronization. In the experimental group, these functions also revealed a significant positive peak at lag 1 during the first and the third week. This phenomenon is not noticeable in the control group.

Figure 2
Figure 2

Evolution of the averaged WDCC function for the experimental group (n = 6; top row) and the control group (n = 5; bottom row) over the 4 weeks of training. A positive peak in lag 0 reveals the presence of a complexity matching effect. The sign of the mean cross-correlation coefficients was tested with two-tailed location t tests, comparing the obtained values with zero. Ctrl. = control group; Exp. = experimental group; WDCC = Windowed detrended cross-correlation. *p < .05.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

This study confirmed that a prolonged experience of close synchronized walking with a young and healthy guide allowed for the restoration of walking complexity in elderly participants and that this effect persisted 2 weeks after the end of the training sessions: The ANOVA revealed a significant interaction between group and week, F(4, 36)  = 4.413, p = .005, ηp2=.329, and the Bonferroni post hoc test showed a significant difference between the mean α-DFA obtained in the experimental group during the fourth week and the posttest on the one hand, and the mean α-DFA in Weeks 1 and 2 (p < .01). We report in Table 1 the averages and SDs of α-DFA estimates for each group and each solo sequence for the 4 weeks of the training protocol and the first posttest (Week 7), and in Table 2, we report the results of the two-factor ANOVA performed on the related samples. These results are illustrated in Figure 3.

Table 1

Mean α-DFA Estimates (SDs) for Each Group and Each Solo Sequence for the 4 Weeks of the Experiment and the First Posttest

GroupWeek 1Week 2Week 3Week 4Posttest
Experimental group0.7770.7980.8630.9590.964
(n = 6)(0.073)(0.109)(0.087)(0.097)(0.075)
Control group0.8450.8100.8080.8370.827
(n = 5)(0.063)(0.145)(0.043)(0.047)(0.048)

Note. An α-DFA close to 1 corresponds to an optimal complexity; any decrease in this exponent reveals a loss of complexity. A total loss of complexity should be revealed by α-DFA close to 0.5. DFA = detrended fluctuation analysis.

Table 2

Results of the Two-Factor Analysis of Variance 2 (Group) × 5 (Week)

SourcedfFpηp2
Group11.715.222.160
Error9
Week44.816.003.349
Week × Group44.413.005.329
Error36
Figure 3
Figure 3

Evolution of α-DFA exponents computed for participants in solo sequences (black, experimental group; gray, control group) over the 4 training weeks and the posttest (Week 7). The interaction Group × Week was significant, F(4, 36)  = 4.413, p = .005, ηp2=.329. This figure highlights the significant improvement in α-DFA exponents in the experimental group, revealing a restoration of the complexity of walking from the fourth week of the experiment up to 2 weeks postprotocol. This graph presents a combination of boxplots and scatter plots (Campbell, notBoxPlot https://www.github.com/raacampbell/notBoxPlot). DFA = detrended fluctuation analysis. *p < .05. **p < .01.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

We report in Figure 4 the evolution of the mean α-DFA exponents during the 4 weeks of the training protocol, for both solo and duo sequences. The left graph represents the results for the experimental group, and the right one for the control group. As the “guide” data were obtained from a unique individual, statistical analyses were not applicable. These graphs, however, highlight the lower level of complexity of participants during the solo sequences, especially at the beginning of the experiment, as compared with that of their guide. They also illustrate the attraction of participants’ complexity toward that of their guide during duo sequences in the experimental group. This convergence appears less noticeable in the control group. Finally, at the beginning of the fourth week, the level of the average α-DFA exponent in the experimental group reached the level of the guide in the solo test, which was not the case for the control group.

Figure 4
Figure 4

Average α-DFA exponents computed for guides (circles) and participants (squares) in solo sequences (black) and duo sequences (white), over the 4 training weeks. Results are displayed for the experimental group (left) and the control group (right). Error bars represent SD. This figure notably illustrates the attraction of participants’ complexity toward that of their guide during duo sequences in the experimental group. This convergence appears less noticeable in the control group. DFA = detrended fluctuation analysis.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

The analysis of the evolution of the α-DFA exponent in the experimental group during the 4 weeks of the training protocol and the three subsequent posttests showed a persistence of the complexity restoration effect up to 6 weeks after the end of the training (Figure 5). The ANOVA revealed a significant effect, F(6, 30) = 6.119, p = .0002, ηp2=.550), and the post hoc test showed that the mean α-DFA exponent was higher during the solo test performed at the beginning of the fourth week and during the posttests performed at Weeks 7 and 11, as compared with the first week of the protocol (p < .05).

Figure 5
Figure 5

Average α-DFA exponents computed for the experimental group in solo sequences over the 4 training weeks and the posttests (Weeks 7, 9, and 11). This graph highlights the persistence of the complexity restoration effect in the experimental group up to 6 weeks posttraining. Error bars represent SD. DFA = detrended fluctuation analysis. *p < .05.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

Finally, merging our data with those obtained by Almurad et al. (2018), we evidenced a similar evolution of mean α-DFA in the two experiments (Figure 6). The ANOVA did not reveal a main effect for the guide factor, nor any interaction between guide and the other factors. The analysis just showed a significant interaction between group and week, F(4, 72) = 8.454, p = .0000, ηp2=.319), and the post hoc test revealed a significant difference between the mean α-DFA obtained in the experimental groups during the fourth week on the one hand, and those obtained during the three first weeks of the training protocol (p < .01). It also showed a significant difference between the means α-DFA of the experimental groups and those of the two first weeks of the training protocol. The complete results of the three-factor ANOVA (2 guide × 2 group × 5 week) are displayed in Table 3.

Figure 6
Figure 6

Average α-DFA exponents computed for participants in solo sequences over the 4 training weeks and the posttest. Results are displayed for the two experimental groups (black circles—Guide 1, present experiment; black squares—Guide 2, Almurad et al.’s [2018] experiment) and for the two control groups (white circles—Guide 1; white squares—Guide 2). This figure highlights the similarity of the results obtained by Almurad et al. (2018) and those of the present study. The interaction Group × Week was significant, F(4, 124)  = 8.620, p = .0000, ηp2=.218. Error bars represent SD. DFA = detrended fluctuation analysis.

Citation: Motor Control 2021; 10.1123/mc.2020-0052

Table 3

Results of the Three-Factor Analysis of Variance 2 (Group) ×5 (Week) × 2 (Guide)

SourcedfFpηp2
Group15.438.026.149
Guide10.043.837.001
Group × Guide10.039.845.001
Error31
Week410.668.000.256
Week × Group48.620.000.218
Week × Guide40.328.859.010
Week × Group × Guide40.960.432.030
Error124

Discussion

The hypotheses of this study were clearly validated: When we invite an older person to walk in synchrony, arm-in-arm, with a young and healthy partner, synchronization is mainly dominated by a complexity matching effect. We were able to check this hypothesis by the WDCC analysis. This function revealed a positive peak at lag 0, showing an immediate synchronization between systems, which represents the typical signature of complexity matching. This peak at lag 0 appeared in both groups, but was higher in the experimental group, confirming that the complexity matching effect is related to the strength of coupling between partners. We noticed, however, that the peak at lag 0 for the control group was weaker than that reported in Almurad et al. (2018). This could be due to an experimenter effect influencing the strength of synchronization in the dyad. Finally, the complexity matching effect was present from the very first duo sequences performed during the training protocol, indicating that this effect appeared spontaneously and was not the result of a specific learning. These results confirm those of Almurad et al. (2018). The positive peak that appears at lag 1 in the WDCC functions for the experimental suggests that participants tended, in addition to the complexity matching synchronization, to correct their steps on the basis on the preceding asynchrony. This discrete correction process remains marginal, however, and synchronization seemed mainly achieved through complexity matching.

The attraction of participants’ complexity toward that of their guide represents a nice experimental validation of the formal result of Mahmoodi et al. (2020), stating that when two systems of different levels of complexity interact, the most complex tends to attract the less one. The present results confirm that this effect depends on the strength of coupling between the two systems in interaction.

We confirm that the prolonged experience of complexity matching allows for the restoration of walking complexity in elderly people, as evidenced by the increase of DFA exponents at the beginning of the fourth week and during the posttest. This experiment and the previous work by Almurad et al. (2018) indicate that 3 weeks of intensive practice could be sufficient for obtaining a significant restoration of complexity. The absence of any significant evolution of scaling exponents in the control group shows that the complexity matching effect is essential in the restoration of complexity and that physical activity alone is not sufficient for inducing any effect on the complexity of stride dynamics (the control group having carried out the same training load as the experimental group).

Several experiments were recently conducted for exploring the effects of walking in synchronization with artificial devices, mimicking the complexity of healthy natural gait (fractal-like metronomes), for walking rehabilitation in older people (e.g., Kaipust, McGrath, Mukherjee, & Stergiou, 2013; Vaz, Knarr, & Stergiou, 2020). Generally, one effectively observes an increase of walking complexity during synchronization. However, is synchronization with an irregular, fractal-like metronome really equivalent to synchronization with a human partner? Delignières and Marmelat (2014) showed that walking in synchronization with a fractal metronome was essentially performed through discrete step-to-step asynchrony corrections (revealed by positive peaks at lag 1 and lag 2 in the WDCC function), but they did not observe any evidence of complexity matching effect. As stated previously, our results suggest that a prolonged experience of complexity matching represents the key of a lasting complexity restoration. We are not sure that the use of a metronome, even perfectly mimicking natural variability, can give the same result. Interestingly, however, Vaz et al. (2020) observed an increase in walking complexity in older participants walking in synchrony with a visual fractal-like metronome and showed that this improvement was retained, as least for some minutes, when the stimulus was turned off. Further research efforts are necessary for testing the durability of this apparent restoration of complexity and to reveal the nature of the processes at work for ensuring synchronization.

We have shown that our results are similar to those obtained by Almurad et al. (2018): This effect of complexity restoration was not related to a particular guide’s behavior, and it could be replicated with another guide. This represents an essential point, allowing one to pursue with some confidence the rehabilitation perspectives offered by this kind of protocol. Finally, our results show that the effect of restoration is preserved up to 6 weeks postprotocol. This medium-term persistence, which was not tested in the previous experiment, represents a very encouraging result for rehabilitation purposes.

The number of participants in this experiment remained modest, but our main goal was to replicate the experiment and check for the absence of any artifact, in order to validate the protocol. Note that, despite the loss of one participant during the experiment, the effect size remains quite satisfactory (d = 0.62), of medium level according to the classification of Cohen (2013).

We are aware, however, that the scope of these experiments remains limited to a quite fundamental issue: is it possible, by means of the complexity matching effect, to restore complexity in deficient systems? The protocol we tested is very challenging for participants (approximately 36 km of walking during the 4 weeks). For evident reasons, we recruited participants in a population of elderly people with nonpathologic aging and a moderate loss of complexity. Additional efforts are obviously needed for adapting and testing this kind of protocol with frailer patients (older individuals, Parkinsonian patients, etc.).

The present results show that a restoration of complexity is conceivable, and one could think that this enhancement in complexity should induce more stable and adaptable walking and a reduction of falling propensity in elderly people. Clinical evidence is still lacking, however, and we have engaged a new experiment for studying the impact of this arm-in-arm synchronized walking protocol on clinical measures assessing the risk of falling.

Finally, we have no indication about the durability of this restoration beyond the 6 weeks, which separates the end of the training and the third posttest. Our current projects aim at checking this durability, with delayed posttests up to 2 months postprotocol.

Conclusion

This work remains fundamental in nature but obviously opens essential perspectives at the clinical level. The present results allow one to pursue these clinical perspectives with more confidence. We would like to emphasize that the operational implementation of this protocol remains very affordable since it does not require any expensive technology but simply young volunteers in good health. We believe that this intergenerational link could be an effective and inexpensive way to prevent the fall of our seniors.

Acknowledgment

This research was supported by a grant from the Union Sportive Léo Lagrange, awarded to the first author. The authors declare no conflict of interest.

References

  • Abney, D.H., Paxton, A., Dale, R., & Kello, C.T. (2014). Complexity matching in dyadic conversation. Journal of Experimental Psychology: General, 143(6), 2304. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., & Delignières, D. (2016). Evenly spacing in detrended fluctuation analysis. Physica A: Statistical Mechanics and Its Applications, 451, 6369. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., Roume, C., Blain, H., & Delignières, D. (2018). Complexity matching: restoring the complexity of locomotion in older people through arm-in-arm walking. Frontiers in Physiology, 9, 1766. PubMed ID: 30564149 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., Roume, C., & Delignières, D. (2017). Complexity matching in side-by-side walking. Human Movement Science, 54, 125136. PubMed ID: 28460275 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., & Heidel, J. (2003). Nonlinear dynamics indicates aging affects variability during gait. Clinical Biomechanics, 18(5), 435443. PubMed ID: 12763440 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Coey, C.A., Washburn, A., Hassebrock, J., & Richardson, M.J. (2016). Complexity matching effects in bimanual and interpersonal syncopated finger tapping. Neuroscience Letters, 616, 204210. PubMed ID: 26840612 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cohen, J. (2013). Statistical power analysis for the behavioral sciences. New York, NY: Academic Press.

  • Delignières, D., & Marmelat, V. (2014). Strong anticipation and long-range cross-correlation: Application of detrended cross-correlation analysis to human behavioral data. Physica A: Statistical Mechanics and its Applications, 394, 4760. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delignières, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., & Ninot, G. (2006). Fractal analyses for ‘short’ time series: A re-assessment of classical methods. Journal of Mathematical Psychology, 50(6), 525544.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175191. PubMed ID: 17695343 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Goldberger, A.L., Peng, C.K., & Lipsitz, L.A. (2002). What is physiologic complexity and how does it change with aging and disease? Neurobiology of Aging, 23(1), 2326. PubMed ID: 11755014 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harrison, S.J., & Stergiou, N. (2015). Complex adaptive behavior and dexterous action. Nonlinear Dynamics, Psychology, and Life Sciences, 19(4), 345394. PubMed ID: 26375932

    • Search Google Scholar
    • Export Citation
  • Hausdorff, J.M. (2007). Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride fluctuations of human walking. Human Movement Science, 26(4), 555589. PubMed ID: 17618701 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hausdorff, J.M., Edelberg, H.K., Mitchell, S.L., Goldberger, A.L., & Wei, J.Y. (1997). Increased gait unsteadiness in community-dwelling elderly fallers. Archives of Physical Medicine and Rehabilitation, 78(3), 278283. PubMed ID: 9084350 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Herman, T., Giladi, N., Gurevich, T., & Hausdorff, J.M. (2005). Gait instability and fractal dynamics of older adults with a “cautious” gait: why do certain older adults walk fearfully? Gait & Posture, 21(2), 178185. PubMed ID: 15639397 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kaipust, J.P., McGrath, D., Mukherjee, M., & Stergiou, N. (2013). Gait variability is altered in older adults when listening to auditory stimuli with differing temporal structures. Annals of Biomedical Engineering, 41, 41(8), 15951603. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurz, M.J., Markopoulou, K., & Stergiou, N. (2010). Attractor divergence as a metric for assessing walking balance. Nonlinear Dynamics, Psychology, and Life Sciences, 14(2), 151164. PubMed ID: 20346260

    • Search Google Scholar
    • Export Citation
  • Lipsitz, L.A., & Goldberger, A.L. (1992). Loss of ‘complexity’ and aging: Potential applications of fractals and chaos theory to senescence. Jama, 267(13), 18061809. PubMed ID: 1482430 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mahmoodi, K., West, B.J., & Grigolini, P. (2018). Complexity matching and requisite variety. arXiv preprint arXiv:1806.08808. [accessed October 30, 2018]

    • Search Google Scholar
    • Export Citation
  • Mahmoodi, K., West, B.J., & Grigolini, P. (2020). Complex periodicity and synchronization. Frontiers in Physiology, 11. doi:

  • Marmelat, V., & Delignières, D. (2012). Strong anticipation: complexity matching in interpersonal coordination. Experimental Brain Research, 222(1–2), 137148. PubMed ID: 22865163 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nessler, J.A., Gonzales, T, Rhoden, E., Steinbrick, M., & De Leone, C.J. (2011). Stride interval dynamics are altered when two individuals walk side by side. Motor Control, 15(3), 390404. PubMed ID: 21878691 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., & Goldberger, A.L. (1994). Mosaic organization of DNA nucleotides. Physical Review E, 49(2), 16851689. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Roume, C., Almurad, Z.M.H., Scotti, M., Ezzina, S., Blain, H., & Delignières, D. (2018). Windowed detrended cross-correlation analysis of synchronization processes. Physica A: Statistical Mechanics and Its Applications, 503, 11311150. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stephen, D.G., Stepp, N., Dixon, J.A., & Turvey, M.T. (2008). Strong anticipation: Sensitivity to long-range correlations in synchronization behavior. Physica A: Statistical Mechanics and Its Applications, 387(21), 52715278. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stergiou, N., & Decker, L.M. (2011). Human movement variability, nonlinear dynamics, and pathology: Is there a connection?. Human Movement Science, 30(5), 869888. PubMed ID: 21802756 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vaz, J.R., Knarr, B.A., & Stergiou, N. (2020). Gait complexity is acutely restored in older adults when walking to a fractal-like visual stimulus. Human Movement Science, 74, 102677. PubMed ID: 33069099 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • West, B.J., Geneston, E.L., & Grigolini, P. (2008). Maximizing information exchange between complex networks. Physics Reports, 468(1-3), 199. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitacre, J.M. (2010). Degeneracy: A link between evolvability, robustness and complexity in biological systems. Theoretical Biology and Medical Modelling, 7(1), 6. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitacre, J., & Bender, A. (2010). Degeneracy: A design principle for achieving robustness and evolvability. Journal of Theoretical Biology, 263(1), 143153. PubMed ID: 19925810 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

Ezzina, Roume, Pla, Blain, and Delignières are with EuroMov Digital Health in Motion, University of Montpellier, IMT Mines Ales, Montpellier, France. Ezzina is also with the Union Nationale Sportive Léo Lagrange, Paris, France. Blain is also with the Montpellier University Hospital, Montpellier, France.

Delignières (didier.delignieres@umontpellier.fr) is corresponding author.
  • View in gallery

    Illustration of the arm-in-arm walking condition. The participants are explicitly instructed to synchronize their steps with those of their guide.

  • View in gallery

    Evolution of the averaged WDCC function for the experimental group (n = 6; top row) and the control group (n = 5; bottom row) over the 4 weeks of training. A positive peak in lag 0 reveals the presence of a complexity matching effect. The sign of the mean cross-correlation coefficients was tested with two-tailed location t tests, comparing the obtained values with zero. Ctrl. = control group; Exp. = experimental group; WDCC = Windowed detrended cross-correlation. *p < .05.

  • View in gallery

    Evolution of α-DFA exponents computed for participants in solo sequences (black, experimental group; gray, control group) over the 4 training weeks and the posttest (Week 7). The interaction Group × Week was significant, F(4, 36)  = 4.413, p = .005, ηp2=.329. This figure highlights the significant improvement in α-DFA exponents in the experimental group, revealing a restoration of the complexity of walking from the fourth week of the experiment up to 2 weeks postprotocol. This graph presents a combination of boxplots and scatter plots (Campbell, notBoxPlot https://www.github.com/raacampbell/notBoxPlot). DFA = detrended fluctuation analysis. *p < .05. **p < .01.

  • View in gallery

    Average α-DFA exponents computed for guides (circles) and participants (squares) in solo sequences (black) and duo sequences (white), over the 4 training weeks. Results are displayed for the experimental group (left) and the control group (right). Error bars represent SD. This figure notably illustrates the attraction of participants’ complexity toward that of their guide during duo sequences in the experimental group. This convergence appears less noticeable in the control group. DFA = detrended fluctuation analysis.

  • View in gallery

    Average α-DFA exponents computed for the experimental group in solo sequences over the 4 training weeks and the posttests (Weeks 7, 9, and 11). This graph highlights the persistence of the complexity restoration effect in the experimental group up to 6 weeks posttraining. Error bars represent SD. DFA = detrended fluctuation analysis. *p < .05.

  • View in gallery

    Average α-DFA exponents computed for participants in solo sequences over the 4 training weeks and the posttest. Results are displayed for the two experimental groups (black circles—Guide 1, present experiment; black squares—Guide 2, Almurad et al.’s [2018] experiment) and for the two control groups (white circles—Guide 1; white squares—Guide 2). This figure highlights the similarity of the results obtained by Almurad et al. (2018) and those of the present study. The interaction Group × Week was significant, F(4, 124)  = 8.620, p = .0000, ηp2=.218. Error bars represent SD. DFA = detrended fluctuation analysis.

  • Abney, D.H., Paxton, A., Dale, R., & Kello, C.T. (2014). Complexity matching in dyadic conversation. Journal of Experimental Psychology: General, 143(6), 2304. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., & Delignières, D. (2016). Evenly spacing in detrended fluctuation analysis. Physica A: Statistical Mechanics and Its Applications, 451, 6369. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., Roume, C., Blain, H., & Delignières, D. (2018). Complexity matching: restoring the complexity of locomotion in older people through arm-in-arm walking. Frontiers in Physiology, 9, 1766. PubMed ID: 30564149 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Almurad, Z.H.M., Roume, C., & Delignières, D. (2017). Complexity matching in side-by-side walking. Human Movement Science, 54, 125136. PubMed ID: 28460275 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Buzzi, U.H., Stergiou, N., Kurz, M.J., Hageman, P.A., & Heidel, J. (2003). Nonlinear dynamics indicates aging affects variability during gait. Clinical Biomechanics, 18(5), 435443. PubMed ID: 12763440 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Coey, C.A., Washburn, A., Hassebrock, J., & Richardson, M.J. (2016). Complexity matching effects in bimanual and interpersonal syncopated finger tapping. Neuroscience Letters, 616, 204210. PubMed ID: 26840612 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cohen, J. (2013). Statistical power analysis for the behavioral sciences. New York, NY: Academic Press.

  • Delignières, D., & Marmelat, V. (2014). Strong anticipation and long-range cross-correlation: Application of detrended cross-correlation analysis to human behavioral data. Physica A: Statistical Mechanics and its Applications, 394, 4760. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delignières, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M., & Ninot, G. (2006). Fractal analyses for ‘short’ time series: A re-assessment of classical methods. Journal of Mathematical Psychology, 50(6), 525544.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39(2), 175191. PubMed ID: 17695343 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Goldberger, A.L., Peng, C.K., & Lipsitz, L.A. (2002). What is physiologic complexity and how does it change with aging and disease? Neurobiology of Aging, 23(1), 2326. PubMed ID: 11755014 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Harrison, S.J., & Stergiou, N. (2015). Complex adaptive behavior and dexterous action. Nonlinear Dynamics, Psychology, and Life Sciences, 19(4), 345394. PubMed ID: 26375932

    • Search Google Scholar
    • Export Citation
  • Hausdorff, J.M. (2007). Gait dynamics, fractals and falls: Finding meaning in the stride-to-stride fluctuations of human walking. Human Movement Science, 26(4), 555589. PubMed ID: 17618701 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hausdorff, J.M., Edelberg, H.K., Mitchell, S.L., Goldberger, A.L., & Wei, J.Y. (1997). Increased gait unsteadiness in community-dwelling elderly fallers. Archives of Physical Medicine and Rehabilitation, 78(3), 278283. PubMed ID: 9084350 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Herman, T., Giladi, N., Gurevich, T., & Hausdorff, J.M. (2005). Gait instability and fractal dynamics of older adults with a “cautious” gait: why do certain older adults walk fearfully? Gait & Posture, 21(2), 178185. PubMed ID: 15639397 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kaipust, J.P., McGrath, D., Mukherjee, M., & Stergiou, N. (2013). Gait variability is altered in older adults when listening to auditory stimuli with differing temporal structures. Annals of Biomedical Engineering, 41, 41(8), 15951603. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kurz, M.J., Markopoulou, K., & Stergiou, N. (2010). Attractor divergence as a metric for assessing walking balance. Nonlinear Dynamics, Psychology, and Life Sciences, 14(2), 151164. PubMed ID: 20346260

    • Search Google Scholar
    • Export Citation
  • Lipsitz, L.A., & Goldberger, A.L. (1992). Loss of ‘complexity’ and aging: Potential applications of fractals and chaos theory to senescence. Jama, 267(13), 18061809. PubMed ID: 1482430 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mahmoodi, K., West, B.J., & Grigolini, P. (2018). Complexity matching and requisite variety. arXiv preprint arXiv:1806.08808. [accessed October 30, 2018]

    • Search Google Scholar
    • Export Citation
  • Mahmoodi, K., West, B.J., & Grigolini, P. (2020). Complex periodicity and synchronization. Frontiers in Physiology, 11. doi:

  • Marmelat, V., & Delignières, D. (2012). Strong anticipation: complexity matching in interpersonal coordination. Experimental Brain Research, 222(1–2), 137148. PubMed ID: 22865163 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nessler, J.A., Gonzales, T, Rhoden, E., Steinbrick, M., & De Leone, C.J. (2011). Stride interval dynamics are altered when two individuals walk side by side. Motor Control, 15(3), 390404. PubMed ID: 21878691 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., & Goldberger, A.L. (1994). Mosaic organization of DNA nucleotides. Physical Review E, 49(2), 16851689. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Roume, C., Almurad, Z.M.H., Scotti, M., Ezzina, S., Blain, H., & Delignières, D. (2018). Windowed detrended cross-correlation analysis of synchronization processes. Physica A: Statistical Mechanics and Its Applications, 503, 11311150. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stephen, D.G., Stepp, N., Dixon, J.A., & Turvey, M.T. (2008). Strong anticipation: Sensitivity to long-range correlations in synchronization behavior. Physica A: Statistical Mechanics and Its Applications, 387(21), 52715278. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Stergiou, N., & Decker, L.M. (2011). Human movement variability, nonlinear dynamics, and pathology: Is there a connection?. Human Movement Science, 30(5), 869888. PubMed ID: 21802756 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vaz, J.R., Knarr, B.A., & Stergiou, N. (2020). Gait complexity is acutely restored in older adults when walking to a fractal-like visual stimulus. Human Movement Science, 74, 102677. PubMed ID: 33069099 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • West, B.J., Geneston, E.L., & Grigolini, P. (2008). Maximizing information exchange between complex networks. Physics Reports, 468(1-3), 199. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitacre, J.M. (2010). Degeneracy: A link between evolvability, robustness and complexity in biological systems. Theoretical Biology and Medical Modelling, 7(1), 6. doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Whitacre, J., & Bender, A. (2010). Degeneracy: A design principle for achieving robustness and evolvability. Journal of Theoretical Biology, 263(1), 143153. PubMed ID: 19925810 doi:

    • Crossref
    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 92 92 36
PDF Downloads 42 42 11