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The authors have declared that no competing interests exist.

Conceived and designed the experiments: CH BB. Performed the experiments: CH GT. Analyzed the data: CH GT BB. Contributed reagents/materials/analysis tools: CH GT BB. Wrote the paper: CH GT BB.

A natural synchronization between locomotor and respiratory systems is known to exist for various species and various forms of locomotion. This Locomotor-Respiratory Coupling (LRC) is fundamental for the energy transfer between the two subsystems during long duration exercise and originates from mechanical and neurological interactions. Different methodologies have been used to compute LRC, giving rise to various and often diverging results in terms of synchronization, (de-)stabilization via information, and associated energy cost. In this article, the theory of nonlinear-coupled oscillators was adopted to characterize LRC, through the model of the sine circle map, and tested it in the context of cycling. Our specific focus was the sound-induced stabilization of LRC and its associated change in energy consumption. In our experimental study, participants were instructed during a cycling exercise to synchronize either their respiration or their pedaling rate with an external auditory stimulus whose rhythm corresponded to their individual preferential breathing or cycling frequencies. Results showed a significant reduction in energy expenditure with auditory stimulation, accompanied by a stabilization of LRC. The sound-induced effect was asymmetrical, with a better stabilizing influence of the metronome on the locomotor system than on the respiratory system. A modification of the respiratory frequency was indeed observed when participants cycled in synchrony with the tone, leading to a transition toward more stable frequency ratios as predicted by the sine circle map. In addition to the classical mechanical and neurological origins of LRC, here we demonstrated using the sine circle map model that information plays an important modulatory role of the synchronization, and has global energetic consequences.

In order to produce the mechanical energy vital to move, the organism of humans and other animals must furnish to the muscles the chemical energy that they need. Respiration does the job, and brings dioxygen (O_{2}) from the environment into the organism. One fundamental issue raised by the requirement of breathing during moving is thus the subtle coupling between the respiratory system and the locomotor system, a phenomenon known as the Locomotor-Respiratory Coupling (LRC). Several studies

Bramble & Carrier _{2} consumption (VO_{2}). This assumption was investigated by several authors, occasionally confirmed _{2} found with stable LRC values, (ii) the difference in workload or populations, or as evidenced below, by (iii) the contribution of other uncontrolled variables.

In bimanual coordination, the

We pointed out several discrepancies in the LRC literature originating from the use of different methodologies. Our method uses the non-linear coupled-oscillators model showing that when two oscillators with different eigenfrequencies are coupled, their interaction results in attraction to a certain frequency ratio, which depends on the ratio between the eigenfrequencies (i.e., the

This equation maps the state of the forced oscillator onto itself from three factors: the phase at the previous cycle (

In order to evaluate if the two oscillators adjust their eigenfrequencies to each other, one can examine how an external force influences the forced oscillator after a great number of rotations, and guides it toward the synchronization region. The iteration of the map is usually described by the

A. Regime diagram: For a particular coupling strength (

Even if the sine circle map is an elegant model to understand the coordination between two coupled oscillators, its use for the experimental characterization of LRC remains very difficult. The Farey tree is a simplified way to represent the coordination expressed by the Arnold tongues regarding the stability of

Several authors have used the sine circle map model to better capture complex human bimanual coordination. For instance, Peper _{2}).

Sixteen male athletes (age 24.63±2.47 years, body mass 72.63±12.15 kg, height 177.94±5.28 cm) not specialized in endurance sports participated in the experiment. None reported any health problem and all were enrolled in a sport association. They all signed an informed consent before participating in the experiment, and received a financial compensation of 30€ for their participation. The experiment was approved by the regional ethic review board (

Experimental trials were performed on cycle ergometer (Ergoline 800S, Hoechberg, Germany) at the EuroMov centre in Montpellier. Gas exchanges and ventilation were recorded and analyzed continuously breath by breath by an automated system (ZAN 600 Ergo test, ZAN Messgerate GmbH, Oberthukba, Germany). The gas analyzers were calibrated before each test according to manufacture specifications (the ZAN instruction manual) using room air and gas concentrations (16% O_{2} and 5% CO_{2}). The air volume was calibrated with a 1-L syringe. Breathing kinematics was measured by a thermocouple sensor (SS6L Temperature Transducer BSL, Biopac Systems Inc., Santa Barbara, USA) placed in front of the mouth into the mask worn by the participant. An electromagnetic sensor located on the left side of the ergometer calculated the pedaling rhythm. It detected the passage of a magnet fixed on the left crank. A pressure sensor was placed in the right shoe of the participant to determine the moment at which the maximum force was exerted on the right pedal. Temperature, electromagnetic and pressure signals were collected simultaneously at 1000 Hz through a data acquisition board (NI USB-6009, National Instruments, Austin, Texas, USA).

The experiment included five sessions, spread evenly over a period of 14 days. In session 1, participants completed a maximal incremental exercise under medical supervision (VO_{2} max), in order to determine their individual power output at their anaerobic threshold (PAT) _{fpref}) and locomotor (L_{fpref}) frequencies were determined. In sessions 3 and 4, participants performed the same constant-load exercise at PAT in two different conditions: 5 minutes with acoustic stimulation (Sound-On) and 5 minutes without stimulation (Sound-Off). In one of these last two sessions, the periodic externally paced acoustic stimulation was presented at R_{fpref} to the participants who were instructed to exhale (RESP) in synchrony with the stimulation. In the other session, the acoustic stimulation was presented at L_{fpref} and participants were requested to cycle (LOC) in synchrony with the stimulation. Both the 5-minutes periods (Sound-Off or Sound-On) and the synchronization instructions (RESP or LOC) were counterbalanced over all participants.

All analyses presented below were realized using softwares MATLAB 7.10.0 (R2010a) (Copyright © The MathWorks, Inc., 1984–2010) and SCILAB 5.2.2 (Consortium Scilab (DIGITEO), Copyright © 1989–2010 (INRIA), ^{nd}-order Butterworth with a 2 Hz cutoff frequency. After having determined the instant in time of each expiration and pedaling strike, we calculated the frequency ratio for each respiratory cycle. We obtained a time-series of real quotients for each participant in each experimental condition. We used the Farey tree to label the observed frequency ratios through their property to bind any Farey ratio at level n by the adjacent Farey ratios at level n+1

Formally, the calculation of the relative phase was based on the equation used by previous authors

The grey curve represents the respiration signal (Exp_{i} corresponds to the beginning of the expiration i) and the black peaks represent the pedalling strikes.

The equation proposed by McDermott

We also computed the average number of consecutive cycles (in percentage of total cycles) spent on the modal frequency ratio (

In sum, to assess the stability of LRC across participants and conditions, five variables were computed, i.e., (i) the percentage of occurrence of the most frequent Farey ratio %

A: modal frequency ratio distribution extracted from the Farey tree (mode = 1/4, level 4). B: 5-minutes evolution of relative phase values where red dots represent the relative phase values for the modal frequency ratio and black dots represent the relative phase values for the remaining frequency ratios. C. Return Map assessing the strength of the phase-coupling by measuring the dispersion of the relative phase values around the identity line.

The data were analyzed statistically using the software Statistica 7.1 (Statsoft, France). A two-way repeated-measures ANOVA was performed on LRC variables and concomitant energy expenditure, with sound (Sound-Off vs. Sound-On) and synchronization (LOC vs. RESP) as factors. Post-hoc

The variability of the metronome was around 1 ms. We used a t-test to first verified that the mean frequency of the metronome was not different from the preferred respiratory (RESP) or locomotor (LOC) frequencies assessed during the FREE session for all participants, and that was indeed the case (RESP: t(15) = 0.23, p>.05; LOC: t(15) = 0.19, p>.05). We then tested whether participants followed the instruction to synchronize their breathing or cycling with the metronome. We found unimodal relative phase distributions between each expiration (RESP) or pedal stroke (LOC) and the metronome beat (respectively, 68.14±12.20 and 39.78±61.91 degrees) (all Raleigh tests p<.05 except for one participant p>.05). We thus concluded that all participants but one correctly performed the task.

We observed a large inter-individual variability on the different variables capturing LRC stability.

The left three panels correspond to S05 and the right three panels correspond to S15. The first two panels (A. Frequency Ratios Distribution) represent the distribution of the frequency ratios. The next two panels (B. Return Maps) represent the dispersion of the relative phase in the return map. The last two panels (C. Relative Phases) represent the relative phase values during five consecutive minutes where red dots represent the relative phase values for the modal frequency ratio and black dots represent the relative phase values for the other expressed frequency ratios.

The left two panels represent three minutes of relative phases for subject S05 (Sound-off at the top and Sound-On at the bottom). The right two panels represent three minutes of relative phases for subject S15. Red dotes represent the relative phase values for the modal frequency ratio and black dotes represent the relative phase values for the other expressed frequency ratios.

Performing individual analyses, we found two additional participants resembling S15 above, exhibiting stability reduction in LRC variables with sound, and twelve participants for which sound stabilized LRC variables in both synchronization conditions. Generally, we observed a negative linear correlation between %FRmode in the Sound-Off condition and the difference in %FRmode between both sound conditions (Sound-On - Sound-Off) (RESP R(13) = −0.71, p<.01; LOC R(13) = −0.79, p<.001). We observed similar results for PC (RESP R(13) = −0.77, p<.001; LOC R(13) = −0.75, p = .001) and for LockTime (RESP R(13) = −0.77, p<.001; LOC R(13) = −0.55, p<.05). Together, these results indicate that the stabilizing effect of sound decreased when intrinsic (Sound-Off) LRC stability was high. The three sound-destabilized participants (S02, S15 and S16) were thus distinguished from the other participants on the basis of their behavior during FREE and Sound-Off conditions. Moreover, each of these participants exhibited values of LRC variables (%FRmode, PC and LockTime) higher than the

Following this first analysis, we decided to remove participants S02, S15 and S16 from subsequent analyses. The main results reported in _{2} (F(1, 14) = 6.23, p<.05); %FRmode (F(1,14) = 5.07, p<.05;) and PC (F(1,14) = 22.05, p<.001).

Experimental Conditions | |||||

RESP | LOC | ||||

Sound-Off | Sound-On | Sound-Off | Sound-On | ||

Ventilatory variables | VO_{2} (L·min^{−1}) |
2.38±0.37 | 2.31±0.40 | 2.35±0.37 | 2.33±0.36 |

VCO_{2} (L·min^{−1}) |
2.37±0.34 | 2.30±0.36 | 2.34±0.36 | 2.33±0.37 | |

FR (breath·min^{−1}) |
29.37±4.85 | 30.25±4.98 | 28.91±5.57 | 31.50±4.19 | |

StdFR (breath·min^{−1}) |
2.25±0.82 | 1.38±0.65 | 2.32±0.66 | 2.27±0.87 | |

VT (L·min^{−1}) |
2.16±0.30 | 2.13±0.34 | 2.19±0.36 | 2.02±0.32 | |

VE (L·min^{−1}) |
62.09±8.50 | 62.61±9.10 | 61.50±8.89 | 62.27±8.24 | |

VE/VO_{2} |
24.61±2.68 | 25.61±3.44 | 24.74±3.24 | 24.99±2.25 | |

VE/VCO_{2} |
24.65±2.75 | 25.60±3.17 | 24.70±3.03 | 24.96±2.20 | |

LRC variables | %FRmode (%) | 56.44±16.03 | 72.04±16.96 | 53.39±23.39 | 67.51±20.10 |

PC (%) | 12.11±4.25 | 9.51±6.18 | 10.21±5.59 | 18.80±7.93 | |

LockTime (%) | 3.89±1.38 | 6.70±5.34 | 4.43±3.03 | 7.05±5.93 | |

Level | 2±1 | 1±1 | 2±1 | 1±1 | |

Sound Effect | VO_{2}, VCO_{2}, StdFR, %FRmode |
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Synchronization Effect | StdFR | ||||

Sound x Synchronization Interaction Effect | FR, VT, PC |

Values for the Level variable correspond to the median ± inter-quartile interval. All variables significantly affected by the Sound and Synchronization conditions are summarized in the bottom part of

We noticed a significant reduction of VO_{2} (F(1,11) = 6.10, p<.05) and VCO_{2} (F(1,11) = 7.17, p<.05) with Sound-On, suggesting that the auditory stimulation induced a diminution of energy consumption. A significant sound by synchronization interaction was found on the respiratory frequency (FR) (F(1,11) = 4.14, p<.05), and post-hoc

We observed a sound-induced increase in %FRmode (F(1,11) = 8.20, p<.05), suggesting that the modal frequency ratio between breathing and cycling was more frequent when paced by the sound, for both synchronization instructions. We also observed a significant interaction between sound and synchronization for PC (F(1,11) = 15.87, p<.01). Post-hoc

Distributions of the LRC frequency ratios: without sound we observed a trimodal distribution in the two synchronization conditions where 1/4, 1/3 and 1/2 occurred more frequently than any other ratios (

The first two panels represent the RESP condition and the two last correspond to the LOC condition. For each condition the upper panel corresponds to the No-Sound condition and the lower panel corresponds to the sound condition.

In this study, we proposed to investigate LRC using the sine circle map, considering the two systems as non-linear coupled oscillators. This method gave us new insights into the LRC dynamics. The first finding is the observation of complex frequency ratios p/q different from the usually 1/q reported ratios. Because the choice of boundaries affects the occurrence of each frequency ratio, it also affects their distribution and limits the comparison between studies. Our main computational novelty is the systematic test of frequency ratios ranked in a hierarchical order in the Farey tree. An important issue not discussed in previous studies concerning LRC is the size of the Farey tree used to assess the observed coordination. Here we proposed a two-steps method: (i) the determination of the modal Farey ratio in a Farey tree composed of 10 levels (from 1 to 10) and (ii) the choice of the level for which the modal Farey ratio exhibits the highest stability, assessed by the phase coupling PC. One other critical point is the labeling of observed frequency ratios. We used here a Farey tree of n+1 level providing precise boundaries to a Farey tree of n level within which each frequency ratio can be rigorously tested (see Data Analysis and Variables) _{2}). We discuss these results in turn.

In average, the modal value of the frequency ratio %FRmode was 55% without sound and 70% with sound. This result is in line with previous work

Interestingly, we noticed a change in the frequency ratios landscape particularly when participants were instructed to cycle in synchrony with the tone, in the form of a transition from higher-level Farey ratios to a lower Farey ratio (i.e. 1/2), a gain of stability for the 1/2 Farey ratio, accompanied by an increase in all criterion used to assess LRC stability. This crucial observation reinforces the importance to assess the complete ratio distribution and not only the mean ratio of the two frequencies

The results analyzed above – stabilizing effect of the sound, asymmetry, change with sound in the Farey tree dynamics, increase in stability – are very much in line with the music-at-sport and music-at-work literatures _{2} (see section 3.3). Interestingly, we observed two different adaptations to the auditory stimulation (see

To the mechanical and neurological origins of LRC, we thus have to add an informational origin, exploiting (in our case) auditory and kinesthetic perceptual channels, acting as a LRC modulator. However, the contribution of these different components to the observed LRC is difficult to assess. One interesting perspective would be to manipulate these properties and evaluate their respective influence on coupling strength in a variety of physical activities.

Concerning the ventilatory parameters, our expectation derived from the Farey tree analysis was a significant reduction in VO_{2} concomitant to the sound-induced stabilization of LRC. Given that participants had to maintain the power output PAT at all time, a reduction of VO_{2} would reflect higher energy efficiency. On average, the value of VO_{2} in the Sound condition corresponded to a decrease around 4% when compared to its value with no sound. Our results demonstrated a significant reduction in oxygen consumption from 2.36 l.min to 2.32 l.min when participants were instructed to breathe or cycle in synchrony with the metronome, accompanying the improvement in LRC stability. This result confirms previous findings _{2} during periods of strong coupling, but with a more powerful methodology defining precisely the periods of synchronization. However, our results are in contradiction with other observations _{2} due to coordination. In our case, the decrease in VO_{2} with sound should be the result of a better mobilization during cycling of the muscles involved in respiration, which are known to mechanically affect LRC _{2} with rhythmic sound is particularly interesting for amateur athletes practicing a sport at intensity below the anaerobic threshold (e.g., footing, cyclotouring, marathon,

In the field of bimanual coordination, a double-metronome situation in which each finger reversal (flexion and extension) occurred simultaneously with the auditory stimulus resulted in a significant increase of the local and global stabilization of coordination pattern when compared to a single-metronome condition

The Locomotor-Respiratory Coupling is a universal phenomenon underlying the production and supply of energy. Rhythmic activities such as walking, running, swimming, or rowing all exhibit LRC. However classical work points out the difficulty to describe LRC through the simple observation of frequency mode locking. Our results confirm this difficulty. Discrepancies in the results published so far on LRC are due to a lack of consensus about the definition and the methodology to investigate it. Here we extended the methodology adopted by previous authors

We thank Sébastien Villard for his contribution to a preliminary version of this work, and Stéphane Perrey for helpful exchanges about the experimental protocol.