Participants
Initially eight males, all novices in breaststroke swimming, voluntarily participated in this study (mean age = 18.4 years, SD = 0.7). Participant 3 was not able to compete all the protocols, leaving 7 participants for analysis. Each participant signed an informed consent form after receiving oral and written descriptions of the procedures, which were approved by the university ethics committee. The two exclusion criteria were principally related to the validity of subject’s initial breaststroke technique. Importantly, they had to be able to: (a) perform a symmetrical leg kick; and (b) perform leg and arm movements at the same frequency. The swimmers were characterized as being in the first stage of learning (i.e., coordination stage), during which learners still have to establish the basic coordination of the key components of the behaviour40,41. All participants had the same goal of learning without any information on how to perform.
General goal of learning and practice sessions
All participants participated in 16 learning sessions. The entire program lasted 8 weeks, with 2 sessions per week. All participants performed at a different time during the day/week, in order to avoid any interaction between participants during the protocol. During each session, in a 25 m indoor pool, participants had to complete 10 x 25 m at sub-maximal speeds (5 trials at 70% of their personal maximal speed (i.e., a comfortable speed, low constraining environment of practice) and 5 trials at 90% of their personal maximal speed (i.e., a high speed, highly constraining environment of practice). Those sub-maximal speeds, based on the maximal speed performed by each participant during the first session, corresponded to the working speed throughout the learning process (i.e., the speed was constant during all the practice sessions). Each session lasted approximatively 35 min per participant and included a 10 min of warm-up followed by the 10 trials with a start every 2 min 30 s (a trial lasted 30 s followed by a 2 min rest period). Participants were asked to avoid practicing breaststroke during the entire experiment (from the first learning session to the retention test), except during the experimental sessions.
For all the participants, the general goal of learning was to decrease their stroke rate (i.e., number of cycles per second, in Hz) while maintaining the same sub-maximal speed – therefore, increasing their efficiency42. Learners were informed of this general goal at the beginning of each session. The basic rules of breaststroke swimming were provided to the participants (as a reminder) during the first session, and only if necessary, thereafter. The speed was self-paced by the learner during the trials based on a target time of competing 25 m. The average speed of performing each trial was measured at the end of each trial and the trial was validated if the actual average speed was within the target speed ± 5% of that speed. If learners failed to follow the rules or the targeted speed, they were stopped by the experimenter and had to perform the trial again. After each trial, learners were informed of their mean stroke rate values (i.e., informed of their performance in the task). No other information was given to the learners during the 16 sessions.
Data collection
During every sessions, participants were equipped with inertial sensors including 3-D accelerometers, 3-D magnetometers and 3-D gyroscopes (MotionPod3, Movea, Grenoble, France)43. The acquisition frequency of the sensors was 100 Hz. Four motion sensors were positioned on the left side of the swimmers, respectively on the forearm (posterior surface of the proximal portion), the arm (posterior surface of the distal portion), the thigh (anterior surface of the distal portion), and the leg (anterior surface of the proximal portion), in order to have the sensors in direct contact with a bony part of the limb. At the beginning of each session, the position of the motion sensors was placed on a black marker, which defined the location of the sensor from the last session. Swimsuit was also worn on the two limbs where sensors were placed in order to limit resistances due to the presence of the sensors. Once the swimmer was ‘suited-up’, he entered the second lane in the pool (i.e., at least 2 m far from the wall to avoid any magnetic disturbance) and performed the 10 trials. Once the trials were completed, the data were uploaded and synchronized a posteriori with Matlab r2015a (version 8.5.0, The MathWorks Inc., Natick, MA, USA).
Processing of behavioural data
Thereafter, elbow and knee angles were computed for each trial by calculating the relative angle between two sensors. Time series representing knee and elbow angles were then computed. These time series were filtered with a low-pass Fourier filter (cut-off frequency 8 Hz44) and partitioned cycle per cycle (i.e., one cycle beginning with a maximal knee flexion and finishing with the next maximal knee flexion). The first cycle as well as the last cycle were removed to account for acceleration of swimming speed due to push-off the wall or deceleration when approaching the wall. For each trial, knee and elbow angular positions for 3 to 17 cycles were normalized between 0 and 1 and used for characterizing the inter-limb coordination of the swimmer (see Figure 7 for an example). The nature of the behaviour was derived from the arm-leg coordination and was assessed by the Continuous Relative Phase (CRP) between knee and elbow angles. The CRP was computed based on elbow and knee angles in the same way as previous experiments44,45, which has been shown to be an effective parameter to quantify the nature of swimmer’s behaviour. A typical in-phase behaviour (i.e., a relative phase value close to 0°) represents an inefficient swimming technique consisting in simultaneous arm propulsion and leg recovery, and conversely an effective coordination pattern has been characterized by a more complex coordination within a single cycle, with fluctuations of relative phase between anti-phase (i.e. a value of -180°) to in-phase coordination mode to anti-phase (see44 for a precise description of an effective coordination pattern).
Processing of Performance Data
During each trial, the instantaneous stroke frequency (Hz) was recorded for each cycle from the cycle duration (measured with the motion sensors) following the equation f = 1 / cycle duration (s). Therefore, changes in performance were actually defined by the decrease in stroke frequency cycle after cycle. The average frequency value per session was computed for each individual and modelled with an exponential function46. The exponential function used to fit the movement performance over sessions is shown in Equation 1:
f(t) = a * exp(-b*t) + c (1)
Where t is the practice time (session number), c represents the asymptotic performance, a represents the initial performance (when t = 0, exp(-b*t) = 1 and a + c represents the performance level before practice), finally b represents the learning rate. From this model, the higher is the value of b, the faster is the learning rate (i.e., representing an early rapid increase in performance followed by a later slow increase). The quality of the fitting is presented by r2 and Root Mean Square Error (RMSE)47.
Profiling motor coordination
An unsupervised cluster analysis procedure was used in order to differentiate the patterns of coordination exhibited by the learners19. The time series of CRP from the cycles of all the seven participants in both speed conditions were used to compute the cluster analysis (i.e., all the cycles, all the sessions, all the participants). Such a cluster analysis allows partitioning the entire set of cycles into meaningful sub-groups or clusters, whereas the “real” number of groups is unknown a priori. The Fisher-EM algorithm has been used for the present experiment48,49. The Fisher-EM algorithm is an iterative cluster algorithm that projects the data in a new subspace at each iteration in such way that emerging clusters maximize the Fisher information (i.e., maximize the inter-cluster distance while minimizing the intra-cluster distance). The final number of emerging clusters was selected based on the Bayesian Information Criterion (BIC) for model selection50 with the first value of the plateau representing the model that best represents the data (also known as the “elbow” method).
Quantifying motor exploration
From the clustering, each trial was labelled with a specific exhibited coordination profile. The time series of those exhibited coordination were re-constructed putting one cycle after the previous one in the chronological order they were performed, representing the successive behaviours that were exhibited by a learner18,25. Those time series of labelled behaviours were thereafter modelled using Drifting Markov Models (DMM)51. DMM have been used in the genome literature and are a relevant tool for modelling how qualitative patterns are organized in time. Specifically, modelling DNA sequences with stochastic models and developing statistical methods to analyse the DNA sequencing have been challenging questions for statisticians, and the most popular model in this domain is the Markov model on the nucleotides (i.e., the modelling of a c g t nucleotides).
Although a traditional Markov model gives a broad overview of the main transitions occurring between the different patterns within the whole time-series of patterns (i.e. a transition matrix), Vergne51 developed a DMM that varies the transition matrix between the different patterns on the basis of a predetermined polynomial model with a degree n that is set by the experimenter. In other words, this modelling is meant to provide the evolution in time of the original transition matrix (e.g., the change in time in the probability of the transitions converging towards a specific pattern). Eventually, using the DMM to model the evolution of the transitions between patterns provides a modelling that accounts for the probability of appearance of any potential patterns as learning operates. DMM are applied individually on the time series of clusters of each participant, for modelling the probability of appearance of one pattern of coordination regarding the previous pattern.
The order of the Markov chain was set to one (i.e., only one previous cycle was considered for the transition), however, two models with different degrees were applied for the modelling of the evolution of the transitions between patterns of coordination. On one hand, a model of degree one was applied, representing a linear trend of the transition matrix. Indeed, a linear trend represents linear increase or decrease of the probability of transition from one coordination toward the final to-be-learned coordination. In other words, with such a linear model, each time a learner practices, he increases the probability of appearance of the to-be-learned coordination.
A model of degree three was applied on the time series. This model, based on the use of the BIC and Akaike Information Criteria (AIC) for model selection50, represented the model that best fitted the actual data and was considered as representing the actual fluctuations that occurred during learning (see Appendix A for the BIC and AIC values for all possible models). Eventually, the average Euclidean distance between those two models (i.e., degree 1 and degree 3) was computed for each coordination pattern for each learner. This distance between the degree one model (i.e., linear modelling of the probabilities of transition) and the degree three model (i.e., the actual evolution of the probabilities) therefore quantify the nature of the exploration. A large distance between the linear and the degree three model reflects the presence of high exploration of a specific pattern at some point during learning (or specific patterns successively), in other words strong anchor points in the learning dynamics. A small distance between both models represents an absence of anchor points in the learning dynamics and a more linear appearance/disappearance of the coordination patterns (i.e., the probability of appearing/disappearing of the pattern evolving linearly).
Summary of dependent variables
With reference to the performance indicator, both the decrease in stroke frequency between the first session and the last session as well as the rate of learning (i.e., value of b from the exponential models) were considered. In regard of the analysis of motor exploration, after the clustering and the DMM application, the quantity of exploration in percentage (i.e., the distance between two models) for each cluster is considered per participant and speed condition. The standard deviation (SD) of the quantity of exploration both within participants and within speed conditions is presented in order to reflect the within individual and within speed amount of variability of the exploratory processes.
Statistical Analysis
With reference to the analysis of the increase of performance between the first session and the last session, after normality and homogeneity of variance were checked, a two-way ANOVA (within-subject effects: session time [first; last] * speed condition [low; high]) was performed on the stroke frequency values between the time of testing and the two speeds of swim. Concerning the rate of learning (i.e., the value of b from the exponential models) and the variability in exploration, a paired sample t-test was performed to compare the rate of learning and the SD of the quantity of exploration between the two speed conditions. When the difference was significant, Cohen’s d was computed as a measure of the size of the effect, with d = 0.2 representing a small effect, d = 0.5 representing a medium effect and d = 0.8 representing a large effect52.
With regard to the motor exploration quantity, a two-way ANOVA was performed (within-subject effects: cluster [1,2,3,4,5,6,7,8,9,10,11] * speed condition [low; high]) in order to investigate any difference in the quantity of exploration of a specific cluster and any difference due to the speed condition. When necessary, the p values were corrected for possible deviation from sphericity using the Greenhouse-Geisser correction when the mean epsilon was lower than 0.75. Otherwise, the Hyun-Feld procedure was used. When a significant effect appeared, post-hoc test using Bonferroni correction were used. Partial eta squared (ηP2) was calculated as an indicator of effect size, considering that ηP2 = 0.02 represents a small effect, ηP2 = 0.13 represents a medium effect and ηP2 = 0.26 represents a large effect 52.
Concerning the relationship between the emerging number of patterns, average individual quantity of exploration and performance improvement and rate, Pearson correlations were used when the assumption of normality was met, otherwise Spearman’s rho correlation was computed. All tests were performed using JASP Statistics V0.13.1 (July 2020 – www.jasp-stats.org), with a level of statistical significance fixed at p =< .05.