Sixteen Japanese darbuka players (all male) participated in the study; eight professional players (mean age = 37.25 years, standard deviation [SD] = 5.66, range = 30–50 years) and eight amateur players (mean age = 41.50 years, SD = 9.34, range = 28–55 years). Professional players earned a living through musical performances and instructing students, whereas amateur players did not. Professional players began playing darbuka at an earlier age (mean = 24.38 years, SD = 4.27, range = 21–34 years) than amateur players (mean = 35.63 years, SD = 10.80, range = 25–51 years; t (14) = 2.74, p = 0.02), and darbuka training duration was significantly longer in professional players (mean = 13.75 years, SD = 2.77, range = 8–17 years) than in amateur players (mean = 6.38 years, SD = 4.92, range = 4–18 years; t (14) = − 3.69, p = 0.02). We used the Edinburgh Handedness Inventory to determine participants’ handedness (Oldfield 1971). Professional players’ mean laterality quotient was 92.37 (SD = 10.54, range = 80–100) and that of amateur players was 92.37 (SD = 15.01, range = 60–100). Professional and amateur players were matched for sex, age, and handedness. The experimental procedure was approved by the Ethics Committee of the Keio University Shonan Fujisawa Campus (No. 161), and all participants provided informed consent.

Drumming chair height and position were adjusted for each participant to ensure comfort. Participants were asked to hold a darbuka (aluminum die-cast model, Egygawhara) under their left upper arm. We placed a camera (FDR-AX700, Sony Corporation) in front of participants, and recorded video at 59.94 Hz. We also recorded sound data; the tapping performances analyzed from the sound data were published in our previous study (for details, see Honda and Fujii (2022)). One professional player’s data were excluded from the analysis because of recording errors.

We used a double-finger coordination task (Honda and Fujii 2022) for which participants were asked to coordinate their right and left ring fingers alternately and as fast as possible (Fig. 1). They started tapping from their preferred hand for 12 s after a start call from the experimenter. As we were investigating actual darbuka playing natural performance, participants were asked to tap with their fingers, but were allowed to use upper arm joint movements without any constraints, as they usually play the instrument. Participants performed three trials each with a one-minute rest between trials to prevent fatigue. We decided to conduct three trials based on previous fast tapping studies (Aoki et al. 2005; Fujii and Oda 2006). The right and left fingers were used because darbuka is primarily played using bimanual finger coordination.

Darbuka player. Players use their right and left ring fingers to play the darbuka

The recorded video data were trimmed to 12-s clips for each trial. DeepLabCut (Mathis et al. 2018) was then used to annotate right and left ring fingers’ positions. After manually annotating the right and left ring fingertips for 20 images in each trial, the network was trained for 500000 iterations using ResNet 50. Finally, we checked the outlier data, and manually fixed and retrained the network for 398000 iterations.

First, we truncated the first and last 1-s data to eliminate the effects of startup and final slowing and used the data during the middle 10-s period. To remove outliers from the annotation data, we used likelihood, computed using DeepLabCut with a threshold of 0.95. Points with likelihood less than threshold were replaced with missing data, and the missing data were incorporated with a piecewise cubic Hermite interpolating polynomial. The fingertips’ trajectory during fast tapping was elliptical. During darbuka play, each ring finger moves to tap a point near the long axis. Therefore, the trajectory was rotated such that the positive direction of the x-axis coincided with the tapping points. To observe coordination during fast movement, we removed the drifts or irrelevant noise caused by the position drifts by applying a 4–9 Hz bandpass filter (4th order Butterworth). The filter cutoff frequency was set to include the Q1–Q3 range of the tapping frequency [6.07–7.64 Hz] calculated from all ITIs shown in our previous study. We used the x-axis position of the rotated trajectory for relative phase analysis and CRQA.

To quantify the degree of variability in bimanual coordination, we evaluated the relative phase distribution by calculating the length of the resultant vector (LRV) (see a paper by Berens (2009) for more details about the circular statistics). The LRV was calculated from the synthesized vector of the relative phases between the two time series’ (i.e., right and left ring fingers’ positions), indicating how the relative phases are aligned and related to the standard deviation of the relative phase (\(SD\phi\)). The LRV is defined as follows:

$$}\frac\left| ^ } } \right|,$$

(2)

where \(_\) is a relative phase at the \(t\)th frame and \(N\) is number of frames. The LRV ranges from 0 to 1; higher values indicate that the participant tapped consistently in a certain relative phase. To calculate the LRV, we used the Hilbert transformation to calculate the instantaneous relative phase between right and left ring fingers’ positions. We also calculated the angle of the resultant vector for each observation.

We conducted the CRQA using the following process: first, we converted the position data into unit interval ranges to remove scale differences (for more detailed information, see a tutorial by Webber and Zbilut (2005)). Three parameters were used to obtain the CRQA results: time delay \(\tau\), embedded dimension \(d\), and threshold radius \(r\), which were determined using the bimanual coordination method described by Richardson et al. (2007).

Time delay, the temporal offset between copies of the time series, was used to reconstruct the higher dimensional phase space. In stationary periodic or oscillatory systems, a quarter cycle of frequency is appropriate for the time delay. Therefore, we set the time delay of \(\tau = 2\) because the average frequencies of the fast bimanual coordination were 7.19 Hz (SD = 1.24) in the professional players and 6.67 Hz (SD = 2.30) in the amateur players. We also calculated the average mutual information (Wallot and Mønster 2018) and confirmed that the optimal value of the time delay was \(\tau = 2\) (see also Fig. S1).

The embedding dimension was the number of dimensions used to determine the reconstructed system trajectory. Related studies found that the appropriate number of embeddings to capture the movement dynamics in rhythmic limb movement was five (Mitra et al. 1997; Goodman et al. 2000). Therefore, we used an embedding dimension \(d= 5\) for the analysis (see also Fig. S2).

Finally, the last parameter, the threshold radius \(r\) was used to determine whether states \(_\) and \(_\) were recurrent in phase space. If \(r\) is too small, only the noise is measured; conversely, an excessively large value of \(r\) results in recurrent points that do not reflect the local structure of the focused dynamical system. To determine the value of \(r\), we used the following criteria: first, the selected \(r\) should result in values of %REC and Lmax that are included in open intervals with floor and ceiling (i.e., %REC ∈ (0, 100) and Lmax ∈ (0, maximum possible line length)). Second, the value of \(r\) should be chosen from a region of \(r\) values that result in linear scaling of %REC values on a log–log plot. We confirmed that r took linear scaling in an interval from 0.09 to 0.21 with a participant’s data. Based on the above criteria, we chose the threshold radius \(r\) (0.2) to calculate %REC and Lmax. CRQA was performed using “crqa” R package (Coco and Dale 2014; Wallot and Mønster 2018).

We then pooled LRV, %REC, and Lmax from the three trials for each participant and created linear mixed-effects models (LMMs) to test hypotheses regarding tapping variability. We first entered group (professional/amateur) as a fixed effect in the LMM to test whether group differences affected tapping variability. Participants and trials were entered as random effects to account for interindividual and intertrial differences. Statistical analyses were performed using R software. The LMMs were performed using the “lmer” function in the “lme4” R package (Bates et al. 2015). Type-3 Wald Chi-Square tests were used to test significant main effects in the LMMs. The results of each statistical analysis were deemed significant at p < 0.05. We calculated the partial eta squared (\(_^\)) values as the effect sizes. To evaluate the robustness of our data with a small sample size, we used an additional k-fold cross-validation analysis when we obtained a significant main effect (de Rooij and Weeda 2020). We then compared the prediction performance of two regression models: Model 1, with only the intercept, and Model 2, with the intercept and group. We set the number of folds to five because of the small sample size (de Rooij and Weeda 2020), and repeated the validation 2000 times. Each observation was not independent in the LMMs; thus, we applied blocking cross-validation to put all data from the same participant into the same fold (Roberts et al. 2017).