Reliable estimation of internal oscillator properties from a novel, fast-paced tapping paradigm

Abstract

Rhythmic structure in speech, music, and other auditory signals helps us track, anticipate, and understand the sounds in our environment. The dynamic attending framework proposes that biological systems possess internal rhythms, generated via oscillatory mechanisms, that synchronize with (entrain to) rhythms in the external world. Here, we focused on two properties of internal oscillators: preferred rate, the default rate of an oscillator in the absence of any input, and flexibility, the oscillator's ability to adapt to changes in external rhythmic context. We aimed to develop methods that can reliably estimate preferred rate and flexibility on an individual basis. The experiment was a synchronization-continuation finger tapping paradigm with a unique design: the stimulus rates were finely sampled over a wide range of rates and were presented only once. Individuals tapped their finger to 5-event isochronous stimulus sequences and continued the rhythm at the same pace. Preferred rate was estimated by assessing the best-performance conditions where the difference between the stimulus rate and continuation tapping rate (tempo-matching error) was minimum. The results revealed harmonically related, multiple preferred rates for each individual. We maximized the differences in stimulus rate between consecutive trials to challenge individuals' flexibility, which was then estimated by how much tempo-matching errors in synchronization tapping increase with this manipulation. Both measures showed test-retest reliability. The findings demonstrate the influence of properties of the auditory context on rhythmic entrainment, and have implications for development of methods that can improve attentional synchronization and hearing.

Figure 1

The experiment procedure of a single session, and illustration of the methods used to estimate preferred rate and flexibility. The horizontal lines represent the temporal structure of events in two consecutive trials. Dots on the top line represent the isochronous stimulus; audible sounds in black, and inaudible, theoretical timestamps in transparent gray. Dots on the bottom line represent synchronization (red) and continuation (blue) taps. Tempo-matching errors (TME) were calculated as the difference between stimulus IOI and tapping rate, separately from synchronization and continuation sections of the trials. Individuals’ preferred rates were estimated by fitting curves to their |TME_continuation| values over the IOI range and obtaining their local minima. Flexibility was estimated by fitting linear models to individuals’ single-session data where |TME_{synchronization}| was predicted by |− ΔIOI|, and obtaining slopes, with a smaller value indicating more flexibility.

Figure 2

Main findings. (A) General tapping patterns across the range of stimulus IOIs. Left: an example participant’s session 1 raw data and the fitted least-squares line. Right: β estimates (slopes) obtained from the linear models (TME_all = α + β × IOI), in individual sessions. Box plots show median (black horizontal line), 25th and 75th percentiles (box edges) and extreme datapoints (whiskers). Each circle represents a single participant. Black circle represents β of the line in the example fit on the left. (B) Preferred rate estimates. Left: an example participant’s raw data and the fitted curves in session 1 (top) and session 2 (bottom). Individuals’ preferred rates were the stimulus IOI at the local minima of the fitted curves, where the predicted |TME_continuation| was minimum. Among the local minima, the ones with less predicted TME were the global minimum, obtained within and across sessions. Right: 21 participants’ local minimum estimates and the resulting distributions in session 1 (dark) and session 2 (light). Each circle represents a local minimum and the black circles represent the example participant’s estimates. (C) Harmonic structure in preferred rates. The plot shows normalized local minima, where each individual’s estimate was divided by the smallest value obtained within session 1 and 2 (top and middle, respectively) and across sessions (bottom). Dotted curves represent the gaussian curves, fitted to the kernel density of the bimodally distributed data. Single circles and the same-color whiskers represent mean and standard deviation of the fitted theoretical distributions. Each circle represents a normalized local minimum and the black circles represent the example participant’s estimates. Box plots show median (black vertical line), 25th and 75th percentiles (box edges) and extreme datapoints (whiskers). (D) Flexibility estimates. Left: an example participant’s session 1 data and the linear fits where |TME_{synchronization}| was predicted by absolute negative ΔIOI (top) and by positive ΔIOI (bottom). Right: β estimates (slopes) obtained from the linear models where |TME_{synchronization}| was predicted by absolute negative ΔIOI (red) and by positive ΔIOI (green), in individual sessions. Each circle represents a single participant. Black circles represent β of the lines in the example fit on the left.

Figure 3

Results of the drift analysis. Left, top: Linear fits to an example trial’s tapping data. Connected dots represent the continuation ITIs, dashed lines are stimulus IOI (grey), and participant’s within-session global minimum estimate, i.e., preferred rate (blue). Drift on individual trials was quantified as the slopes (β) of the fitted least-squares lines (black). Left, bottom: β estimates obtained from an example participant’s single-session data. Gray dots represent β from individual trials. The dashed blue line represents the participant’s preferred rate estimate (within-session global minimum). Circle on the left (red) is the mean of β values obtained from the trials where IOI was faster than the preferred rate estimate, circle on the right (green) is the mean of β values obtained from the slower-than-preferred rate trials. Right: Distribution of β estimates, averaged within individual sessions. Each circle represents an individual’s average β. Average β from trials where stimulus IOI was faster than the individual’s preferred rate estimate are marked in red, average β from trials where stimulus IOI was slower than the individual’s preferred rate estimate are marked in green. Box plots show median (black vertical line), 25th and 75th percentiles (box edges) and extreme datapoints (whiskers).

Figure 4

Test–retest reliability of the (A) IOI slopes, (B) flexibility and (C) preferred rate estimates, and (D,E) the relationship between preferred rate estimates from the current paradigm and SMT tapping rates. In (A) and (B), each circle represents a single participant’s β estimates from the equations TME_all = a + β × IOI and |TME_{synchronization}|= a + β × |− ΔIOI|, respectively. In (C), each circle represents a participants’ within-session global minimum estimate (local minimum of the fitted curves where predicted |TME_continuation| was minimum within a single session). Straight line represents 1:1 correspondance, and the dashed lines represent 2:1 and 1:2 ratio between x and y axes. The histogram on the right shows the permutation test results: the distribution of summed residuals (distance of the data points to the closest y = x, y = 2 × x and y = x / 2 lines) with shuffled data over 1000 iterations, and the summed residual from original data (dashed line). On the bottom panel, within-session global minima in (D) session 1 and (E) session 2 were plotted against the mean SMT tapping rate of the respective sessions. As in (C), the histograms show the permutation test results for each session.

Figure 5

(A) “A theoretical detuning function” for an entraining oscillator, its preferred rate and the entrainment region. Adapted from Ref.. At fast rates outside of the entrainment region, TME_continuation is positive since tapping intervals are longer than the stimulus intervals due to drift to the preferred rate. At slow rates, the opposite is true. (B) An illustration of the hypothesized contraction of the entrainment region due to the ΔIOI manipulation. The lower plot illustrates how preferred rate was captured by our estimation method, curve fitting to the absolute values of TME_continuation; and that the contraction would leave preferred rate unaffected.