Response to perturbations as a built-in feature in a mathematical model for paced finger tapping

Paced finger tapping is one of the simplest tasks to study sensorimotor synchronization. The subject is instructed to tap in synchrony with a periodic sequence of brief tones, and the time difference (called asynchrony) between each response and the corresponding stimulus is recorded. Despite its simplicity, this task helps to unveil interesting features of the underlying neural system and the error-correction mechanism responsible for synchronization. Perturbation experiments are usually performed to probe the subject's response, for example, in the form of a “step change,” i.e., an unexpected change in tempo. The asynchrony is the usual observable in such experiments and it is chosen as the main variable in many mathematical models that attempt to describe the phenomenon. In this work we show that although asynchrony can be perfectly described in operational terms, it is not well defined as a model variable when tempo perturbations are considered. We introduce an alternative variable and a mathematical model that intrinsically takes into account the perturbation and make theoretical predictions about the response to novel perturbations based on the geometrical organization of the trajectories in phase space. Our proposal is relevant to understand interpersonal synchronization and the synchronization to nonperiodic stimuli.

Figure 1

(a) Schematic description of the task, variables, and parameters. A perturbation occurs at step $n$. (b) Isochronous sequence from one subject (single trial, no averaging). $\mathrm{Mean}=3.9\mathrm{ms}$, $\mathrm{SD}=12.1\mathrm{ms}$.

Figure 2

Experimental time series and phase-space reconstruction (all experimental data shown in this work were published in Ref. [17]). Perturbation sizes are in milliseconds and the preperturbation period is $T_0=500\mathrm{ms}$. (a) Negative step changes. (b) Positive step changes. (c) Phase-space reconstruction by embedding the time series above. Note that there is a common region of phase space shared by trajectories from large opposite perturbations (most notably $+50$ and $-20$ trajectories). Black curves are the estimated return points.

Figure 3

Model fitting. (a) Experimental $\pm 50$-ms time series and model fitting results. (b) Model phase space. Response to perturbations occuring at $n=0$, ${\mathrm{\Delta }}_0=\pm 10$, $\pm 20$, $\pm 30$, $\pm 40$, $\pm 50$, $\pm 60$, and $\pm 70\mathrm{ms}$. Larger perturbations begin farther away from the origin (preperturbation steps not shown for visual clarity). Note the region of phase space that is shared between trajectories corresponding to positive and negative perturbations (upper right and lower right quadrants).

Figure 4

Variety of solutions from the fitting procedure. (a) Type-I phase space and time series (58% of all solutions; the fittest solution of all, shown in Table 1 and Figs. 3, 5, 6, and 7, is this type). (b) Type-II phase space and time series (40% of all solutions; parameter values in Appendix pp1-s5). (c) Type-III phase space and time series (2% of all solutions; parameter values in Appendix pp1-s5).

Figure 5

Three common experimental manipulations of the stimulus period (top row) and response of our model (second row). (a) Step change in stimulus period. (b) Sinusoidal change. (c) Pseudorandom change. For comparison, the third row shows the behavior of the model when no distinction between predicted $p_n$ and observed $e_n$ is made, which is the traditional way of modeling where by-hand modification of $e_n$ is needed (shown as orange filled circles when $e_n$ must be increased and open circles when it must be decreased); fourth row shows a “naive” approach where neither distinction $p_n/e_n$ nor by-hand modification are made.

Figure 6

Model predictions (I). (a) Response to a step change perturbation and choice of points for the second perturbation. Points labeled “B” and “C” have similar asynchrony values and correspond to the perturbations in panels (b) and (c), respectively. (b) Response to the proposed manipulation: Both conditions of perturbation 1 ($-40$- and $+70$-ms step changes) give similar responses to perturbation 2 ($-40$-ms variable only). The perturbed points are labeled with stars and correspond to the “B” label in panel (a). (c) Response to consecutive perturbations as in (b) but the second perturbation takes place in different points of phase space: The response after perturbation 2 depends on the condition of perturbation 1, despite having similar initial asynchrony values (positive control). The perturbed points are labeled with stars and correspond to the “C” label in panel (a).

Figure 7

Model predictions (II). Response to perturbations to the variable only (large: $\pm 60\mathrm{ms}$; small: $\pm 20\mathrm{ms}$). The response to large perturbations is asymmetric (i.e., it overshoots after the negative perturbation only); the response to small perturbations is mostly symmetric. (a) Asynchrony time series. (b) Trajectories in phase space.

Figure 8

Estimation of return points. (a) Return points of $e_n$ (top panel; labeled “horiz” in Fig. 2) and corresponding $z_n$ value (bottom panel). (b) Return points of $z_n$ [bottom panel; labeled “vert” in Fig. 2] and corresponding $e_n$ value (top panel).

Figure 9

Model fitting: Experimental time series vs. fitted time series. The obtained set of coefficient values is displayed in Table 1.

Figure 10

Distribution of parameter fitted values.

Figure 11

Joint distributions of linear parameter values.

Figure 12

Joint distributions of nonlinear parameter values.

Figure 13

Joint distributions of linear vs. nonlinear parameter values.