Dynamical incompatibilities in paced finger tapping experiments

The gist: Responses to period perturbations in paced finger tapping tasks are dynamically incompatible when they occur in different experiments, but they become compatible and can be described by a single underlying dynamical system when both perturbation types are presented randomly within the same experiment.

Background and Problem

Paced finger-tapping tasks are used to probe the error correction mechanism underlying sensorimotor synchronization Despite their century-long history, fundamental contradictions persist in the literature. One such contradiction arises when comparing the two most common types of period perturbation: step change and phase shift. The stimulus sequence is exactly the same up to and including the (unexpected) perturbed stimulus. Why then would the timing of the next response be different between perturbation types, as observed? We show, both experimentally and theoretically, that responses to both types of perturbation are dynamically incompatible when recorded in separate experiments; that is, they cannot be described by a single underlying dynamical system due to the build-up of different temporal contexts. In contrast, when both types of perturbation are presented randomly within the same experiment, the responses become compatible and can be explained by a single mechanism. We conclude that a single underlying dynamical system can represent the response to all perturbation types, signs, and sizes, which is nevertheless calibrated by temporal context. Our results challenge the established idea of phase and period correction processes that are separately activated for different perturbation types.

Experimental Design

The experimental data analyzed were previously published and involved participants asked to synchronize to an external metronome. A trial consisted of a sequence of 35 brief tones with a baseline interstimulus interval T = 500 ms and their responses. Upon the appearance of a period perturbation, participants were instructed to regain synchrony without interrupting the tapping. Two perturbation types were used (Repp, 2005): step changes (SC), where the stimulus period is abruptly modified by an amount ∆T at a random step of the sequence; and phase shifts (PS), where the period changes at two consecutive steps, first by +∆T and then by −∆T, thus returning to its original value. Perturbations could be positive (∆T > 0, “pos”) or negative (∆T < 0, “neg”), with a magnitude of either 20 or 50 milliseconds. Participants were grouped into two contexts: the “pure” context participants were exposed to only one type of perturbation (Group 1, SC perturbation; Group 2, PS perturbation; both groups with 20 and 50 ms perturbation magnitudes) and the “combined” context participants were exposed to both types of perturbation, randomly interspersed on a trial-by-trial basis (Group 3, 50 ms magnitude; Group 4, 20 ms magnitude). The experimental design was fully factorial, with factors Context (pure and combined) x Type (SC and PS) x Sign (pos and neg), with additional isochronous control conditions (∆T = 0).

Phase Space Analysis

The reconstruction of the experimental phase space via embeddings was performed using a difference embedding technique. The variable en, however, is ill-defined when the interstimulus interval changes. Based on previous work, we switched to a variable that is more appropriate in the presence of period perturbations: predicted asynchrony pn, defined as pn = en + ∆T. This simple relationship between en and pn is best conceptualized by realizing that, when an unexpected period perturbation ∆T occurs at step n in the sequence, the asynchrony value predicted by the participant pn will not be the same as the actually observed value en due to the experimental manipulation. The difference embedding was defined as predicted asynchrony (pn) on the horizontal axis vs difference between consecutive predicted asynchronies (pn − pn−1) on the vertical axis, averaged across subjects.

Model Fitting and Incompatibility

The experimental data were modeled with a single nonlinear dynamical system proposed in a previous work. The model included two nonlinear terms only: pn+1 = aen + b(xn − Tn) + βen(xn − Tn) 2 and xn+1 = cen + d(xn − Tn) + δe2n + Tn. The model was fitted to the data using the Differential Evolution function from the SciPy library in Python. To test incompatibility, two independent fittings were performed, grouping experimental data according to perturbation type (SC pure vs PS pure; see Methods). In the pure context, the fitted values of parameter a displayed a bimodal distribution that we associate with two distinct subpopulations. The Oobs index was defined to measure the percentage of common values between two populations of the same parameter, belonging to different types of perturbation within the same context.

Context Effects and Conclusion

The results show that responses after SC and PS perturbations in pure contexts are incompatible with each other. In contrast, when these perturbations occur in a combined context instead, they are indeed compatible, as shown experimentally in Figure 1(d). This means that we can find a single set of model parameter values that reproduce both types of perturbation at once, with no fine tuning needed. Our main result is not quantitative but qualitative in nature: we plot trajectories from different perturbations in pure contexts and show that they cross each other in phase space. This crossing points to a fundamental difference between conditions beyond the statistically significant difference observed in our previous work. We propose that during resynchronization after either perturbation type both processes are simultaneously engaged, with a fixed set of parameter values that is chosen among three alternatives depending on the condition: combined context vs phase shift pure context vs step change pure context.

Implications

Our results challenge the proposed idea of “different mechanisms for different perturbation types” (Repp, 2005; Repp & Su, 2013): that is, the proposed existence of two different processes, named phase correction and period correction, that are selectively engaged depending on the perturbation type. When participants are exposed to both perturbation types at random in the same experiment, they don’t seem to select different strategies but a single, intermediate strategy (Silva & Laje, 2024). We propose that during resynchronization after either perturbation type both processes are simultaneously engaged, with a fixed set of parameter values that is chosen among three alternatives depending on the condition: combined context vs phase shift pure context vs step change pure context. This suggests that the putative process being studied in a specific experiment is not to be identified with the perturbation type used (phase shift or step change)—if the putative phase and period correction processes have distinct neural correlates, an association with a particular perturbation type would rather confound them instead of isolating them.

Clinical Relevance

Our work represents one more step towards understanding the underlying mechanism so appropriate quantitative measures of behavior can be defined to distinguish health from disease. Models are even less used (Lainscsek et al., 2012). Our work shows that sensory expectations can be explicitly and quantitatively described by a single set of parameter values of a behavioral mathematical model that reproduces all data in an experiment.

Constraints

We expect that our main result, that is the dynamical incompatibility between responses from different temporal contexts created by step changes and phase shifts, will hold when considering event-onset shifts, the third most used perturbation type in the literature (Bavassi et al., 2013; Repp, 2005).

1. textbf Fundamental Model Selection for Sensorimotor Synchronization (SMS) Models with Contextual Tuning: The improved AI system can distinguish between incompatible and compatible dynamical systems based on experimental context, allowing it to select a single underlying nonlinear mechanism calibrated by temporal context rather than assuming separate processes for different perturbation types.

2. textbf Robust Error Correction Mechanism Identification: The system can accurately determine if the observed response is due to phase correction or period correction, noting that when perturbations occur in the same experiment, they are compatible and can be explained by a single mechanism.

3. textbf Context-Aware Response Prediction: The AI system will predict the outcome of a period perturbation by considering the specific experimental context (e.g., 'pure' vs. 'combined'), as it learns that "responses to different types of perturbation are dynamically incompatible when recorded in separate experiments; that is, they cannot be described by a single underlying dynamical system due to the build-up of different temporal contexts."

4. textbf Model Fitting Quality Assessment: The system can assess the plausibility of a model fit by comparing fitting loss functions across contexts, noting that the joint fitting of both types of perturbation in the combined context is closer to the experimental time series than in the pure context (Figure 6).

Paced finger-tapping tasks are used to probe the error correction mechanism underlying sensorimotor synchronization. Despite their century-long history, fundamental contradictions persist in the literature. One such contradiction arises when comparing the two most common types of period perturbation: step change and phase shift. The stimulus sequence is exactly the same up to and including the (unexpected) perturbed stimulus. Why then would the timing of the next response be different between perturbation types, as observed? We show, both experimentally and theoretically, that responses to both types of perturbation are dynamically incompatible when recorded in separate experiments; that is, they cannot be described by a single underlying dynamical system due to the build-up of different temporal contexts. In contrast, when both types of perturbation are presented randomly within the same experiment, the responses become compatible and can be explained by a single mechanism. We conclude that a single underlying dynamical system can represent the response to all perturbation types, signs, and sizes, which is nevertheless calibrated by temporal context. Our results challenge the established idea of phase and period correction processes that are separately activated for different perturbation types.