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

Claudia R. Gonz\'alez; M. Luz Bavassi; Rodrigo Laje

REVIEW 3 major objections 5 minor 55 references

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

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that the standard asynchrony variable is ill-defined under tempo perturbations and that distinguishing predicted from observed asynchrony yields a closed model with perturbations built in.

desk verdict A genuinely new distinction between predicted and observed asynchrony, but the central 'model-free' relation p_n = e_n + Δ_n is actually an untested one-step-extrapolation assumption that the predictions depend on. read the letter →

arxiv 1908.03615 v1 pith:UTJH5GVU submitted 2019-08-09 q-bio.NC nlin.CD

Claudia R. González , M. Luz Bavassi , Rodrigo Laje This is my paper · ORCID

classification q-bio.NCnlin.CD

keywords sensorimotorsynchronizationpacedfingertappingasynchronytempoperturbationpredictedphasespacemathematicalmodelerrorcorrection

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Paced finger tapping is the standard task for studying how people keep time with a periodic beat, and the asynchrony between tap and tone is the usual observable. This paper argues that under tempo perturbations the asynchrony is ill-defined as a model variable: at the moment the period changes, the observed asynchrony jumps by an arbitrary amount that has nothing to do with the subject's correction dynamics. The proposed fix distinguishes the predicted asynchrony $p_n$ from the observed asynchrony $e_n$, linked by $p_n = e_n + \Delta_n$ with $\Delta_n$ the change in the inter-stimulus interval, and builds a two-variable nonlinear map in which the perturbation is intrinsic to the dynamics. Fitted to step-change data, the closed model reproduces resynchronization without any by-hand adjustment and yields phase-space predictions for novel perturbations. If correct, this re-frames how resynchronization experiments are modeled and interpreted.

What carries the argument

The load-bearing identity is Eq. (5), $p_n = e_n + \Delta_n$, where $p_n$ is the asynchrony the subject would predict before the perturbed stimulus arrives and $e_n$ is the asynchrony actually measured after it. With $\Delta_n = T_n - T_{n-1}$, the closed model is the two-variable nonlinear map of Eq. (11), $p_{n+1} = a(p_n - (T_n - s_n)) + b(x_n - T_n) + F(\cdot)$, $x_{n+1} = c(p_n - (T_n - s_n)) + d(x_n - T_n) + T_n + G(\cdot)$, $s_{n+1} = T_n$, with $F = \alpha e_n^3 + \beta e_n (x_n - T_n)^2 + \gamma (x_n - T_n)^3$ and $G = \delta e_n^2$. The auxiliary variable $s_n$ stores the previous interval, so the perturbation $\Delta_n$ enters the autonomous dynamics rather than being imposed as an external reset.

What would settle it

Run a step-change experiment in which preceding rhythmic context biases the tempo expectation, and measure the asynchrony at the perturbed stimulus; if the observed $e_n$ matches a prediction based on a longer-term tempo estimate rather than on $\Delta_n$ against $T_{n-1}$, Eq. (5) is false. Alternatively, deliver a variable-only perturbation in the shared phase-space region labeled B after both positive and negative step changes; if the subsequent evolution differs by condition, the paper's main prediction fails.

Watch

Extended reading notes

Core claim

The central claim is that the asynchrony $e_n = R_n - S_n$, although operationally well defined, is ill-defined as the state variable of a difference-equation model when the tempo changes: a period change $\Delta_n = T_n - T_{n-1}$ shifts the stimulus $S_n$ by an arbitrary amount, so $e_n$ changes instantaneously without reflecting the correction mechanism. The paper's resolution is to distinguish the predicted asynchrony $p_n$ from the observed asynchrony $e_n$ through $p_n = e_n + \Delta_n$, write the correction map for $p_{n+1}$ as a function of $e_n$ and $T_n$, and close the system with the auxiliary variable $s_{n+1}=T_n$ (Eq. (11)). The paper claims that this closed model fits step-change data with a single parameter set and no manual reset, and that its phase-space geometry gives testable predictions: identical post-perturbation evolution from a shared phase-space region, and asymmetric responses to large perturbations applied to the variable alone.

Load-bearing premise

The load-bearing premise is that the subject predicts each stimulus time by extrapolating only the most recent interval; if the brain instead uses a longer-term tempo estimate, the predicted-versus-observed relation at the heart of the model breaks.

Editorial extensions

If this is right

Step-change resynchronization can be reproduced with one fitted parameter set and no manual reset of the asynchrony at the perturbation step.
Two trajectories from opposite perturbations that pass through the same phase-space region should respond identically to a second perturbation delivered there.
Large perturbations applied only to the variable, with the stimulus period unchanged, should produce asymmetric overshoot, while small ones should be nearly symmetric.
A saddle node in the phase space separates resynchronizing from diverging trajectories, predicting that sufficiently large perturbations cause loss of synchronization.
Past and future asynchrony play different causal roles, giving indirect support for separate neural processing of predicted versus observed timing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: if prediction relies on a tempo estimate longer than one interval, Eq. (5) should be replaced by $p_n = e_n + \Delta_n^{\mathrm{est}}$ against that estimate; the paper's framework would still work but with a different reference.
Editorial extension: the predicted-versus-observed split suggests neural markers such as evoked responses locked to predicted stimulus times should shift relative to actual stimulus times at a tempo step, which is testable with EEG or MEG.
Editorial extension: because the model is now autonomous in $p_n$, bifurcation analysis becomes possible; increasing perturbation size should reveal the saddle-node separatrix as the boundary of successful resynchronization, and possibly new regimes beyond it.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main thing you should know: this paper's key move—splitting the asynchrony into a predicted part p_n and an observed part e_n—is genuinely useful and, as far as I can tell, new in the map-modeling literature. The authors show that e_n alone is a bad state variable when the stimulus sequence is perturbed, because a shift in the stimulus changes e_n without any dynamics. Then they close a map on p_n with an auxiliary variable s_{n+1}=T_n, which makes the system autonomous. That is a clean solution. The paper ships code and data, and the fit to their ±50 ms step data looks good.

The soft spot is Eq. 5: p_n = e_n + Δ_n. The paper calls this "the actual relationship between the variables." It is not. It holds only if the subject's predicted stimulus time is a one-step extrapolation of the immediately preceding interval. That is a plausible account for a step change, but for the sinusoidal and random sequences featured in Fig. 5, and for the two-step perturbation predictions, human prediction probably uses a longer-term tempo estimate, so Eq. 5 may be false. Under that alternative, the closed model of Eq. 11, plus the predictions in Section II E, lose their grounding. The stress-test note is correct on this.

Other soft spots, in order of severity: the nonlinear terms F and G are admittedly non-unique, and the return-point geometry doesn't pin them down; the model is fit only to ±50 ms step changes, and two post-perturbation baselines are imported from the data, so the fit is more in-sample than it looks; the "first model" claims are overstated given that oscillator models are naturally well-defined under tempo changes; and the experimental reference [21] that motivates the predictions is unpublished.

None of this sinks the paper. The core conceptual point is sound, and the proposed closed form gives a concrete way to build perturbations into map models. The main fix is to treat Eq. 5 as a modeling assumption, define p_n operationally, and test the reference assumption—ideally by fitting to one perturbation size and predicting another. The paper deserves a serious referee; it should be sent out, but with a request to push on Eq. 5 and the baseline import. I'd expect a substantial revision.

Referee Report

3 major / 5 minor

Summary. This paper addresses a conceptual issue in mathematical models of paced finger tapping: when the interstimulus interval is perturbed, the observed asynchrony e_n = R_n - S_n changes at the perturbation step solely because the stimulus time S_n shifts, even if the subject's response is unchanged. The authors argue that e_n is therefore not a well-defined state variable for a map model and introduce a predicted asynchrony p_n related to e_n by p_n = e_n + Δ_n (Eq. 5), where Δ_n = T_n - T_{n-1}. They reformulate their earlier two-variable model in terms of p_n, obtaining a closed three-variable system (Eq. 11) that takes tempo perturbations as input. The model is fit to ±50 ms step-change data with eight parameters plus a post-perturbation baseline offset per condition, and the fitted phase space contains a saddle-node separatrix. The paper makes two sets of predictions: (i) a perturbation delivered when two different trajectories occupy the same phase-space region should produce identical subsequent dynamics, and (ii) variable-only perturbations of large amplitude should elicit asymmetric overshoot. The manuscript includes code and data for reproduction.

Significance. The conceptual distinction between predicted and observed asynchrony, if justified, would be a useful contribution to the sensorimotor-synchronization modeling literature, and the closed formulation in Eq. (11) is a clean algebraic construction that allows tempo sequences to be treated as inputs without ad-hoc intervention at the perturbation step. The paper ships reproducible code and data and states several falsifiable predictions, which is commendable. However, the central step—Eq. (5)—is presented as model-free when it in fact encodes a one-step-extrapolation assumption about the subject's prediction; and the nonlinearities in Eq. (9), acknowledged as non-unique, were chosen to reproduce the asymmetry that the paper later 'predicts.' These issues currently limit the strength of the claims.

major comments (3)

[Section II A, Eq. (5)] The identity p_n = e_n + Δ_n is presented as an 'actual relationship' but it holds only under the assumption that the subject's predicted stimulus time is S_n^pred = S_{n-1} + T_{n-1}, i.e. one-step extrapolation of the immediately preceding interval. The manuscript never defines p_n operationally or tests this reference assumption; if subjects use a longer-term tempo estimate, the difference S_n - S_n^pred is not Δ_n and Eq. (5) fails. Because Eq. (7), the closed model Eq. (11), and all predictions in Sections II D and II E rely on Eq. (5), this assumption is load-bearing and should be stated as a modeling assumption and validated (or its robustness to alternative prediction rules demonstrated).
[Section II B, Eq. (9)] The nonlinear functions F and G are not unique, as the authors admit in Section II B, and the selection was made after testing many combinations to reproduce the observed asymmetric overshoot. Consequently, the prediction in Section II E 2 that sufficiently large variable-only perturbations produce asymmetric responses is not an independent consequence of the data; it may be an artifact of the particular cubic and quadratic terms chosen. The authors should test whether this prediction is robust across the family of nonlinear terms that are compatible with the step-change data, or provide a normal-form argument justifying the choice.
[Appendix A 2] The fitting procedure adds a post-perturbation constant baseline to p_n with a fixed value 'equal to the experimental post-perturbation baseline of the corresponding perturbation size.' This is an additional data-derived offset not included in Table I and not predicted by the model dynamics. As a result, the claim that perturbations are 'built in' is weakened: for novel perturbation sizes (e.g., the ±10 ms and variable-only perturbations in Figs. 5-7), it is unclear how the baselines are set, and the model's autonomous evolution is not fully specified by Eq. (11) alone.

minor comments (5)

[Throughout] There are several typographical errors, including 'mantain' for 'maintain' and 'hypotetically' for 'hypothetically' in Section I, and 'asociated' for 'associated' in Section II E 3.
[Figure 1 and Section II A] The variable p_n is shown in the schematic but is not defined in the caption or in the text before Eq. (5); a definition in terms of the predicted stimulus time (e.g., p_n = R_n - S_n^pred) would make the modeling assumption explicit.
[Figure 5] The 'by hand' simulation in the third row is described only vaguely; the exact update rule (when and how e_n is adjusted at each step) should be stated in the text or in an appendix so that the comparison with the proposed model is reproducible.
[References] Reference [21] is cited as 'to be published elsewhere'; since the experimental feasibility of variable-only perturbations is central to the predictions in Section II E, a preprint or published version should be cited if available.
[Section II E 3] The threshold of 'half period' for the valid range of the model is introduced but is not used in the simulations; the authors should specify whether any trajectory in Figs. 6-7 crosses this threshold.

Circularity Check

2 steps flagged · score 4.0 of 10

Eq. 5's 'actual relationship' is an implicit one-step-extrapolation definition of p_n; the built-in perturbation claim and the model closure inherit it.

self definitional [Section II A, Eq. (5) and Fig. 1(a)]
"If a change in period occurs at step n: Δn = Tn − Tn−1 (4) then the predicted asynchrony pn and the actually observed asynchrony en are related by the following expression: pn = en + Δn (5). It is important to note that Eq. 5 is not a theoretical assumption but the actual relationship between the variables."

The paper introduces p_n as 'predicted asynchrony' but never defines it operationally. Eq. 5 is true only if the subject's predicted stimulus time is a one-step extrapolation, S_n^pred = S_{n-1} + T_{n-1}; only then does the difference between actual and predicted stimulus times equal Δ_n. If prediction uses a longer-term tempo estimate, the difference is not Δ_n and Eq. 5 fails. Therefore Eq. 5 is either an implicit definition of p_n or a substantive assumption, not the 'actual relationship' asserted. Since Eqs. 6, 7, 10 and 11 all close through Eq. 5, the built-in-perturbation result is loaded into the definition of p_n.
self citation load bearing [Section II E 1 (predictions) and Conclusions]
"In a recent work, however, we showed the experimental feasibility of such manipulations [21]. ... Experimental perturbations like the ones proposed in [21] but with larger magnitudes are needed."

The proposed two-step perturbation experiment, which the paper offers as the way to test its phase-space prediction, is justified by citation [21], a work 'to be published elsewhere' by the same corresponding author. The current paper treats this self-citation as external empirical support for feasibility and for the assumption that one can perturb the variable without changing the stimulus period. Because [21] is not independently available, machine-checked, or reproduced, the testability claim rests on the authors' own unpublished claim. This is a load-bearing self-citation for the experimental agenda, though not for the algebraic derivation.

full rationale

The main derivation of the ill-definedness of e_n is self-contained and correct as a critique of using e_n as a map variable under tempo changes. However, the proposed cure—splitting observed and predicted asynchrony—rests on Eq. 5, which the paper labels model-free but which is actually an implicit definition/assumption about how the subject predicts the next stimulus time. Under a one-step extrapolation the equality holds; under longer-term tempo estimation it does not. Because p_n is never operationally defined elsewhere, Eq. 5 is definitional, and Eqs. 7 and 11 simply build that definition into the dynamics. The quantitative predictions in Sec. II E are simulations of the same model fitted to ±50 ms step changes; they are genuine extrapolations to novel perturbations, but their asymmetry content partly re-expresses the even-order nonlinear terms chosen to fit the step-change data, so they are not independent confirmations. The paper's admission that the nonlinear terms are non-unique (Sec. II B) limits the strength of the predictions but is not itself a circularity. Citation [21] is a self-citation used to support the feasibility of the proposed experiments; it is minor compared with the Eq. 5 issue. Overall, the central conceptual move is circular/definitional at one point, while the rest of the modeling is transparent.

Assumptions & free parameters 9 free parameters · 4 assumptions · 1 invented entities

The conceptual point about re-referencing is self-contained, but the quantitative model rests on eight fitted coefficients plus two baseline offsets and an untested definition of predicted asynchrony. The predictions in Section II E are simulations of the fitted system, not independent tests.

free parameters (9)

a = -0.0485
Linear coefficient in the p_{n+1} update equation, fitted by genetic algorithm to ±50 ms step-change data.
b = 0.467
Linear coupling of the second variable into the p_{n+1} equation, fitted to ±50 ms data.
c = -0.491
Linear coefficient in the x_{n+1} update equation, fitted to ±50 ms data.
d = 0.987
Linear coefficient for x_n in the x_{n+1} update equation, fitted to ±50 ms data.
alpha = 5.67e-05 ms^-2
Cubic coefficient in F, one of the nonlinear terms chosen to reproduce the overshoot asymmetry.
beta = 7.71e-05 ms^-2
Cubic coefficient in F, fitted to ±50 ms data.
gamma = 9.74e-05 ms^-2
Cubic coefficient in F, fitted to ±50 ms data.
delta = 4.61e-03 ms^-1
Quadratic coefficient in G, fitted to ±50 ms data.
post-perturbation baseline offsets = Not reported; two values for ±50 ms conditions
Appendix A2 adds a post-perturbation constant baseline to p_n with a value equal to the experimental post-perturbation baseline for each fitted condition. These offsets act as fitted constants drawn from the data but are not counted among the eight fitted parameters.

assumptions (4)

ad hoc to paper The predicted asynchrony is anchored to one-step extrapolation, so p_n=e_n+Δ_n holds as an identity (Eq. 5).
The paper calls this an actual relationship, but it presumes the predicted stimulus time is S_{n-1}+T_{n-1}; no behavioral evidence is given for this reference rule.
ad hoc to paper The specific nonlinear terms F=αe^3+βe(x-T)^2+γ(x-T)^3 and G=δe^2 reproduce the data (Eq. 9).
The authors state the choice is not unique, normal form theory cannot be used because no bifurcations are observed, and several term combinations gave qualitatively similar results.
domain assumption The deterministic noiseless map is a sufficient description of resynchronization after perturbations.
Noise is discarded after Eq. 2, and the model is fitted as a deterministic map to experimental time series from previous work.
domain assumption The origin of the asynchrony does not matter for resynchronization dynamics.
Stated in the Conclusions, Section III; required so that perturbations to the variable and to the stimulus period are equivalent in the model.

invented entities (1)

p_n, the predicted asynchrony variable
purpose: Removes the reference-frame jump from the state dynamics by separating expected from observed asynchrony.
Not directly measured in this paper; Eq. 5 defines it in terms of e_n and Δ_n only under an assumed one-step reference. The cited experimental feasibility reference [21] is unpublished and from the same group.

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Cite this review

Pith. "Pith review of Response to perturbations as a built-in feature in a mathematical model for paced finger tapping." pith.science (2026). https://pith.science/paper/UTJH5GVU

@misc{pith2026190803615,
  author       = {Pith},
  title        = {Pith review of: Response to perturbations as a built-in feature in a mathematical model for paced finger tapping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTJH5GVU}},
  note         = {Machine review of arXiv:1908.03615}
}

read the original abstract

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 non-periodic stimuli.

Figures

Figures reproduced from arXiv: 1908.03615 by the authors.

**Figure 2.** FIG. 2. Experimental time series and phase space reconstruction (all experimental data shown in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Model fitting. (a) Experimental [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4. Variety of solutions from the fitting procedure. (a) Type-I phase space and time series [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. Three common experimental manipulations of the stimulus period (top row) and re [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6. Model predictions (I). (a) Response to a step change perturbation and choice of points [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FIG. 7. Model predictions (II). Response to perturbations to the variable only (large: ∆ [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: FIG. 8. Estimation of return points. (a) Return points of [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: FIG. 9. Distribution of parameter fitted values. [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: FIG. 10. Joint distributions of linear parameter values. [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: FIG. 11. Joint distributions of nonlinear parameter values. [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: FIG. 12. Joint distributions of linear vs nonlinear parameter values. [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Geometrical organization of phase space The geometrical arrangement of the trajectories in the experimental phase space is re- markable (Fig. 2(c)). The trajectories corresponding to the largest positive perturbations share a region of phase space during the overshoot with some of the trajectories of the neg- ative perturbations (that do not have overshoo...

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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A. D. Patel, J. R. Iversen, M. R. Bregman, and I. Schulz, Curr. Biol. 19, 827 (2009)

work page 2009

[2] [1]

Geometrical organization of phase space The geometrical arrangement of the trajectories in the experimental phase space is re- markable (Fig. 2(c)). The trajectories corresponding to the largest positive perturbations share a region of phase space during the overshoot with some of the trajectories of the neg- ative perturbations (that do not have overshoo...

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[3] [2]

ﬁrst a traditional step change perturbation (in two conditions, positive and negative)

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[4] [3]

B” and “C

second, and while the resynchronization from the ﬁrst perturbation takes place, a perturbation to the variable without changing the stimulus period. Our prediction is that the time evolution following the second perturbation will be the same for both conditions (positive or negative ﬁrst perturbation), provided the second per- turbation is performed when ...

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[5] [4]

This can be seen in the phase space of Fig

Perturbations to the variable only Our second prediction is that the response to large enough symmetric perturbations to the variable might be asymmetric. This can be seen in the phase space of Fig. 7, after noting that large negative perturbations (i.e., jump to the left) have a larger overshoot than large positive perturbations (i.e., jump to the right)...

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[6] [5]

dynamical systems

Large perturbations We note the saddle node appearing in the upper right quadrant of the phase space, Fig. 3(b). Its stable manifold is a separatrix between bound trajectories (i.e., converging to the origin) and unbound trajectories (i.e., diverging). The existence of the saddle node was not explicitly considered in the modeling eﬀort; a post-hoc interes...

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[7] [6]

In the time series en the return points appear as local maxima or minima

Estimation of return points The return points of a map en+1 = f(en) are the values en such that en+1 = en (the analogous concept in a continuous-time ﬂow is the nullcline, that is the points in phase space where the rate of change associated to a given variable is zero). In the time series en the return points appear as local maxima or minima. 18 In a 2D ...

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[8] [7]

Deﬁne a 5-point time window from n = 0 through n = 4

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