{"id":"294bc5d3-cda0-4a77-a829-7ca5164e44a2","arxiv_id":"1908.03615","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"The paced finger-tapping asynchrony e_n is ill-defined as a map variable under tempo perturbations, and the paper fixes this by modeling a predicted asynchrony p_n=e_n+Δ_n inside a closed three-variable map.","lead":"This paper argues that the standard asynchrony variable in finger-tapping models becomes ill-defined when the pacing sequence changes tempo, and introduces a corrected variable plus a closed dynamical model that treats perturbations as built-in. A generalist reader may care because the correction matters for studying synchronization to music, other people, and non-periodic rhythms.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 5's 'actual relationship' between predicted and observed asynchrony holds only under a one-step extrapolation reference; this untested assumption is the load-bearing link for the closed model and its predictions.","rationale":"The reader's weakest assumption—that Eq. 5 depends on a specific, untested prediction reference—is precisely the most load-bearing concern I find. The conceptual claim that e_n is ill-defined as a map variable is somewhat overstated, since a map with T_n as an input (Eq. 2) can still describe e_n; however, the concrete falsifiable content of the paper, including the closed model and the Section II E predictions, depends on Eq. 5 holding as an actual behavioral relationship. If a subject uses a longer-term tempo estimate, p_n ≠ e_n + Δ_n and the claimed model-free link breaks. This does not make the mathematical construction invalid as a model, but it makes the paper's stronger empirical claims conditional on an assumption that could be tested directly. Since the reader already issued a CONDITIONAL verdict, my stress-test does not move the verdict; it reinforces the condition and sharpens the required test. I therefore recommend UNCHANGED.","tokens_in":13256,"tokens_out":7552,"duration_ms":81317,"concrete_test":"Collect (or use existing unpublished data from reference [21]) paced finger-tapping time series with sinusoidal and pseudorandom stimulus period sequences. Fit two variants to held-out trials: (i) the proposed model with p_n = e_n + Δ_n, and (ii) a generalized variant where p_n = e_n + Σ_{k≥1} w_k (T_{n-k+1} − T_{n-k}), with weights w_k estimated from data (e.g., an exponential or finite-memory tempo tracker). Compare one-step-ahead predictive accuracy for observed e_n. If the generalized variant fits significantly better, or if the estimated weights deviate substantially from w_1 = 1, w_k = 0 for k > 1, then Eq. 5 is not the actual relationship and the model's predictions are conditional on an untested reference assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central innovation—distinguishing predicted asynchrony p_n from observed asynchrony e_n—rests entirely on Eq. 5, p_n = e_n + Δ_n, which is presented as an 'actual relationship' rather than a modeling assumption. That equality follows only if the subject's predicted stimulus time is obtained by extrapolating the immediately preceding interval T_{n-1}, i.e., S_n^pred = S_{n-1} + T_{n-1}. The paper never defines p_n operationally and never tests this reference assumption. For step-change perturbations, a one-step extrapolation may be a reasonable account. But for the sinusoidal and random T_n sequences featured in Fig. 5, and for the two-step perturbation predictions in Section II E, human temporal prediction is commonly thought to rely on a longer-term estimate of tempo (e.g., a weighted average of recent intervals). Under that alternative, the difference between the actual stimulus time and the predicted stimulus time is not exactly Δ_n, so Eq. 5 fails. If Eq. 5 fails, the closed model Eq. 11, the claim that perturbations are 'built-in' without by-hand adjustment, and the phase-space predictions in Section II E lose their empirical grounding. The model can of course define p_n internally via Eq. 5, but then it is a modeling assumption, not a model-free identity, and it must be validated against behavioral data before the paper's stronger claims are accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13575,"tokens_out":6234,"duration_ms":65118,"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":[{"comment":"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":"Section II A, Eq. (5)"},{"comment":"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.","section":"Section II B, Eq. (9)"},{"comment":"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.","section":"Appendix A 2"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Figure 1 and Section II A"},{"comment":"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.","section":"Figure 5"},{"comment":"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":"References"},{"comment":"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.","section":"Section II E 3"}],"recommendation":"major_revision","confidential_remarks":"This is a borderline case: the conceptual point about predicted versus observed asynchrony may be more novel than the modeling results themselves. The central relationship (Eq. 5) is an assumption rather than a derived identity, and the nonlinear terms are non-unique, so the claimed predictions need to be reframed as conditional on those choices. The paper also relies on an unpublished companion paper for the feasibility of variable-only perturbations, and the total number of free parameters is understated because the post-perturbation baselines are not included in Table I. It would be helpful to verify that the contribution is sufficiently distinct from the authors' previous work [17], which used the same data and a similar two-variable nonlinear model."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nOther 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.\n\nNone 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.","headline":"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.","tokens_in":14092,"tokens_out":5741,"would_cite":true,"duration_ms":53368,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["sensorimotor synchronization","paced finger tapping","asynchrony","tempo perturbation","predicted asynchrony","phase space","mathematical model","error correction"],"falsifier":"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.","tokens_in":12996,"feed_emoji":"🎵","tokens_out":9325,"duration_ms":80249,"temperature":0.7,"pith_summary":"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.","feed_headline":"Tempo shifts break the standard tap-timing model; a new variable fixes it","feed_subtitle":"Separating predicted from observed asynchrony lets the model absorb tempo changes instead of patching them by hand.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Documents the asynchrony as the standard quantitative observable in paced finger tapping.","marker":"[6]"},{"why":"Reviews the asynchrony as one of the most used variables for quantifying sensorimotor synchronization.","marker":"[7]"},{"why":"Supplies the experimental step-change time series used for fitting and the earlier two-variable nonlinear model this work extends.","marker":"[17]"},{"why":"Shows that perturbations to the variable without changing the stimulus period are experimentally feasible, grounding the paper's novel predictions.","marker":"[21]"},{"why":"Justifies the need for a second variable by showing that a one-variable deterministic model cannot produce the observed overshoot.","marker":"[22]"},{"why":"Provides the phase-space embedding method used to reconstruct return points and infer the nonlinear terms.","marker":"[23]"}],"fun_headline_variants":["Tempo shifts break tap-timing models; a new variable absorbs them","Why asynchrony is the wrong variable for tapping with tempo changes","New tapping model builds in tempo perturbations without resets","Tempo step changes expose a flaw in tap-timing asynchrony","Paced tapping: asynchrony isn't a state variable under tempo changes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Tempo shifts break tap-timing models; a new variable absorbs them","Why asynchrony is the wrong variable for tapping with tempo changes","New tapping model builds in tempo perturbations without resets","Tempo step changes expose a flaw in tap-timing asynchrony","Paced tapping: asynchrony isn't a state variable under tempo changes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000665,"raw_usage":{"total_tokens":3052,"prompt_tokens":981,"completion_tokens":2071,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":597,"completion_tokens_details":{"reasoning_tokens":1979}},"tokens_in":597,"tokens_out":2071,"duration_ms":14330,"temperature":1.0,"reasoning_tokens":1979,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:07:59.935306+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"In the time series en the return points appear as local maxima or minima","cited_arxiv_id":null,"evidence_quote":"Documents the asynchrony as the standard quantitative observable in paced finger tapping."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reviews the asynchrony as one of the most used variables for quantifying sensorimotor synchronization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the experimental step-change time series used for fitting and the earlier two-variable nonlinear model this work extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that perturbations to the variable without changing the stimulus period are experimentally feasible, grounding the paper's novel predictions."},{"cited_title":"Hasegawa, K","cited_arxiv_id":null,"evidence_quote":"Justifies the need for a second variable by showing that a one-variable deterministic model cannot produce the observed overshoot."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the phase-space embedding method used to reconstruct return points and infer the nonlinear terms."}],"review_version":1}