Fig 1.
Illustration of the synchronization tasks and corresponding simulation experiments.
(A) The task simulated in Experiment 1, in which a person synchronizes with a metronome (top). Illustration of our simulation, in which our model synchronizes with an external sinusoidal stimulus (bottom). (B) The first task simulated in Experiment 2, in which one musician taps to every other metronome beat while listening his or her own taps (top). Illustration of our simulation in which a SAPPA model synchronizes with an external sinusoidal stimulus (bottom). Blue colored part of the model’s activity indicates that the model is receiving its own non-delayed activity as input in addition to the external sinusoid, and gray colored part indicates that the model only receives the external sinusoid as input. (C) The second task simulated in Experiment 2. This task is the same as the first one described in (B), except that the musician did not hear his or her own taps (top). Illustration of our corresponding simulation (bottom). The gray lines indicate that the model only receives the external sinusoid as input. (D) The third task simulated in Experiment 2, in which two musicians alternately tap with a metronome while listening to their own taps and the other musician’s taps (top). Illustration of our simulation where two models synchronize with an external sinusoidal stimulus (bottom). Blue and red colored parts of the model’s activity indicate the time window where the model’s non-delayed activity is used as input for both models in addition to the external sinusoid, while grayed part indicates the time window when the model receives the non-delayed activity of the other model in addition to the external sinusoid as input. (E) The fourth task simulated in Experiment 2. This task is the same as the third one described in (D), except that the musicians did not hear their own or each other’s taps (top). Illustration of our corresponding simulation (bottom). The grayed cycles indicate that the models only receive the external sinusoid as input. (F) The task simulated in Experiment 3, in which two musicians clapped a rhythm alternately (top). Illustration of our simulation where two models oscillate while alternately receiving each other’s activity as input (bottom). Blue and red cycles indicate the model whose activity is received by both models as input, while gray cycles indicate that the model’s activity is not received as input by either model. TL stands for transmission latency.
Fig 2.
Dynamical systems model of anticipation when musicians and non-musicians synchronize with an isochronous stimulus.
(A) The anticipation (mean values with error bars representing the standard error of the mean) in musicians and non-musicians tapping with an isochronous metronome while listening their own taps. The regression lines for the mean values are also shown. (B) The anticipation obtained when the musician (green dots) and non-musician (yellow dots) SAPPA models were stimulated by an external sinusoid while also receiving their own non-delayed activity as input (A = -0.5; see model definition in the methods section). (C) The anticipation obtained when the musician (gray-green dots) and non-musician (gray-yellow dots) SAPPA models were only stimulated by an external sinusoid and did not receive their own non-delayed activity as input (A = 0; see model definition in the methods section). In all simulations τ = 0.222 seconds. The D parameter differentiates the musician and non-musician models. The same regression lines for the behavioral data are shown in both (A), (B) and (C) for comparison purposes (see Supplementary S1 Fig for the model’s behavior with a square wave input).
Fig 3.
The effect of auditory feedback on anticipation when musicians synchronize with an isochronous metronome alone or with a musician partner.
(A) Behavioral measurements when a single musician taps every other beat in synchrony with a metronome for the feedback-on and feedback-off conditions (mean asynchronies with error bars representing the standard error of the mean). (B) The musician SAPPA model’s anticipation when synchronizing with an external sinusoid, while receiving (feedback-on) or not receiving (feedback-off) its non-delayed activity as input every other beat. (C) The non-musician SAPPA model’s anticipation when synchronizing with an external sinusoid, while receiving (feedback-on) or not receiving (feedback-off) its non-delayed activity as input every other beat. Dotted contours around circle data points indicate that this is a prediction that can be tested with behavioral data from non-musicians. (D) Behavioral measurements when two musicians tap every other beat in synchrony with a metronome for feedback-on and feedback-off conditions (mean asynchronies with error bars representing the standard error of the mean). (E) The anticipation when two musician SAPPA models synchronize with an external sinusoid, while alternating (feedback-on) or not receiving at all (feedback-off) each other’s non-delayed activity as input every beat. (F) The anticipation when two non-musician SAPPA models synchronize with an external sinusoid, while alternating (feedback-on) or not receiving at all (feedback-off) each other’s non-delayed activity as input every beat. Dotted borders around data points indicate that this is a prediction that can be tested by collecting behavioral data from non-musicians.
Fig 4.
The effect of transmission latencies on the anticipation of pairs of musicians alternatively clapping a rhythm.
(A) Illustration of the dynamics observed during our simulations in the presence of a TL between two synchronizing SAPPA models. The alternating blue and red background colors indicate which model’s activity is used as input to both models. The arrows show the end points of cycles for both models. Note how the passive model lags the active model in turn at the end of the cycle, due to the presence of the TL. (B) The lead and lag between musicians (measured as the percentage of a 90 bpm beat) as a function of TLs (mean values with error bars representing the 95% variance), with the linear regression on the behavioral data. (C-D) The lead and lag between pairs of musician (C) and non-musician (D) models, with the linear regression from the behavioral data (B) for comparison purposes.
Fig 5.
The effect of modified f (Eq (1)) on our model’s simulation in Experiment 3 where pairs of musicians alternatively clapping a rhythm in the presence of transmission latencies.
The lead and lag between pairs of musician (A) and non-musician (B) models, measured as the percentage of a 90 bpm beat as a function of TLs, with f modified as described in Eq (1). The linear regression on the behavioral data from Fig 4B is shown for comparison purposes.
Fig 6.
Analysis of the effect that different parameters in the SAPPA model have on its anticipation tendency.
(A) Illustration of what the asynchrony between the SAPPA model and the external sinusoidal stimulus can look like, and how it’s measured. (B-F) Analysis of the anticipation as a function of D and τ in Eq (5), and A in Eq (7): (B) A = 0, (C) A = -1.0, (D) A = -0.5, (E) A = 0.5, (F) A = 1. In these analyses (B-F) the parameter f = 1. The numbers in each cell indicate the anticipation (in ms) observed when the SAPPA model synchronized with the external sinusoidal stimulus. A black cell indicates that the SAPPA model did not synchronize with the external sinusoidal stimulus and hence the asynchronies could not be computed. In the analyses (B-F), the asynchrony quickly moves away from zero as 0 <τ <0.5, especially when D = 1. Additionally, we explored how different initial conditions affect the model’s asynchrony and discontinuities, described in the Supplementary S2 Fig which contains the bifurcation diagram for the SAPPA model when D = 1.
Fig 7.
Analysis of the effect that different frequencies in the SAPPA model have on its anticipation.
(A) Analysis of the anticipation as a function of D and τ in Eq (5) when A = -0.5 and f = 1. (B) Analysis of the anticipation as a function of D and τ in Eq (5) when A = -0.5 and f = 0.2857. The numbers in the cells indicate the anticipation (in ms) observed when the SAPPA model synchronized with the external sinusoidal stimulus.