Data for Fig 2A,2B and 2C.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model–both widely used for temporal estimation and prediction–and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors–in a Bayesian sense–that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.</div

Oscillatory behavior of the WC model.

Running numerical simulations of the WC equations with no stimulus (Stim = 0), we estimated the period of the excitatory population activity for different values of ρE (gray dots). A saddle node in limit cycle bifurcation takes place around ρ0 = −3.16 (dashed line). Such a bifurcation gives birth to very slow oscillations rapidly increasing its frequency as the relevant parameter departs from the bifurcation point. We fitted a rational function to the numerical data to get an analytic parametrization ofρE as a function of the natural period of the system. Blue traces depict the activity of the excitatory population in the different regimes. Data for Fig 5 can be found in S5 Data.</p

Data for S2 Fig.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model–both widely used for temporal estimation and prediction–and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors–in a Bayesian sense–that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.</div

Experimental design and analysis framework.

A. Schema reflecting the trials design and the different plausible perceptual timing computations used to estimate the expected final location. The position of the tones is jittered from an isochronous underlying structure (gray vertical lines) according to different probability functions (gray traces): a gaussian distribution was chosen for the cue tones (black) and a uniform distribution for the final probe tone (green). According to relative perceptual timing (blue), expectation reconstructs the underlying isochronous structure. According to the absolute perceptual timing (magenta), predictions rely on the estimation of the mean absolute duration of the intervals. B. Computations used to calculate relative and absolute estimations. Relative timing uses linear regression, fitting a line to predict timing of the next tone based on its position in the sequence. Absolute stores the intervals between tones, averages and adds this interval to the final tone location. C. Participants’ psychometric functions were fitted according to the two perceptual timing frames. Analysis of one exemplar participant. Responses (coded 0 for early, 1 for late) are compared to both the relative (left) and the absolute computations (right). Logistic model is fitted to the results and the slope of the fit is used as a measure of consistency with the given computational mechanism. This example shows a participant more consistent with the absolute computation of time. Dots represent single trial answers with a small vertical jitter added for visualization purposes. The data set used to generate these two panels can be found in S1 Data. The music note icons is used from openclipart.org.</p

Data for S1 Fig.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model–both widely used for temporal estimation and prediction–and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors–in a Bayesian sense–that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.</div

Bayesian model yields human-like performance.

A. Example trial for temporal estimation of a series of tones in a single trial. The prior (left), likelihood (middle) and posterior (right) of each tone in the sequence. Distributions are color-coded from first to last tone in the sequence from blue to yellow. BC. The differences in slope for the model when the prior is estimated using the relative timing algorithm (B) or the absolute one (C). DE. Performance of the model for different stimulus rates (corresponding to our human experiments) where stimulus noise is relative to the stimulus rate (D) or the same across rates (E). F. Performance of the model when the temporal jitter in the tone sequences is altered. Red and black show the high and low jitter conditions, respectively. G. Performance of human subjects with lower temporal jitter. Red shows the original 1 Hz condition participants, black shows new participants with reduced jitter. Data for Fig 3B,3C,3D,3E,3F and 3G can be found in S4 Data.</p

Performance of the Sensory Anticipation Module.

A. Model schematic of the ramping model (adapted from Egger and colleagues23). The model contains two competing units us and vs whose values decay to a stable point, driven by current I controlling the speed of this decay. Their difference yields ys, a ramping value which is compared with threshold, y0, at the time of a stimulus. The difference d = ys—y0 at the time of each tone is used to adjust I controlling the speed of the ramp to reduce d on the next interval. In our case, d is also used at the time of the probe tone to output a response to the behavioral trial. If d > 0, the model responds “late”; if dB. The model responses are then treated as behavioral data. Late and early responses are coded as 1 and 0 respectively and a logistic function is fitted, and the slope extracted to identify the precision of the responses relative to the relative and duration algorithms. Same procedure as the one applied to the behavioral data. Slope differences are shown here by stimulus rates at 1.2 Hz (red), 2 Hz (blue) and 4 Hz (gold). C. Slope difference relative to overall performance (slope sum) for the same stimulus rates. Black line represents polynomial fit, while the gray area represents the 95% confidence interval of the mean. Third order polynomial selected through AIC model selection. Data for S3 Fig can be found in S8 Data. (PNG)</p

Data for Fig 5.

Humans excel at predictively synchronizing their behavior with external rhythms, as in dance or music performance. The neural processes underlying rhythmic inferences are debated: whether predictive perception relies on high-level generative models or whether it can readily be implemented locally by hard-coded intrinsic oscillators synchronizing to rhythmic input remains unclear and different underlying computational mechanisms have been proposed. Here we explore human perception for tone sequences with some temporal regularity at varying rates, but with considerable variability. Next, using a dynamical systems perspective, we successfully model the participants behavior using an adaptive frequency oscillator which adjusts its spontaneous frequency based on the rate of stimuli. This model better reflects human behavior than a canonical nonlinear oscillator and a predictive ramping model–both widely used for temporal estimation and prediction–and demonstrate that the classical distinction between absolute and relative computational mechanisms can be unified under this framework. In addition, we show that neural oscillators may constitute hard-coded physiological priors–in a Bayesian sense–that reduce temporal uncertainty and facilitate the predictive processing of noisy rhythms. Together, the results show that adaptive oscillators provide an elegant and biologically plausible means to subserve rhythmic inference, reconciling previously incompatible frameworks for temporal inferential processes.</div

Classic oscillator simulations.

A. Set up of a Wilson-Cowan oscillator model (see Methods) with parameters set at: a = b = c = 10, d = -2 and (E,I) = (1.6, -2.9). The acoustic envelope of a stimulus trial drives the excitatory population with coupling determined by parameter k. The phase of the oscillator at the expected time of the last tone according to absolute timing (ABS) or the relative timing algorithm (REL) is computed on each trial. B. Phase concentration of predicted phases, ABS in purple and REL in blue across trials at a restricted range of stimulus rates (240 to 260 ms). Better phase concentration would lead to a more accurate prediction of the probe time relative to the corresponding perceptual timing mechanism. Shaded areas mark significant differences using the circular K test to test for significant differences in concentration (correcting for multiple comparisons using the false discovery rate Benjamini & Hochberg, 1995). Insets represent example concentrations at = 0.15, left, and = 2.0, right. C. Same as b but with a range of stimulus rates that reflects the statistics of the experiment (210 to 290 ms). D. Model task performance compared between Absolute and Relative algorithms in the restricted range of stimulus rates (240 to 260 ms). The difference in slope parameters fitting a Logistic regression between phase of the oscillator at probe time and the correct response defined either by absolute or relative perceptual timing mechanisms (see Methods). Polynomial fit and confidence shown in black line and gray patch respectively. Second order determined through AIC Model selection. E. Same as D with the broader range of stimulus rates (210 to 290 ms). F. Phase Response Curve of the Wilson-Cowan oscillator in response to a single 100 ms tone at a range of coupling parameters (from 0.02 to 1). Blue lines refer to coupling constants that lead to significantly higher concentration for the relative prediction, Purple to those significantly higher for the absolute one and gray to those with no significant difference between them. Phase response curves consistent with relative timing show a topologically distinct behavior than the other two categories. Data for S2 Fig can be found in S7 Data. (PNG)</p

Reducing temporal jitter increases relative alignment by strengthening the prior.

A. Bayesian simulation of the low jitter (black) and high jitter (magenta) conditions. Equivalent to Fig 4F. B. Comparison of the Sensory Precision with the Prior Precision in the low jitter (black, solid line) and high jitter (magenta, solid line) conditions. Horizontal dashed lines refer to for each experimental jitter: σ = .1 for high jitter (magenta) and σ = .075 for low jitter (black). These are the limit of possible precision of the prior if it were 100% confident where the expected tone location would be (considering the unpredictable variance of the experiment). Gray line dashed line refers to the identity line. C. The low and high jitter conditions are fully aligned when explained by sensory precision relative to the prior . Data for S4 Fig can be found in S9 Data. (PNG)</p