A Generative Model for Measuring Latent Timing Structure in Motor Sequences

<div><p>Motor variability often reflects a mixture of different neural and peripheral sources operating over a range of timescales. We present a statistical model of sequence timing that can be used to measure three distinct components of timing variability: global tempo changes that are spread across the sequence, such as might stem from neuromodulatory sources with widespread influence; fast, uncorrelated timing noise, stemming from noisy components within the neural system; and timing jitter that does not alter the timing of subsequent elements, such as might be caused by variation in the motor periphery or by measurement error. In addition to quantifying the variability contributed by each of these latent factors in the data, the approach assigns maximum likelihood estimates of each factor on a trial-to-trial basis. We applied the model to adult zebra finch song, a temporally complex behavior with rich structure on multiple timescales. We find that individual song vocalizations (syllables) contain roughly equal amounts of variability in each of the three components while overall song length is dominated by global tempo changes. Across our sample of syllables, both global and independent variability scale with average length while timing jitter does not, a pattern consistent with the Wing and Kristofferson (1973) model of sequence timing. We also find significant day-to-day drift in all three timing sources, but a circadian pattern in tempo only. In tests using artificially generated data, the model successfully separates out the different components with small error. The approach provides a general framework for extracting distinct sources of timing variability within action sequences, and can be applied to neural and behavioral data from a wide array of systems.</p> </div

The influence of additional global factors on model fit and song timing.

<p>A, Goodness of model fit with additional global factors included (black) vs. 1 global factor (white) for the 5 birds for which BIC was lowest for factor. Numbers above black bars indicate the number of factors associated with the lowest BIC for that bird. B, Distribution of global parameters for the 2nd factor (sign-normalized, see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0037616#s4" target="_blank">Methods</a>), separated by whether the song segment was a syllable or silent gap. Across birds syllable and gap parameters tended to be of opposite sign. C, Timing covariance matrices generated for the 2nd global factor in 3 representative birds. D, Average changes in the 2nd global variable in the 3 birds that showed significant circadian variation for that factor. The circadian pattern showed that syllables tended to elongate at the expense of gaps over the afternoon. E, Example of day-to-day drift in the 2nd global variable for one bird over a 1-month period. As in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0037616#pone-0037616-g004" target="_blank">figure 4</a>, hour-of-day and daily averages in D and E have been adjusted for unequal sampling of factor combinations while dotted lines in E represent adjusted means ± standard error.</p

Results of Monte Carlo simulations based on real parameter distributions in the song data.

<p>For plots A-D, F-I and K-N, parameter type is designated by labels on the far left. A-D, Real parameter values (black lines), along with parameter estimates from 20 randomly selected simulations (blue) for one bird. E, Median SRMR across simulations for each bird (black bars, errorbars indicate median absolute deviation) vs. SRMR values from the real data (white bars). F-I, Median absolute deviation in parameter estimates from real parameters across all birds. J, Parameter error and SRMR as a function of sample size for simulations selected from 3 birds. Here, parameter error was taken as the median MAD across simulations and birds. Solid lines represent a regression of error and SRMR with the square root of sample size. K-N, Distribution of Pearson’s correlation between the real and simulated latent variables. O, Median latent variable correlation from the same 3 birds taken as a function of sample size. Color legend is the same as in panel J.</p

Timing variability and its decomposition.

<p>A, Spectrogram of a single song motif from one bird, with syllables indicated above by letters and silent gaps by “-”. B, timing covariance matrix for the same bird, in which the color of each square indicates the pairwise covariance between the durations of two intervals denoted along the rows and columns. C, Schematic of the different noise sources captured in the model. Below each noise source is a covariance matrix generated by the model of that source.</p

Scaling of timing parameters with average length.

<p>A, Scatter plot of global tempo variability by average interval duration across all syllables and silent gaps. Silent gaps within motifs and between motifs have been separated out. B, C, Analogous plots for the independent and jitter parameters respectively. In each plot solid lines come from a regression of timing parameters on average interval duration, calculated across birds. Overall, plots show that global and independent variability scale with average duration, while jitter does not. The plots also show systematic differences among interval types with respect to how much each parameter scales with average duration.</p

Circadian rhythm and day-to-day drift in the latent timing variables.

<p>A, Average tempo by hour of the day (defined from lights on) by bird (blue lines) and averaged across birds (black). B, example of day-to-day drift in tempo for one bird over a 1-month period. C, example of drift in independent variability for two different song segments in the same bird. All data in plots A-C are group means from a two-factor ANOVA that have been adjusted for unequal samples of factor combinations (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0037616#s2" target="_blank">Results</a>), while dotted lines in B and C indicate adjusted means ± standard error. D, The estimated amount of variance in tempo explained by both hour of the day (black) and day (white) across all 11 birds. E and F, histograms of the amount of variance in the independent and jitter variables respectively, explained by both hour of the day and day (same color code). Generally the data show significant circadian variation in tempo only, and day-to-day drift in all three latent timing variables.</p

Goodness of fit between the timing model and data from 11 zebra finch birds.

<p>A, SRMR by bird (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0037616#s2" target="_blank">Results</a>). B, Color plots of the timing covariance matrices from the data and model for 3 representative birds.</p