Figure 3. A The summed PSTH was convolved represented by an asterisk with a Gaussian-shaped temporal smoothing window with a standard deviation of 40 ms see Materials and Methods. Then the Fourier transform of the smoothed PSTH was used to compute the vector strength of the neural activity, which quantifies the strength of synchronization, at each tempo.
B The Fourier transform of the temporal smoothing window. The temporal smoothing window smooths the PSTH and suppresses the vector strengths at high tempi. In the Fourier domain the temporal smoothing window imposed a low-pass filter on the vector strength and thus suppressed the vector strength of fast tempi Figure 3B. Several studies have demonstrated that the upper limit of the human perception of isochrony occurs at inter-onset intervals around ms for review see Repp, ; London, The temporal window width of 40 ms was used for all SFIE models examined.
Throughout, all stimuli were set to 70 dB SPL and were up-sampled to a kHz sampling rate, which was required for the AN fiber model. For stimuli that started or ended with a non-zero signal for example, amplitude modulated noise , 15 ms raised-sine ramps were applied to the start and end of the stimulus.
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Stimuli were 10 s long and consisted of either 1 ms clicks 0. The phase of the stimulus modulation was randomized for each presentation. The preferred tempo was determined for each type of stimulus using quadratic interpolation. Several studies have demonstrated that humans' ability to perceive and reproduce regular events is optimized for inter-onset intervals around ms, corresponding to a tempo of BPM London, We hypothesized that the modulation filtering of the SFIE model and the temporal smoothing window could produce a vector strength maximum around BPM.
Additionally, Henry et al.
To identify a frequency bias in tempo induction that could result from subcortical processing, we presented the model with stimuli consisting of two stimulus trains of ms raised-sine tone pips presented at two different tempi from the range 60 to BPM and two different frequencies from the range to 8, Hz an example stimulus can be found in Figure 6A. The tempi, frequencies, and phases of the two tones were randomly selected to generate different stimuli, and each stimulus was presented once.
The frequencies of the two tone pips were spaced at least one octave apart to reduce AN adaptation effects Zilany et al. For each stimulus, we computed the normalized synchronization tempo NST :. We expected the synchronization tempo to be close to the tempo of either the tone pips with the low-frequency carrier or the high-frequency carrier for most of the stimuli, resulting in an NST near either zero or one, respectively.
Of those stimuli, we next examined how the other factors, the tempi of the two tone pips and their carrier frequencies, affected the NST. The significance of this difference was assessed using a likelihood ratio test. The significance of the individual coefficients in the model was also assessed using a likelihood ratio test comparing the full model to a reduced model with each component removed individually. These datasets are standards for assessing the performance of tempo-induction and beat-detection algorithms Gouyon et al.
We determined the synchronization tempo based on the tempo between 30 and BPM with the maximum vector strength. Often, the peak vector strength occurred at a multiple of the ground truth tempo rather than at the actual ground truth tempo. To understand the importance of speed and rhythm on tempo induction, we used regularized multi-class linear discriminant analysis mcLDA fitcdiscr. The first classifier used the synchronization tempo alone to classify the scaling factor; faster synchronization tempi were more likely to have higher scaling factors.
For the second classifier, we reasoned that, if the model neurons were synchronizing to events in the music, then the rhythm of the music could be quantified by the number of times certain intervals appear between simulated spikes. All stimuli from both datasets were included in this analysis, and the ratios were rounded to closest integer between 1 and 4. We determined whether the second classifier performed significantly better than the first by testing the null hypothesis that the distribution of differences in performance between the two classifiers for the re-samplings was no greater than 0.
Firstly, we examined if the vector strength of the model PSTH was maximal over a specific range of tempi. We hypothesized that sub-cortical processing could contribute to this biasing, which has been observed around BPM.
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The vector strength as a function of tempo was computed using three different midbrain models Table 1 that were tuned to different best modulation frequencies Figure 2. For comparison, the vector strength was also computed based on the unfiltered summed AN fiber output. While the temporal smoothing window suppressed vector strengths at high tempi Figure 3B , there was also a reduction in vector strengths at low tempi due to an intrinsic property of the auditory nerve model.
For click trains at 30 BPM Figure 4A , SFIE model A generated the largest firing rates in response to a click, but it also produced the highest spontaneous rate, resulting in the lowest vector strength of the three midbrain models. In contrast, SFIE A showed a stronger onset response during the rising phase of the stimulus modulation followed by a reduction in firing during the rest of the cycle of the modulation.
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Figure 4. The corresponding stimulus is shown above each plot of the firing rate. All stimuli were presented at 70 dB SPL. The firing rates were summed across CF and averaged across 10 repetitions of each stimulus with different noise tokens. Spontaneous firing during silences A,C and saturating firing rates during continuous noises A,B contributed to a falloff in vector strength at lower tempi see Figure 5. In contrast, peak vector strengths occurred at a much wider range of tempi for the other two SFIE models and for the AN fiber activity. Since human perception of musical beats is invariant to the envelope of the stimulus Henry et al.
Such neurons would produce strong onset firing and reduced sustained firing necessary for creating salient beats. We also found empirically that vector strengths were larger for musical recordings using SFIE A than the other two models Supplementary Figure 2. For these reasons, SFIE A was used when simulating sub-cortical neural activity in the following experiments. Figure 5. Error bars designate interquartile ranges for 10 repetitions of each stimulus. The preferred tempos were determined by quadratic interpolation.
The black dashed line in the inset in A shows the quadratic fit to the points surrounding the maximum vector strength for SFIE A. The preferred tempo is equal to the peak of the quadratic fit. Preferred tempos and peak vector strengths are quantified in Table 2. Table 2. There is some evidence that human perception of musical beats may be biased to particular frequency ranges, but the strength of this effect and the underlying mechanism are unclear.
We hypothesized that subcortical processing may produce a frequency bias for tempo induction. Specifically, when multiple carrier frequencies are present with temporal modulations at distinct tempi, we expected the synchronization tempo to equal the tempo of the lowest carrier frequency. For each stimulus, the synchronization tempo was normalized relative to the tempos of the two tone pips to get the NST Figure 6B.
An NST of zero means that the synchronization tempo was closer to the tempo of the tone pip with the low-frequency carrier, and an NST of one means that it was closer to the tempo for the high-frequency carrier. On average, synchronization tempi were biased to lower audio frequencies. Figure 6. A To test for a frequency bias in tempo induction, stimuli consisted of two sets of tone pips at two different carrier frequencies and different tempi.
B The vector strength as a function of tempo for the stimulus in A is shown. Dashed lines mark the tempi for the tone pips with the low-frequency carrier blue and the high-frequency carrier red. Figure 7. A Distribution of the NSTs for all randomly generated stimuli consisting of two tone pips.
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On average, the synchronization tempi were closer to T L. Each bin shows the marginal probability given f L and f H. Each showed a monotonic relationship with the proportion of NSTs equal to zero. To quantify these dependences and assess their significance, we fit a logistic generalized linear model to the individual NSTs with the low-frequency carrier f L , high-frequency carrier f H , and the tempi of those tone pips T L and T H respectively as dependent variables see Materials and Methods.
Overall, synchronization tempi were biased to the tempo for the tone pips with the lower carrier frequency, but the biasing was weakest when the interfering modulations from the higher carrier frequency was close to the lower carrier frequency. Both low-CF and high-CF responses resulted in similar vector strengths for broadband stimuli with tone-pip-like modulations, suggesting that the biasing observed here was due to the spread of excitation in the basilar membrane and not due to differences in the response properties of different CFs Supplementary Figure 3.
However, the tempi of the tone pips had a stronger influence on the synchronization tempo than the carrier frequency, and the synchronization tempo was more likely to equal the fastest tempo. This was contrary to our earlier finding that the vector strength was maximized around BPM for salient, isochronous stimuli.
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When multiple competing modulations are present in complex stimuli, the faster modulations dominate in the summed synchronized activity, primarily because faster modulations produce more events and are more likely to mask slower modulations Supplementary Figure 4. We lastly evaluated tempo-induction performance using two datasets widely used for testing tempo-induction algorithms Gouyon et al. For each stimulus the synchronization tempo was computed and compared to the ground-truth tempo for the recording. The synchronization tempo was equal to the ground-truth tempo for only More often, the synchronization tempo was twice the ground-truth tempo Figure 8.
The histogram of the ratio between the synchronization tempo and the ground truth tempo is plotted for the Ballroom dataset A and the Songs dataset B without the temporal Gaussian window applied black and with the temporal Gaussian window red.