Stimulus generation and data collection.

Sinusoidal amplitude-modulated (AM) stimuli were created with a “frozen” broadband noise carrier and were 400 ms in duration. AM stimuli were created at a variety of modulation frequencies (5, 10, 15, 20, 30, 60, and 120 Hz) and modulation depths (6%, 16%, 28%, 40%, 60%, 80%, and 100% depth). The modulation envelope was determined by

where m is the modulation index (ranges from 0 to 1; % modulation depth is m × 100), f_m is the modulation frequency in hertz, and t is the waveform time in seconds.

The sound signals were generated by a digital signal processor (AT&T DSP32C) and a digital-to-analog converter (TDT Systems DA1) and then passed through programmable and passive attenuators (TDT Systems PA4, Leader LAT-45). The signal was amplified (Radio Shack MPA-200) before being delivered to a speaker (Radio Shack PA-110, 10-in. woofer and piezo-horn tweeter, 10-dB cutoff: 0.038–27 kHz) positioned at ear level 1.5 m in front of the subject.

The stimuli were presented at a sampling rate of 50 kHz (2.5- to 25-kHz digital bandwidth, analog bandwidth limited by 27-kHz 10-dB cutoff point of the speaker) and were cosine ramped at onset and offset (5.0-ms rise/fall time). Stimulus intensity was adjusted to ~65 dB SPL (<2-dB variation). Extracellular potentials were amplified and filtered (0.3–5 kHz; A-M Systems 1800), sampled at 50 kHz, and stored on hard disk for later analysis. Spikes were resorted off-line with Spike2 software (CED). All numerical analysis was done with custom software written for MATLAB (MathWorks). For all calculations, only spikes occurring between 70 and 400 ms after the onset of the stimulus were included, to eliminate any contribution of an onset response. Including the onset response did not substantially alter the results.

At each recording site, neurons were assessed with at least two different batteries of stimuli. To determine the best modulation frequency (BMF) of the multiunit, 100% depth AM stimuli were presented at each modulation frequency and modulation transfer functions (MTFs) were created by taking the mean spike count (SC) or phase-projected vector strength (VS_PP) across all trials. MTFs were calculated separately for rate and temporal measures. We defined the temporal BMF (tBMF) as the point with the highest mean VS_PP. Because of the number of cells that decreased their activity in response to AM relative to their noise response, the rate BMF (rBMF) was defined as the modulation frequency that evoked the mean SC furthest from the noise response [as measured by the distance of the receiver operator curve (ROC) area from 0.5; ROC area is described in the next section], regardless of whether that modulation frequency resulted in an increase or decrease relative to the noise response. A second battery of stimuli was used to determine the depth sensitivity of the same cells for a single modulation frequency. During each recording we attempted to measure depth sensitivity at several modulation frequencies: the multiunit (MU) tBMF, the MU rBMF, and 15 Hz. For some units we were able to obtain sensitivity functions at all three modulation frequencies, but for many we were not able to maintain a stable recording and only obtained data for one or two modulation frequencies. Because the BMF was not identical for all single units (SUs) within a MU and because most cells were tested at 15 Hz, a substantial number of recordings were not at BMF.

Depth sensitivity: neurometric analysis.

One hundred twenty-three unique cells were tested with a neurometric analysis for AM depth sensitivity. Because many cells were tested at more than one modulation frequency, we obtained a total of 249 measurements of depth sensitivity. Cells were tested at modulation depths of 0%, 6%, 16%, 28%, 40%, 60%, 80%, and 100%. Modulation depths were chosen to conform to the log-scaled depths in O'Connor et al. (2000), with points added near the expected psychophysical threshold (28%) and at high depths (60%, 80%) in case we encountered cells with high thresholds. For most cells, 50 blocks of trials were presented, with each block consisting of 8 stimuli, one presentation of each of the 8 modulation depths in a randomized order. (For 2 of 249 recordings there were 100 blocks, and for 2 of 249 recordings there were only 30 blocks.)

At each nonzero modulation depth, we calculated the area under the receiver operating curve (ROC area; Green and Swets 1966), for both SC and VS_PP (defined in the next section), comparing responses to the modulated stimulus to responses to the unmodulated stimulus. The ROC area measures how well the neural response on a trial-by-trial basis can distinguish between two stimuli (in our case a noise at a particular modulation depth vs. the unmodulated noise) for a given metric (SC or VS_PP). An ROC area of 1 means that for every stimulus presentation (trial) the value of the metric was larger for the modulated than the unmodulated sound—simply by observing the value of the metric on a given trial, an ideal observer would predict with 100% accuracy whether a noise was modulated or unmodulated. An ROC area of 0.5 indicates that an ideal observer would perform at chance. ROC area is symmetrical around 0.5, such that an ROC area of 0 indicates that for every trial the value of the metric was smaller for the modulated than the unmodulated sound. This will also result in 100% accuracy in predicting which of the two sounds was presented, as long as the ideal observer associates small values of the metric with the modulated sound.

The ROC is a plot of hit probability (y-axis) against false alarm probability (x-axis), which we calculated at each of 100 equally spaced decision criteria ranging between the lowest and highest observed response values (SC or VS_PP) for the two stimuli being compared. At each criterion level the proportions of hits and false alarms were calculated from the neural responses, where hits are modulated-sound trials on which the metric gives a greater value than the criterion, and false alarms are unmodulated-sound trials on which the metric gives a greater value than the criterion. ROC area was calculated as the trapezoidal area (MATLAB: “trapz”) under this ROC. Mathematically, the ROC area is equivalent to the probability that a randomly selected trial from the modulated-sound distribution will have a response value (SC or VS_PP) larger than a randomly selected trial from the unmodulated-sound distribution. In Fig. 1 we plot the single-trial distributions of both SC and VS_PP values for unmodulated, 16% depth, and 100% depth stimuli from a single example recording (rasters for this recording are shown in Fig. 2A).

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Fig. 1.

Illustration of distributions underlying receiver operator (ROC) area calculations. Left: spike count. Bottom: histogram of spike count values for 50 trials of unmodulated noise for a sample recording. Middle: 16% modulation depth. Top: 100% modulation depth. Right: same as left, using phase-projected vector strength (VS_PP). Sample distributions are taken from cell V2475-2, which is fully illustrated in Fig. 2A.

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Fig. 2.

Example neurons. Left: raster plots show responses to modulation depths between 0% (unmodulated) and 100% (fully modulated). Right: ROC area plots (right) show the ROC area measured for each modulated vs. unmodulated comparison. Top: spike count. Bottom: VS_PP (see E for labels). Heavy dashed line is best sigmoid or Gaussian fit to the data, marked in black if the fit is significant at the 0.01 level and gray if not. Threshold (0.75, or 0.25 for declining functions) is indicated by horizontal dashed lines. Vertical dashed lines mark threshold level. Th, % modulation at threshold. Test frequencies: 60 Hz (A), 30 Hz (B), 10 Hz (C), 30 Hz (D), 15 Hz (E), and 120 Hz (F).

ROC area was plotted as a function of modulation depth for each unit. Neurometric depth sensitivity functions were created with an automated curve-fitting procedure on the obtained ROC area vs. depth functions. Because some cells were nonmonotonic to depth (i.e., showed higher firing rates or better phase locking for intermediate modulation depths than for fully modulated stimuli; cf. Middlebrooks 2008a) our curve-fitting procedure included both logistic (Eq. 2) and Gaussian (Eq. 3) functions:

Both curves used four free parameters determining the y-offset (a), the height (b), the x-center (μ), and the slope (s). Fitting was performed with MATLAB's “fmincon” function, constraining the slope (s) of the logistic between 2 and 20 and constraining the height (b) of the Gaussian to six times the difference between the ROC area at the lowest modulation depth and the ROC area at the most responsive modulation depth. These constraints were sufficient to eliminate the small number of spurious fits found by manual examination in the unconstrained case. To avoid overfitting with the Gaussian, we calculated an absolute ROC value (|ROC area − 0.5|) for each modulation depth. If the value at 100% depth was more than 7/8 of the cell's maximal value, only the logistic function was used in fitting. In the case where both Gaussian and logistic fits were attempted, the final fit was chosen on the basis of the highest correlation coefficient between the two curve fits and the data. Threshold was taken as the point where the fitted curve crossed an ROC area of 0.75 (or 0.25 for a declining function). In the case of a Gaussian curve, only the first threshold crossing was used. Using the relationship between the ROC area and the Mann-Whitney U (e.g., Hand and Till 2001; Hanley and McNeil 1982), we determined the P value corresponding to our ROC area threshold of 0.75 (0.25) where the null hypothesis is that the test statistic (SC or VS_PP) does not allow us to distinguish between modulated and unmodulated trials. Our threshold corresponds to P = 8.3 × 10⁻⁶ for cases where 50 trials were collected, and for the rare cases (n = 4) where either 30 or 100 trials were collected the threshold corresponds to P = 4.5 × 10⁻⁴ and P = 5.1 × 10⁻¹⁰, respectively.

Phase-projected vector strength.

VS_PP is a trial-by-trial vector strength measure that compares the mean phase angle on each trial with a global reference phase angle and penalizes single trials that are not in phase with the global response by multiplying the vector strength value by the cosine of the phase difference (Yin et al. 2011). It is calculated as follows:

where n is the number of spikes, θ_i is the phase of each spike in radians, and and _c are the trial-by-trial and global mean phase angles, respectively. For each trial, the resulting VS_PP value is always less than or equal to a non-phase-corrected vector strength value. Except for trials with low spike counts, VS and VS_PP are typically similar. We applied VS_PP specifically to reduce spuriously high VS values that can occur for trials with low spike counts, so that we could use all trials in neurometric analysis rather than discard them. Trials with no spikes were assigned a VS_PP of zero.

Increasing and decreasing spike count functions.

At a given modulation frequency, cells could either increase or decrease their responses with increasing modulation depth. To classify depth sensitivity curves as increasing (SC_inc) or decreasing (SC_dec), we took the average of the calculated ROC areas at each modulation depth. If this average was >0.5 the cell was classified as increasing and if it was <0.5 the cell was classified as decreasing, regardless of whether the cell ever reached the 0.75 (or 0.25) threshold.

Weighted mean of ROC area.

To determine the effect of testing away from the BMF we calculated the mean ROC area for VS_PP, SC_inc, and SC_dec as a function of distance (in octaves) from the cell's BMF. Because this octave space was not evenly sampled, the number of observations at each distance from the BMF was quite variable. Consequently, ROC area as a function of distance from the BMF was smoothed as follows:

where w_i is a Gaussian weighting function

and x_i is a discrete list of sampled points on the x-axis, x_j is the x-location of the point in question, n_i is the number of observations at each x value, and μ_i is the mean ROC area at each x value. We selected σ to be 1 octave. For this analysis, points at extreme distances from the BMF were considered to be outliers and not included if they were at least 1 octave away from the next-nearest data point. Three of 249 measurements were excluded from the VS_PP functions, 2 of 140 from the increasing SC functions, and 4 of 109 from the decreasing SC functions, and the points included in the plots reflect the extent of the nonoutlying points.

The general shape resulting from this weighted mean of ROC area analysis was Gaussian-like. Therefore, to assess whether these curves showed significant structure in the data (i.e., whether they were different from a flat function), we designed the following ad hoc Monte Carlo analysis. Each resulting curve was fit with a Gaussian function, and the height parameter of the curve fit was recorded. Then, for each of 10,000 repeats we randomly scrambled the relationship between the ROC area value and the distance from BMF value and repeated the calculation in Eq. 5, again fitting with a Gaussian and recording the height parameter of the curve fit. The resulting P value was taken as the probability that the curve fit of the scrambled data produced a larger height on the Gaussian curve fit than that produced by the actual data.

Trial pooling: within cells.

For some analyses, we pooled trials to simulate the convergence of multiple presynaptic cells with similar response properties onto a single postsynaptic cell (cf. Schneider and Woolley 2010). For within-cell pooling, trials from each cell's set of responses were pooled together under the premise that reduced noise in pooled trials might result in greater discriminability. Within-cell pooling results in an inherent trade-off between the number of trials available for ROC analysis and the amount of pooling—if the cell starts with 50 trials, pooling by twos will result in 25 trials, pooling by threes will result in 16 trials, and so on.

A pooled trial was defined as the union of the spike times recorded in two or more individual trials:

where P_x is a list of spike times in pooled trial x and T_i is a list of spike times in individual trial i. For within-cell pooling each individual trial was used exactly once without replacement. Trials were distributed in the temporal order of collection—conceptually equivalent to dealing out a deck of cards (individual trials) to each player (pooled trial) until the deck is out. The pooling number n_p then corresponds to the smallest number of individual trials dealt to any pooled trial—some pooled trials will have one extra individual trial if the number of pooled trials does not divide evenly into the number of individual trials collected. This arrangement ensures that the trials comprising each pooled trial are evenly spaced over the time of collection as nearly as possible. Pooled trials were then analyzed in the same manner as individual trials. This method is similar to that used in Middlebrooks (2008b), where four trials per data point were pooled in order to mitigate low spike count issues associated with vector strength analyses.

Depth sensitivity functions: population statistics.

Overall, 84 of our 249 depth sensitivity functions (34%) reached the ROC area threshold of 0.75 (or 0.25 for decreasing functions) for SC, while 128 (51%) reached threshold for VS_PP (0.75; only increasing functions were observed). It is important to note that each depth sensitivity function was tested with both SC and VS_PP measures—56 depth sensitivity functions (22%) reached threshold for both measures, while 93 (37%) did not reach threshold for either measure. A significantly higher proportion of depth sensitivity functions reached threshold for VS_PP than SC measures (P = 6.7 × 10⁻⁵, z-test for 2 independent proportions). Figure 3 shows the distribution of depth sensitivity functions reaching threshold, broken down by the modulation frequency tested. At most tested modulation frequencies, VS_PP resulted in a greater percentage of depth sensitivity functions reaching threshold than SC, although there is a notable exception at a test frequency of 120 Hz (VS_PP 0/7, compare SC at 4/7), which is expected because of the decreased ability of A1 neurons to strongly phase lock to higher modulation frequencies and the increased percentage of nonsynchronized responders at these frequencies (Lu et al. 2001; Yin et al. 2011). As an estimate of exclusively nonsynchronized responders, we found that 28 of our 84 depth sensitivity functions reached threshold for SC but not for VS_PP (11% of all functions).

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Fig. 3.

Population statistics. A: total number of cells reaching detection threshold (ROC area >0.75 or <0.25), broken down by amplitude-modulated (AM) test frequency, with spike count as a test measure. B: as in A, with VS_PP as the test measure. C: data in A and B presented in terms of % cells reaching threshold.

Many cells in A1 respond to increasing modulation depth by decreasing rather than increasing SC. Both increasing and decreasing rate functions for modulation depth encoding have been reported previously for AM tones in gerbil IC (Krishna and Semple 2000). We attempted to test each unit at the modulation frequency that best distinguished AM from unmodulated noise, regardless of whether that was due to an increase or a decrease in SC. We classified each of our depth sensitivity functions as increasing or decreasing on the basis of the mean ROC area across all modulation depths tested (depth sensitivity functions with a mean ROC area >0.5 were increasing and <0.5 decreasing). Of 140 increasing functions, 55 (39%) reached threshold (threshold ROC area = 0.75). Of 109 decreasing functions, 29 (27%) reached threshold (threshold ROC area = 0.25). The proportion of increasing depth sensitivity functions (55/140) reaching threshold significantly differs from the proportion of decreasing functions (29/109) reaching threshold (z-test for 2 independent proportions, P = 0.002), suggesting that increasing SC functions are more sensitive to modulation depth than decreasing SC functions.

However, increasing and decreasing depth sensitivity functions do not appear to differ in mean threshold (Fig. 4A). There was no significant difference between the mean thresholds of increasing and decreasing functions (increasing 65%, decreasing 62%, 2-sided t-test, P = 0.45, not significant). Accordingly, all analyses include both increasing and decreasing depth sensitivity functions except where noted.

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Fig. 4.

Threshold distributions. Gray dots indicate mean thresholds. Central horizontal lines of box-and-whisker plots indicate medians, upper and lower edges of boxes indicate quartiles, and whiskers indicate most extreme data points. The notch is a comparison interval; 2 medians are significantly different if the notches do not overlap; the notch may extend beyond the quartile. A: spike count thresholds of increasing (n = 55) and decreasing (n = 29) cells. B: spike count thresholds for all (n = 84) recordings, recordings made off the rate best modulation frequency (rBMF) (n = 63) and on the rBMF (n = 21). C: VS_PP thresholds for all (n = 128) recordings, recordings made off the temporal BMF (tBMF) (n = 36) and on the tBMF (n = 92).

The sensitivities of the A1 neurons that reached threshold for SC and VS_PP are compared in Fig. 4, B and C, left. If we look at all cells that reached threshold regardless of the modulation frequency of the test stimulus, we find a mean threshold of 64% depth for SC, while for vector strength the mean threshold is 52% (2-sample t-test, P = 7 × 10⁻⁴). These mean neurometric thresholds are relatively high compared with the macaque's psychophysical threshold of ~20–30% across the range of frequencies tested. If we look only at SC sensitivity for cells that do not reach threshold for VS_PP (an estimate of the exclusively nonsynchronized population), we find that the mean threshold is 70%, which is not statistically different from the mean SC threshold of 61% for the cells that also reached threshold for VS_PP (P = 0.06, t-test), suggesting that SC sensitivity is not strongly different in cells that synchronize and cells that do not.

Depth sensitivity functions: relation to modulation transfer functions.

It is reasonable to expect that neurons might be more sensitive to depth at the modulation frequency to which they respond most strongly. To examine this, we looked separately at cells that were recorded at their best modulation frequency (BMF), and off their BMF. A total of 46 recordings were made at the rate BMF (rBMF), and 203 recordings were made off the rBMF. On-rBMF recordings showed 21 cells (46%) reaching threshold, while off-rBMF recordings revealed 63 cells (31%) reaching threshold. These proportions are not significantly different (z-test for independent proportions, P = 0.06). The mean threshold measured off rBMF was 64%, while the mean on-rBMF threshold was 65% (Fig. 4B). These mean values are not significantly different (2-sample t-test, P = 0.78). Looking at temporal measures, 36/51 (71%) recordings made at the temporal BMF (tBMF) reached threshold while only 92/198 (46%) recordings off the tBMF did. These proportions are different (z-test, P = 0.002); therefore cells are more likely to reach threshold for depth encoding with VS_PP if they are tested at their tBMF. Despite the fact that significantly more cells reach threshold when measured at their tBMF, the mean thresholds in the on-tBMF and off-tBMF cases (Fig. 4C) do not significantly differ (54% off-tBMF vs. 45% on-tBMF, 2-sample t-test, P = 0.09).

Similar population thresholds for on- and off-BMF recordings initially suggest that cells have little advantage for encoding depth if the modulation frequency of the signal is at their preferred frequency. However, an alternate possibility is a recruitment effect—it might be that some cells that would not reach threshold if tested off BMF (so they would not contribute to the off-BMF threshold calculation) do reach threshold on BMF. Because depth encoding is weaker in these cells, their on-BMF thresholds would be relatively high and counteract on-BMF improvements in cells that also reach threshold off BMF. If this were the case, we would expect that a paired within-cell comparison would show better thresholds for on-BMF recordings. Figure 5 shows a scatterplot of thresholds taken from on-BMF recordings against thresholds taken off BMF in the same cells. There were 41 cells (Fig. 5A) that had depth thresholds measured both at the rBMF and at least one other frequency (47 total paired thresholds because some cells were recorded at more than one off-rBMF frequency) and 51 cells (Fig. 5B) that had depth thresholds measured both at the tBMF and another frequency (67 total paired functions). Points plotted outside and to the right of the box in Fig. 5 represent cells that reached threshold for the on-BMF test but not for the off-BMF test; points plotted outside and above the box represent cells that reached threshold for the off-BMF test but not for the on-BMF test (note that 20 recordings for rBMF and 14 recordings for tBMF did not reach threshold for either test and are plotted on top of each other at top right). If there were a trend toward lower thresholds at the BMF, it would result in points being clustered below the unity line. In the case of “recruited” cells that reach threshold at the BMF but not at another test frequency, the points would cluster to the right of the box. We performed a binomial sign test with the null hypothesis that the median change in threshold is not different from zero in the on-BMF and off-BMF cases. A binomial test (in contrast to, for instance, a paired t-test) allowed us to consider cells that only reached threshold in one of the two cases as having a threshold improvement (or decrement) without assigning an arbitrary threshold value to cells that did not reach threshold. For SC we were unable to reject the null hypothesis (P = 0.70), but for VS_PP cells showed a clear improvement (P = 8 × 10⁻⁴) when tested on rather than off BMF.

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Fig. 5.

Changes in threshold for recordings made at and away from BMF. For A and B, only cells that had recordings made both at the BMF and away from the BMF are included. A: spike count thresholds, 47 pairs. B: VS_PP thresholds, 67 pairs. Cells that lie above the horizontal line at 100% did not reach threshold when tested on BMF. Cells that lie to right of the vertical line at 100% did not reach threshold when tested off BMF. C: mean ROC area measured for the comparison of 40% modulation depth against the unmodulated case as a function of distance of test frequency from BMF in octaves. Lines are weighted running averages (see methods). Values for decreasing spike count curves (SC-Dec) have been reflected about ROC area = 0.5 to bring them into register with the other 2 measures. BMFs for SC-Dec correspond to the minimum of the modulation transfer functions. SC-Inc, increasing spike count curve.

Several factors could lead to a weak SC effect. One possibility is an unintentional sampling bias since rBMFs tend to be at higher modulation frequencies than tBMFs. A second possibility is that the available statistical power is reduced because we have few neurons that reach threshold for both rBMF and tBMF. To alleviate these problems and to look at the effect of testing on or off BMF across our entire data set, we recoded the test modulation frequency in terms of octave distance and direction from the BMF for all neurons and then calculated the mean ROC area for the 40% depth stimuli for each distance from the BMF, smoothing with a Gaussian-weighted average (see methods); 40% depth stimuli were chosen for this analysis because they were slightly below the population threshold and above typical behavioral thresholds and we were looking for effects that resulted in an improvement of threshold. The result is shown in Fig. 5C, with separate results for cells with increasing or decreasing SC functions. For decreasing SC functions only, the value plotted is 1 − ROC area in order to bring the values into register with the other functions and the BMF was considered to be the minimum, rather than the maximum, in the modulation function. Significance of the curve fits (i.e., whether there was a dependence on distance from BMF) was assessed by a Monte Carlo analysis (see methods). For VS_PP, mean ROC area at 40% depth shows a significant peak value near the BMF (P < 0.02) and rolls off on either side over a range of several octaves, suggesting that there is a decline in depth encoding as test frequencies move away from the BMF. However, the SC functions are quite flat compared with the VS_PP function (neither SC function significant; increasing: P > 0.2, decreasing: P > 0.05), suggesting that testing at or near the BMF should result in more threshold improvement relative to off-BMF tests for temporal measures than for SC measures, just as we find in Fig. 5, A and B. Together these results indicate that for VS_PP there is decreased neural sensitivity to modulation off BMF that declines as a function of distance. Little or no BMF relationship is found with SC.

Depth sensitivity functions: pooling/multiunit population data.

We have shown, using SC and VS_PP metrics, that mean individual single units are less sensitive to depth than the behaving animal. Since the animal has a large pool of neurons to draw on to detect modulation, we examined the effects of several types of pooling. In the first case we looked at the multiunit (MU) rather than the single-unit (SU) response. In a typical recording we found two to four isolatable cells. Although they did not always have identical properties, the columnar organization of A1 suggests that they should make similar connections, so the MU recording may effectively simulate one type of (postsynaptic) neural pooling. Alternatively, the auditory system may pool neurons that are more alike in response properties than those found in a MU cluster. To simulate this possibility, we took multiple spike trains produced on different trials by the same neuron and pooled them (“within-cell pooling,” see methods). Indiscriminate pooling (“across-cell pooling”) will be considered later.

Both MUs and within-cell pooling result in significant increases in the proportions of units reaching threshold relative to single units (SUs) for both rate (z-test: multiunit, P = 0.01; pooled, P = 0.002) and temporal (z-test: multiunit, P = 0.002; pooled, P = 0.005) measures. This can be seen by comparing Fig. 6 to Fig. 3. For VS_PP the increase in the number of units reaching threshold from SU to MU can be seen at every modulation frequency except 120 Hz.

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Fig. 6.

Population statistics (left and center) and thresholds (right) for 2 types of pooling. Details of population histograms are as in Fig. 3. Details of threshold distributions are as in Fig. 4. Top: spike count. Bottom: VS_PP. Multiunits are the combined activity of all isolated cells (usually 2 or 3) recorded simultaneously at a single site. Pooled trials represent the combined activity of 2 separate trials taken without replacement from the same cell (see methods).

A higher percentage of pooled-unit responses reached threshold for VS_PP than for SC, as was the case with SUs. The ROC area threshold (0.75) was reached for 42 of 85 MUs (49%) with SC, while 60 MUs (71%) reached threshold for VS_PP. The numbers of MUs reaching threshold for the two measures are significantly different (P = 0.005, z-test for two independent proportions). Figure 6, left, shows the distribution of MUs reaching threshold, as a function of modulation frequency. Aside from 120 Hz, where none of the three MUs reached threshold for VS_PP (and all 3 reached threshold for SC), temporal pooling resulted in a higher percentage of cells reaching threshold than rate pooling, a pattern that is similar to that seen for SUs. The same trend holds for cells with trials pooled in pairs (Fig. 6, center). Here, 118 of 249 pooled-trial responses (47%) reached threshold for SC and 159 (64%) reached significance for VS_PP. Again, temporal measures result in significantly higher numbers of pooled-trial responses reaching threshold than rate measures (P = 0.0002, z-test for 2 independent proportions), with similar advantages for all modulation frequencies except for 120 Hz.

The mean MU SC threshold was 61%, and the mean MU VS_PP threshold was 45% (Fig. 6, right). As for single units, the VS_PP mean is significantly lower than the SC mean (2-sample t-test, P = 0.002). For pooled trials, the mean SC threshold was 61% and the mean VS_PP threshold was 49%. Again, the means were significantly different (2-sample t-test, P = 4 × 10⁻⁴). However, none of the three SC thresholds (SU, MU, pooled) is significantly different from any other, nor are any of the three VS_PP thresholds different from each other.

Thus, although threshold is more commonly reached under these two pooling methods, this pooling does not result in a lower mean threshold. To determine whether pooling results in lower thresholds in individual cells, we created paired-threshold plots in Fig. 7. Figure 7, left, shows the MU threshold plotted against each SU threshold for simultaneously recorded units. Here gray dots indicate cells/MUs that reached threshold for both SU and MU analysis, and black dots indicate cells that only reached threshold in one of the two cases (cells to the right of the graph only reached threshold in the MU case; cells above the graph only reached threshold in the SU case). For both SC and VS_PP, the majority of points lie below the unity line (or on the right side of the graph), indicating that thresholds of individual SUs improve when pooled as MUs. This improvement is statistically significant for both rate and temporal measures (Wilcoxon signed-rank test: SC, P = 9.0 × 10⁻⁷; VS_PP, P = 3.5 × 10⁻¹²).

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Fig. 7.

Threshold scatterplots. Plot conventions are similar to Fig. 5, A and B. Left: multiunit threshold plotted against single-unit threshold. Right: 2-trial within-cell pooled threshold plotted against unpooled threshold. Top: spike count. Bottom: VS_PP. Gray dots, cases in which both measures reached threshold; black dots, cases in which only 1 or neither measure reached threshold.

Threshold improvement for individual cells is even more apparent for two-trial within-cell pooling (Fig. 7, right). Almost all points lie below the unity line for both SC and VS_PP, and the improvement compared with SUs was significant (Wilcoxon signed-rank test: SC, P = 1.3 × 10⁻¹³; VS_PP, P = 6.3 × 10⁻²⁰). These results demonstrate that pooling methods do improve thresholds relative to those of individual cells on a cell-by-cell basis despite the lack of an improvement in overall mean threshold across all cells. As seen in the comparison of on-BMF and off-BMF thresholds above, the lack of improvement in overall mean threshold is due to a recruitment effect: Many cells that did not reach threshold (and were therefore not included in mean threshold calculations) in the SU case did reach threshold and were included in the mean threshold calculations after pooling. The addition of these pools, which generally have high thresholds, counteracts the general improvement seen for individual cells, washing this improvement out of the overall average.

Depth sensitivity functions: limits of within-cell pooling.

Because pooling resulted in an improvement of threshold on a cell-by-cell basis, we decided to test the limits of our within-cell pooling method by increasing the number of pooled trials from the original 2 to a maximum of 25, and found improvement approaching behavioral thresholds with increased pooling. We found that as the number of trials pooled increased both the percentage of pooled responses reaching threshold and the mean threshold improved (Fig. 8A).

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Fig. 8.

Within-cell and across-cell pooling. A: within-cell pooling; pooling is done without replacement (see methods). All values are averaged across all tested modulation frequencies. Left: % of cells reaching threshold as a function of number of trials pooled. Unconnected symbols are unpooled values. Connected symbols are pooled values; between 2 and 25 trials pooled. Right: mean threshold as a function of number of trials pooled. For the mean threshold plot, only cells that reach threshold in the unpooled case (unconnected symbols) are used to calculate subsequent thresholds to avoid recruitment effects (see text). B: across-cell pooling; pooling is done with replacement (see methods). All values are averaged across all modulation frequencies used in the analysis (10–60 Hz). For VS_PP and spike count, pooling was done from a pool of all cells. For SC-Inc and SC-Dec, spike count pooling was done from a pool of only cells with increasing (decreasing) spike count functions. Left: % of pools reaching threshold as a function of number of cells pooled. Right: mean threshold as a function of number of cells pooled.

Trials were pooled without replacement, so as the number of trials pooled increased the number of “trials” available for ROC area calculations decreased. Data showing the number of pooled responses reaching threshold as a function of the number of trials pooled together are plotted in Fig. 8A, left. The unconnected square (SC) and circle (VS_PP) represent the number of responses reaching threshold without pooling. Both SC and VS_PP measures show drastic increases in the number of cells reaching threshold as the number of trials pooled increases to about six to eight and a plateau thereafter (although SC peaks at 12 trials pooled). The decrease after 15 trials pooled for SC probably results from the reduction of “trials” available to carry out the ROC area calculations (3 “trials” when pooling 16 trials together and only 2 when pooling 25 together). Although only 34% (SC) and 51% (VS_PP) of our cells reach threshold in the unpooled conditions, on average between 65% and 75% of cells reach threshold with a reasonable amount of within-cell pooling (equivalent to 6–12 cells with similar responses).

In addition to investigating the effect of within-cell pooling on the number of pooled responses reaching threshold, we also asked what effect such pooling had on mean thresholds. These data are shown in Fig. 8A, right. We limited the mean threshold analysis to only those cells that reached threshold without pooling (SC: n = 84, VS_PP: n = 128) to avoid recruitment effects that would obscure improvement in the mean threshold (cf. Fig. 6 and Fig. 7). Although we found that the majority of recruitment of previously nonsensitive cells occurs within the first 5–10 trials pooled (Fig. 8A, left), individual cells continue to see threshold improvements across the range of pooled trials tested—suggesting that recruitment is not limited by the pooling process but by exhausting cells whose sensitivity will improve with pooling from our sample. It is notable that as pooling increases VS_PP thresholds remain better than SC thresholds throughout. The mean VS_PP threshold drops below 20% for a pooling of 25 trials, demonstrating that modest pooling of responsive cells (i.e., those reaching threshold without pooling) results in neuronal thresholds roughly equivalent to psychophysical thresholds in the behaving macaque (~20–30% modulation depth across a relatively wide bandwidth; O'Connor et al. 2011). Thus pooling may provide an effective strategy of increasing behavioral sensitivity for measures based on either rate or temporal (so long as phase relationships are consistent) aspects of the neural response in A1.

Depth sensitivity functions: across-cell pooling.

While within-cell pooling assumes that neurons with similar response properties converge to segregated postsynaptic neurons, it is possible that this assumption is not justified and the code for modulation might rest with a less selective model. Under the assumption that convergence mechanisms might not be highly specific with respect to response properties, we developed an across-cell pooling technique that modeled the indiscriminate convergence of between 2 and 50 cells as described in methods. Cells tested at modulation frequencies between 10 and 60 Hz (5 and 120 Hz were excluded because of low numbers of tested cells) were grouped by the modulation frequency of the test and were randomly sampled for each pool size for 1,000 iterations. Mean threshold and percentage of pools reaching threshold were calculated for each set of 1,000 iterations and then averaged across tested modulation frequencies (weighted by the number of cells at each modulation frequency) to arrive at a single summary value for each pool size. An important property of this model is that responses from different neurons are pooled (i.e., the union of all spike times is collected into a single pooled spike train) before the pooled spike train undergoes neurometric analysis (as opposed to taking a measurement from each cell and then pooling the analyzed metric).

For across-cell pooling, as the number of cells pooled increased almost all pools reached threshold (Fig. 8B, left). Despite the fact that the population of cells tested were not likely to have identical or even similar temporal characteristics (particularly in the phase of response), we found that the percentage of pools reaching threshold for VS_PP exceeded 90% for pools of 20 cells or larger. Slightly higher percentages of pools reached threshold for SC, as long as cells with increasing functions and cells with decreasing functions were kept segregated. If increasing and decreasing cells were not kept segregated, the percentage of pools reaching threshold for SC measures dropped drastically to between 50% and 60% for pool sizes of 20 and above (this means it might be necessary to have separate postsynaptic targets for neurons with increasing and decreasing functions).

Across-cell pooling improved VS_PP-based thresholds, but not as much as within-cell pooling, while across-cell pooling improved SC thresholds (for increasing-function cells) more than within-cell pooling (Fig. 8B, right). When fewer than 20 cells are pooled, VS_PP provides more sensitivity than spike count, but temporal pooling thresholds appear to asymptote at ~30%. Rate measures, particularly those from segregated increasing-function cells, continue to improve their sensitivity up to 50 cells per pool. The mean SC threshold for nonsegregated cells starts high with low numbers of pooled cells but approaches the sensitivity of VS_PP by 50 pooled cells.

Pooling of auditory responses.

In our method of across-cell pooling, the actual spike trains from individual trials were pooled together in a literal fashion. If one imagines that all of the pooled cells were to synapse (with identical weights) onto the same dendritic compartment of a target neuron, the pooled spike train that we create would represent the input to that dendritic compartment. From this point of view, our pooling method is an empirical first-pass attempt to estimate the sensitivity that could be obtained in a downstream neuron by convergent inputs. While this approach has rarely been taken, in a study of the encoding of marmoset twitter calls in ferret A1, Walker et al. (2008) used an aggregate signal similar to our across-cell pooling. Although the natural twitter calls do not have precise regular periodic AM, the envelope modulates roughly periodically at ~7–9 Hz (Wang et al. 1995; Wang and Kadia 2001). Walker et al. investigated how well the neurons could distinguish natural twitters from manipulated twitters and found that at the single-unit level temporal coding more accurately reflected human psychophysical thresholds than rate coding. They used a principal component coding algorithm that did not necessarily measure phase locking in the same way that vector strength does, but the examples and the aggregate activity shown suggest that phase locking contributes to the temporal coding. As was the case with our study when the across-cell pooling method was used, temporal-based neurometric thresholds improved (rate codes were not tested with this kind of pooling).

The mechanics of our pooling method differ somewhat from most previous pooling studies, which have tended to use simulated responses based on draws from an approximation of the observed distribution rather than pooling actual responses (e.g., Rosen et al. 2010; Shadlen et al. 1996; but see Schneider and Woolley 2010). For SC methods (e.g., Shadlen et al. 1996) there should be little difference between using simulated responses and actual responses, but our method allowed us to perform temporal analysis on pooled raw data. In comparison, Rosen et al. (2010), who investigated AM tone responses in gerbils, obtained their temporal neurometric measures on mean vector strength values drawn from an estimated distribution. This difference is crucial because when precalculated vector strength values are pooled any phase difference between pooled responses has already been removed. In our pooling method, spike phase information is preserved up to the point of the final phase-locking calculation, just as it would be in an in vivo pooling, and differences in response phase will be reflected in the resulting VS_PP values. We find that for the across-cell condition pooling resulted in improvement of thresholds—even at 60 Hz—a result we would not expect to see were the phase relationships between different cells random. At the same time, we note that our pooled VS_PP thresholds are not as sensitive in the across-cell case as the within-cell case, which suggests that inconsistent phase relationships do contribute to the difference. Still, even indiscriminate pooling results in a notable increase in temporal information, which is consistent with the finding that phase-locked AM responses in A1 generally synchronize to the same phase in the stimulus cycle (Bendor and Wang 2008; Yin et al. 2011).

One important issue to keep in mind when pooling is the possible response correlations between neurons. In the case of SC, across-cell correlation has been noted to reduce the effectiveness of pooling (Alves-Pinto et al. 2010; Zohary et al. 1994; for a counterexample, see Romo et al. 2003). Since our pooling methods (except in the case of the multiunit considered as a pool) combine trials that were collected at different times, any correlation in SC that might have been present in an in vivo pooling operation will be missing in our simulation, and our SC threshold estimates may be too low (i.e., too sensitive) to the extent that we have failed to capture existing correlations. However, interinput correlation only reduces the effectiveness of pooling when the inputs have identical tuning curves (Abbott and Dayan 1999), a condition that is only true for our within-cell pooling method—our across-cell method encompasses cells with diverse tuning properties and should be relatively robust to SC correlations.

For measures of phase locking, such as VS_PP, the influence of across-cell correlation is more complicated. Whereas for SC it is clear that high intercell correlation decreases the benefits of pooling, for temporal-based measures it is not clear that higher intercell correlation will always result in decreasing the benefits of pooling, nor is it clear that lower intercell correlation will benefit pooling (Elhilali et al. 2009; Walker et al. 2008). The ideal method to determine the effect of potential correlation issues for pooling with temporal-based measures would be multiunit array recording (e.g., Bizley et al. 2010 in ferrets). Unfortunately, such arrays have yet to be employed successfully in the macaque model, and direct correlation analysis has to date proven elusive.

Implications of pooling for rate and temporal coding.

The results of this study have interesting implications for the usefulness of rate versus temporal coding of temporal envelope modulation. Previous studies with AM (e.g., Malone et al. 2010; Nelson and Carney 2007) and communication vocalizations with strong AM (e.g., Huetz et al. 2009; Schnupp et al. 2006; Walker et al. 2008) indicate that temporal codes (including VS) represent the sounds better than rate codes. This could mean that temporal codes are preferentially used to encode AM, but our results suggest that a preference for rate or temporal codes should depend on how the brain integrates information across the neuronal population. For example, if the brain were able to select only the most sensitive A1 units, a rate code would fairly accurately predict performance but a phase locking code would not, suggesting performance better than that obtained psychophysically. The inability to perform better than the best phase-locking units from A1 suggests that the brain is unable to simply isolate those units.

An alternate population decoding strategy is neuronal pooling. Indiscriminate pooling (pooling without respect to coding efficiency, best modulation frequency, etc.) would be the easiest form of pooling to implement. For truly indiscriminate pooling, phase locking measures outperform rate-based measures through pools of at least 50 neurons. However, our pooling simulations find that thresholds based on SC are lower and more accurately reflect behavior across modulation frequency than thresholds based on phase locking measures—but require one restriction, that only cells with increasing depth sensitivity functions are included in the pool, because at the pooling level excited and suppressed responses counteract (e.g., Oshurkova et al. 2008). This sort of segregation seems to represent an intermediate level of specificity in convergence and might be biologically plausible in several ways. Phase locking strength can vary with laminar depth (Wallace et al. 2011), so it is not unlikely that other AM response properties might as well. Because we do not have information about the laminar location of our recorded neurons it is possible that cells with increasing and decreasing depth functions differ in their laminar location and have different synaptic targets. Hebbian mechanisms or other physiological factors (decreasing cells could be different cell types, for instance, local interneurons) could also allow these two classes of cells to be segregated onto separate targets.

For within-cell pooling, which is analogous to pooling from neurons with identical properties, phase locking measures appear to give even lower detection thresholds, which would allow smaller pool sizes to account for behavior. This result is consistent with the observation that phase locking is better for multiunits than single units (Oshurkova et al. 2008), and if such a strategy were to be implemented the selection of cells with similar properties might well be based on columnar tuning. It would be interesting to see whether pooling can improve detection thresholds found with other temporally based codes, such as those based on overall spike timing (Furukawa and Middlebrooks 2002; Kajikawa et al. 2008; Malone et al. 2007; Wang et al. 2007) and interspike interval (ISI) distributions (Imaizumi et al. 2010), and to what extent pooling results using alternate codes depend on whether neurons with similar properties are sampled.

Comparison to previous studies.

The neurometric thresholds that we found for single cells were somewhat higher (less sensitive) than those found in previous studies of AM depth sensitivity in midbrain and in cortex, where AM tones rather than AM noise were used. AM noise has often been used in psychophysical studies because spectral and temporal cues in AM tones can be confounded, a problem that is alleviated with a noise carrier. Therefore, in addition to determining modulation encoding for tone carriers, it is important to determine the relationship of neuronal to behavioral sensitivity for noise carriers.

In the IC, Nelson and Carney (2007) recorded from awake rabbits while presenting 2-s AM tones (further broken down into nine 500-ms segments, discarding onset) with the carrier at the cell's best frequency. When using a signal detection method as a neurometric, they found temporal-based thresholds to be lower than rate-based thresholds—the median of single-cell rate thresholds was ~30% modulation depth, and synchrony thresholds were somewhat lower with a median below ~20% modulation depth. Krishna and Semple (2000) recorded from IC of anesthetized gerbils; they also report that the median cutoff depth for significant phase locking (using the Rayleigh statistic) is ~20% and do not report rate-based thresholds. The better ability to phase lock to lower depths compared with our findings (particularly for temporal measures) is likely to be a consequence of recording in IC, where phase locking is stronger than in cortex, but may reflect species differences in AM sensitivity as well.

In auditory cortex AM sensitivity has been found to be worse. Eggermont (1994) found that most units did not phase lock to 25% or lower modulation depth with AM noise. This is consistent with our results. Malone et al. (2010) recorded from A1, R, and three belt areas in macaque while presenting very long (10 s) AM tones and focusing on lower modulation frequencies, making only 11 recordings with modulation frequencies above 20 Hz. For spike count they found that 41% of their cells could detect modulation (using a comparison to the unmodulated tone) with a median threshold of 50% modulation depth. While these spike count data were similar to ours, for VS they found a remarkably high percentage of single cells that were capable of detecting modulation (99%, compared with 51% for the data we report here) with a median threshold of 20% modulation depth. Several factors may contribute to the difference between their result and ours. Their long stimulus duration provides 25 times more cycles per stimulus than ours (10 s compared with 400 ms in this study) and so approximates our within-cell pooling condition. When we pooled 25 trials within cell, our mean VS_PP thresholds were also ~20% modulation depth and we found ~75% of pools to reach threshold. In addition, their VS detection thresholds were based on a significant Rayleigh statistic, testing for nonuniformity in the cycle histogram instead of a comparison to an unmodulated tone, and did not exclude onset responses. We have previously shown that the Rayleigh statistic is prone to producing false alarms at low modulation frequencies (Yin et al. 2011). The presence of onset responses may also result in spurious nonuniformity in the cycle histogram, particularly for low modulation frequencies (long cycles) and cells with low sustained responses.

Electrical stimulation studies using cochlear implants in guinea pigs (Middlebrooks 2008b) find more sensitive rate and temporal multiunit responses than our study (the median for both is slightly greater than 10%). This cannot be accounted for by their using essentially four-unit within-cell pooling (see Fig. 8A). One possibility is that electrical stimulation is more effective at entraining neural responses. The electrical stimulus consists of pulse trains with a constant interpulse interval, and the amplitude of each pulse is adjusted to create the AM envelope. Because A1 neurons are most sensitive for carriers with lower interpulse intervals (Middlebrooks 2008b), the pulsing carrier might aid in phase locking to the envelope. Middlebrooks (2008a) also shows that many units have nonmonotonic depth functions and there is a tendency to see more nonmonotonic functions at higher modulation frequencies, while we find no strongly nonmonotonic depth functions for VS_PP and only 3 of 55 for increasing SC, consistent with Liang et al. (2002). This suggests that the high proportion of nonmonotonic depth functions found by Middlebrooks may be a factor of electrical stimulation and that nonmonotonic depth functions for AM are not common for acoustical stimulation.