Fusion

Fig 3A–3D plot against delay the proportion of trials in which subjects reported an additional image in Experiments 1–4. Since fusion is the tendency to report a single auditory image, a low proportion along the ordinate suggests strong fusion; a high proportion suggests weak fusion. Ten subjects participated in each experiment, and each data point represents the average across the ten subjects (vertical lines show ± 1 standard deviation [s.d.]).

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Fig 3. Contributions to fusion from the alone and superposed segments.

(A-D) Markers indicate the proportions of trials in which subjects indicated a second auditory image at a given delay (abscissa) when the noise-pairs were 200 ms (A), 30 ms (B), 10 ms (C), or 30 ms with synchronized onsets and offsets (D). Error bars indicate variation across subjects (±1 s.d.). Solid lines and filled circles show the results for the identical noise-pairs. Dashed lines and open circles show the results for the independent noise-pairs. (E-H) Markers indicate how the proportions differed for the independent noise-pairs relative to when there was no delay and thus the contributions to fusion from the alone segments. (I-L) Markers indicate how the proportions differed for the independent and identical noise-pairs and thus the contributions to fusion from the superposed segment.

https://doi.org/10.1371/journal.pone.0137900.g003

Results obtained with the independent noises (open circles) are first considered, from which contributions to fusion from the lead- and lag-alone segments are inferred. Contributions to fusion from the superposed segments are then inferred by comparing results for the independent (open circles) and identical (filled circles) noises.

Localization dominance

Fig 4A–4D and 4E–4H plot, respectively, the average locations indicated (μ) on the upper and lower arcs against delay. Each data point represents the average of the 10 subjects in each experiment (vertical lines show ±1 s.d.). When fusion is strong, an average near 0° indicates the presence a single mode near 0° (e.g., filled circles in Fig 2A, 2C and 2M). When two sources are reported, an average near 0° indicates the presence of two equal-sized modes straddling 0° (e.g., open circles in Fig 2A–2D and 2M). Positive μ values therefore indicate a higher incidence of responses on the leading side of an arc (e.g., circles in Fig 2I and 2K). Negative values indicate a higher incidence of responses on the lagging side (e.g., circles in Fig 2J and 2L). Angles corresponding to the distribution peaks (modes; Fig 2) are not reported because they lack precision when the auditory images were poorly circumscribed.

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Fig 4. Contributions to localization dominance from the alone and superposed segments.

Markers indicate the mean angles indicated by subjects on the upper (A-D) and lower (E-H) arcs at a given delay (abscissa) when the noise-pairs were 200 ms (A and E), 30 ms (B and F), 10 ms (C and G), or 30 ms with synchronized onsets and offsets (D and H). Error bars indicate variation across subjects (±1 s.d.). Solid lines and filled circles show the results for the identical noise-pairs. Dashed lines and open circles show the results for the independent noise-pairs. (I-L) Markers indicate the sums of the angles reported on the upper and lower arcs and thus the contributions to localization dominance from the alone segments. (M-P) Markers indicate the differences in the angles reported for the independent and identical noise-pairs on the upper (filled squares) and lower (open squares) arcs and thus the contributions to localization dominance from the superposed segment.

https://doi.org/10.1371/journal.pone.0137900.g004

Results obtained with the independent noises (open circles) are first considered, from which contributions to localization dominance from the lead- and lag-alone segments are inferred. Contributions to localization dominance from the superposed segments are then inferred by comparing results for the independent (open circles) and identical (filled circles) noises.

Contributions to localization dominance from the alone segments.

Contributions to localization dominance from the lead- and lag-alone segments may be inferred from the psychometric functions obtained with independent noises (open circles in Fig 4), since a superposed segment consisting of independent sounds is devoid of cues that could bias spatial perception toward either source. As shown below, evidence of localization dominance was therefore absent in Experiment 4 when the alone segments were excised from the independent noises.

In Experiment 1, where the independent noises were 200 ms (open circles, Fig 4A), the average locations of the reports on the upper and lower arcs change in opposite directions with delay, with the dominant image being localized increasingly closer to the lead, (positively) and the additional image closer to the lag (negatively). Thus, as the alone segments lengthen, subjects indicate the two images at increasingly eccentric average loci. Importantly, the average location indicated for nonzero delays on the upper arc (μ_upper; Fig 5A) is, on average, 4.1° farther from the midline (0°) than the average location indicated on the lower arc (μ_lower; Fig 4E). In other words, both the dominant and additional images are biased toward the leading source. This is shown in Fig 4I, which plots against delay the sums of μ_upper and μ_lower, thus: μ_{upper arc} + μ_{lower arc} = 4.1° (± 1.8°). As shown, the points are slightly shifted toward the lead (upward), suggesting weak localization dominance due to the presence of the alone segments.

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Fig 5. Demonstration of how ITD may be shifted toward the leading source (positively).

(A) Amplitude envelopes of identical, 500 Hz, signals in the left and right ears when the lead-lag delay was 2.5 ms. (B) Measurements of ILD at the times when the signals were sampled, colored according to weights that were attributed to them by the envelopes of the left and right signals. (C) Distributions of weighted (solid line) and non-weighted (dashed line) ILDs measured when the leading and lagging noises pairs were superposed for 500 ms. (D) Measurements of ITD at the times when the signals were sampled, colored as in (B). (E) Distributions of weighted (solid line) and non-weighted (dashed line) ITDs measured when the leading and lagging noises pairs were superposed for 500 ms.

https://doi.org/10.1371/journal.pone.0137900.g005

The alone segments contributed considerably more to localization dominance when the (independent) noises were shortened to 30 or 10 ms in Experiments 2 and 3. Comparing the open circles in Fig 4B and 4F, it is clear that the average loci indicated on the upper arc at nonzero delays are, on average, farther from the midline than observed with the 200 ms noises, whereas those for the lower arc are closer to the midline. As a result, the dominant and additional images are shifted more toward the lead than was observed with the 200 ms noises. The sums of the distances from the midline, μ_upper + μ_lower, are plotted against delay in Fig 4J and 4K. Thus, the contributions of the alone segments are greater when the noises are shorter. As expected, there were no biases toward either source when the alone segments were excised in Experiment 4 (Fig 4L).

Envelopes and spatial cue distributions

Studies of the barn owl have suggested that low-frequency (40–150 Hz) envelopes imposed on lead-lag stimuli, lacking lead or lag-alone segments, can invoke neural responses and behaviors consistent with localization dominance [27, 36]. A key aspect of these findings was that spatially selective neurons were found to respond more frequently when the experimenter-imposed envelopes of the leading and lagging stimuli (Ē) were both rising (dĒ/dt > 0) than when they were declining (dĒ/dt < 0). Similar findings have since been observed for binaural beat stimuli [37–39].

Here, a similar model is proposed for noises bursts that contain only intrinsic amplitude modulations and no experimenter-imposed modulations. Even for such noises, the outputs of low-frequency (<1.5 kHz) cochlear filters are slowly (< ~150 Hz) and deeply amplitude-modulated. Shown in Fig 5A, for instance, are the envelopes of representative broadband noises passed through filters simulating the left (l) and right (r) cochlea at 500 Hz [40]. The envelopes of the signals, obtained using the Hilbert transformation, differ in level and shape because a 2.5 ms delay was introduced between the leading and lagging stimuli that were spatialized with ITDs of 250 and -250 μs, respectively, but were otherwise identical. As noted by Dizon and Colburn [26], this stimulus configuration, at 500 Hz, is interesting because significant ILDs are produced over time (Fig 5B) that favor the lagging source’s location on average. This can be seen in Fig 5C, which shows the distribution of ILDs accumulated over the stimulus’ duration (mean ILD ≈ -6.5 dB). Yet subjects in the study of Dizon and Colburn [26] did not always lateralize an intracranial image on the side of the lagging stimulus. Instead, they frequently lateralized the image on the opposite side, as if another cue were biasing their perception toward the leading source. Similar biases were shown for other frequency bands (< ~2 kHz) and other delays within the precedence range, regardless of ILD. Thus, modulations intrinsic to cochlear filters may be responsible for some aspects of the precedence effect when the bandwidths of superposed stimuli are wide enough.

An analysis of ongoing ITDs (Fig 5D) and ILDs (Fig 5B), obtained from Hilbert transformed signals in the ears, provides a possible explanation if spatially selective neurons in humans, as in the owl, are presumed to discharge when the envelopes of the signals in the ears are rising. ITDs were obtained by subtracting the phases of the signals after dividing them by frequency. ILDs were obtained by subtracting the envelopes of the transformed signals. When the envelopes of the signals are rising (upward arrows in Fig 5A), representative ITDs in Fig 5D are mostly positive (toward the leading source) and neurons responsive to these ITDs may be more likely to fire. In contrast, when the envelopes are declining (downward arrows in Fig 5A), the representative ITDs are mostly negative (toward the lagging source), and neurons responsive to these ITDs may be less likely to fire.

How ITDs are distributed over time and possibly ‘weighted’ by amplitude modulations further demonstrates why ITD may favor the leading source under precedence conditions. The dashed line in Fig 5E shows the average distribution of ITD over a time period of 15 seconds (30 different stimuli that were each 500 ms). As described by Dizon and Colburn [26], the distribution’s central tendency is close to zero. However, when weights representing envelope derivatives are applied (see below), as shown by the solid line, the distribution’s central tendency and skew shifts toward the leading source (positively). In contrast to ITD, distributions of ILD remain roughly the same whether they are weighted or not (overlapping solid and dashed lines in Fig 5C). Note that ILD would be biased in a way resembling ITD if the stimuli were spatialized using ILD instead of ITD, and that distributions for both cues may often be biased if the stimuli were spatialized using both cues (not shown).

To obtain envelope weights across sampled points (44.1 kHz), the leading and lagging signals were first summed individually for the left (l) and right (r) ears (earphones). The resulting signals were then passed through a bank of 64 gammatone filters with center frequencies (f) between 0.2 and 1.5 kHz [40]. Envelopes (E) for each ear and frequency band were next obtained and derivatives (dE/dt) were measured across adjacent points of the sampled envelopes. Weights (Wt), which are presumed to represent the probability of a neuron firing (see below), were then calculated as: (1) where μ was the average derivative observed while the stimuli were superposed within a given frequency band (horizontal dashed line in Fig 6A; usually very close to zero) and σ was the standard deviation (shaded region in Fig 6A). Weights calculated for the 500 Hz band, in the left ear, are shown for a continuous range of dE/dt values by the sigmoidal function in Fig 6C (solid line). Analogous weights were calculated for the right ear using the same equation and values of μ and σ that were specific to this ear’s envelope (not shown). Weights derived for the left and right ears were then multiplied and are shown by hot colors in Figs 5B, 5D and 6B. For comparison, the dashed line in Fig 6C shows the analogous sigmoidal function (Fig 3B in [36]), derived for the barn owl from the responses of space-map neurons to the derivatives of deeply amplitude-modulated noise bursts [27]. To our knowledge, this implementation is novel in that no experimenter-imposed modulations are necessary and no modulations were imposed on the stimuli.

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Fig 6. Demonstration of how envelope weights were obtained from band-filtered noise-pairs.

(A) Envelope derivatives measured from the envelope of the signal in the left ear. A horizontal dashed line shows the mean derivative (μ) whereas gray shading shows ±1 standard deviation (±σ). (B) Envelope of the signal in the left ear from which the derivatives were measured. Shading shows the weights that were attributed to the signal using Eq 1. (C) Weights calculated using Eq 1 (solid line) for a continuous range of dE/dt values (abscissa). The dashed line shows normalized weights derived for the barn owl from the responses of space-map neurons to the derivatives of deeply amplitude-modulated noise bursts. (D) Spectra obtained from the envelopes of band-filtered noises (0.5–8 kHz; solid lines). A single dashed line, resembling the 500 Hz band, shows the average spectrum of the low-pass envelopes that were examined by Nelson and Takahashi [27, 36]. Note that weights used in the owl study (dashed line in C) are similar to those at 500 Hz (solid line in C) because their envelope spectra (D) are similar.

https://doi.org/10.1371/journal.pone.0137900.g006

Selectivity for the rising edges of envelopes may be limited by several factors, including a decline in the responses of inferior colliculus (IC) neurons to high-frequency amplitude modulations. The envelope of a noise filtered at 2 kHz, for instance, can contain significant energy above ~200 Hz (Fig 6D) but these components may be removed by a typical ITD modulation transfer function [41], causing the envelope to effectively flatten. The envelopes of identical lead-lag stimuli are also expected to become effectively dissimilar (independent) when they occur with a period that is shorter than the stimulus’ lead-lag delay [27, 36]. Significant envelope energy above 100 Hz (Fig 6D), for instance, may cause dissimilar envelope features to overlap in time, instead of similar features, when the lead-lag delay is longer than ~10 ms (~100⁻¹ Hz). In terms of the stimuli after they sum in the ears (Fig 5A), this means that the envelopes of filtered signals may be declining (downward arrows in Fig 5A) as often as they are rising (upward arrows in Fig 5A) when spatial cues favor one source or the other (e.g., Fig 5D). Consistent with this interpretation, precedence phenomena weaken at short delays when high frequency noise-bands are tested [26].

Frequency-specific distributions of ITDs, attributed to the superposed and alone segments of noise-pairs as described in S1 Fig, are shown in Fig 7 with (A-F) and without envelope weights (G-L). Gray-scale bars show the frequency of occurrence of ITD values in a given frequency band, averaged over 100 different noises pairs.

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Fig 7. Frequency-specific distributions of ITD when the lead-lag delay was 4 ms.

(A-F) Distributions of envelope-weighted ITDs attributed to the superposed and alone segments when identical noise-pairs were 200 or 10 ms (A-B and C-D, respectively) or when the noise-pairs were 30 ms and independent (E-F). Each distribution shows the average of 100 different noise-pairs and reflects cues observed only during the superposed segment (A, C, and E) or only during the alone segments (B, D, and F). Note that the same gray-scale is applied to each stimulus and pair of plots. (G-L) Distributions of unweighted ITDs generated by the same noise-pairs and segments as in (A-F) (see labels above and to the left of the plots).

https://doi.org/10.1371/journal.pone.0137900.g007

Comparing plots on the left (A-F) and right-hand (G-L) sides of Fig 7 shows the effects of envelope weighting when the lead-lag delay is 4 ms. As in Fig 5E, ITDs arising during the superposed segments of the identical noise-pairs shift and skew toward the leading source (positively; e.g., compare Fig 7A and 7G). This effect is absent, however, when independent noise pairs are considered (e.g., compare Fig 7E and 7K).

During the alone segments, envelope weights similarly emphasize the ITDs of the leading source over those of the lagging source. This is most visible for the shorter noise-pairs. Thus, distributions of ITDs attributed to the lead-alone segment (distribution peaks ≈ 0.25 ms) in Fig 7D are higher (darker) than those that oscillate with frequency and are attributed to the lag-alone segment (distribution peaks < 0 ms). Without weights, the distributions have similar heights, as one can see in Fig 7J. Under our model, this difference occurs because amplitude always rises at the onset of the lead-alone segment but generally declines during the lag-alone segment (S1 Fig). This effect also occurs for independent noise-pairs (e.g., compare Fig 7F and 7L).

Importantly, cues accumulated during the alone segments are unrelated to a given noise pair’s length and are determined instead by the stimulus’ lead-lag delay. Thus, differences in cues observed as the noise-pairs are lengthened, as well as differences in perception (Figs 2–4), may not be attributable to the alone segments, but instead, to differences in cues accumulated across the superposed segment.

Under our model, weights were accumulated over the entire lengths of the various stimulus segments because stimulus duration was found to have a considerable influence on subject’s responses (e.g., Fig 3). Our application of envelope weights (Figs 5 and 6) also means that cues were effectively integrated over much shorter, frequency-specific, time-scales (see Discussion section: Effects of time, envelopes, and spatial cue distributions).

Edge hypothesis.

When there is no delay between a pair of identical noises, the stimuli are diotic and spatial cues equal the averages of the left and right sources. At delays less than approximately 1-ms, subjects experience summing localization. As the delay increases, distributions of envelope-weighted ITDs during the superposed segment not only shift and skew toward the leading source (positively), but also broaden in a delay-dependent manner. These differences are shown in Fig 8A for the superposed segments of 200-ms noise-pairs lacking alone segments. To simplify presentation, frequency-specific distributions, as shown for a 4-ms delay in Fig 7A, 7C and 7E, were averaged across frequency and 100 different noise-pairs.

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Fig 8. Distributions of ITD averaged across frequency and the superposed segments of noise-pairs.

(A) Distributions of ITD, averaged across frequency (0.5–1.5 kHz) and across the superposed segments 100 different, 200-ms, noise-pairs, when the lead-lag delay was 0, 0.5, 1, 4, or 16 ms. Because each distribution is scaled to a maximum of 1.0, open circles indicate the ITDs that occurred at the 50% point along the edges of the distributions. (B) Distributions of ITD as described in (A) but for independent noise-pairs.

https://doi.org/10.1371/journal.pone.0137900.g008

The distributions broaden with delay (Fig 8A) and each stimulus generates only a single mode. Why then do subjects often report one image (Fig 3A) when identical noise-pairs are tested at short delays (blue, green, and orange lines in Fig 8A) but two images (Fig 3A) at long delays (red line in Fig 8A) or when independent noise-pairs are tested (Fig 8B)? One possibility is that the subjects’ reports may reflect the neural representations of the edges of the distributions, not their central tendencies. Specifically, ITDs corresponding to the edges of the distributions, defined operationally as the 50% point on the ordinate of (Fig 8), for instance, are identical when there is no delay (dark vertical line at 0-ms), a condition that could lead to summing localization were subjects reporting ITDs corresponding to the distribution’s edges. This would also be the case when there is but a single source. The edges (open circles in Fig 8A) then diverge as the delay increases until points corresponding to 50% are separated by ~0.4 ms of ITD (16 ms delay; red line) or when the noise-pairs are independent (Fig 3B), at which point subjects’ responses evince “fission” i.e., a tendency to report two sources (Fig 3A and 3I). The edges are also seen to shift toward the ITD of the leading source (positively) at intermediate delays, a finding consistent with localization dominance. Lastly, note that distributions obtained at the longest delay (16 ms) are nearly identical to those generated by independent noise-pairs, regardless of delay (Fig 8B).

We next consider how the edges of ITD distributions may be represented across subpopulations of ITD selective neurons and the degree to which these representations, when combined with envelope weighting, may explain spatial perception and reproduce our psychophysical findings (Figs 3 and 4).

Neural Representations of Cue-Distributions.

How might the edges of ITD distributions be represented in the auditory system? One possibility is that spatial perception could correspond to intermediate or sub-maximal activities evoked across spatially selective neurons, as suggested in a number of previous studies [10–13]. Although these studies only addressed single sources, when spatial cues would be narrowly distributed, they pointed out that the ipsilateral slopes of individual tuning curves often correspond to frontal locations, where spatial acuity in humans is greatest, whereas their maxima correspond to contralateral locations [4, 5]. In addition, ipsilateral slopes may be steeper than their contralateral counterparts [3] allowing them to convey greater information than their peaks [42, 43]. Note that this sub-maximal hypothesis [10–13] differs markedly from the hypothesis that spatial locations may be represented by differences in activities evoked across neurons throughout the left and right hemispheres [1, 4, 5].

The model only consider frequencies below 1.5 kHz since effects of envelope weights are expected to diminish at higher frequencies and because earlier studies have shown that precedence phenomena are strongest at low frequencies [26, 35, 44, 45]. Frequency-specific neurons with best interaural phase differences (IPD) beyond the “π-limit” [4, 5] are also considered, so that the implications of this limit may be examined under the edge model.

To simulate activities across each IC, stimulus-specific distributions of IPD were convolved with the “spike probability function” (SPF), which represents the probabilities that an IPD (Φ) at a given moment in time would evoke a single spike in a subpopulation of IPD-selective neurons. Fig 9A shows three such subpopulations. Each thin gray line is the IPD “tuning curve” of a single neuron in the IC. Heavy black lines represent their averages, where the maximum probability of a spike corresponds, for instance, to an IPD of ~0 cycles (left), ~0.15 cycles (center), or ~0.2 cycles (right). Subpopulations of neurons are considered because the firing rates of individual neurons are too low to represent IPD on the time-scales of amplitude modulations intrinsic to cochlear filters (e.g., Fig 6B). Note that it is not necessary to assume any topography, in the representation of IPD, such as on an auditory space map. The shape of the SPF is assumed to resemble a typical IPD tuning curve, as described by Harper and McAlpine [43]: (2)

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Fig 9. Description of the edge model.

(A) Thick black lines show three different spike probability functions (SPF), each reflecting the average preference of a subpopulation of integrated neurons (thin gray lines). (B) Hypothetical array of SPFs in the left IC (thin lines). Open circles indicate the probabilities of spiking across the SPFs when the stimulus’ IPD is 0 cycles. (C) Array of SPFs in which downward arrows indicate the IPD at each SPF’s maximum. (D) Probability of spiking across as array of SPFs, where the abscissa indicates the IPD at each SPF’s maximum. (E) Contributions to perception from an activity maximum centered at an IPD of 0 cycles. (F) Array of SPFs in which downward arrows indicate the IPD at each SPF’s half-maximum. (G) Probability of spiking across as array of SPFs, where the abscissa indicates the IPD at each SPF’s half-maximum. (H) Contributions to perception from an activity half-maximum centered at an IPD of 0 cycles.

https://doi.org/10.1371/journal.pone.0137900.g009

To describe the ‘edge’ model and distinguish it from the more conventional ‘maximum’ model (e.g., [2]), an array of SPFs is first considered (lines in Fig 9B). The peaks of the SPFs correspond to positive IPDs within the π-limit (<0.25 cycles) because they represent the activities of neurons in the left IC. Open circles indicate the probabilities of spiking across the SPFs when the stimulus’ IPD is 0 cycles.

Under the 'maximum' model, spatial perception is assumed to correspond to activity maxima across the IC. Thus, Fig 9D reflects the output of an array of SPFs in which the ordinate indicates the probability of a spike (see Fig 9B) and the abscissa indicates the IPD that corresponds to each SPF’s maximum (downward arrows in Fig 9C). Accordingly, perception would be based on the single activity maximum centered at an IPD of 0 cycles (Fig 9E).

In contrast to the ‘maximum’ model, spatial perception under the ‘edge’ model is determined by neurons firing at sub-maximal values. Thus, Fig 9G reflects the output of the same array of SPFs as in Fig 9D except that the abscissa indicates the IPD that corresponds to each SPF’s half-maximal spike probability (downward arrows in Fig 9F), as opposed to their maxima (Fig 9C). Maximal spike probabilities may not contribute as much to spatial perception under the edge model and are therefore reflected downward in Fig 9H, resulting in a single half-maximal ‘peak’ at 0 cycles (at 0.5 along the ordinate).

To infer activities across both ICs in a computationally efficient manner, SPFs and stimulus-specific distributions of IPD were convolved (Fig 10A–10C). Fig 10A and 10B show, respectively, an SPF and a narrow distribution IPDs (mean, standard deviation = 0). Fig 10C shows the result of the convolution. Note that the traditional ‘maximum’ model is assumed in this example and that similar results are expected for the left and right ICs.

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Fig 10. Computationally efficient implementation of the edge model.

(A) Spike probability function (SPF) corresponding to the maximum model. (B) Narrow distribution of IPDs. (C) Results of convolving the SPF in (A) with the distribution of IPDs in (B). (D) Spike probability function for the left IC under the edge model. (E) Spike probability function for the right IC under the edge model. (F) Narrow distribution of IPD for which an open circle indicates the IPD that arose half-maximally. (G) Modeled population activities when the distribution of IPD is narrow (F). Cross-hatching indicates activities of neurons with best IPDs beyond the ‘pi-limit’ (< -0.13 or > 0.13 cycles) which may rarely occur (e.g., [3, 4]) and therefore contribute little to spatial perception. (H) Relative contributions of population activities to spatial perception under the edge model. (I) Modeled activities obtained after converting IPD to ITD and averaging across frequency to simplify presentation (0.2–1.5 kHz). (J-Q) Distributions of IPD and modeled activities as described in (F-I) but when the distributions of IPD (J and N) were broader.

https://doi.org/10.1371/journal.pone.0137900.g010

To implement the edge model, SPFs representing the left and right ICs were phase-shifted (±φ) by an amount equal to the distances from their maxima to their ipsilateral (inner) half-maxima (φ = ±0.13 cycles) thus: (3) (4)

These phase-shifts indicate neither a shift of activity within each IC nor differences in spatial “tuning” per se. Instead, they represent the notion that spatial locations may be represented by subpopulations of neurons (SPFs) whose firing probabilities are half of other neurons throughout the left and right ICs (Fig 9F–9H). This is conceptually similar to labeling a single neuron’s “tuning curve” according to the cues at which its firing rates are half-maximal (Fig 9F), as opposed maximal (Fig 9C), except that the SPF reflects the momentary probability of spiking across many neurons (Fig 9A).

When distributions of IPDs are narrow, as in Fig 10F, modeled population activities (Fig 10G) mirror the SPFs (Fig 10D and 10E) and half-maximal activities in each IC correspond to the same IPD (open circle in Fig 10G). Note that the two distributions in Fig 10G would have both been centered on 0 cycles, as shown in Fig 10C, were the SPFs referenced to their maxima (Eq 3) rather than their ipsilateral half-maxima (Eqs 4 and 5).

The relative contributions of ITD to spatial perception are assumed to be greatest when population-wide activities are half-maximal. Conversely, the contributions are assumed to be lowest when activities are either lower (activity < 0.5) or higher (activity > 0.5) than the half-maximum. To better visualize contributions to spatial perception, activities greater than the half-maximum (>0.5) are therefore reflected downward in Fig 10H. Thus, half-maximal “peaks” near 0.5 in Fig 10H indicate the IPDs that are expected to contribute most to spatial perception. Note that significant half-maximal side-peaks are evident near ±0.25 cycles but may be diminished under at least two conditions. The first is if listeners were to integrate across frequency, as shown in Fig 10I, where IPD was converted to ITD and frequency-specific activities, analogous to those in Fig 10H, were averaged (0.2–1.5 kHz). Half-maximal side peaks could also be diminished if neurons “tuned” (maximally) to ipsilateral IPDs were excluded from the convolutions along with neurons tuned (maximally) to IPDs beyond the π-limit, both of which are equivalent to excluding half-maximal activities less than approximately -0.13 or greater than 0.13 cycles (cross-hatching in Fig 10).

Besides offering a higher spatial resolution when spatial cues are narrowly distributed over time, owing to the steep slopes of the modeled SPFs, distinct images are predicted in the left and right ICs when IPDs (or ITDs) are broadly distributed. This is shown in Fig 10 where IPD distributions having standard deviations of 0.1 cycles (Fig 10J–10M) or 0.2 cycles (Fig 10N–10Q) were considered. After convolving these broader distributions (Fig 10J and 10N) with the SPFs (Fig 10B and 10C), activities in both ICs are predicted to broaden (Fig 10K and 10O), causing half-maximal values to be achieved at distinct IPDs in each IC (Fig 10L and 10P). Fig 10M and 10Q show again that ITD half-maximal side-peaks can be significantly diminished if listeners are able to integrate across frequency (0.2–1.5 kHz) or if half-maximal activities are constrained to between approximately -0.13 cycles and 0.13 cycles (cross-hatching in Fig 10), due to a paucity neurons tuned (maximally) to ipsilateral IPDs or IPDs beyond the π-limit.

Distributions of envelope-weighted ITDs can broaden significantly when independent noise-pairs are considered or when long delays are introduced between identical lead-lag noise-pairs (Figs 7 and 8). Figs 11 and 12, reflecting independent and identical noise-pairs, respectively, therefore show modeled results as a function of lead-lag delay (ordinate axis), when distributions of ITD over time were: (1) convolved with SPFs representing ITD as opposed to IPD, (2) reflected so that greater values (warmer colors) indicate half-maximal activities, and (3) integrated across frequency to simplify presentation, as illustrated in Fig 10. In addition, the modeled results were obtained for 100 different noise-pairs and averaged. Warm colors along each row represent ITDs expected to contribute most heavily to spatial perception, on average, across listening trials in a given IC (A, C, E, G = left IC; B, D, F, H = right IC). For each row and lead-lag delay, a circle indicates the ITD at which activities were half-maximal.

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Fig 11. Modeled contributions to spatial perception for independent noise-pairs of various lengths and lead-lag delays.

Warm colors along each row represent ITDs expected to contribute most heavily to spatial perception, on average, across listening trials in a given IC. For each row and lead-lag delay, a circle indicates the ITD at which activities were half-maximal. (A and B) Contributions to perception in the left and right ICs, respectively, when the noise-pairs were 200 ms. (C-H) Contributions as indicated for the left and right ICs in A and B but when the noise-pairs were 30 ms (C and D), 10 ms (E and F) or 30 ms and lacking alone segments (G and H). (I and P) Signal-to-noise ratios (SNR), computed as the ratio of half-maximal activity over one standard deviation, when the noise-pairs were 200 ms (I and J), 30 ms (K and L), 10 ms (M and N) or 30 ms and lacking alone segments (O and P). Warm colors indicate a higher SNR.

https://doi.org/10.1371/journal.pone.0137900.g011

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Fig 12. Modeled contributions to spatial perception for identical noise-pairs.

See Fig 11 and the text for descriptions of the various plots.

https://doi.org/10.1371/journal.pone.0137900.g012

Signal-to-noise ratios (SNR), computed as the ratio of half-maximal activity (warm colors in Figs 11A–11H and 12A–12H) over one standard deviation (100 different noise-pairs), are shown in Figs 11I–11P and 12I–12P. A high SNR (warm color) indicates a pattern of activity that, on any single trial, is likely to resemble the average pattern (Figs 11A–11H and 12A–12H). In contrast, a low SNR indicates a pattern of activity that, on any single trial, may only rarely resemble the average pattern. In other words, the numerator (signal) represents the pattern of activity observed across trials whereas the denominator (noise) represents variation across trials. High SNRs are generally observed when ITDs are consistent over time, such as when identical noise-pairs with short lead-lag delays are considered (Fig 12I–12P) or when the lead or lag-alone segments are long (Figs 11I–11P and 12I–12P). SNRs also increase with stimulus duration. Thus, at short and intermediate delays, SNRs were higher for the longer, 200 ms, stimuli than for the shorter, 30 or 10 ms, stimuli.

Fusion

Fusion, which is the tendency of listeners to report a single auditory image, is strongest when stimulus pairs are identical and the lead-lag delay is short. Our model is consistent with this characterization in that similar half-maximal activity patterns are predicted in the left and right ICs, both corresponding to ITDs near 0 ms (Fig 12). In contrast, activity patterns in the left and right ICs are often distinct, corresponding to different ITDs, when the lead-lag delay is long (Figs 11 and 12) or when the stimulus pairs are independent (Fig 11).

To model our empirical results (Fig 3A–3D), half-maximal activities derived for the left and right ICs were averaged across frequency (see Figs 11 and 12) and cross-correlated. The degree to which activities in each IC correspond to distinct ITDs was then inferred from the time lag at the correlogram’s maximum (D_max, Fig 13A–13D). Sigmoidal functions (Fig 13E–13H) were next used to predict the proportions of trials in which an additional image was reported (Fig 13I–13L), thus: (5) where values of M and R were chosen to provide reasonable fits to our psychophysical results (Fig 3; Exp. 1: M = 0.00016, R = 0.000015; Exp. 2: M = 0.00019, R = 0.000015; Exp. 3: M = 0.00018, R = 0.00001; Exp. 4: M = 0.000125, R = 0.000055). Different values of M and R were used for each experiment because different subjects were tested (with one exception). Criteria used by the subjects to judge when an additional image was heard may also have differed across the four experiments. In comparison to Experiment 2 (Fig 3B), for example, subjects in Experiment 4 (Fig 3D) were more likely to report an additional image at short delays. We presume this difference is because the alone segments were excised in Experiment 4 and subjects may therefore have used less stringent criteria in deciding when an additional image was heard.

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Fig 13. Empirical predictions of fusion.

(A-D) Correlogram maxima (D_max) describing the degree to which activities in the left and right ICs corresponded to distinct ITD. (E-H) Sigmoidal functions used to predict the proportions of trials (I-L) in which an additional image was reported.

https://doi.org/10.1371/journal.pone.0137900.g013

In most major respects, the modeled proportions (Fig 13I–13L) resemble our behavioral results (Fig 3A–3D). When the noise-pairs were identical, for instance, both proportions rose for delays longer than ~2 ms as is evident by comparing the filled circles and solid lines in Fig 3A–3C with those in Fig 13I–13K. Both proportions also remained consistently high for the longer (200 ms) independent noise-pairs, regardless of delay (open circles and dashed lines in Figs 3A and 13K) but declined at shorter delays (<~4 ms) when shorter noise-pairs were tested or modeled (30 or 10 ms; Figs 3B, 3C, 13J and 13K).

Contributions to fusion from the alone and superposed segments may also be inferred. Contributions from the alone segments, for instance, may be deduced from the proportions obtained with independent noises (open circles in Fig 13I–13K). Similarly, contributions from the superposed segment may be deduced by comparing the proportions obtained for the identical (filled circles) and independent noises (open circles). For example, comparing Figs 13I and 3A in which the noise-pairs were 200 ms shows that in both cases the proportions are considerably higher for the independent noises (open circles and dashed lines) than for the identical noises (filled circles and solid lines), suggesting that corresponding features during the identical stimulus’ superposed segment (~200 ms) contributed substantially to fusion. In contrast, the proportions for the identical and independent noise-pairs are more similar when shorter noise-pairs were tested in both the modeled and empirical results (Figs 13J, 13K, 3B and 3C), suggesting that the superposed segment contributes less to fusion as it is shortened.

Inter-subject variability (vertical error bars in Fig 3) may also be accounted for if we consider inter-stimulus variability (vertical error bars in Fig 13) and assume that subjects may have differed in their propensities to report an additional image when half-maximal activities in the left and right ICs differed. To simulate this latter effect, the midpoint at which subjects were predicted to indicate an additional image in 50% of the trials (M in Eq 6) was varied in a uniform manner (between ± 0.05), resulting in the gray-shaded regions shown in Fig 13E–13H. Lightly colored shading in Fig 13I–13L show the results. When D_max values are consistently large (independent noises, open circles in Fig 13A) or small (identical noises and short delays, filled circles in Fig 13A–13D), the model predicts only a moderate degree of inter-subject variation, as was observed when similar noise-pairs were tested (Fig 3A–3D). When D_max values are intermediate, however, greater inter-subject variation is predicted, as shown by broader shading in Fig 13I–13L.

Localization dominance

Localization dominance, which is the tendency of listeners to report a single image or pairs of images closer to the source of the leading sound, is strongest when identical stimuli are considered and the lead-lag delay is short. Our model is consistent with this characterization in that distributions of ITDs (Figs 7A–7F and 8A) and half-maximal activity patterns (Figs 11 and 12) are predicted to ‘shift’ toward the leading source (positively) under such conditions. In contrast, the shift is diminished at long delays (Fig 12) or when noise pairs are independent (Fig 11).

To quantify these shifts, we computed lateralization indices (LI), which indicate the degree to which half-maximal activities in the left and right ICs correspond to leading or lagging (positive or negative) ITDs after averaging across frequency. Each index quantifies the proportion of half-maximal activities corresponding to positive ITDs (red areas in Fig 14A), thus: (6) where act represents neural activity relative to half-maximum (as plotted in Fig 10H, 10L and 10P). Indices for the left and right ICs (dashed and solid lines respectively) are plotted in Fig 14 for identical (B-E) and independent (F-I) noise pairs.

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Fig 14. Empirical predictions of localization dominance.

(A) Modeled half-maximal activities, averaged across frequency, in the left and right ICs when the lead-lad delay was 4 ms and the noise-pairs were identical. (B-I) Lateralization indices (LI) indicating the degree to which half-maximal activities in the left and right ICs correspond to leading or lagging (positive or negative) ITDs after averaging across frequency. (J-Q) Empirical predictions of localization dominance obtained as a product factors described by Eq 7.

https://doi.org/10.1371/journal.pone.0137900.g014

Because localization dominance can apply to conditions when one or two images are reported, our empirical results (Fig 4A–4H) must be simulated as the product of several additional factors: (7)

The first term on the right side of the Eq 7, LI_fusion, takes into account that the left and right ICs may represent distinct auditory locations when fusion is weak but a single location when fusion is strong. Values of LI_fusion are thus obtained by weighting LI (Eq 6) with the probability of an additional image having been reported (P, Fig 13I–13L): (8) where LI_μ is the average of the left and right IC lateralization indices (lines without symbols in Fig 14B–14I). When P is very low (strong fusion), left and right LI_fusion values equal LI_μ and each other. When P is high, LI_fusion values differ for the left and right ICs and equal their respective LI values (circles in Fig 14B–14I).

The second term in Eq 7, uncertainty, considers that subjects may have been uncertain as to which of two images dominated when two were reported. Responses in our psychophysical experiments that were bimodally distributed (e.g., dashed lines in Fig 2E and 2F), for instance, resulted in mean angles (Fig 4A–4H) that were close to 0°. Small angles corresponding to such bimodal distributions were, in turn, interpreted as evidence that localization dominance was weak or did not occur. Thus: (9) where P is the proportion of trials in which subjects were predicted to indicate an additional image (Fig 13I–13L). The term is limited to values above 0.5 because some subjects continued to experience localization dominance even when the alone segments were long and fusion was weak (Fig 4B, 4C, 4F and 4G).

The last term, symmetry, assumes that localization dominance should not occur if the images that subjects perceive are symmetric around the midline: (10) where dom_left and dom_right are measures of dominance obtained, respectively, for the left and right ICs by Eqs 8 and 9. This term differs from the previous two terms (Eqs 8 and 9) in that it does not directly involve fusion (P). The term’s greatest effect is when an intermediate level of fusion was predicted (P ≈ 0.5, Fig 13I–13L). Symmetry values approached or equaled zero, for instance, when the stimuli were independent and there was no lead-lag delay (leftmost points, blue lines, Fig 14J–14O) or when the stimuli were independent and the alone segments were excised (blue lines, Fig 14P and 14Q).

In most major respects, predictions obtained in this way (Fig 14J–14Q) resemble our behavioral results if we assume that LI (Fig 14B–14I) and dominance (Fig 14J–14Q) values of ~0.5 correspond to a reported angle of approximately 45 degrees (Fig 4A–4H), i.e., if we assume that indications on the arcs (Fig 1C) averaged ~45° when ~75% of half-maximal activities corresponded to positive (or negative) ITD. When the noise-pairs were identical, for instance, both our predictions and behavioral results are highly positive for delays shorter than ~4 ms, as is evident by comparing the filled circles and solid lines in Fig 4A–4H with those in Fig 14J–14Q. Both values also generally increased for independent noise-pairs having longer alone segments (i.e., longer lead-lag delays; open circles and dashed lines in Figs 4A–4C, 4E–4G and 14J–14O).

Contributions to localization dominance from the alone and superposed segments may also be inferred. Contributions from the alone segments, for instance, may be deduced from the values obtained with independent noises (open circles in Fig 14J–14Q), whereas contributions from the superposed segment may be deduced by comparing values obtained for the identical (filled circles, red lines) and independent noises (open circles, blue lines).

Empirical predictions and signal-to-noise ratios

One aspect of our behavioral results that is only partially accounted for by our empirical predictions is the effect of the superposed segment’s length, especially on the independence ratio, obtained when the noise-pairs were independent and there was no delay (0 ms). An additional image was nearly always reported when this stimulus was long (200 ms; leftmost open circles in Fig 3A), for example, but was rarely reported when the stimulus was short (10 ms; leftmost open circles in Fig 3C). Our empirical measures also do not consider that reports on the arcs for the longer stimuli were bimodal (open circles in Fig 2A and 2B) but reports for the shorter stimuli were widely scattered and unimodal (open circles in Fig 2M and 2N). Thus, how ITDs were distributed across trials, when gauged by subjects’ reports on the arcs, provides little indication as to why additional images were reported.

Signal-to-noise ratios (SNR), computed as the ratio of half-maximal activities, summed across frequency and ITD, over the sums of their standard deviations (Fig 15A–15D), provide a possible explanation. As previously described (see Figs 11I–11P and 12I–12P), a high SNR indicates that the pattern of activity obtained on any single trial was likely to resemble the pattern obtained, on average, across trials (i.e., for individual noise-pairs). In contrast, a low SNR indicates that the patterns were likely to vary from trial-to-trial.

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Fig 15. Signal-to-noise ratios of half-maximal activity patterns.

(A-D) Signal-to-noise ratios (SNR) when noise-pairs were (A) 200 ms, (B) 30 ms, (C) 10 ms, or (D) 30 ms but lacking alone segments. Distributions of ITD (black bars), for the 500 Hz band, when a single pair of independent noises was superposed for (E) 200 ms, (F) 30 ms, or (G) 10 ms. Red lines show how ITDs were distributed, on average, across 100 different noise-pairs.

https://doi.org/10.1371/journal.pone.0137900.g015

To appreciate why these SNR values may have influenced when additional images were reported, it is necessary to first consider that ITD measurements were often clustered during individual carrier cycles (e.g., Fig 5D). Which, as an aside, is why ITD measurements obtained from zero-crossings, and thus from cycle-to-cycle, can be accumulated into frequency-specific distributions of ITDs similar to those obtained in the current study (not shown). As illustrated for the 500 Hz band in Fig 15E–15G, this clustering means that distributions of ITDs could be quite narrow when ITDs were accumulated for only a handful of carrier cycles (e.g., ~10 ms or 5 cycles at 500 Hz; Fig 15G) or rather broad when accumulated for as long as 200 ms (~100 cycles; Fig 15E). Indeed, frequency-specific distributions of ITDs for the 10 ms condition (Fig 15G) were often as narrow as when the noise-pairs were identical, which may explain why reports of additional images were infrequent under both conditions. For comparison, red lines in Fig 15E–15G show how ITDs were distributed, on average, across 100 different noise-pairs.

Additional or alternative spatial representations

To describe the “edge” model, comparisons are drawn with a more conventional “maximum” model, in which spatial perception is viewed as corresponding to activity maxima (Fig 9C–9E) and thus the peaks of spatial cue distributions (e.g., Fig 8). The maximum model provides little indication as to why subjects would report additional images, when spatial cues are broadly distributed across low frequency bands, and is therefore at least partially rejected. Yet, under conditions when sub-maximal activities may represent the edges of spatial cue distributions, and possibly the edges of auditory objects, maximal activities may indicate their centers of mass. In this way, the peaks of spatial cue distributions and activity maxima may still provide important spatial information under the edge model.

Without invoking additional unexplored mechanisms, the inter-hemispheric hypothesis also provides little indication as to why subjects would report additional images when spatial cues are broadly distributed across low frequency bands [1, 4, 5]. Moreover, support for this hypothesis lies mainly in the argument that neurophysiological findings in mammals are incongruent with the maximum model under a narrow range of experimental conditions, when fluctuations in spatial cues are small [3–5].

In contrast to the maximum and inter-hemispheric hypotheses, neurophysiological findings in mammals [3–5] and owls [47, 48] are not only consistent with the edge model but required. The edge model, for instance, requires neurons with best ITDs that, in some animals, may extend well beyond their “physiological ranges” (e.g., [3, 4]). This requirement is evident in Fig 10N–10Q, where half-maximal activities near ± ~0.125 cycles (green circles in Fig 10O and 10P) would not have been obtained had neurons with best IPDs as large as ± ~0.25 cycles not been considered by our model. The significance of this is that a best IPD of 0.25 cycles corresponds to an ITD of 0.5 ms, at 500 Hz, which is several times larger than the “physiological range” of an animal like the gerbil (±0.135, [4, 49]). In other words, the best IPDs of neurons may often greatly exceed those that are generated by an animal’s head-related transfer function (HRTF) so that their slopes can represent the edges of spatial cue distributions under naturalistic conditions (e.g., [50]).

In addition to requiring at least some neurons with best IPDs as large as ± ~0.25 cycles, ambiguous half-maxima may be avoided, under the edge model, if best IPDs are also limited to values between approximately 0 and ± ~0.25 cycles (i.e., the π-limit) (e.g., [3, 4]). Ambiguous half-maxima corresponding to IPDs beneath the cross-hatchings in Fig 10H, near ±0.3 cycles, for instance, may contribute less to spatial perception if there are few or no neurons with best IPDs beyond ± ~0.25 cycles. Neurons with best IPDs between approximately 0 and ± ~0.25 cycles (Fig 9C) are also consistent with the edge model in the sense that their medial slopes and half-maxima (Fig 9F) span the midline [3, 4, 13] where spatial acuity is best in humans [51]. Thus, in our view, the best IPDs of laboratory mammals [3, 4] provide considerable support for edge model, as opposed to evidence against a narrow interpretation of the maximum model.

Effects of frequency

High frequency bands (> 1.5 kHz) were excluded from our model because results from previous psychophysical studies [26, 35, 44, 45], in addition to results from our own model, suggest that contributions to human precedence phenomena may weaken as frequency increases. In Fig 10Q, for example, listeners are predicted to perceive two edges in directions corresponding to IPDs near ±0.125 cycles (green circles), which should correspond to substantially larger ITDs in lower frequency bands (e.g., ITDs ≈ ±0.125, ±0.25, and ±0.5 ms, respectively, at 1, 0.5 and 0.25 kHz). The envelopes of frequency bands above ~1.5 kHz may also generate significant energy above ~150 Hz (Fig 6D) and thus either lack representation [41] or cause dissimilar envelope features to overlap, as opposed to similar features [27, 36].

Another reason for excluding high frequency bands (> 1.5 kHz) is because representations derived from our model are averaged across frequency to simplify presentation (0.2–1.5 kHz; Figs 11 and 12). The strongest effects of precedence, across low frequency bands, are therefore already obscured and would have been further obscured had higher frequencies been incorporated. Indeed, our modeled results were averaged across frequency mainly to simplify presentation and because subjects in our behavioral experiments were not asked to provide separate reports for different frequency bands.

Lastly, while high frequencies are neither expected [26] nor predicted to contribute to localization dominance, except for when the lead-lag delay is very short, envelopes generated by high frequency filters may still contribute substantially to reports of additional images (i.e., fission). Neurons selective for envelope ITDs [52, 53], for instance, may generate distributions of ITDs that are just as broad as those by generated by neurons responsive to low frequency carriers. This could explain why additional images have been reported when independent noise-pairs are high-pass filtered [46].

Effects of time, envelopes, and spatial cue distributions

Investigations of temporally superposed or binaurally independent stimuli have typically used noise-pairs that are quite long (e.g., 150–500 ms, [26, 46, 54]) or practically continuous (e.g., [6, 25]), suggesting that spatial cues may be “integrated” for considerable lengths of time. Our results further support this interpretation in that “independence proportions” increased considerably as our noise-pairs were lengthened (horizontal lines in Fig 3A–3C). For this reason, subjects’ perceptions and behavioral responses were modeled after weighting and accumulating spatial cues over the entire lengths of our stimuli.

Because our model attributes weights to spatial cues based on frequency-specific fluctuations in amplitude (Figs 5 and 6), it could be argued that our model also integrates spatial cues on much finer time-scale. The envelope of a noise filtered at 500 Hz, for instance, contains significant energy up to ~50 Hz (black line in Fig 6D). As a result, weights are attributed across the peaks and troughs of envelopes at a similar rate (50 Hz) and thus with an average period of approximately 20 ms (50 Hz^-1 = 20 ms; see Fig 6A). One result of this envelope weighting is that cues near the onsets of our stimuli were almost always heavily weighted, since this was the only time when the envelopes of the filtered noise-pairs rose in unison. When averaged across a broad enough range of frequencies, weights following a transient even resemble the onset sensitivity functions of previous studies [55, 56]. In contrast, envelopes generally declined and were lightly weighted near the offsets of the noise-pairs.

Despite our use of envelope weights, one might still argue that the use of a single “time-window” is unreasonable since the lead and lag-alone segments of our longer, 200-ms, noise-pairs may have caused subjects to perceive two or more temporally distinct auditory events. We do not exclude this possibility, although how these events may have contributed to our behavioral results is unclear. When the lead and lag-alone segments were as long as 8 or 16 ms, for instance, reports of additional images could not have become much more frequent (Fig 3E) and their contributions to localization dominance were small (Fig 4I).

Because spatial cues were weighted and accumulated over the entire lengths of our stimuli, distributions of ITD were often bimodal, as opposed to unimodal, especially when the lead and lag-alone alone segments of our experimental stimuli were long. Even without these alone segments, the distributions were sometimes bimodal within some frequency bands (e.g., Fig 7A). These differences in modality, however, appear to be of little importance in the present study. Empirical measures of fusion and localization dominance obtained for stimuli whose ITD distributions were almost entirely unimodal, for instance, were similar to those obtained when the distributions were strongly bimodal (e.g., independent, 200 ms, noise-pairs versus identical, 10 ms, noise-pairs with 16 ms delay). One reason for these results is that distinct modes, which are closely spaced (e.g., separated by < ~0.25 cycles), may no longer be distinct when represented by neurons whose selectivity for ITD is broad across low frequency bands (Fig 9A). Empirical measures of fusion and localization dominance may also simply fail to detect differences in perception arising from distributions that are unimodal or multimodal in a given frequency band.

Implications for other experimental and naturalistic sounds

Fluctuations in spatial cues are typically inferred by cross-correlating the signals in a listener’s ears. How spatially selective neurons may respond to a given stimulus, and how the stimulus may be perceived, is then inferred from the correlation (ρ). One such inference, suggested by the “maximum” model, is that a high correlation may evoke a narrow pattern of activity across a small number of similarly “tuned” neurons, whereas a low correlation may evoke a broad pattern across many different neurons. This inference is attractive in that auditory “images” could possibly resemble, or even mirror, how spatial cues are distributed (e.g., Figs 7 and 8). On the other hand, humans do not perceive a single intracranial event that broadens as interaural correlation is lowered but seem to perceive an auditory event that “splits,” so that one event is perceived in the left ear and an additional event is perceived in the right ear [6, 9]. We suggest that subjects in our experiments may have experienced a similar “splitting” effect when spatial cues were broadly distributed and that the edge model provides a possible explanation.

The edge model may also explain why humans are highly sensitive to differences in correlation (Δρ) [7, 8]. Specifically, it may be that listeners can readily detect when maximal and sub-maximal activities correspond to slightly different ITDs or that listeners can detect when sub-maximal activities in the left and right hemispheres correspond to slightly different ITDs. In other words, listeners may readily detect when auditory objects acquire edges and detect when existing edges expand or contract, as may occur under most naturalistic listening conditions.