Distribution of best delay depends on best frequency
Rate-ITD curves were measured for 107 single units in the IC from 13 anesthetized cats. Physiologically relevant parameters were estimated by fitting a cross-correlation model to the data using nonlinear estimation techniques. The model gave an excellent fit to the data, on average accounting for 93% of the variance in the neural responses. As shown in , the overall distribution of best delay estimates was essentially the same as the distribution reported previously for the anesthetized cat (Yin et al., 1986
Figure3 Distributions of best ITD in the anesthetized cat. Grayshading, Present study. Black line, Data from Yin et al.(1986). Vertical lines indicate approximate natural ITD range for the cat. In both studies, best ITD is estimated from rate-ITD curves measured (more ...)
shows that the BD distribution depends on BF, in which the BF for each unit is the CF parameter of the best-fitting single-neuron model. Best delays greatly exceed the natural ITD range (~300 μsec; horizontal dashed line) for many low-BF units. For higher BFs, however, the best delays seem to cover this range only partially. The dependence on BF was quantified by dividing the units into four quartiles based on BF and computing the median BD for each quartile (filled squares). The BF ranges for each quartile are <450, 451-615, 616-855, and >856 Hz. demonstrates that the median BD in the cat IC (squares) and the mean BD in the guinea pig IC (circles) (McAlpine et al., 2001
) decrease with BF in a similar manner. Furthermore, the interquartile deviation of BD in the cat and the SD of BD in the guinea pig are similar in magnitude and also decrease with increasing BF (, inset).
As shown in , when the BD is instead expressed as a phase relative to BF (BP = BD × BF), the distribution becomes more nearly independent of BF. In the cat IC, the median and interquartile deviation of best phase increase only slightly with BF in a manner quantitatively consistent with the best phase mean and SD in the guinea pig (; squares, cat; circles, guinea pig). Thus, in the ITD processing mechanism of the cat as well as the guinea pig, best phase, rather than best delay, is distributed independently of BF.
In the guinea pig, the maximum slopes of the rate-ITD curves tend to occur within the range of naturally occurring ITDs and are centered around the midline (McAlpine et al., 2001
). This is because, as the peaks shift to higher BDs for lower BFs, the peaks also become wider because the rate-ITD curves are quasiperiodic at BF. The same is true in the cat, as shown in . The ITD corresponding to the maximum slope was found for each unit from the best fit of the cross-correlation model. The distribution of these ITDs (gray bars) is centered near ITD = 0 and primarily restricted to the natural ITD range (vertical dashed lines). This is consistent with the notion that a change in stimulus ITD is coded not by a change in the locus of peak activity but by changes in firing rate within the population. The results from the cat, including both this study and that of Joris et al. (2004)
, extend the results of McAlpine et al. (2001)
by demonstrating that these characteristics are not limited to rodents or to animals with small heads but may be a general property of mammalian ITD coding.
Distribution of the ITD corresponding to maximum slope of rate-ITD curve. Values are clustered around the midline, within the natural ITD range for the cat (dashed lines).
Simulations of ITD discrimination for broadband noise
The population model of is an implementation of an ITD processor consistent with the physiological results of . The model was constrained using the parameters obtained by fitting the cross-correlation model to the rate-ITD data for noise stimuli and tested by predicting human psychophysical ITD discrimination performance for both tones and noise.
shows simulations of ITD discrimination using broadband noise as the stimulus. Human performance is characterized by a greater than twofold increase in JND between the midline and ITD = 600 μsec (Mossop and Culling, 1998
). The model, however, predicts a nearly constant JND for all ITDs (triangles). This behavior is a direct result of the symmetry of the model ITD curves with respect to best ITD, as illustrated in . An individual model element is maximally sensitive to changes in ITD along the rising slope of the central peak but is equally sensitive to ITD changes along the falling slope. The ITD acuity of the model is thus improved by the existence of falling slopes that lie within the physiological ITD range. Such slopes arise from all high-CF neurons regardless of best ITD and also from the small minority of low-CF neurons having best ITDs near the midline.
Figure 6 a, Population model predictions of JNDs as a function of ITD for broadband noise stimulation, with across-BF integration (circles) and without (triangles). Mean human performance from Mossop and Culling (1998) shown by black line. b, Top, Rate versus (more ...)
This argument suggests that a mechanism that suppresses the falling slopes while preserving the rising slopes will lead to decreased sensitivity away from the midline. One such mechanism is simply to average rate responses across BF at each best phase. The midline slopes of the central peaks tend to align across BF, but the large variation in falling slope location with BF causes these slopes to misalign across BF (). Therefore, the result of pooling across BF is to preserve the large rising slope near the midline while reducing the magnitude of the falling slope, as shown by the heavy black line in . The effect on ITD discrimination was computed in the model by averaging firing rates of neural elements across BF. The firing rate of a model element in the i
th position on the CF
-axis and the j
th position on the BP
-axis was computed as follows:
(= 15) is the length of the CF
-axis. The ideal observer model was then applied to the averaged firing rates, retaining the assumption that the variance of the firing rate is proportional to the mean. No change was made to the proportionality constant k0
because any such change can be offset by an appropriate change in the efficiency parameter to maintain quantitative consistency with psychophysical data. The circles in demonstrate that this configuration of the model accurately predicts ITD JNDs for human listeners in response to broadband noise.
Simulations of ITD discrimination for pure tones
The model was further tested by applying it to ITD discrimination for 500 Hz tones (). Across-BF integration was used because doing so provides the most accurate predictions for broadband noise (circles, repeated from ). Two configurations of the model were tested differing only in the implementation of the best phase assigned to each model neuron (). In the “pure delay model,” the mechanism takes the form of a pure delay (i.e., BP = CD/CF), whereas in the “pure phase shift model,” it takes the form of a pure phase shift (BP = CP). Both configurations reproduce the observed dependence of BD on BF for broadband noise. The pure delay model is intuitively appealing because of the long-standing idea that spike conduction delays underlie ITD tuning. The notion of a pure phase shift model requires additional comment. It is not intended to suggest that internal delays are identically zero; the existing evidence clearly indicates otherwise. Rather, the pure phase shift model is a more abstract construction whose properties overcome certain limitations of the pure delay model. Possible physiological interpretations of the pure phase shift model will be discussed below.
Figure 7 Predictions of ITD acuity for 500 Hz tones (upward triangles) from pure delay model. Average human data from Domnitz and Colburn (1977) (bottom black line) is much smaller in magnitude than model predictions. Increasing model efficiency to = (more ...)
In response to 500 Hz tones, human listeners exhibit JNDs that are nearly constant as a function of ITD and ~10 μsec in magnitude (bottom black line) (Domnitz and Colburn, 1977
). In contrast, the pure delay model (upward triangles) predicts that JNDs increase slightly with ITD up to ~200 μsec and decrease for higher ITDs, attaining their overall minimum value at ITD = 600 μsec, contrary to the psychophysical data. Furthermore, the predicted JNDs are >50 μsec, more than five times larger than the values measured experimentally.
The ITD discrimination curve for 500 Hz tones can be brought in line with psychophysical observations by increasing the efficiency value
from 1/18 to 1 (downward triangles). As discussed in Materials and Methods, this parameter may reflect any number of inefficiencies in the auditory system not explicitly incorporated into the model. The discrepancy in acuity between tones and noise is a problem that afflicts not just our model but is a property of the psychophysical data for which there is no satisfactory explanation in the literature of which we are aware. One possibility is that the difference lies in the temporal details of the responses to each stimulus. ITD discrimination may be less efficient for noise because, unlike the model, the biological system cannot average over long times and is subject to short-term fluctuations in estimates of interaural correlation. A quantitative treatment of such short-term variability requires a specifically designed study and is not practical with the data at hand. As stated above, however, the value of the efficiency parameter affects only the absolute value of the simulated JNDs, whereas this study is primarily concerned with trends in the data with changes in ITD.
As shown in , increasing the value of
also flattens the ITD discrimination curve for 500 Hz tones. However, when the JND
-axis is expanded, the nonmonotonic character of the response is still apparent (). The underlying cause for this behavior is illustrated in , which shows rate-ITD curves for all of the model elements at a single best phase (BP
= 0.19, the median physiological value). For clarity, only one best phase is represented; elements at all best phases contribute to the data shown in . In the pure delay model, the best delay varies inversely with BF (BD
). Hence, the peaks of the rate-ITD curves shift toward zero with increasing BF. For broadband noise, the curves are quasiperiodic at BF so that the peaks narrow as they shift (), causing the midline slopes to align across BF. For tonal stimuli, however, the curves are periodic at the stimulus frequency, and hence the peaks just shift without becoming narrower. Consequently, the midline slopes do not align (), leading to a shallower slope after pooling across BF.
Figure 8 Predictions of ITD acuity for 500 Hz tones with = 1 (triangles). Curve is non-monotonic and is minimal off the midline. Pure phase shift model (squares) more accurately predicts the shape of the discrimination curves for tones. Black lines, (more ...)
Figure 9 Comparison of pure delay and pure phase shift models. In response to broadband noise, rate-ITD curves across BF for BP = 0.19 align on the rising slopes of the central peaks for both the pure delay model (a) and the pure phase shift model (b). c, For (more ...)
For the pure phase shift model, the peripheral filter outputs in response to noise are quasiperiodic at BF, and the central peaks shift with BF exactly as in the case of the pure delay model (). The pure phase shift causes an asymmetry in the magnitudes of the side lobes. This asymmetry does not affect the ITD discrimination curve because the side lobes are smaller in magnitude than the central peaks, and most occur outside the range of base ITDs tested. In response to tones, the rate-ITD curves are periodic at the stimulus frequency, and, because the best delay is determined by the characteristic phase, the best delay also depends on the stimulus frequency (BD = BP/f). This means that, for tonal stimuli, the rate-ITD curves are identical regardless of BF and the midline slopes are perfectly aligned ().
The pure phase shift model thus provides alignment of the rate-ITD curve central slopes in response to both tones and noise. As a result, the predicted ITD JNDs for 500 Hz tones are minimal on the midline and increase only slightly as the base ITD increases (, squares), consistent with the psychophysical data. The results of Figures - suggest that an ITD processing mechanism that incorporates an internal phase shift better accounts for some aspects of psychophysical ITD discrimination than does an internal time delay alone.