Responses of ITD-sensitive neurons in the DNLL of gerbils change with the frequency of a pure tone stimulus, similar to all other ITD sensitive neurons in the brainstem 
. Here, we have evaluated this frequency-dependent modulation in terms of its influence on the encoding of ITD by firing rate patterns of the neuronal population. For the 153 recorded cells we have characterized the frequency dependence by the two parameters characteristic phase (CP) and characteristic delay (CD) 
. We found that the two parameters are significantly negatively correlated, as has also been reported for the midbrain and DNLL of guinea pigs 
, although there DNLL data did not reveale negative CDs. Also consistent with these and several other studies in various binaural brainstem nuclei and animals, we found that CPs are broadly distributed over almost the whole phase cycle 
Analysis of single-cell mutual information revealed that the observed distribution of CPs and CDs performs slightly better than a distribution with shuffled CPs and CDs. Furthermore, the single-cell mutual information strongly depends on the noise level. We found that for high noise levels peak-based codes are advantageous in terms of mutual information. For low and moderate noise levels we find mixed coding schemes to be viable: Both slopes and peaks can be used to extract information and should be located in the physiological range. These results are consistent with theoretical work comparing slope and peak-based coding schemes 
. There it is generally shown that for high noise levels strong signal changes are preferred and thus binary-like (i.e., peak-based) codes are beneficial. For low noise, slope-based codes are preferred since only then can continuous rate changes be sampled well enough.
The statistical model allows derivation of hypothetical distributions of CP and CD for different head sizes. As expected, mutual information grows with increasing inter-ear distance. Also the regions of highest mutual information move towards smaller CPs as we increased the inter-ear distance. Interestingly, this effect corresponds well to the finding that for large mammals the medial superior olive (MSO; with most CPs between
cycles) is generally larger than for smaller mammals 
An increase in the inter-ear distance can alternatively be interpreted as an increase of best frequency. In both cases tone delay functions with peaks in the physiological range exhibit increased mutual information. With this interpretation we can also assess the situation when phase-locking is present up to several kHz, as found in the barn owl 
. There, as well as for a large head diameter, the two regions of high mutual information merge into one cluster centered about CP
0. As a consequence, a Jeffress-like coding strategy with CP
0 would be sufficient for achieving high single-cell mutual information.
The variability of phase delay functions in the DNLL provides the basis for a frequency-invariant population representation of ITDs. We find that both of two readout strategies, a linear classifier and a bilateral rate difference signal (two channel code) can explain a coding acuity of down to
. For the linear classifier, however, the observed distribution in CP-
space with BP clustered about
cycles is suboptimal in that a Jeffress-type representation with CP
0 would account for a better acuity. For the bilateral difference code the observed distribution of CPs and CDs seems appropriate, particularly because of the linearity and the frequency invariance of the difference signal. There, a Jeffress-like representation would yield a much lower acuity.
The behavioral acuity of gerbils at midline (
) has been estimated as 
and thus is worse than the acuity of about
derived from the bilateral difference model. Such hyperacuity of the estimator is not surprising as the relative noise decreases with the size of the population. In general, hyperacuity has two possible explanations. First, it may hint at several noisy downstream readout stations before the localization signal is translated to a behavioral response. As a second possibility, however, it could also hint at hidden stimulus dimensions that are not taken into account by the decoding model. As for the frequency dependence discussed in our paper, one could also ask for a code to be invariant with respect to intensity, background noise etc.. Each of these additional dimensions, hence, reduces the predictive value of single neurons. The real psychophysical acuity should then be achieved by a decoding model that takes into account all possible invariances assuming no further noise in the readout.
Another possible discrepancy to psychophysical data is that the acuity of the bilateral rate difference model is independent of stimulus ITD. In humans the minimal audible angle at lateral (azimuth
) positions is up to 10 times worse than at frontal positions (
. However, the transformation from angle to ITD only accounts for a factor of about 2 (see Materials and Methods
). Indeed, the psychophysical ITD resolution for low-frequency pure tones is about 2 to 5 times worse for lateral positions as for frontal ones 
. In gerbils, localization acuity has not yet been determined at locations different from midline. However, the bilateral difference model predicts that in gerbils the just noticeable ITD difference is independent of azimuth and conversely the acuity in terms of azimuthal angle should be about 2 times worse for lateral positions than at midline. This feature could be a specialty of animals with small head size, because if the inter-ear distance gets larger more peaks of the phase delay functions move into the physiological range and impair decoding via a difference rate particularly for more lateral positions.
Non-zero CPs are most often thought to originate in the lateral superior olive where neurons receive inhibition from contralateral and excitation from ipsilateral. The combination of these antiphasic signals is able to explain CPs around 0.5 and low CDs. Such cells are generally called troughers. For neurons that receive bilateral excitation (as in the MSO) CPs different from zero still pose a major problem for mechanistic models of ITD sensitivity as the physiological mechanisms that give rise to them are not fully identified, yet. The classical Jeffress model 
in which the best ITD is solely determined by temporal latency differences predicts constant CP
0. Cells with small CPs are generally called peakers. There are several candidate models for non-zero CPs in binaurally excited neurons. 1) Ipsi- and contralateral input fibers might have mismatched center frequencies and thus a mismatch of phases might be induced by the preprocessing of different cochlear filters 
. 2) Morphological asymmetries 
of the coincidence detecting neuron can induce distinct temporal filtering of the ipsi- and contralateral inputs. 3) Phase-locked inhibition 
that differs between ipsi- and contralateral input can induce asymmetric phase shifts. 4) Phase disparities may be a direct consequence of asymmetries in the ipsi- and contralateral excitatory synaptic kinetics 
. The present study shows that generating specific CPs may not just be an epiphenomenon of the physiological mechanisms that underlie ITD-sensitive responses in the brainstem but may be required for an optimal neuronal representation of ITD. Thus the physiological mechanisms underlying ITD sensitivity should allow the deliberate tuning of CPs, which argues against hard-wired solutions as (1) and (2) and favors synaptic mechanisms like (3) and (4).
A problem in the interpretation of our data is that the DNLL is not a primary nucleus in which the ITD-sensitive responses are computed. The ITD representation in the DNLL might already be imposed by secondary processing steps. Instead one would rather want to compare population responses in the MSO (for low CPs) and the low-frequency region of the lateral superior olive (for high CPs). Single units in the MSO are, however, difficult to record from. Data from a few tens of gerbil MSO units also shows negatively correlated CP and CD with a broad distribution of CPs (unpublished observation about data from 
). The DNLL, however, is a particularly good place to study ITD population codes, since it is much easier to record from than the MSO and, moreover, it is the first station in which genuine ITD-sensitive responses from MSO (peakers) and lateral superior olive (troughers) are combined 
. The only major computation occurring at the synapses from the superior olivary complex to the DNLL seems to be noise reduction 
Most theoretical analyzes of neuronal representations deal with only one or two stimulus dimensions as e.g. the frequency of a tone or the loudness of a sound. In the example discussed in the present paper the two stimulus dimensions ITD and frequency are both physically and statistically independent since sound position and sound spectrum are generally unrelated. Here, we have shown that considering population responses across an invariant dimension (frequency) of the stimulus not only allows the assessment of the neuronal population representation in terms of coding acuity, but also allows to evaluate, how different hypothetical invariant read-out strategies fit to the population representation.