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Estimation of the deleterious genomic mutation rate
If we use 1(K a/K s) as a measure of the fraction of mutations in protein-coding genes that are deleterious (for Caenorhabditis, K a/K s < 0.06)24 and multiply this by the genomic rate of nonsynonymous base substitution mutations in exon sequences in the MA lines (0.18 nonsynonymous mutations per genome per generation, given that ,26% of the C. elegans genome is exon, ,75% of exon nucleotides are nonsynonymous sites, and our total mutation rate of 0.9 base substitutions per genome per generation), we get 0.17 as an estimate of U d for C. elegans. Adding the rate of indel mutations in exon sequence (0.31 indel mutations in exons per genome per generation, given that ,26% of the C. elegans genome is exon and our total mutation rate of 1.2 indels per genome per generation) and assuming all exon indels are deleterious, yields 0.48 as an estimate of U d for C. elegans.
Received 3 February; accepted 1 June 2004; doi:10.1038/nature02697.
1. Drake, J. W., Charlesworth, B., Charlesworth, D. & Crow, J. F. Rates of spontaneous mutation. Genetics 148, 16671686 (1998). 2. Nachman, M. W. & Crowell, S. L. Estimate of the mutation rate per nucleotide in humans. Genetics 156, 297304 (2000). 3. Kumar, S. & Subramanian, S. Mutation rates in mammalian genomes. Proc. Natl Acad. Sci. USA 99, 803808 (2002). 4. Robertson, H. M. The large srh family of chemoreceptor genes in Caenorhabditis nematodes reveals processes of genome evolution involving large duplications and deletions and intron gains and losses. Genome Res. 10, 192203 (2000). 5. Witherspoon, D. J. & Robertson, H. M. Neutral evolution of ten types of mariner transposons in the genomes of Caenorhabditis elegans and Caenorhabditis briggsae. J. Mol. Evol. 56, 751769 (2003). 6. Vassilieva, L. L., Hook, A. M. & Lynch, M. The tness effects of spontaneous mutations in Caenorhabditis elegans. Evolution 54, 12341246 (2000). 7. Denver, D. R., Morris, K. & Thomas, W. K. Phylogenetics in Caenorhabditis elegans: an analysis of divergence and outcrossing. Mol. Biol. Evol. 20, 393400 (2003). 8. Denver, D. R., Morris, K., Lynch, M., Vassilieva, L. L. & Thomas, W. K. High direct estimate of the mutation rate in the mitochondrial genome of Caenorhabditis elegans. Science 289, 23422344 (2000). 9. Ochman, H. Neutral mutations and neutral substitutions in bacterial genomes. Mol. Biol. Evol. 20, 20912096 (2003). 10. Petrov, D. A., Lozovskaya, E. R. & Hartl, D. L. High intrinsic rate of DNA loss in Drosophila. Nature 384, 346349 (1996). 11. Petrov, D. A. & Hartl, D. L. Pseudogene evolution and natural selection for a compact genome. J. Hered. 91, 221227 (2000). 12. Hirotsune, S. et al. An expressed pseudogene regulates the messenger-RNA stability of its homologous coding gene. Nature 423, 9196 (2003). 13. Yamada, K. et al. Empirical analysis of transcriptional activity in the Arabidopsis genome. Science 302, 842846 (2003). 14. Balakirev, E. S. & Ayala, F. J. Pseudogenes: are they junk or functional DNA? Annu. Rev. Genet. 37, 123151 (2003). 15. Charlesworth, B. The changing sizes of genes. Nature 384, 315316 (1996). 16. Lynch, M. & Conery, J. S. The origins of genome complexity. Science 302, 14011404 (2003). 17. Marais, G., Mouchiroud, D. & Duret, L. Does recombination improve selection on codon usage? Lessons from nematode and y complete genomes. Proc. Natl Acad. Sci. USA 98, 56885692 (2001). 18. Hahn, M. W., Stajich, J. E. & Wray, G. A. The effects of selection against spurious transcription factor binding sites. Mol. Biol. Evol. 20, 901906 (2003). 19. Langley, C. H. & Ito, K. Spontaneous mutability in Drosophila melanogaster, in natural and laboratory environments. Mutat. Res. 36, 385386 (1976). 20. Gunsalus, K. C., Yueh, W. C., MacMenamin, P. & Piano, F. RNAiDB and PhenoBlast: web tools for genome-wide phenotypic mapping projects. Nucleic Acids Res. 32, D406D410 (2004). 21. Naclerio, G. et al. Molecular and genomic organization of clusters of repetitive DNA sequences in Caenorhabditis elegans. J. Mol. Biol. 226, 159168 (1992). 22. Keightley, P. D. & Ohnishi, O. EMS-induced polygenic mutation rates for nine quantitative characters in Drosophila melanogaster. Genetics 148, 753766 (1998). 23. Davies, E. K., Peters, A. D. & Keightley, P. D. High frequency of cryptic deleterious mutations in Caenorhabditis elegans. Science 285, 17481751 (1999). 24. Stein, L. D. et al. The genome sequence of Caenorhabditis briggsae: a platform for comparative genomics. PLoS Biol. 1, 166192 (2003). 25. Kondrashov, A. S. & Houle, D. Genotypeenvironment interactions and the estimation of the genomic mutation rate in Drosophila melanogaster. Proc. R. Soc. Lond. B 258, 221227 (1994). 26. Higgins, D. G., Thompson, J. D. & Gibson, T. J. Using CLUSTAL for multiple sequence alignments. Methods Enzymol. 266, 383402 (1994). 27. Hill, F., Gemund, C., Benes, V., Ansorge, W. & Gibson, T. J. An estimate of large-scale sequencing accuracy. EMBO Rep. 1, 2931 (2000). 28. Richterich, P. Estimation of errors in raw DNA sequences: a validation study. Genome Res. 8, 251259 (1998). 29. Denver, D. R. et al. Abundance, distribution, and mutation rates of homopolymeric nucleotide runs in the genome of Caenorhabditis elegans. J. Mol. Evol. 58, 584595 (2004).
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Optimal neural population coding of an auditory spatial cue
Nicol S. Harper1,2 & David McAlpine1
1 Department of Physiology and UCL Ear Institute and 2CoMPLEX, University College London, London WC1E 6BT, UK
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Supplementary Information accompanies the paper on www.nature.com/nature. Acknowledgements We thank L. L. Vassilieva, S. Estes, V. Katju and C. Steding for their respective roles in propagating and maintaining the MA lines over the past 5 years; D. Ash for help with primer sequence design and DNA sequencing; and the Caenorhabditis Genetics Center for providing the C. elegans natural isolates. This work was supported by a University of Missouri Research Board grant to W.K.T., and an NIH grant to M.L. and W.K.T. Competing interests statement The authors declare that they have no competing nancial interests. Correspondence and requests for materials should be addressed to D.R.D. ([e-mail]).
A sound, depending on the position of its source, can take more time to reach one ear than the other. This interaural (between the ears) time difference (ITD) provides a major cue for determining the source location1,2. Many auditory neurons are sensitive to ITDs3,4, but the means by which such neurons represent ITD is a contentious issue. Recent studies question whether the classical general model (the Jeffress model5) applies across species6,7. Here we show that ITD coding strategies of different species can be explained by a unifying principle: that the ITDs an animal naturally encounters should be coded with maximal accuracy. Using statistical techniques and a stochastic neural model, we demonstrate that the optimal coding strategy for ITD depends critically on head size and sound frequency. For small head sizes and/or low-frequency sounds, the optimal coding strategy tends towards two distinct sub-populations tuned to ITDs outside the range created by the head. This is consistent with recent observations in small mammals6,7. For large head sizes and/or high frequencies, the optimal strategy is a homogeneous distribution of ITD tunings within the range created by the head. This is consistent with observations in the barn owl810. For humans, the optimal strategy to code ITDs from an acoustically measured distribution depends on frequency; above 400 Hz a homogeneous distribution is optimal, and below 400 Hz distinct sub-populations are optimal. The ability to localize sound sources has obvious survival value, whether for prey or predator. Many vertebrates, including humans, make use of ITDs to localize sounds in the horizontal plane. These ITDs can be in the order of just a few tens of microseconds. In the Jeffress model5, these minute ITDs are encoded by an array of coincidence-detector neurons. Each neuron is tuned for (responds maximally to) an ITD within the range created by the head size (the physiological range), the precise tuning being determined by the difference in axonal conduction time from each ear. The Jeffress model, developed with human spatial hearing in mind, has received extensive support from studies in barn owls810, and until recently was presumed to apply to mammals11. However, the preferred ITD tuning of neurons recorded in small mammals that use ITDs for sound localization seems to lie outside the physiological range6,7. This is markedly different to the preferred ITD tuning observed in the barn owl and suggested by the Jeffress model. It is now debated whether a single unifying principle or model can any longer explain the means by which ITD is encoded across the wide range of species in which ITD sensitivity is observed. We investigated the possibility that different coding strategies observed in different species can be explained by the demand for accurate ITD coding. For simplicity, we considered rst how to encode most accurately the ITDs in the ongoing ne structure of a pure tone using a population of ITD-tuned model neurons. In mammals, ne-structure ITD sensitivity is restricted to sound frequencies below ,1,500 Hz. At higher frequencies, some ITD information can be conveyed by the envelope structure of complex sounds12; however, this seems to have relatively little impact on localization judgements2,13, and thus we only consider nestructure ITDs. Here, we present the optimal coding strategies for four species with different head sizes and/or different sound-frequency ranges
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over which ITD sensitivity is observed: gerbil, a species with lowfrequency (,,1,500 Hz) ITD sensitivity and a small head; barn owl, a high-frequency (,,10 kHz) ITD specialist with a small head; human, low-frequency ITD sensitivity with a large head; and cat, low-frequency ITD sensitivity with an intermediate head size. We use the simplest neural model, the ITD tuning curve (that is, a function describing a neurons mean ring rate in response to different ITDs). For pure tones, each ITD corresponds to an ongoing interaural phase difference (IPD). IPDs up to half the pure-tone period in magnitude constitute ITD as a proportion of that period. Because pure tones are periodic, IPDs beyond half a cycle of the period are subject to phase wrapping. For example, an IPD of 0.6 cycles is equivalent to an IPD of 20.4 cycles. Although measured ITD tuning curves scale with frequency, when considered in terms of IPD they are largely invariant with frequency, being bellshaped around a best IPD (that is, the IPD that evokes maximum discharge rate; Fig. 1a). Thus, for convenience, all ITD tuning curves are represented in terms of the equivalent IPD. We approximate the IPD tuning curves with a cosine-based function (see Methods). We assume that IPD is encoded in the neural populations spike counts within a time window, with intrinsic Poisson noise producing variability in the spike counts. The overall measure of the populations coding error is the sum of the error for each IPD, weighted by the probability of occurrence of that IPD. The measure of coding error for each IPD is the variance of the populations representation of that IPD, as calculated from Fisher information (see Methods). For each species, for each pure-tone frequency, we determine the optimal coding strategy (optimal distribution of best IPDs) by shifting systematically the best IPD of each neuron (in a population of 200) until the overall coding error is minimized. For the probability distribution of the IPDs of a pure tone we observe that the maximum ITD an animal experiences under natural listening conditions is limited by the size of its head. The maximum IPD corresponds to the maximum ITD as a proportion of the pure-tone period, up to a limit of half a cycle. We assume that only IPDs with magnitude smaller than the maximum IPD (that is, within the physiological range) are encountered, and that all IPDs within this range are equally likely. The effect of phase wrapping on the IPD probability distribution is ignored, as including it in the model has little qualitative effect on the results. The physiological range for each tone frequency is illustrated in Fig. 1b for the cat. The rst species considered, the gerbil, has a relatively small head with a maximum ITD of approximately ^120 ms. Consequently, for every frequency at which IPD sensitivity is observed, the gerbils physiological range of IPDs is narrow relative to the IPD tuning curve. The optimal coding strategy for a 500-Hz pure tone (Fig. 2a) takes the form of two distinct neural sub-populations, with best IPDs positioned either side of zero and beyond the gerbils physiological range. These two sub-populations are apparent over the full range of ITD-sensitive tone frequencies (Fig. 2c), with only a slight divergence at the highest frequencies. This is consistent with recent electrophysiological recordings made in the brainstem of the gerbil7 and the midbrain of the guinea-pig6,14,15, another small mammal with low-frequency IPD sensitivity. It is clear that this solution is optimal from the Fisher information, a local measure of coding accuracy, which is highest over the slopes of the IPD tuning curves rather than at their peaks. To position the peak Fisher information within the physiological range it is therefore necessary to position peaks of IPD functions beyond the physiological range. The solution for the gerbil could be taken to suggest that the optimal coding strategy is largely independent of frequency and involves distinct neural sub-populations. However, the solution for the barn owl indicates that this is not the case. Barn owls, with a similar maximum ITD (^180 ms) to the gerbil, make use of ne-structure ITD cues over a much higher frequency range (3 10 kHz810) than mammals. Over this range, the physiological range of IPDs (Fig. 2b) is wide relative to the IPD tuning curves, and the optimal coding strategy involves, instead, a homogeneous distribution of neurons tuned to IPDs within the physiological range.
Figure 1 Key features of model. a, Illustration of the IPD tuning curve used in the model. b, Physiological range (dark grey) of IPD in a cat, showing the limit of IPD sensitivity (white line) and the maximum IPD for each frequency (black line).
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Figure 2 Optimal distributions of tuning curves for coding IPDs. a, Optimal distribution of tuning curves to code the IPD of 500-Hz tones in gerbil for left (red) and right (blue) sub-populations. Dashed lines show Fisher information curves (scaled by dividing by duration of the spike-count window, T). The grey region indicates the physiological range of IPDs, whereas vertical black lines indicate the maximum IPD. b, Optimal distribution for coding IPDs of 5,000-Hz tones in barn owl (sample only shown). Red lines show tuning curve (solid) and scaled Fisher information (dashed) for one neuron. cf, Optimal distributions for coding pure tone IPDs in gerbil (c), barn owl (d), human (e) and cat (f). The abscissa and ordinate indicate best IPD and pure-tone frequency, respectively. For each frequency, the colour indicates the number of neurons (of 200) with that best IPD. White lines indicate frequency limit for IPD sensitivity, and black lines maximum IPD. Black arrows show how a and b illustrate the distribution at particular frequency bands in c and d, respectively.
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This is consistent with physiological recordings made at all levels of the barn owl binaural pathway810. When the effects of phase wrapping on the IPD distribution are included in the analysis, the homogeneous distribution is maintained, save for slight uctuations in neuron densities. The homogeneous distribution seems to be consistent with the basic tenets of the classical Jeffress model5. The solution is optimal because the slopes of IPD tuning curves, where Fisher information is maximal, always reside within the physiological range, and homogeneity is preferred because of the reciprocal term in equation (2) (see Methods). Previous general theoretical studies using relatively narrow tuning curves also demonstrate that the optimal distribution of preferred tuning follows the probability distribution of the stimulus16. The homogeneous solution departs from the Jeffress model in the form of the presumed coding strategy. The Jeffress model posits local coding for ITD, whereas we assume a population code, in which the slopes of functions are critical. This is consistent with a recent study of the barn owl midbrain17, which reports neural resolution of a parameter related to ITD (azimuthal location of broadband noise) to be highest over the slopes, rather than the peaks, of functions. Furthermore, although the homogeneous distribution is the optimal strategy above 3 kHz in the barn owl, below 3 kHz the solution is more complex, and distinct subpopulations are observed (Fig. 2d). Below approximately 800 Hz, the solution is essentially the same as for the gerbil. There is a paucity of data for barn owl neurons with sound frequency tuning below 3 kHz. However a recent study18 reports a population of ITD-sensitive neurons with tuning for sound frequency extending down to the sub-kHz range. Although this study indicates a subpopulation of neurons tuned around zero ITD, a distinct subpopulation also exists with response maxima to ITDs signicantly beyond the barn owls physiological range. Distinct sub-populations and tuning beyond the physiological range are inconsistent with the Jeffress model; however, they are consistent with the optimal coding model. The optimal coding strategies described thus far consist largely of either two distinct, opposed sub-populations tuned to ITDs beyond the physiological range, or a single homogeneous lying distribution entirely within the physiological range. For humans, however, with their relatively large head size and maximum ITD in excess of ^600 ms, the solution is more variable (Fig. 2e). For the lowest frequencies (,250 Hz) the optimal coding strategy is the same as for the gerbil, a distribution consisting of two distinct sub-populations. For the highest frequencies (.700 Hz) the homogeneous distribution is optimal. For intermediate frequencies (250700 Hz), however, a transition occurs between the two coding strategies, with the optimal strategy consisting of several distinct subpopulations. Similarly, the solution for the cat (Fig. 2f), with a physiological range of ITDs intermediate to that of gerbil and humans, emphasizes the transition distributions. The model indicates that a critical factor in determining the strategy for optimal coding is the relative width of the physiological range of IPDs compared with the width of individual IPD tuning curves. The model assumes the shape of the IPD tuning curve to be similar across species. Therefore, the exact form of the optimal coding strategy depends only on the maximum IPD at that frequency. As maximum IPD scales with frequency and/or head size, the panels of Fig. 2cf are simply scaled and shifted versions of each other. However, it is likely that the optimal strategy will also depend on the exact shape of the IPD probability distribution, and to this end we measured the probability distribution of IPDs for human listeners and applied this distribution to the model. Small microphones, positioned at the entrance to each ear canal, were used to make stereo sound recordings from three human subjects interacting in a range of acoustic environments. IPD probability distributions were calculated for each frequency component in recordings from each subjectenvironment combination (Figs 3ad, see Methods). The pure-tone model can be used to nd the optimal coding strategy for these IPD distributions if it is assumed that ITD coding in each frequency band is optimized independently, a reasonable assumption as auditory processing is carried out within nite frequency channels up to a very high level in the auditory pathway. Figure 3eh illustrates the corresponding solutions for each subjectenvironment combination. Above 400 Hz, the optimal coding strategy is a homogeneous distribution, consistent with the previous assumption of equal probability IPDs. Despite the IPD distributions being subject to the effects of phase wrapping, only small uctuations in the homogeneous distribution were observed.
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Figure 3 Probability distributions of IPDs at different frequencies for three human subjects. ad, Subjects in a room (ac), and outdoors (d; same subject as in c). The IPD distribution at each frequency is normalized to maximum. The abscissa and ordinate indicate IPD and frequency, respectively. Colour indicates the normalized probability. eh, Optimal distributions of best IPDs for ad. The abscissa and ordinate indicate best IPD and frequency, respectively. For each frequency, the colour indicates the number of neurons (out of 200) with that best IPD. White lines indicate the frequency limit of IPD sensitivity.
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Below 400 Hz, however, a modication of the two sub-population solution is optimal, with two central sub-populations and two additional, less populous, anking sub-populations. Although we assume ITD coding to be optimized independently for each frequency band, similar solutions are found without this assumption. If we optimize a population of model neurons to encode most accurately the ITD of a broadband noise, with the ITD tuning of each neuron modelled by a Gabor function with randomly assigned frequency, the solution is consistent with the pure-tone model (data not shown). The two sub-population solution was obtained for low-frequency-tuned neurons and small head sizes. For larger head sizes and higher frequency tuned neurons, the solution tended towards the homogeneous distribution. Furthermore, if the measure of coding accuracy is based on mutual information16 rather than expected variance, it has little qualitative effect on the solutions of the models. Finally, as long as spike-rate variance is approximately proportional to mean spike rate, the same coding strategies result. A question arises as to the developmental mechanism by which each optimal distribution could be achieved. Where the model suggests only one coding strategy across the entire frequency range at which ITDs are used (for example, 310 kHz in the barn owl) the optimal distribution could be hardwired, possibly by means of neural delay lines 19. Another, more dynamic, mechanism is suggested by recent evidence from the gerbil of a fast glycinemediated input that modulates the best ITD of medial superior olive neurons7. Evidence suggests that this inhibitory input can be modulated over development by the sound environment20. Such a regulatory mechanism for ITD tuning would be useful where developmental changes in head size inuence which coding strategy is optimal, or where, as we predict for humans, different coding strategies are optimal for different sound frequencies. The relatively simple model we describe provides an explanation for reported differences in neural coding of ITDs between different mammals, and between mammals and barn owls610,18. Assuming only the requirement of accurate coding of this auditory spatial cue, the model suggests a general framework in which to predict the coding strategy adopted by any species, depending on head size and ITD-sensitive sound frequency range. Two general categories of coding strategy were apparent: one with distinct sub-populations of ITD-sensitive neurons, and one with a homogeneous population. The model also predicts that some species, humans for example, may use different ITD-coding strategies over different sound frequency ranges. The assumption of accurate coding provides a unifying principle by which future investigations into the anatomical and physiological mechanisms responsible for ITD sensitivity may be guided. A
population Fisher information is given by: FvjJ
N X i1
f vjJi ; with f vjJi T
v gvjJi
2 3
gvjJi
V(J) is minimized using a conjugate-gradient-based algorithm (Minimize; http:// www.kyb.tuebingen.mpg.de/bs/people/carl/code/). It nds the minimum within 1,000 line searches, and usually fewer than 50. J is initially normally distributed around 0 IPD, however the same solutions result when other initial distributions are used.
IPD tuning curves
For the IPD tuning curve we chose the function g(vjJ i) to provide a reasonable t to the inner slope of a number of tuning curves recorded from the guinea-pig inferior colliculus. The exact function is: !4 11 gvjJi rmax cosv 2 Ji 4 22 The maximum ring rate r max, like the window length T, is a constant that can be ignored in the optimization so long as r maxT is sufciently large23. The maximum ring rates tend to be high in the auditory brainstem (.100 spikes per second), and psychophysically derived binaural integration times are long (,100 ms)24.
Measuring the probability distribution of IPDs
Small microphones (Knowles FG3452) were xed at the entrance to each ear canal in three human subjects and ambient sounds recorded over approximately 2 min. The subject environment combinations were male 1 (Fig. 3a) or female 1(Fig. 3b) in conversation with the experimenter walking around a room (4 4 3 m), male 2 walking in the same environment, but in the absence of conversation (Fig. 3c), and male 2 seated in the front quadrangle (100 50 m) of University College London, with multiple other persons in the quadrangle (Fig. 3d). For each subjectenvironment combination the recorded stereo sound was partitioned into sections using a 100-ms (seven standard deviations) gaussian window, shifted in 50-ms increments. A Fast Fourier transform was applied to each section for each ear, to give the phase for each frequency band. From this, the IPD for each frequency band in each section was calculated as the difference in phase between the ears. The probability distribution of IPDs across all of the sections was then obtained for each frequency band.
Received 25 February; accepted 18 June 2004; doi:10.1038/nature02768.
1. Rayleigh, L. On our perception of sound direction. Phil. Mag. 13, 214232 (1907). 2. Wightman, F. L. & Kistler, D. J. The dominant role of low-frequency interaural time differences in sound localisation. J. Acoust. Soc. Am. 91, 16481661 (1992). 3. Kuwada, S. & Yin, T. C. T. Binaural interaction in low-frequency neurons in inferior colliculus of the cat. I. Effects of long interaural delays, intensity, and repetition rate on interaural delay function. J. Neurophysiol. 50, 981999 (1983). 4. Yin, T. C. T. & Chan, J. C. K. Interaural time sensitivity in medial superior olive of cat. J. Neurophysiol. 64, 465488 (1990). 5. Jeffress, L. A. A place theory of sound localisation. J. Comp. Physiol. Psychol. 41, 3539 (1948). 6. McAlpine, D., Jiang, D. & Palmer, A. R. A neural code for low-frequency sound localization in mammals. Nature Neurosci. 4, 396401 (2001). 7. Brand, A., Behrend, O., Marquardt, T., McAlpine, D. & Grothe, B. Precise inhibition is essential for microsecond interaural time difference coding. Nature 417, 543547 (2002). 8. Takahashi, T. & Konishi, M. Selectivity for interaural time difference in the owls midbrain. J. Neurosci. 6, 34133422 (1986). 9. Wagner, H., Takahashi, T. & Konishi, M. Representation of interaural time difference in the central nucleus of the barn owls inferior colliculus. J. Neurosci. 7, 31053116 (1987). 10. Coles, R. B. & Guppy, A. Directional hearing in the barn owl (Tyto alba). J. Comp. Physiol. A 163, 117133 (1988). 11. Joris, P. X., Smith, P. H. & Yin, T. C. T. Coincidence detection in the auditory system: 50 years after Jeffress. Neuron 21, 12351238 (1998). 12. Henning, G. B. Detectability of interaural delay in high-frequency complex waveforms. J. Acoust. Soc. Am. 55, 8490 (1974). 13. Bernstein, L. R. & Trahiotis, C. Lateralization of sinusoidally amplitude-modulated tones: Effects of spectral locus and temporal variation. J. Acoust. Soc. Am. 78, 514523 (1985). 14. Skottun, B. C., Shackleton, T. M., Arnott, R. H. & Palmer, A. R. The ability of inferior colliculus neurons to signal differences in interaural delay. Proc. Natl Acad. Sci. USA 98, 1405014054 (2001). 15. Shackleton, T. M., Skottun, B. C., Arnott, R. H. & Palmer, A. R. Interaural time difference discrimination thresholds for single neurons in the inferior colliculus of Guinea pigs. J. Neurosci. 23, 716724 (2003). 16. Brunel, N. & Nadal, J. P. Mutual information, Fisher information, and population coding. Neural Comput. 10, 17311757 (1998). 17. Bala, A. D., Spitzer, M. W. & Takahashi, T. T. Prediction of auditory spatial acuity from neural images on the owls au...