Characterization of the decline in auditory nerve phase locking at high frequencies

The central prerequisite for encoding temporal fine structure (TFS) in the auditory system is the phase locking of auditory nerve (AN) fibers to the frequency of a stimulus (Moore, 2008). This phase-locking is usually quantified by vector strength (VS) (Goldberg and Brown, 1969). The detailed behavior of vector strength across frequency in humans remains unclear and estimates of a so called “upper frequency limit” vary substantially from 1.5 to 10 kHz (Verschooten , 2019). Establishing the vector strength of phase locking in the AN as a function of frequency would be highly informative for elucidating various auditory coding mechanisms from pitch perception to binaural hearing.

It is well known that vector strength in single fibers of the AN declines above a certain frequency (Rose , 1967; Johnson, 1980; Joris , 1994) that varies across species (Weiss and Rose, 1988a). The frequency above which vector strength becomes insignificant is often referred to as the upper frequency limit (Palmer and Russell, 1986; Verschooten , 2015; Joris and Verschooten, 2013). Weiss and Rose (1988a) criticized the comparison of the upper frequency limit among different species stating: “This metric obviously depends upon the method of detection and is generally the highest frequency for which the experimenter detected synchronization in the measurements.” The upper frequency limit is therefore ill-defined. Any attempt to define the upper limit for this continually decreasing function depends entirely on the noise floor [Michael Heinz in Verschooten (2019)]. Despite these shortcomings, the ill-defined characterization of the upper frequency limit is still often used in the auditory literature (Liu , 2006; Verschooten , 2015; Joris and Verschooten, 2013; Verschooten , 2019).

To estimate the filter order, Weiss and Rose (1988a) compensated for the different corner frequencies across species and fitted a regression line to the five highest frequency points [two from Johnson (1980) and three from Weiss and Rose (1988b)], resulting in a decline of $≈ 106$ dB/decade. This line is reproduced as the dotted black line in Fig. 1. Overall, this decline was related to a filter order of 4–6 by Weiss and Rose (1988a). However, using a cascaded low-pass filter of order n = 5 with a corner frequency $f corner = 2500$ does not account for the vector strength, e.g., of the cat data from Johnson (1980). Figure 1 clearly shows that the transition band of the filter is too wide and the nominal decline of 100 dB/decade is reached only at much higher frequencies.

Despite these discrepancies to physiological data, the filter is prevalent in computational auditory models (see Table 1 for an overview). Some of the studies [e.g., Bernstein and Trahiotis (1996)] motivate their filter order based on the nominal order suggested by Weiss and Rose (1988a), arguably requiring a very low corner frequency of 425 Hz to obtain a sufficiently steep roll-off at higher frequencies. Others, e.g., Heinz (2001), instead use a higher filter order for the effect of the inner hair cell (IHC), whereby the combination with their other stages (outer hair cell, synapse) reproduces the phase-locking roll-off from Johnson (1980).

The two goals of the current study were therefore: (1) To revisit the fitting of a low-pass filter to vector strength data and (2) to quantify the decline of AN phase locking as a function of frequency.

As data sets, we used vector strength derived from single-unit neuron recordings: from AN fibers in Johnson (1980) from cats, as well as from AN fibers in Heeringa (2020) from gerbils and in Palmer and Russell (1986) from guinea pigs. As the error metric, we chose the root mean square error (RMSE) in the logarithmic space. The RMSE for all three functions describing the vector strength in the three data sets was calculated as a function of order n, gain g, and corner frequency $f corner$⁠. The minimum RMSE fit was derived by a grid search, the range and step size of the individual parameters are shown in Table 2. For the calculations, matlab (MathWorks, Natick, MA) was used. By definition, the filter order is an integer number but the filter equations produce meaningful output also for any positive non-integer number n. As the focus of the present study is on describing the data, some non-integer values for n were included.

For eight of the nine fitted functions (3 types × 3 data sets), a global best fit was obtained within the abovementioned parameter range, i.e., not at the lowest or highest parameter values. The only exception is CASCADE, which showed no local minimum for the order in Johnson (1980). The best solutions are shown above the respective data sets in the left column in Fig. 3. The right column of Fig. 3 shows the RMSE across order n for the three functions and three data sets. LINEAR results in the lowest decline estimates followed by BUTTERW, followed by CASCADE. For CASCADE the best fit low-pass filter order is 3 for Palmer and Russell (1986) and 4 for (Heeringa , 2020). For these two data sets the roll-off within the shown range is about 40 dB/dec, as the cascade filters have not reached their maximum roll-off. For CASCADE, the RMSE declines across n for the Johnson (1980) data set and shows no local minimum for $n ≤ 16$⁠. This decrease is caused by the data point with the highest frequency and lowest vector strength. For all data sets BUTTERW provides the lowest RMSE solutions. Fits were similar when only data with frequencies > 1.8 kHz or > 3 kHz were considered to estimate the roll-off.

The decline in vector strength at high frequencies is much flatter than implied by the nominal low-pass filter order. Especially for the cascade filter, the slope at intermediate frequencies is different from the asymptotic values (cf. Fig. 2). Linear fits of the high-frequency roll-off suggest a steepness of about 40 dB/dec, corresponding to the asymptotic steepness of a second order low-pass filter.

As the low-pass characteristic of the phase locking decline is the outcome of a sequential interaction of several processing stages, the use of a cascade filter is a logical means for a biologically feasible simulation of the underlying processes. The low-pass filter order of this type is at least 3 and even much larger orders provide good fits. The roll-off within the range shown is generally no more than about 60 dB/dec, still considerably lower than the 106 dB/dec derived by Weiss and Rose (1988a). While it produced generally accurate fits, the biologically less plausible filters had lower RMS errors in all three data sets.

While the corner frequency appears to differ between different species, Palmer and Russell (1986) and Weiss and Rose (1988a) concluded that there is a similar decline across species. The similarity in decline across species is unsurprising, considering that the free parameters of the resistor-capacitor (RC) low-pass filter can only influence the corner frequency but not the filter order (Lopez-Poveda and Eustaquio-Martín, 2006). On the other hand, Altoè (2018) showed that the voltage-dependent activation of the K⁺ channels in the IHC enhances the phase-locking properties within a small resonant frequency range. The decline above the resonant frequency range is consequently steeper than with a conventional RC low-pass filter.

Without investigating specific mechanisms in the IHC-AN complex, this study only analyses the function of VS with respect to frequency that can be used in functional models. Thus, we do not claim that any of the filters we employ correspond to the physiological processes. From the fitted parameter (see Fig. 2), a small difference in slope can be deduced, with the corner frequency varying between species, as was reported by Weiss and Rose (1988a). This finding suggests that the common practice of using the same filter order in models of the human auditory system as determined for animal models [e.g., Bernstein and Trahiotis (1996)] is plausible.

The decline of vector strength across frequency is particularly important for understanding the interaural phase difference (IPD) encoding mechanism. Phase-locking information is a strict prerequisite, as it is the only available cue for IPD encoding. In normal hearing humans and for pure tones, it is well established that the sensitivity to IPD decreases rapidly at frequencies above $≈$ 1300 Hz (Brughera , 2013; Klug and Dietz, 2022). A common hypothesis is that this decline in IPD sensitivity is caused by the decline in phase locking of the AN fibers (Joris and Verschooten, 2013; Verschooten , 2019). In computational models of human IPD perception, the decline of AN phase locking usually contributes considerably, or even exclusively, to the decline in IPD sensitivity (Breebaart , 2001b; Brughera , 2013; Bouse , 2019). However, the decline in vector strength in the AN (⁠ $m ≤ 60$ dB/dec) is not sufficient to account for the dramatic decline of IPD sensitivity in humans occurring in the narrow range between 1300 and 1500 Hz (⁠ $m ≥ 150$ dB/dec), as reported in Klug and Dietz (2022). Therefore, the high-frequency limit of IPD sensitivity is a less suitable correlate for the AN phase locking “limit” than previously thought (Verschooten , 2019; Bernstein and Trahiotis, 1996) and another effect must contribute to limiting IPD sensitivity.

The nominal decline of 80 to 120 dB/dec reported by Weiss and Rose (1988a) does not accurately describe the decline of vector strength in AN fibers across frequency. In the biophysically plausible and widely used cascade filter, the slope changes slowly with frequency, so the slope at intermediate frequencies differs substantially from its asymptote. The relevant vector strength values decline with approximately 40 dB/dec—the asymptotic decline of a second order filter.

We thank Amarins Heeringa for sharing her data. Furthermore, we are grateful to Henri Pöntynen for his assistance with formulations. This project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation Programme (Grant Agreement No. 716800).