Rate-intensity Functions Research Articles

A study of a survival data using kernel estimates of hazard rate and aging intensity functions

Analyzing survival (life-testing) data and drawing inferences about them is a part of engineering and health sciences. So far, various statistical tools, e.g., survival (reliability) function (s f ), probability density function (pd f ), and hazard rate function (HR) were available among decision-making scientists to handle time-to-event data (complete or censored). But because functions (pd f ) estimators were interval (window) based, they mostly gave qualitative ideas having pictorial representation resembling step functions, ordinate remain constant when abscissa vary over an interval, thereby giving incomplete information. However, it can be sorted out with the use of kernel estimates of the above mentioned functions, resulting into smooth estimators. Moreover, the metric based on aging intensity function (AI) gives an alternative way of studying lifetime or clinical datasets as it is a quantitative measure (not interval-based), thereby depicting a broader view of a given data. In our study, we primarily focus on AI and HR functions estimated using four different kernels. We apply them to a case study of patients with primary malignant tumors of sternum (cf. Daniel and Cross, 2014) with the right-censored data. Our result shows that kernel estimates of HR and AI functions for patients with high grade tumor (HGT) are higher than for patients with low grade tumor (LGT), as expected. Thus, the study opens up a new direction for applying AI and HR functions in health sciences and engineering studies.

On ageing intensity function of some Weibull models

There are many measures for qualitative and quantitative analysis of ageing phenomena of a system with a well defined statistical distribution, namely, hazard rate, ageing intensity (AI) function etc. The extensive usage of Weibull distributions in various fields has prompted us to explore the ageing property of these models. The present work derives the AI function of recent continuous Weibull distributions. In the process, we obtain generalizations of a few recent Weibull models. As noted, the characteristics of a distribution described by hazard rate and ageing intensity function differ altogether in many cases. Some of the results presented in this work have been analyzed through a simulation study using R software.

Cross-modality modulation of auditory midbrain processing of intensity information

In nature, animals constantly receive a multitude of sensory stimuli, such as visual, auditory, and somatosensory. The integration across sensory modalities is advantageous for the precise processing of sensory inputs which is essential for animals to survival. Although some principles of cross-modality integration have been revealed by many studies, little insight has been gained into its functional potentials. In this study, the functional influence of cross-modality modulation on auditory processing of intensity information was investigated via recording neuronal activity in the auditory midbrain (i.e., inferior colliculus, IC) under the conditions of visual, auditory, and audiovisual stimuli, respectively. Results demonstrated that combined audiovisual stimuli either enhanced or suppressed the responses of IC neurons compared to auditory stimuli alone, even though the same visual stimuli alone induced no response. Audiovisual modulation appeared to be strongest when the combined audiovisual stimuli were located at the best auditory azimuth of neurons as well as when presented with intensity at near-threshold levels. Additionally, the rate-intensity function of IC neurons to auditory stimuli was expanded or compressed by audiovisual modulation, which was highly dependent on the minimal threshold (MT) of neurons. Lowering of the MT and greater audiovisual modulation for the neuron indicated an intensity-specific enhancement of auditory intensity sensitivity by cross-modality modulation. Overall, evidence suggests a potential functional role of cross-modality modulation in IC that serves to instruct adaptive plasticity to enhance the auditory perception of intensity information.

The Interplay Between Spike-Time and Spike-Rate Modes in the Auditory Nerve Encodes Tone-In-Noise Threshold.

Auditory nerve fibers (ANFs) encode pure tones through two modes of coding, spike time and spike rate, depending on the tone frequency. In response to a low-frequency tone, ANF firing is phase locked to the sinusoidal waveform. Because time coding vanishes with an increase in the tone frequency, high-frequency tone coding relies on the spike rate of the ANFs. Adding a continuous broadband noise to a tone compresses the rate intensity function of ANFs and shifts its dynamic range toward higher intensities. Therefore, the ANFs with high-threshold/low-spontaneous rate (SR) are thought to contribute to behavioral tone detection in noise. However, this theory relies on the discharge rate of the ANFs. The direct comparison with the masking threshold through spike timing, irrespective of the spontaneous rate, has not so far been investigated. Taking advantage of a unique proxy to quantify the spike synchrony (i.e., the shuffle autocorrelogram), we show in female gerbils that high-SR ANFs are more adapted to encode low-frequency thresholds through temporal code, giving them a strong robustness in noise. By comparing behavioral thresholds measured using prepulse inhibition of the acoustical startle reflex with population thresholds calculated from ANFs pooled per octave band, we show that threshold-based spike timing provides a better estimate of behavioral thresholds in the low-frequency range, whereas the high-frequency behavioral thresholds rely on the spiking rate, particularly in noise. This emphasizes the complementarity of temporal and rate modes to code tone-in-noise thresholds over a large range of frequencies.SIGNIFICANCE STATEMENT There is a general agreement that high-threshold/low-spontaneous rate (SR) auditory nerve fibers (ANFs) are of prime importance for tone detection in noise. However, this theory is based on the discharge rate of the fibers. Comparing the behavioral thresholds and single ANF thresholds shows that this is only true in the high-frequency range of tone stimulations. In the low-frequency range of tones (up to 2.7 kHz in the gerbil), the most sensitive ANFs (high-SR fibers) carry neural information through a spike-timing mode, even for noise in which tones do not induce a noticeable increment in the spike rate. This emphasizes the interplay between spike-time and spike-rate modes in the auditory nerve to encode tone-in-noise threshold over a large range of tone frequencies.

The Effect of Complex Acoustic Environment during Early Development on the Responses of Auditory Cortex Neurons in Rats

Acoustical environment plays an important role during the maturation of the auditory system. It has been shown that the sensory inputs to the developing centres influence the development of the structure of projections, neuronal responsiveness, excitatory-inhibitory balance, or tonotopical arrangement, throughout the auditory pathway. Our previous study (Bures et al., 2014) showed that rats reared in a complex acoustic environment (spectrally and temporally modulated sound reinforced by an active behavioural paradigm with a positive feedback) exhibit permanently improved response characteristics of the inferior colliculus (IC) neurons. Extending these results, the current work provides evidence that the changes occur also at the level of auditory cortex (AC). In particular, the enriched animals have lower excitatory thresholds, sharper frequency selectivity, and a lower proportion of non-monotonic rate-intensity functions. In contrast to the changes observed in the IC, the cortical neurons of enriched animals have lower response magnitudes. In addition, the enrichment changed the AC responsiveness to frequency-modulated and also to a lesser extent, amplitude-modulated stimuli. Significantly, the alterations span the entire hearing range and may be regarded as general and not directly linked to the characteristics of the acoustical stimulation. Furthermore, these developmentally induced changes are permanent and detectable in adulthood. The findings indicate that an acoustically enriched environment during the critical period of postnatal development influences basic properties of neuronal receptive fields in the AC, which may have implications for the ability to detect and discriminate sounds.

English

Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided.

Sound coding in the auditory nerve of gerbils

Gerbils possess a very specialized cochlea in which the low-frequency inner hair cells (IHCs) are contacted by auditory nerve fibers (ANFs) having a high spontaneous rate (SR), whereas high frequency IHCs are innervated by ANFs with a greater SR-based diversity. This specificity makes this animal a unique model to investigate, in the same cochlea, the functional role of different pools of ANFs. The distribution of the characteristic frequencies of fibers shows a clear bimodal shape (with a first mode around 1.5 kHz and a second around 12 kHz) and a notch in the histogram near 3.5 kHz. Whereas the mean thresholds did not significantly differ in the two frequency regions, the shape of the rate-intensity functions does vary significantly with the fiber characteristic frequency. Above 3.5 kHz, the sound-driven rate is greater and the slope of the rate-intensity function is steeper. Interestingly, high-SR fibers show a very good synchronized onset response in quiet (small first-spike latency jitter) but a weak response under noisy conditions. The low-SR fibers exhibit the opposite behavior, with poor onset synchronization in quiet but a robust response in noise. Finally, the greater vulnerability of low-SR fibers to various injuries including noise- and age-related hearing loss is discussed with regard to patients with poor speech intelligibility in noisy environments. Together, these results emphasize the need to perform relevant clinical tests to probe the distribution of ANFs in humans, and develop appropriate techniques of rehabilitation.This article is part of a Special Issue entitled <Annual Reviews 2016>.

Neural coding of monaural and binaural intensity at low stimulus frequencies

At low sound frequencies, spikes in the auditory nerve (AN) are phase-locked to the stimulus waveform. For sinusoidal stimuli, spike occurrences are restricted to the positive half-waves of the sound cycles, which constraints possible values of spike rates and their variability. As the spike rates are used for representation of sound intensity, the just noticeable differences (JNDs) of sound level and of inter-aural level difference (ILD) will be influenced. Due to the lack of appropriate AN data, we explore the topic using a computational model [1]. Three possible types of spike trains are considered: a synchronous, an asynchronous spike train with variable intensity, and a periodically restarting (with sound period) asynchronous process. Generally, the synchronous process leads to several times lower JNDs than the other processes. The JNDs depend on further parameters: longer counting windows or higher number of converging fibers result in smaller JNDs, larger spike timing jitter (random displacement of spikes from their ideal position) slightly increases the JNDs. Notably, the dependence of JNDs on mean spike rate is in contradiction to psychophysical observations that report monotonic decrease of JND of intensity with increasing sound pressure level, the so called near-miss to Weber's law [2]. Even with ideal phase-lock and no jitter, the simulated JND of the synchronous algorithm has a bell shape, the other two algorithms give monotonically increasing function (Figure ​(Figure1A,1A, colored lines). Recruitment of more AN fibers with rising intensity overcomes this discrepancy; a simulation of this mechanism yields results well matching the psychophysical results (Figure ​(Figure1A,1A, black line). Figure 1 Just-noticeable differences of sound level depending on spike rate (A) and sound frequency (B). Another discrepancy of the modeling results with psychophysics is the simulated JND increasing with frequency, given a constant spike rate (Figure ​(Figure1B),1B), while psychophysical JNDs decrease with frequency. However, if we assume that maximum spike rate is in some way limited by the frequency (e.g., at most one spike can occur in each sound cycle) and that a given intensity range (e.g., 30 dB) is always mapped to the available range of spike rates (e.g., between zero and sound frequency), then the slope of the rate-intensity function increases and the JND decreases with frequency. The results show that intensity coding has specific properties at low sound frequencies, deserving more detailed electrophysiological studies than what is available in experimental literature today.

Acoustic signal characteristic detection by neurons in ventral nucleus of the lateral lemniscus in mice.

Under free field conditions, we used single unit extracellular recording to study the detection of acoustic signals by neurons in the ventral nucleus of the lateral lemniscus (VNLL) in Kunming mouse (Mus musculus). The results indicate two types of firing patterns in VNLL neurons: onset and sustained. The first spike latency (FSL) of onset neurons was shorter than that of sustained neurons. With increasing sound intensity, the FSL of onset neurons remained stable and that of sustained neurons was shortened, indicating that onset neurons are characterized by precise timing. By comparing the values of Q10 and Q30 of the frequency tuning curve, no differences between onset and sustained neurons were found, suggesting that firing pattern and frequency tuning are not correlated. Among the three types of rate-intensity function (RIF) found in VNLL neurons, the proportion of monotonic RIF is the largest, followed by saturated RIF, and non-monotonic RIF. The dynamic range (DR) in onset neurons was shorter than in sustained neurons, indicating different capabilities in intensity tuning of different firing patterns and that these differences are correlated with the type of RIF. Our results also show that the best frequency of VNLL neurons was negatively correlated with depth, supporting the view point that the VNLL has frequency topologic organization.

Acoustical enrichment during early postnatal development changes response properties of inferior colliculus neurons in rats.

The structure and function of the auditory system may be influenced by acoustic stimulation, especially during the early postnatal period. This study explores the effects of an acoustically enriched environment applied during the third and fourth week of life on the responsiveness of inferior colliculus neurons in rats. The enrichment comprised a spectrally and temporally modulated complex sound reinforced with several target acoustic stimuli, one of which triggered a reward release. The exposure permanently influenced neuronal representation of the sound frequency and intensity, resulting in lower excitatory thresholds at neuronal characteristic frequency, an increased frequency selectivity, larger response magnitudes, steeper rate-intensity functions and an increased spontaneous activity. The effect was general and non-specific, spanning the entire hearing range - no changes specific to the frequency band of the target stimuli were found. The alterations depended on the activity of animals during the enrichment - a higher activity of rats in the stimulus-reward paradigm led to more profound changes compared with the treatment when the stimulus-reward paradigm was not used. Furthermore, the exposure in early life led to permanent changes in response parameters, whereas the application of the same environment in adulthood influenced only a subset of the examined parameters and had only a temporary effect. These findings indicate that a rich and stimulating acoustic environment during early development, particularly when reinforced by positive feedback, may permanently affect signal processing in the subcortical auditory nuclei, including the excitatory thresholds of neurons and their frequency and intensity resolution.

Interdependent effects of sound duration and amplitude on neuronal onset response in mice inferior colliculus

In this study, we adopted iso-frequency pure tone bursts to investigate the interdependent effects of sound amplitude/intensity and duration on mice inferior colliculus (IC) neuronal onset responses. On the majority of the sampled neurons (n=57, 89.1%), sound amplitude and duration had effects on the neuronal response to each other by showing complex changes of the rat-intensity function/duration selectivity types and/or best amplitudes (BAs)/durations (BDs), evaluated by spike counts. These results suggested that the balance between the excitatory and inhibitory inputs set by one acoustic parameter, amplitude or duration, affected the neuronal spike counts responses to the other. Neuronal duration selectivity types were altered easily by the low-amplitude sounds while the changes of rate-intensity function types had no obvious preferred stimulus durations. However, the first spike latencies (FSLs) of the onset response neurons were relative stable to iso-amplitude sound durations and changing systematically along with the sound levels. The superimposition of FSL and duration threshold (DT) as a function of stimulus amplitude after normalization indicated that the effects of the sound levels on FSLs are considered on DT actually.

Inequalities involving expectations to characterize distributions

The characterization of distributions is well known in the field of Statistics and Reliability. This paper characterizes a few distributions with the help of failure rate, mean residual, log-odds rate, and aging intensity functions.

Asymmetric response properties of rapidly adapting mechanoreceptive fibers in the rat glabrous skin

Previous histological and neurophysiological studies have shown that the innervation density of rapidly adapting (RA) mechanoreceptive fibers increases towards the fingertip. Since the psychophysical detection threshold depends on the contribution of several RA fibers, a high innervation density would imply lower thresholds. However, our previous human study showed that psychophysical detection thresholds for the Non-Pacinian I channel mediated by RA fibers do not improve towards the fingertip. By recording single-unit spike activity from rat RA fibers, here we tested the hypothesis that the responsiveness of RA fibers is asymmetric in the proximo-distal axis which may counterbalance the effects of innervation density. RA fibers (n = 32) innervating the digital glabrous skin of rat hind paw were stimulated with 40-Hz sinusoidal mechanical bursts at five different stimulus locations relative to the receptive field (RF) center (two distal, one RF center, two proximal). Different contactor sizes (area: 0.39, 1.63, 2.96 mm2) were used. Rate-intensity functions were constructed based on average firing rates, and the absolute spike threshold and the entrainment threshold were obtained for each RA fiber. Thresholds for proximal stimulus locations were found to be significantly higher than those for distal stimulus locations, which suggests that the mechanical stimulus is transmitted better towards the proximal direction. The effect of contactor size was not significant. Mechanical impedance of the rat digital glabrous skin was further measured and a lumped-parameter model was proposed to interpret the relationship between the asymmetric response properties of RA fibers and the mechanical properties of the skin.

Noise exposure during early development impairs the processing of sound intensity in adult rats

During the early postnatal development of rats, the structural and functional maturation of the central auditory nuclei strongly relies on the natural character of the incoming neural activity. Even a temporary deprivation in the critical period results in a deterioration of neuronal responsiveness in adult animals. We demonstrate that besides the poorer frequency selectivity of neurons in the impaired animals reported previously [Grecova et al. (2009)Eur. J. Neurosci. 29, 1921-1930], the neuronal representation of sound intensity is significantly affected. Rate-intensity functions of inferior colliculus neurons were recorded in anaesthetized adult rats that were exposed to intense noise at postnatal day 14, and compared with those obtained in age-matched controls. Although the response thresholds were similar in the exposed and control rats, the neurons in the exposed animals had a longer first-spike latency, a narrower dynamic range, lower maximum response magnitudes and a steeper slope of the rate-intensity functions. The percentage of monotonic neurons was significantly lower in the exposed animals. The observed anomalies were confined to the mid- and high-frequency regions, whereas no significant changes were found in the low-frequency neurons. The altered parameters of the individual rate-intensity functions led also to differences in the cumulative responses. We conclude that a brief noise exposure during the critical period leads to a frequency-dependent alteration of the sound intensity representation in the inferior colliculus of adult rats. The results suggest that such impairments may appear in individuals with normal hearing thresholds, but with a history of noise exposure very early in childhood.

Threshold and Beyond: Modeling The Intensity Dependence of Auditory Responses

In many studies of auditory-evoked responses to low-intensity sounds, the response amplitude appears to increase roughly linearly with the sound level in decibels (dB), corresponding to a logarithmic intensity dependence. But the auditory system is assumed to be linear in the low-intensity limit. The goal of this study was to resolve the seeming contradiction. Based on assumptions about the rate-intensity functions of single auditory-nerve fibers and the pattern of cochlear excitation caused by a tone, a model for the gross response of the population of auditory nerve fibers was developed. In accordance with signal detection theory, the model denies the existence of a threshold. This implies that regarding the detection of a significant stimulus-related effect, a reduction in sound intensity can always be compensated for by increasing the measurement time, at least in theory. The model suggests that the gross response is proportional to intensity when the latter is low (range I), and a linear function of sound level at higher intensities (range III). For intensities in between, it is concluded that noisy experimental data may provide seemingly irrefutable evidence of a linear dependence on sound pressure (range II). In view of the small response amplitudes that are to be expected for intensity range I, direct observation of the predicted proportionality with intensity will generally be a challenging task for an experimenter. Although the model was developed for the auditory nerve, the basic conclusions are probably valid for higher levels of the auditory system, too, and might help to improve models for loudness at threshold.

Tone and call responses of units in the auditory nerve and dorsal medullary nucleus of Xenopus laevis

The clawed frog Xenopus laevis produces vocalizations consisting of distinct patterns of clicks. This study provides the first description of spontaneous, pure-tone and communication-signal evoked discharge properties of auditory nerve (n.VIII) fibers and dorsal medullary nucleus (DMN) cells in an obligatorily aquatic anuran. Responses of 297 n.VIII and 253 DMN units are analyzed for spontaneous rates (SR), frequency tuning, rate-intensity functions, and firing rate adaptation, with a view to how these basic characteristics shape responses to recorded call stimuli. Response properties generally resemble those in partially terrestrial anurans. Broad tuning exists across characteristic frequencies (CFs). Threshold minima are -101 dB re 1 mm/s at 675 Hz; -87 dB at 1,600 Hz; and -61 dB at 3,000 Hz (-90, -77, and -44 dB re 1 Pa, respectively), paralleling the peak frequency of vocalizations at 1.2-1.6 kHz with approximately 500 Hz in 3 dB bandwidth. SRs range from 0 to 80 (n.VIII) and 0 to 73 spikes/s (DMN). Nerve and DMN units of all CFs follow click rates in natural calls, < or =67 clicks/s and faster. Units encode clicks with a single spike, double spikes, or bursts. Spike times correlate closely with click envelopes. No temporal filtering for communicative click rates occurs in either n.VIII or the DMN.

Deviation from Weber's Law in the Non-Pacinian I Tactile Channel: A Psychophysical and Simulation Study of Intensity Discrimination

This study involves psychophysical experiments and computer simulations to investigate intensity discrimination in the non-Pacinian I (NP I) tactile channel. The simulations were based on an established population model for rapidly adapting mechanoreceptive fibers (Güuçlü & Bolanowski, 2004a). Several intensity codes were tested as decision criteria: number of active neurons, total spike count, maximal spike count, distribution of spike counts among the afferent population, and synchronization of spike times. Simulations that used the number of active fibers as the intensity code gave the most accurate results. However, the Weber fractions obtained from simulations are smaller than psychophysical Weber fractions, which suggests that only a subset of the afferent population is recruited for intensity discrimination during psychophysical experiments. Simulations could also capture the deviation from Weber's law, that is, the decrease of the Weber fraction as a function of the stimulus level, which was present in the psychophysical data. Since the psychophysical task selectively activated the NP I channel, the deviation effect is probably not due to the contribution of another tactile channel but rather is explicitly produced by the NP I channel. Moreover, because simulations with all tested intensity codes resulted in the same effect, the activity of the afferent population is sufficient to explain the deviation, without the need for a higher-order network. Depending on the intensity code used, the mechanical spread of the stimulus, rate-intensity functions of the tactile fibers, and the decreasing spike-phase jitter contribute to the deviation from Weber's law.

Reorganization of receptive fields following hearing loss in inferior colliculus neurons

We explored frequency and intensity encoding in the inferior colliculus (IC) of the C57 mouse model of sensorineural hearing loss. Consistent with plasticity reported in the IC of other models of hearing loss, frequency response areas (FRAs) in hearing-impaired (HI) mice were broader with fewer high-frequency units than normal-hearing (NH) mice. The broad FRAs recorded from HI mice had lower cutoffs on the low frequency edge of the FRA. Characteristic frequency (CF) and sharpness of tuning (Q10) calculated from the FRA were used to divide the sample into four categories: low-CF sharp-FRA, low-CF broad-FRA, high-CF sharp-FRA, and high-CF broad-FRA units. Rate-intensity functions (RIFs) for CF tones and noise were used to determine the minimum and maximum response counts as well as the sound pressure levels resulting in 10%, 50%, and 90% of the maximum spike count. Tone RIFs of broad FRA units were shifted to the right of tone RIFs of sharp FRA units in both NH and HI mouse IC, regardless of the unit CF. The main effects of hearing loss were seen in the noise RIFs. The low-CF broad-FRA units in HI mice had elevated responses to noise, and the high-CF sharp-FRA units in HI mice had lower maximum rates, as compared with the units recorded from NH mice. These results suggest that, as the IC responds to peripheral hearing loss with changes in the representation of frequency, an altered balance between inhibitory and excitatory inputs to the neurons recorded from the HI mice alters aspects of the units’ intensity encoding. This altered balance likely occurs, at least in part, outside of the IC.

Masking effect of different durations of forward masker sound on acoustical responses of mouse inferior collicular neurons to probe sound

To study the effects of different durations of forward masker sound on neuronal firing and rate-intensity function (RIF) of mouse inferior collicular (IC) neurons, a tone relative to 5 dB above the minimum threshold (re MT+5 dB) of the best frequency of recorded neurons was used as forward masker sound under free field stimulation condition. The masker durations used were 40, 60, 80, and 100 ms. Results showed that as masker duration was increased, inhibition in neuronal firing was enhanced (P < 0.000 1, n = 41) and the latency of neurons was lengthened (P<0.01, n = 41). In addition, among 41 inhibited IC neurons, 90.2% (37/41) exhibited narrowed dynamic range (DR) when masker sound duration was increased (P < 0.000 1), whereas the DR of 9.8%(4/41) became wider. These data suggest that masking effects of different durations of forward masker sound might be correlated with the amplitude and duration of inhibitory input to IC neurons elicited by the masker sound.

Dynamic range relations for auditory primary afferents

Dynamic range is one of four attributes typically assigned to the plot of firing rate vs. stimulus level of an auditory primary afferent. Dynamic range is generally understood to be the contiguous range of sound-pressure-level over which the neuron can indicate some small level change. Typically, however, dynamic range has been quantified as the width in decibels between the endpoints of the rate-level plot, which is not a measure of sensitivity to level change. A sensitivity measure is provided here by first deriving an equation for the intensity-difference limen (DL) in terms of attributes of the rate-level curve. The result is a generally U-shaped curve of DL vs. level. Any given criterion DL corresponds to a horizontal line cutting the DL curve at two points, with the separation in decibels between those points providing a dynamic range for that DL criterion. Plotting the dynamic ranges vs. the respective DLs yields a dynamic range curve. These were made for 62 afferents from the cat. The dynamic ranges of sloping-saturating rate-level plots do not exceed those for sigmoidal plots until the DL criterion reaches 50 dB, supporting the conclusion of Palmer and Evans [Cochlear fibre rate-intensity functions: no evidence for basilar membrane nonlinearities, Hearing Research 2 (1980) 319–326] that sloping saturation is not a reflection of cochlear nonlinearity.