 |
INTRODUCTION |
Natural acoustic signals, including many of those used by animals and humans for auditory communication, are spectrally and temporally complex. A recent study has emphasized the importance of the temporal structure of the envelope by showing that it can convey an unexpected amount of information needed for speech recognition (Shannon et al. 1995
). Animal studies have shown that throughout the auditory pathway neurons can be excited by rapid temporal changes in stimulus envelopes, provided that the stimuli have an adequate spectral content. In many studies researchers have used stimuli with repetitive envelope fluctuations, such as periodically amplitude-modulated sinusoids or noise or click trains, and have demonstrated that neuronal responses can be locked to the individual repetitive envelope fluctuations (e.g., auditory nerve: Joris and Yin 1992; cochlear nucleus: Frisina et al. 1985; Rhode and Greenberg 1994; inferior colliculus: Heil et al. 1995; Langner and Schreiner 1988; Rees and Møller 1983; thalamus: Rouiller et al. 1981; cortex: Eggermont 1993; Schreiner and Urbas 1988).
A particularly salient temporal envelope change is the onset of a sound, and nearly all neurons along the auditory pathway respond briskly to such a transient. For example, all physiologically classified neuron types of the cochlear nucleus, with the exception of buildup neurons in the dorsal division, display an initial peak in their poststimulus time histograms recorded in response to short tone bursts (e.g., Rhode and Greenberg 1992). This peak reflects the locking of the neuron's initial spike(s) to the tone's onset, and therefore such responses or response components are sometimes referred to as onset responses. Because of the demonstrated phase-locking of spikes to amplitude-modulated signals or click trains, such signals may constitute a rapid series of like onsets for a neuron. In fact, Rhode and Greenberg (1992) have noted that cochlear nucleus neurons, classified as onset units, phase-lock with high precision also to low-frequency signals (sinusoidal carriers and amplitude-modulated sounds) ". . . responding as if each cycle is an effective excitatory stimulus" (p. 100). Onset response components are also evident in the discharge patterns of neurons in locations higher up the pathway, such as the medial geniculate or the auditory cortex (for review see Clarey et al. 1992). Onset responses appear to be least vulnerable to the effects of anesthesia (Zurita et al. 1994), and the responses of neurons in the auditory cortices of chloralose- and barbiturate-anesthetized animals are dominated by discharges locked to the stimulus onset (e.g., Brugge et al. 1969; Phillips 1988; Zurita et al. 1994).
Although it is widely accepted that the initial discharges of most auditory neurons are evoked by stimulus onset, little attention has been given to the question of which physical parameters of the stimulus onset actually shape a neuron's onset response. When auditory neurons are probed with narrowband stimuli, such as pure tone bursts, the effects of the abruptness of the amplitude change on the short-term frequency spectrum (e.g., Durrant and Lovrinic 1984; Pickles 1988) have been of some concern, and to reduce spectral splatter at signal onset, signals are generally shaped with some finite rise time. The neglect of the physical parameters of sound onsets (other than the general concern about spectral splatter), despite the recognition that the initial discharges of most auditory neurons are evoked by stimulus onsets, has an almost paradoxical consequence: it can be seen in innumerable studies that measures of neuronal properties that were extracted from onset responses (or responses that contained an onset component) are reported and analyzed with respect to stimulus parameters that characterize features of the steady-state or plateau portion of the stimulus. An important case in point is the effect of sound pressure level (SPL) on neuronal onset responses. Alterations of the SPL of a stimulus inevitably coalter features of its onset, particularly when the rise function and the rise time are held constant, as is routinely done. When stimuli are shaped with the widely used linear rise function, for example, the most obvious feature is the slope of the envelope, i.e., the rate at which the peak pressure changes until the plateau value is reached. Any 6-dB increase in SPL will double this rate. A second feature that is coaltered with SPL under such conditions is the quasi-instantaneous acceleration of peak pressure, a parameter whose potential relevance has not been recognized at all. Both stimulus onset parameters are also coaltered when the rise time is altered and the SPL is held constant. Thus it is conceivable that neuronal onset responses might be shaped by factors other than the SPL or the short-term frequency spectrum.
Natural sound onsets will not only vary due to variation of signal SPL, but, because of differences in the manner in which sounds are produced, also due to variation in signal rise time (e.g., Cutting and Rosner 1974; Hall and Feng 1988). In speech sounds, for example, rise time can vary with the manner of articulation (Pickett 1980; Stevens 1980). Rise time can in fact cue perceptual categories in speech (Cutting and Rosner 1974; Stevens 1980), but clearly affects the perception of nonspeech sounds as well (Cutting and Rosner 1974). In humans, the just noticable difference for a change in rise time is ~25% of the duration of the rise time (van Heuven and van den Broecke 1979). Natural signals, including speech sounds, also differ in rise function, but according to our knowledge, in no physiological or psychophysical studies has the potential relevance of this onset feature been investigated. Nevertheless, the auditory system will experience, and may be able to discriminate, a wealth of different sound onsets.
In the present study and the companion paper (Heil 1997) the question of how auditory onset responses code or represent auditory onsets is investigated. This question is addressed by focusing the analysis on onset parameters such as the rate of change or the acceleration of peak pressure. Onset features were varied by varying SPL, rise time, and rise function. In addition to the widely used linear rise function, which is characterized by a constant rate of change of peak pressure during the rise time, cosine-squared rise functions were used. These have the advantage that peak pressure, rate of change, and acceleration of peak pressure are smooth and assessable functions of time that reach their maxima at different points during the rise time and are differentially affected by manipulations of rise time or SPL. Neurons of the primary auditory cortex (AI) are particularly suited to tackle the issue of onset coding because they preferentially respond to sound onsets, and any later discharges, if they occur, can be readily distinguished (e.g., Brugge et al. 1969). Because auditory cortical neurons have complex frequency filters, we have employed simple tonal stimuli to more easily decipher the effects of carrier frequency. A thorough understanding of coding strategies for isolated onsets will also promote our understanding of the coding of envelope transients that occur periodically or aperiodically during the course of complex auditory signals and that are so critical for speech recognition (Shannon et al. 1995). Preliminary reports of some of the findings have been presented (Heil 1996; Heil and Irvine 1996a).
| |
METHODS |
Animal preparation
Seven adult cats (3 females and 4 males, weighing between 2.6 and 3.8 kg) contributed data to this study. All had healthy ears as judged by otoscopic inspections of the tympani and middle ears and by the shapes and sensitivities of the N1 audiogram. Each cat was deeply anesthetized with pentobarbitone sodium (40 mg/kg ip). Atropine (0.3 ml im) was administered to reduce tracheal mucous secretion. A broad-spectrum antibiotic (Amoxil; 0.5 ml im) was also given. The trachea and the radial vein were cannulated and anesthesia was maintained throughout surgery and recordings (up to 30 h) by intravenous injections of pentobarbitone in a physiological saline solution that also contained a few drops of heparin. The electrocardiogram was continuously monitored and rectal temperature was held near 38°C by a thermostatically controlled DC blanket. Surgical procedures have been described in detail elsewhere (Heil et al. 1992b). In brief, the left auditory cortex was exposed by trepanation of the overlying skull and removal of the dura. A specially designed Perspex chamber was mounted to the skull surrounding the opening, filled with warm saline, and sealed with a glass plate on which a small hydraulic microdrive was mounted and that housed the glass-insulated tungsten microelectrode. Each bulla was exposed and a round-window electrode and a length of fine-bore polyethylene tubing, allowing static pressure equalization within the middle ear, were inserted through a small hole. Thereafter the bullae were resealed with dental acrylic. The external meati were also cleared of surrounding tissue and transected to leave only short meatal stubs.
Acoustic stimulation and recording procedures
The cat was located in a sound-attenuating chamber. Stimuli were digitally produced (Tucker Davis Technology) and presented to the cat's ears via precalibrated sealed sound delivery systems. Each system consisted of a STAX SRS-MK3 transducer in a coupler. The sound delivery tube of the coupler fitted snugly into the meatal stub.
During viewing under an operating microscope, the microelectrode (tip diameter ~ 10 µm; impedances ~ 3-5 M
at 1 kHz) was positioned manually close above a chosen point on the cortical surface and was then advanced near-normal to the surface by means of the microdrive. Neural activity was amplified (×1,000) and, for recording of action potentials, also filtered (500-5,000 Hz) and displayed on storage oscilloscopes.
Once a neuron was well isolated, its characteristic frequency (CF; frequency of lowest response threshold) and its preferred laterality of stimulus presentation (viz., monaural ipsilateral, monaural contralateral, or binaural with identical tones to each ear) were determined by manually varying the appropriate stimulus parameters. The discriminator level was set to trigger off either the positive or the negative slope of the filtered action potential waveform, but was not switched between the two during data acquisition. Adjustments of the trigger level during data acquisition were sometimes necessary. However, the effects of this procedure on the trigger instant were very small (<0.1 ms, as judged by inspection of the oscilloscope traces). Event times were stored on disk with 10-µs resolution for off-line analysis.
Under computer control, 20 repetitions of CF tones with a given rise function and a fixed rise time were presented at 1 Hz, at SPLs ranging from below threshold up to 90 dB SPL in 10-dB steps, followed by a measure of spontaneous activity. A different rise time was then selected and the recording procedure was repeated. As many as seven different rise times, covering the range of 1-170 ms, were tested and presented in random sequence. Most neurons were tested with CF tones of their preferred stimulus laterality, but some were tested with other stimulus lateralities and at other frequencies as well. In the latter cases, tones of a given rise function and rise time but of different frequencies and amplitudes were presented pseudorandomly as described in detail elsewhere (Heil et al. 1992b).
All tone bursts were 400 ms in duration including the times comprised by the symmetrical rise and fall functions. Tone bursts were shaped with either linear or cosine-squared rise and fall functions. Because it is the peak pressure PP (measured in Pa, and not the SPL, expressed in dB SPL), that changes according to the rise function, it is thus convenient for the present purpose to express the SPL as the plateau peak pressure PPplateau.
With the cosine-squared rise function used here, peak pressure (in Pa) changes as a function of time t (in s) according to
|
(1)
|
where CRT is the cosine-squared rise time (in s).1
The rate of change of peak pressure RCPP (in Pa/s) varies with time according to
|
(2)
|
Maximum RCPP is reached halfway through the rise time and is given by
|
(3)
|
The acceleration of peak pressure APP (in Pa/s2) varies with time according to
|
(4)
|
Maximum APP occurs at the beginning of the rise time and is given by
|
(5)
|
With a linear rise function, the peak pressure changes according to
|
(6)
|
where LRT is the linear rise time (in s) and PPplateau/LRT identifies the constant rate of change of peak pressure RCPP (in Pa/s). Mathematically, acceleration and deceleration of peak pressure are instantaneous and infinite and occur at the beginning and at the end of the rise time, respectively.
The time courses of peak pressure, rate of change, and acceleration of peak pressure for cosine-squared and for linear rise functions are schematically illustrated in Figs. 1 and 9, respectively.
View larger version (21K):
[in this window]
[in a new window]
| FIG. 1.
Schematics of envelope characteristics of the onsets of tone bursts shaped with cosine-squared rise functions. Left: for 3 different stimuli, time courses of the peak pressure during the rise time are shown. Only the top halves of the symmetrical envelopes are illustrated. Middle and right: resulting time courses of the rate of change of peak pressure and the acceleration of peak pressure, respectively. Signals in the rows from top to bottom are of identical rise time, plateau peak pressure, maximum rate of change of peak pressure, and maximum acceleration of peak pressure, respectively. Note that plateau peak pressure, rate of change of peak pressure, and acceleration of peak pressure reach their maxima at different points during the rise time.
|
|
View larger version (23K):
[in this window]
[in a new window]
| FIG. 9.
Schematics of envelope characteristics of the onsets of tone bursts shaped with linear rise functions. Left and right: for 3 different stimuli (identified by  , - - -, and · · ·), the time courses of peak pressure and of rate of change of peak pressure, respectively. Signals in the top row are of identical rise time, signals in the middle row are of identical plateau peak pressure, and signals in the bottom row are of identical magnitude of rate of change of peak pressure.
|
|
Data analysis
Spikes in response to the 20 presentations of a given stimulus were displayed off-line as a poststimulus time histogram. The histogram was used to select analysis windows that would comprise only onset responses and would discard late discharges, offset responses, and occasionally presumed spontaneous spikes. Spontaneous activity was generally very low (<3 spikes/s) and late discharges, if they occurred at all, were clearly separated in time from onset responses by marked intervals of no activity. Thus the selection of an appropriate onset window was generally straightforward. In most cases, analysis windows used for a given neuron were the same for all rise times and amplitudes studied (e.g., from 5 to 100 ms after tone burst onset). In some instances, however, different windows had to be selected. In these cases, windows for tones of long rise times and low amplitudes were longer or delayed relative to windows for tones of short rise times and high amplitudes, because otherwise onset responses would have been missed or late responses would have been included, respectively. In the present paper aspects of spike timing are analyzed, whereas in the companion report the focus is on response magnitudes. Only the timing of the first (and in many neurons the only) spike will be considered because the interspike intervals of the onset responses of auditory cortex neurons, which discharge more than one spike per stimulus, are very regular and independent of stimulus level (Phillips and Sark 1991). Mean and SD of first-spike latency, measured from stimulus onset, response probability, and number of discharges in the window were computed. As a rule, only means and SDs based on response probabilities of 
0.15 were considered further.
| |
RESULTS |
The results on mean first-spike latency are presented first, and then those on the variability of first-spike latency. In each section, data recorded with cosine-squared rise function tones are presented before those recorded with linear rise function tones, followed by a comparison of the results obtained with the two different rise functions.
Data base
This study is based on 74 well-isolated single neurons, recorded in the left AI, as inferred from the locations of the recording sites with respect to the sulcal pattern, the tonotopic sequence, and the presence of a short-latency strong evoked potential to tone bursts. In only one penetration in one cat did we not see an AI-like evoked potential. The twoneurons recorded in this penetration (95-87/03 and 95-87/04)had very long minimum latencies (>30 ms). A few isolated AI neurons, which were spontaneously active, appeared not to be driven by tone bursts. Sixty-five neurons were studied with tones shaped with cosine-squared rise functions, 39 neurons were studied with tone bursts shaped with linear rise functions, and 30 neurons were studied with both types of tones. Tones were presented with the neuron's preferred stimulus laterality, which was binaural for 31 neurons, contralateral for 40 neurons, and ipsilateral for 3 neurons. Four neurons were in addition studied with several stimulus lateralities. The neurons in the sample had CFs ranging from 1.5 to 35.2 kHz, with most CFs in the octave band from 12 to 24 kHz. Three neurons were also studied at multiple frequencies other than their CFs.
Mean first-spike timing
ASPECTS OF COSINE-SQUARED RISE FUNCTION TONES.
Figure 1, left, schematically illustrates the time courses of the envelopes of the onsets of cosine-squared rise function signals. During the rise time the peak pressure (in Pa), but not the SPL (in dB SPL), of the signal changes according to the rise function (Fig. 1, top left). The rate of change of peak pressure also changes gradually during the rise time (Fig. 1, top middle). It is zero at the beginning and at the end of the rise time and reaches a maximum halfway through the rise time. Acceleration of peak pressure is maximal at the beginning of the rise time and decreases smoothly with time. It is zero halfway through the rise time. From then on acceleration becomes increasingly negative (deceleration) and reaches a negative maximum at the end of the rise time (Fig. 1, top right). Thereafter acceleration is zero. Alterations of both plateau peak pressure and rise time effect the onset of stimuli shaped with cosine-squared rise functions, but in different fashions. A 6-dB increase in the plateau SPL of stimuli with a given rise time will lead to a twofold increase in the maximum rate of change of peak pressure and in the maximum acceleration of peak pressure. Shortening the rise time by a factor of 2 for any given plateau SPL also leads to a twofold increase in the maximum rate of change of peak pressure (Fig. 1, 2nd row, middle), but maximum acceleration of peak pressure increases fourfold (Fig. 1, 2nd row, right). Therefore signals can be grouped to match in rise time, plateau peak pressure, maximum rate of change of peak pressure, or maximum acceleration of peak pressure (Fig. 1, 1st-4th rows, respectively). Signals that share the same value of maximum acceleration of peak pressure differ in rise time and in plateau peak pressure (Fig. 1, bottom row).
MEAN FIRST-SPIKE LATENCY TO COSINE-SQUARED RISE FUNCTION TONES.
Figure 2a shows the mean first spike latencies of one AI neuron (95-95/04) to contralateral CF tone bursts of 22 kHz, all shaped with cosine-squared rise functions. The data are plotted as a function of plateau peak pressure (in Pa). The longest mean first-spike latency of ~100 ms was measured in response to tones with 170-ms rise times and plateau peak pressures of 0.00028 Pa, equivalent to 20 dB SPL. For each rise time, latency declines nonlinearly with increasing plateau peak pressure. For tones of any given plateau peak pressure, latency increases systematically with rise time, although the different functions appear to converge on a single minimum at ~12.3 ms.
View larger version (26K):
[in this window]
[in a new window]
| FIG. 2.
Effects of rise time and plateau peak pressure of cosine-squared rise function tones on latency. Mean 1st-spike latency of neurons 95-95/04 and 95-98/03 (left and right, respectively) to 20 repetitions of characteristic frequency (CF) tones shaped with cosine-squared rise functions of 5 and 6 different rise times (see key). In a and d, mean latency is plotted as a function of plateau peak pressure (in Pa). The range of 5 orders of magnitude is equivalent to a 100-dB range of sound pressure level (SPL) from about  10 to 90 dB SPL. In b and e, mean latency is plotted as a function of the maximum rate of change of peak pressure, and in c and f as a function of the maximum acceleration of peak pressure. Note the close congruence of the latency-acceleration functions. For further details see RESULTS.
|
|
Similar observations can be made when latency is plotted as a function of the maximum rate of change of peak pressure (Fig. 2b). Although the functions relating latency to maximum rate of change of peak pressure obtained with different rise times are closer together than those relating latency to plateau peak pressure, latency still increases with rise time for signals with the same maximum rate of change of peak pressure. Also, for some tones of long rise time the neuron discharges before the maximum rate of change of peak pressure is reached.
In contrast, when latency is plotted as a function of maximum acceleration of peak pressure, all five latency-acceleration functions obtained with different rise times are in close register, i.e., at any given acceleration the functions are within 1 SD of the means (Fig. 2c). For clarity, SDs are not plotted in Fig. 2, but decreased for neuron 95-95/04 from 6.4 to 0.5 ms.
Tones with a common maximum acceleration at their onsets also share a number of other properties. These are the maximum deceleration occurring at the end of the rise time; the mean acceleration and mean deceleration averaged over the first and second half of the rise time, respectively; the ratio of RCPPmax and rise time; and the ratio of PPplateau and the square of the rise time (Fig. 1; Eq. 4 and 5). However, these parameters or any combination thereof can be ruled out as determinants of latency because in response to many tones, particularly of long rise times, the first spike occurs long before the end, or even the midpoint, of the rise time (Fig. 2).
Data from a second neuron (95-98/03) are illustrated inFig. 2, d-f. This neuron's CF was similar to that of 95-95/04(viz., 21 kHz), but the neuron was excited best by tones presented to the ipsilateral ear. The total range of latencies obtained with the six different rise times tested was nearly 100 ms. As was the case for neuron 95-95/04, all latency functions obtained with different rise times were in close register only when plotted as a function of the acceleration of peak pressure (Fig. 2f).
In all 65 neurons studied with cosine-squared rise function tones, mean latencies obtained at any given acceleration of peak pressure with tones of different rise times were within 1 SD of each other. Note that over the range of rise times used (1.7-170 ms), tones with the same maximum acceleration differ in plateau peak pressure by as much as 80 dB, i.e., by a factor of 10,000.
In several cases, some mean latencies could be systematically longer than others recorded to tones of the same maximum acceleration of peak pressure. In all these cases, the extraordinarily long latencies were measured to tones closest to the firing threshold of the neuron (e.g., the mean latencies to the 1.7-ms rise time tones with accelerations between 100 and 1,000 Pa/s2 in Fig. 2, c and f). Neuron 95-98/14 (Fig. 14a) represents the most drastic example of this "near-threshold effect." In this case the means of first-spike latency closest to threshold are based on the same response probability (viz., 100%) as are all the other means. In other cases in which the near-threshold effect was observed, the exceptionally long near-threshold means were mostly based on much lower response probabilities (e.g., neuron 95-98/08 in Fig. 3a).
View larger version (20K):
[in this window]
[in a new window]
| FIG. 14.
Comparison of SD and of mean of 1st-spike latency. Data are from neuron 95-98/14, stimulated binaurally with CF tones of 5.5 kHz. Note the pronounced near-threshold effects for both mean and SD of 1st-spike latency. The 7 near-threshold points were discarded for the fits of the functions relating SD to maximum acceleration of peak pressure and to mean latency in b and c, respectively. All other conventions as in Fig. 13.
|
|
View larger version (22K):
[in this window]
[in a new window]
| FIG. 3.
Comparison of latency-acceleration functions. a: data from neurons from different cats, with different CFs, obtained with different laterality of stimulus presentation are selected to illustrate the similarity in the shapes of the latency-acceleration functions despite differences in their extent. Mean latencies obtained from a given neuron are represented by the same symbols, and latencies obtained from that neuron with tones of the same rise time are connected by solid lines. Note that, as in the cases illustrated in Fig. 2, mean latencies are in close register when plotted as a function of maximum acceleration of peak pressure. b: mean latencies of 2 neurons from a are reproduced. Solid and dashed lines: best fits of Eq. 8 to the data. The 2 fitted functions have identical shape. As can be derived from the differences in the solutions for Lmin and S for the 2 neurons (as specified in the key), the function for neuron 95-98/16 is displaced upward by 8.3 ms and rightward by 0.95 log units of maximum acceleration of peak pressure relative to the function of neuron 95-95/03. For further descriptions see RESULTS.
|
|
Latency-acceleration functions obtained with shorter rise times take up the common course of the latency-acceleration functions obtained with longer rise times at consecutively higher values of maximum acceleration of peak pressure (e.g., Figs. 2, 3, and 13, a and d). Thus a neuron's firing threshold is not determined by the maximum acceleration of peak pressure at signal onset (see also companion paper).
View larger version (24K):
[in this window]
[in a new window]
| FIG. 13.
Comparison of SD and mean of 1st-spike latency. Data from neuron 95-98/11 (a-c), stimulated with contralateral CF tones of 10.3 kHz, and from neuron 95-98/01 (d-f), stimulated binaurally with CF tones of 20.3 kHz, are illustrated. a and c: mean 1st-spike latencies obtained with tones of different cosine-squared rise times (see keys) plotted against maximum acceleration of peak pressure. b and e: corresponding SDs of 1st-spike latency also plotted against maximum acceleration of peak pressure. Note that the functions obtained with different rise times are in close register. Solid lines without symbols: best fits of Eq. 15 to the data sets. The equation assumes that the SD is proportional to the slope of the mean latency-acceleration function. c and f: scatterplots of the SD of 1st-spike latency against the mean. Solid lines: best fits of Eq. 16 to the data sets. Dashed lines: best linear fits. See RESULTS for further explanations.
|
|
COMPARISON OF LATENCY-ACCELERATION FUNCTIONS AMONG DIFFERENT NEURONS.
The latencies of neurons 95-95/04 and 95-98/03 in Fig. 2 are plotted with the same resolution, and comparison of Fig. 2, c and f, reveals that their latency-acceleration functions are very similar in shape. In Fig. 3a, latency-acceleration functions obtained from another five neurons are plotted in a single graph, facilitating a comparison of latency-acceleration functions among different neurons. The data illustrated in Fig. 3a were selected to represent neurons recorded in different cats and with widely different CFs (range 2.3-30 kHz), data obtained with different laterality of presentation, and functions covering very different ranges of latency. The latency-acceleration function of neuron 95-98/08 (
) covered an extensive range of latency (130-15 ms) and of maximum acceleration of peak pressure (>8 orders of magnitude). Because of higher response thresholds, strongly nonmonotonic spike count functions, or both, the latency-acceleration functions of the other neurons were more restricted along the abscissa, but also along the ordinate. However, an inspection of Fig. 3a suggests that the shapes of these more restricted functions closely resemble sectors of the extensive function of neuron 95-98/08. This is most obvious for neuron 95-95/03 (
), which had a threshold slightly higher than that of neuron 95-98/08. Neuron 95-92/21 (
) had a considerably higher threshold, but also slightly longer mean latencies, than neuron 95-98/08. But even the course of the latency-acceleration function of neuron 95-98/16 (
), which is restricted at each end, resembles the course of the extensive function of neuron 95-98/08 in its intermediate part.
All 93 latency-acceleration functions, obtained from the 65 neurons studied with cosine-squared rise functions tones, had strikingly similar shapes. All functions could be brought into very close register by allowing them to be shifted only along the ordinate and along the abscissa, as initially judged by visual inspection. Shifts along the ordinate compensate for differences in the minimum or asymptotic latency, and shifts along the abscissa compensate for differences in sensitivity to acceleration.
MATHEMATICAL DESCRIPTION OF LATENCY-ACCELERATION FUNCTIONS.
To get quantitative measures of the similarity of the latency-acceleration functions of different neurons and of the shifts along the ordinate and abscissa required to obtain congruence, a simple mathematical function was selected that described the form of the latency-acceleration functions, and also allowed quantification of the positional differences along the abscissa and the ordinate
|
(7)
|
The subscript CRF indicates that the measures were obtained with cosine-squared rise function tones. LCRF is a neuron's mean latency as a function of maximum acceleration of peak pressure APPmax. Lmin is the minimum or asymptotic latency against which LCRF converges for acceleration approaching infinity. Lmin is a constant that would include all the delays that are independent of the stimulus magnitude, such as acoustic delays, delays introduced by the traveling wave in the cochlea, the sum of all axonal travel times, and possibly some synaptic factors. The other term describes the inverse dependence of latency on the magnitude of maximum acceleration of peak pressure, where ACRF is a scaling factor and S is the neuron's transient sensitivity, which codetermines the position of the function along the abscissa. The value of S is the logarithm of an acceleration of peak pressure (in Pa/s2). A larger S places the function more to the left and represents a higher transient sensitivity (see also Fig. 3b). The function does not account for the near-threshold effects observed in some neurons and described above. It also does not account for the finding that in a few neurons with nonmonotonic spike-count functions, mean latency could increase slightly but systematically with very high values of maximum acceleration of peak pressure (e.g., 95-95/03 in Fig. 3, a and b).
Iterative curve fitting was performed in the following way. In initial fitting procedures, Lmin, ACRF, S, and the exponent 
were allowed to vary. Each deviation of the fitted function from the measured mean latency was squared and then weighted by multiplying it with the response probability on which the measured mean was based. The smallest sum of the weighted squared deviations, i.e., the best fit, was generally found with <1,000 iterations. In some cases the fit was found to improve with increasing . The improvement, however, was marginal for > 4, and also pushed ACRF into unwieldy dimensions (e.g., years for = 10). For a second fitting step, we therefore selected = 4, and allowed Lmin, ACRF, and S to vary. For the 93 different functions fitted, ACRF showed a unimodal distribution. Figure 4a shows a scatterplot of ACRF against the number of first spikes that had contributed to the fitted function. The figure shows that the width of the distribution of ACRF diminished rapidly with increasing number of first spikes and converged toward theweighted average of ÃCRF = 12,791 ms (Fig. 4a, - - -).In a third and final fitting procedure, ACRF was also kept constant (at 12,791 ms). In this way, a function with a fixed shape, as determined by and ACRF, but free to be placed within the coordinate system of latency and maximum acceleration of peak pressure, was fitted to the data
|
(8)
|
Figure 4b provides a scatterplot of the sums of the weighted least-squared deviations of mean latency obtained with the second and third fitting step, i.e., with ACRF variable and ÃCRF fixed at 12,791 ms, respectively. Only few points are considerably above the line of unity slope (dashed line). The three most deviating points were provided by one neuron tested under different stimulus conditions. In general, the most deviating points were based on low numbers of first spikes. Most points are in relatively close proximity to the dashed line, indicating that the latency-acceleration function with the fixed shape provides nearly as good a description of the data as does a function with an additional free variable.
View larger version (20K):
[in this window]
[in a new window]
| FIG. 4.
Descriptions of neuronal latency-acceleration functions. a: scatterplot of the scaling factor ACRF obtained from fitting the function
to neuronal latency-acceleration functions against the number of 1st spikes contributing to the fit. Note that the distribution converges against the weighted average of ÃCRF = 12,791 ms (- - -) with increasing number of 1st spikes, thus with presumed increasing reliability of the fit. b: scatterplot of the sums of the weighted least-squared deviations of the above functions, fitted to the relationship between mean latency and maximum acceleration of peak pressure, from the actual data. ACRF was either a free parameter or it was fixed at ÃCRF = 12,791 ms, the value of its weighted average. In only a few instances was the quality of the fit notably reduced when ACRF was fixed. Note that most points are close to the line of unity slope (- - -).
|
|
In Fig. 3b, the mean latencies of two of the neurons of Fig. 3a (viz., 95-95/03 and 95-98/16) are reproduced together with the fitted functions (Eq. 8), which are of identical shape. The figure allows a visual assessment of the quality of the fit and the similarity of the fitted function with neuronal latency-acceleration functions. The best solutions for S and Lmin found by the final fitting procedure are 3.96 and 18.8 ms for neuron 95-98/16 and 4.91 and 10.5 ms for neuron 95-95/03. Thus, according to the fitting results, the latency-acceleration function of neuron 95-98/16 is displaced upward by 8.3 ms and rightward by 0.95 log units of acceleration relative to the function of neuron 95-95/03.
COMPARISON OF TRANSIENT SENSITIVITY AND FIRING THRESHOLD.
S is not to be confused with firing threshold, a measure generally expressed in dB SPL and related to peak pressure. To emphasize this point more clearly, note, for example, that in Fig. 3a, the latency functions of neurons 95-98/08 () and 95-95/03 () are in nearly perfect register, without requiring any notable shifts to obtain congruence, i.e., the two neurons have the same S. However, the latency functions do not start at the same point along the abscissa, reflecting differences in their firing thresholds. Figure 5 presents, for all neurons in the sample, a scatterplot of the firing thresholds (in dB SPL) against S. Each neuron contributed multiple data points to the plot, because threshold SPL increased with rise time (see companion paper and also Fig. 6). Although a low transient sensitivity seems to exclude low-threshold SPLs, there is only a loose relationship between the two parameters (r2 = 0.123; n = 319). Threshold SPLs can vary over a range of 100 dB for the same S.
View larger version (13K):
[in this window]
[in a new window]
| FIG. 5.
Scatterplot of neuronal firing thresholds (expressed in dB SPL) against S extracted from latency-acceleration functions. S is the logarithm of acceleration of peak pressure measured in Pa/s2. Note that the 2 measures are only loosely related.
|
|
View larger version (27K):
[in this window]
[in a new window]
| FIG. 6.
Effects of tone burst frequency on latency-acceleration functions. Data from 2 neurons (95-95/18, a-c, and 95-95/09, d-f) are shown. In a and d, mean latency is plotted against maximum acceleration of peak pressure and different symbols identify different frequencies. Mean latencies obtained with tones of the same cosine-squared rise time are connected. Note the different resolutions of the abscissas and ordinates in a and d. b and e: measure of S, obtained from fitting Eq. 8 to the latency-acceleration functions for tones of different frequencies. S reflects the size of the lateral displacement of these functions, and a difference in S of 1 is equivalent to 20 dB. c and f: conventional response threshold curves or tuning curves obtained with tones of different cosine-squared rise times. Threshold was defined by a response probability of 0.1. Note the increase in thresholds with rise time throughout the frequency range (see also companion paper). Note that the transient sensitivity vs. frequency functions share features with the classical tuning curves (cf. b with c and e with f).
|
|
EFFECTS OF STIMULUS LATERALITY ON LATENCY-ACCELERATION FUNCTIONS.
In four neurons latencies to tone bursts were presented with different stimulus lateralities, i.e., binaural, monaural contralateral, and monaural ipsilateral. In general, stimulus laterality had a very small, if any, effect on the shapes of the latency-acceleration functions or their horizontal position within the coordinate systems. In a comparison of stimulus laterality in a given neuron, fitting results yielded differences in S that averaged 0.1 and were all <0.3, ~1/10 of the variation seen across neurons. The largest effect of stimulus laterality was on the estimated Lmin. With monaural ipsilateral stimulation Lmin was consistently 2-3 ms longer than with contralateral or binaural stimulation, whereas differences in Lmin between monaural contralateral and binaural stimulation were <0.9 ms.
EFFECTS OF STIMULUS FREQUENCY ON LATENCY-ACCELERATION FUNCTIONS IN A GIVEN NEURON.
In three neurons latencies were obtained to tone bursts of different frequencies including the CF. Results from two of these neurons (95-95/18 and 95-95/09) are illustrated in Fig. 6. Figure 6, a and d, shows mean latencies plotted against maximum acceleration of peak pressure. The latency-acceleration functions for different frequencies all have similar shape, but are obviously dispersed along the abscissa. The analysis of the fitting results illustrates the systematic nature of this dispersion: in Fig. 6, b and e, the value of S obtained from these fits is plotted against tone burst frequency. For neuron 95-95/18 the highest transient sensitivity is obtained for 26.8 and 24.8 kHz, and S decreases toward higher and lower frequencies, whereas for neuron 95-95/09 the function is more complex.
The transient sensitivity versus frequency functions can be compared with the more conventional threshold or tuning curves based on firing probabilities (Fig. 6, c and f). Tones of the same rise time that differ in peak pressure by 20 dB SPL differ in the acceleration of peak pressure by a factor of 10. This is equivalent to a difference in S of 1, so that the ordinates in Fig. 6, b and c and e and f, have the same relative scaling. The CF of neuron 95-95/18 was near 26.8 kHz when tone bursts with 1.7- and 8.5-ms rise time were used, but shifted to 28.8 kHz with tone bursts having 17-ms rise times. In addition, with prolongation of the rise time systematic elevations in response threshold were observed throughout the excitatory frequency range, when threshold was expressed as a function of the plateau peak pressure or level (in dB SPL; Fig. 6c, see also companion paper). Neuron 95-95/09 had twin-peaked tuning curves with lowest thresholds at 24 and at 14 kHz. Again, threshold SPLs increased systematically with rise time at all frequencies, although not by identical amounts (Fig. 6f). Note that the transient sensitivity versus frequency curves obtained from the analysis of latency-acceleration functions and the tuning curves share common features, but are not identical. For neuron 95-95/09, S and the tuning curves show a dip at 18 kHz (cf. Figs. 6, e and f), but this dip is more pronounced in the tuning curves, particularly those obtained with longer rise times.
View larger version (8K):
[in this window]
[in a new window]
| FIG. 7.
Estimated minimum latency Lmin obtained from fits of Eq. 8 to the latency-acceleration functions plotted against tone frequency. Data obtained at frequencies other than the CF are omitted. Note that neurons of the same CF can differ widely in their minimum latency and that Lmin tends to decrease with frequency.
|
|
For neuron 95-95/18, the estimated values of Lmin obtained from the fits of the latency-acceleration functions varied between 8 and 9 ms and were not systematically related to frequency, whereas for neuron 95-95/09, Lmin varied between ~4.5 and 7 ms with frequency, and its course approximately paralleled the tuning curve, with Lmin being shortest at 14 and 26 kHz (not shown).
EFFECTS OF STIMULUS FREQUENCY ON LATENCY-ACCELERATION FUNCTIONS ACROSS NEURONS.
For a comparison of latency-acceleration functions among different neurons, only measures obtained at CF were considered. Figure 7 provides a scatterplot of Lmin, as obtained from the fits, against frequency. In different neurons, Lmin varied between 5.6 and 37 ms, with most values between 9 and 15 ms. On average, Lmin decreased with increasing CF. This decrease is obvious for the shortest Lmin and a similar trend for the entire data set emerged from a regression analysis. Lmin was closely correlated with the shortest measured latency (r2 = 0.894), but on average was 1.8 ms shorter.
Figure 8 shows a scatterplot of S obtained from the fits over frequency. The least reliable S estimates are shown by open squares. The degree of reliability of S was quantified by the increase in the sum of the weighted least-squared deviations, when S was arbitrarily incremented by 1 after the best fit had been obtained. This increase could be as small as twofold, indicating low reliability for the obtained value of S, and as high as 1,200-fold, with a mean of 60-fold. For the open squares, the increase was <15-fold. The distribution of S, particularly that of the most reliable measures (solid squares), is similar to the cat's compound action potential audiogram. The audiogram shows highest sensitivity at ~10 kHz, and a steeper rolloff for higher than for lower frequencies (see Rajan et al. 1991 for illustrations of audiograms measured under different stimulus conditions). At most frequencies the vertical scatter in the data points of Fig. 8 is in the range of only 0.5, equivalent to 10 dB. Because there may have been differences in hearing sensitivity among the six cats that contributed data to this figure, differences in the sensitivities of the two ears in a given cat, and imprecisions in CF determination (cf. Fig. 6), it is conceivable that some, if not all, of this vertical scatter may be noise due to these factors.
View larger version (7K):
[in this window]
[in a new window]
| FIG. 8.
S obtained from fits of Eq. 8 to the latency-acceleration functions plotted against tone frequency. Data obtained at frequencies other than the CF are omitted and fits with the most reliable S are shown with solid squares.
|
|
ASPECTS OF LINEAR RISE FUNCTION TONES.
With linear rise functions, the rate of change of peak pressure during the rise time is constant (Fig. 9) and, for a given rise time, its magnitude is directly proportional to the plateau peak pressure achieved at the end of the rise time, and, for a given plateau peak pressure, is inversely proportional to rise time. Thus the first derivative of the stimulus envelope has the shape of a rectangle, with its vertical axis proportional to rate of change of peak pressure (expressed in Pa/s) and its horizontal axis equivalent to the rise time (Fig. 9, middle). Signals shaped with linear rise functions can be grouped to match either in rise time (Fig. 9, top), in plateau peak pressure (middle), or in the rate of change of peak pressure (bottom). Acceleration of peak pressure occurs at the beginning of the rise time and deceleration occurs at the end of the rise time. Mathematically, acceleration and deceleration are instantaneous and their magnitudes are infinite.
MEAN FIRST-SPIKE TIMING TO LINEAR RISE FUNCTION TONES.
Figure 10, a and b, shows the mean first-spike latencies of neuron 95-95/04 to linear rise function tones. It is the same neuron for which latencies obtained with cosine-squared rise function tones were illustrated in Fig. 2, a-c. Figure 10a illustrates that for each rise time, latency declines nonlinearly with plateau peak pressure. For tones of a given plateau peak pressure, latency increases with rise time. As was the case with cosine-squared rise function tones, the curves appear to converge on a single minimum and in response to some tones of long rise times the neuron discharges long before the plateau peak pressure is reached.
View larger version (17K):
[in this window]
[in a new window]
| FIG. 10.
Effects of rise time and plateau peak pressure of linear rise function tones on latency. Mean 1st-spike latency of neurons 95-95/04 and 95-87/13 (left and right, respectively) to linear rise function CF tones of different plateau peak pressure and different rise time. In a and c, mean latency is plotted as a function of plateau peak pressure (in Pa). Different symbols identify different rise times (see key) and latencies to tones of the same rise time are connected. In b and d, mean latency is plotted as a function of rate of change of peak pressure. Note the close match of the latency functions obtained with the 5 different rise times.
|
|
Figure 10b shows mean first spike latencies plotted over the rate of change of peak pressure during the rise time. Note that all five functions are now in very close register, i.e., for any given rate of change of peak pressure, mean latencies are within <1 SD of each other.
A second example (neuron 95-87/13) is illustrated in Fig. 10, c and d. This neuron had a CF similar to that of neuron 95-95/04 (viz., 20.5 kHz), but was stimulated binaurally and had a much more restricted range of latencies (~12-18 ms).
In all 39 neurons studied with linear rise function tones, tone bursts characterized by the same rate of change of peak pressure during the rise times elicited a response from a given neuron with the same first-spike latency, i.e., within 1 SD of the mean, irrespective of differences in rise time or plateau peak pressure. Tones of identical rate of change of peak pressure that differ in rise times by a factor of 100 differ in plateau peak pressure by the same factor, i.e., by 40 dB. The finding that tones with rise times of 1-100 ms and possibly beyond those limits initiate spikes with the same latency, provided they have identical rate of change of peak pressure, suggests that the latency of the first spike must be determined very early during the rise time, viz., within <1 ms after stimulus onset.
POST HOC ANALYSIS OF PREVIOUSLY PUBLISHED LATENCY DATA.
There has been one previous report on the effect of varying rise time and level of linear rise function tones on the responses of AI neurons (Phillips 1988). In the following, I present a post hoc analysis of latency data published in that paper, because they showed a behavior that is markedly different from that of all neurons in my sample. Figure 11, left, replots latency of one of the three units (viz., RT206) for which Phillips has presented data, and Fig. 11a does so in the published and conventional form, viz., as a function of plateau peak pressure or tone level (in dB SPL). As noted by Phillips (1988), for each rise time latency declines with increasing level toward asymptotic values, but the functions do not converge on a single minimum.
View larger version (35K):
[in this window]
[in a new window]
| FIG. 11.
Analysis of data presented by Phillips (1988) on effects of rise time and plateau peak pressure of linear rise function tones on latency of 2 neurons from cat auditory cortex (RT206, left, and RT209, right). a and f were taken from this study, and show latency to CF tones with different linear rise times (see key) as a function of SPL. In b and g, latency is plotted as a function of rise time for tones of the same plateau peak pressure (in dB SPL, see key). Note that all functions have slopes >1. Dashed lines have unity slope. In c and h, latency is plotted as a function of the rate of change of peak pressure for tones of the same rise time. Note that the functions are not in register, unlike those of the units illustrated in Fig. 10. In d and i, latency is plotted as a function of rise time for tones of the same rate of change of peak pressure. Note that latency increases directly with rise time, and thus increases with plateau peak pressure. Dashed lines have unity slope. In e and j, the rise time was subtracted from the response latency and values were plotted as a function of rate of change of peak pressure. These corrected latency functions are in close register, suggesting that in these units spikes were triggered by the quasi-instantaneous deceleration at the end of the rise time. See text for further discussion.
|
|
In Fig. 11b, some of these same data are replotted as functions of rise time for tones of specified plateau peak pressure. Note that for every plateau peak pressure, latency increases roughly linearly with rise time. The slopes (as well as the Y-intercepts) decline systematically with increasing level, but unlike those of the neurons in our sample (see Heil and Irvine 1996b), all slopes are 1 (for comparison, unity slope is illustrated by the dashed line in Fig. 11b). Linear regression analysis revealed slopes of 2.94 ± 0.05, 1.91 ± 0.02, 1.54 ± 0.04, 1.46 ± 0.05, and 1.05 ± 0.03 for plateau peak pressures equivalent to 22, 34, 46, 58, and 70 dB SPL, respectively. In other words, the differences in response latencies to tone bursts of the same plateau peak pressure are larger than the differences in rise time. In Fig. 11c, the latency data of Fig. 11a are plotted against the rate of change of peak pressure during the rise time. Note that the functions obtained with different rise times are not in register, quite unlike the behavior of all neurons in our sample (cf. Fig. 10, b and d). Instead, latency for tones of the same rate of change of peak pressure still increases systematically with rise time. This is more clearly illustrated in Fig. 11d, where latency is plotted as a function of rise time, and where each function represents latencies obtained from tone bursts characterized by the same rate of change of peak pressure.
Several points are noteworthy here. First, for any given rate of change of peak pressure latency increases with rise time, and thus increases with the plateau peak pressure of the tone bursts (cf. Fig. 9, bottom left), a result that at first glance may seem paradoxical or at least counterintuitive. Second, all functions can be approximated by linear functions, but their slopes do not vary with rate of change of peak pressure. In fact, the slopes relating latency to rise time were 1.01 ± 0.20, 1.06 ± 0.14, 1.03 ± 0.13, 1.02 ± 0.02, and 0.95 ± 0.06 for the five rates of change of peak pressure in ascending order, i.e., they are very close to, and not significantly different from, 1.
The only reasonable interpretation of this result is that the spikes are in fact triggered at or by the end of the rise time. This point in time is characterized by the quasi-instantaneous deceleration of peak pressure. Figure 11e therefore plots the response latency corrected for the rise time as a function of the rate of change of peak pressure. Now the five functions are in close register, and the corrected latency decreases nonlinearly with this parameter. The same results were obtained for the other two neurons for which Phillips (1988) has published latency data, and are illustrated for RT209 in Fig. 11, f-j.
COMPARISON OF LATENCY-RATE OF CHANGE OF PEAK PRESSURE FUNCTIONS AMONG DIFFERENT NEURONS.
As was the case for latency-acceleration functions obtained with cosine-squared rise function tones, the latency-rate of change of peak pressure functions of different neurons obtained with linear rise functions tones could be brought into very close register by allowing shifts along the ordinate and the abscissa (not shown). The common form of these latency-rate of change of peak pressure functions, which differed from that of the latency-acceleration functions, and the shifts along the coordinates were found with fitting procedures analogous to those described above for cosine-squared rise functions and using the same type of formula
|
(12)
|
The weighted average of the scaling factor ÃLRF found with the 39 neurons studied with linear rise function tones was 1,277 ms.
COMPARISON OF LATENCY WITH LINEAR AND WITH COSINE-SQUARED RISE FUNCTION TONES.
Thirty neurons were studied with both linear and cosine-squared rise function tones of the same frequency and can therefore be used for a direct comparison of the relevant features of latency functions. Figure 12a shows a scatterplot of the estimated minimum latencies obtained with linear and with cosine-squared rise functions tones. As expected, the two estimates are nearly identical. Note that they lie close to the line of unity slope. A linear regression analysis yielded a slope of 0.87 with r2 = 0.973. Exclusion of only the rightmost point increases the slope to 0.94.
View larger version (16K):
[in this window]
[in a new window]
| FIG. 12.
Comparison of estimates derived from latency to linear and to cosine-squared rise function tones. a: scatterplot of minimum latencies. b: scatterplot of S. Note that both types of stimuli yield nearly identical estimates of minimum latency and of transient sensitivity. Dashed lines have unity slope.
|
|
Figure 12b shows a scatterplot of the corresponding estimated S values obtained from the fits. Again, the estimates are nearly identical. A linear regression analysis yielded a slope of 0.91 with r2 = 0.898.
Because the estimates of minimum latency and transient sensitivity are basically independent of the rise function, it is easy to derive the characteristics of linear and of cosine-squared rise function tones that would ideally yield a response from a given neuron with the same first-spike latency. With the formulas used here to describe the neuronal latency functions (Eq. 8 and 12), this is the case when
Rearranging terms yields
and
With ÃCRF = 12,791 ms and ÃLRF = 1,277 ms, it follows
|
(13)
|
Thus, for a given neuron and for tones of the same frequency, the isolatency conditions are described by a linear relationship between the logarithm of the rate of change of peak pressure of linear rise function tones and the logarithm of the maximum acceleration of peak pressure of cosine-squared rise function tones. The ordinate intercept is proportional to the neuron's S. In other words, to yield the same latency from a neuron as a cosine-squared rise function tone with a given acceleration of peak pressure, a linear rise function tone with only a low rate of change of peak pressure is required when the neuron's transient sensitivity is high, whereas a higher rate of change of peak pressure is required when the neuron's transient sensitivity is low.
SD of first-spike timing
The data presented so far have been based on the mean first-spike latency derived from up to 20 individual measures of latency on consecutive stimulus repetitions. However, the timing of the first spike varied from trial to trial. In accordance with previous studies (e.g., Aitkin et al. 1970; Brugge et al. 1969; Kitzes et al. 1978; Phillips and Hall 1990; Phillips et al. 1989), the SD of the first-spike latency around the mean will be used here as a measure of this variability.
COSINE-SQUARED RISE FUNCTIONS.
The finding that with cosine-squared rise function tones a neuron's mean first-spike latency is a function of the maximum acceleration of peak pressure suggests the possibility that the SD of the first-spike latency may also be a function of this parameter.
Figures 13 and 14 present data on SD and its relationship with maximum acceleration of peak pressure for three neurons. Figures 13, a and d, and 14a show the now familiar finding that mean first-spike latency is a unique function of maximum acceleration of peak pressure. Figures 13, b and e, and 14b show the corresponding SDs of first-spike latency, also plotted against this parameter. Several observations are important.
First, SD is also inversely related to maximum acceleration of peak pressure, and consequently increases with mean latency (Figs. 13, c and f, and 14c).
Second, the SD of first-spike latency is also an unambiguous function of maximum acceleration of peak pressure, irrespective of the rise time or of the plateau peak pressure, and also approaches some asymptotic value. SD is more variable than mean latency when the two parameters are compared for stimuli of identical maximum acceleration of peak pressure. Therefore SD-acceleration functions were generally noisier than the mean latency-acceleration functions. As illustrated in Fig. 14b, the near-threshold effect, described above for mean latency, could be quite pronounced for SD.
Third, the shapes of the SD-acceleration functions are distinctly different from the shapes of the mean latency-acceleration functions. In particular, the decline in SD with acceleration is relatively steeper than the decline of mean latency for low magnitudes and relatively shallower for high magnitudes of maximum acceleration of peak pressure.
Such differences in function shape are inconsistent with a linear relationship between SD and mean first-spike latency as
Abstract
Introduction
Abstract
![− sin<SUP>2</SUP>(<IT>t</IT>/<IT>CRT</IT>*π/2 + π/2)]](/content/vol77/issue5/fulltext/2616/img006.gif)
![log<IT>APP</IT><SUB>max</SUB> = (<IT><A><AC>A</AC><AC>˜</AC></A></IT><SUB>CRF</SUB>/<IT><A><AC>A</AC><AC>˜</AC></A></IT><SUB>LRF</SUB>)<SUP>1/4</SUP>*log<IT>RCPP + S</IT>*[(<IT><A><AC>A</AC><AC>˜</AC></A></IT><SUB>CRF</SUB>/<IT><A><AC>A</AC><AC>˜</AC></A></IT><SUB>LRF</SUB>)<SUP>1/4</SUP>* − 1]](/content/vol77/issue5/fulltext/2616/img016.gif)
