The growth of sensation magnitude with stimulus intensity is a fundamental property of sensory systems. Beginning with the classical studies by G.T. Fechner and S.S. Stevens, a wide variety of studies in several sensory systems have revealed that the growth of sensation, including loudness, is fundamentally related to the stimulus intensity by a power law. For a power law, y = a(x)b where in normal acoustic hearing y is perceived loudness, x is the sound pressure level (SPL), and b is an exponent approximately equal to 0.6, (i.e. the function is compressive). However, cochlear implants bypass the auditory periphery and stimulate the auditory nerve directly, and the growth of loudness magnitude with stimulus current for cochlear implant stimulation has been found to be very different from normal hearing individuals. The results typically conclude that loudness grows as an “expansive” nonlinear function of stimulus current. Some studies have shown that loudness growth functions are a power function of current when using magnitude estimation and are bestfit by power functions with an exponent of 2.72 to 3.5 [11,12,30,1,2,18]. Since the exponent range is much higher in electric hearing than acoustic hearing, the dynamic range is much smaller than for normal hearing subjects and hearingimpaired subjects with acoustic stimulation. In modeling the growth of loudness in cochlear implant listeners, there have also been several other proposed models of these expansive functions. The loudness growth functions have also been reported to grow exponentially with stimulus current so that y=ae^{bx}, [30, 33]. Electrical loudness functions have also been described by combinations of both power and exponential functions [16], or by utilizing a forthorder polynomial, which follows the equation y=a_{0} + a_{1}x^{1} + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4}, [8]. The polynomial fit may be a convenient way to approximate individual variations in how loudness grows with current. While a higherorder polynomial may provide the bestfit approximation of individual variations in loudness functions in electric hearing, higherorder polynomial fits are more sensitive to perturbations or noise in the data.
Previous studies have investigated the relationship between loudness growth and the underlying neural mechanisms in processing signals in acoustic hearing and hypothesized that the relationship represents a summation of responses of neural excitation in the peripheral and central auditory system [31, 35]. Electrical stimulation of the auditory nerve, as in cochlear implant listening, bypasses the peripheral sensory organ to reveal the central processes of the auditory system. A loudnesscoding model proposed by Zeng and coworkers [28, 30, 31, 33] suggest that in the periphery, the normal cochlea provides logarithmic compression. Consequently with the loss of the normal compressive peripheral cochlear mechanics and direct stimulation of the auditory nerve in electrical hearing, an “expansive” exponential nonlinear loudness growth in the central auditory system processing stages is revealed [31]. Similar findings have been described for electrical stimulation of skin [25].
It has been suggested that individual variations in the loudness growth functions may be due to variations in neural processes that accompany cochlear implant stimulation. These variations may be the result of spatial distribution of surviving neurons and/ or contributing neurons, variations in the distributions of electrical fields due to electrode placement, and stimulation mode [15].
Not only can the understanding of the growth of sensation magnitude with stimulus intensity provide a better understanding of neural processes, it also may have clinical implications. Knowing the loudness growth functions in cochlear implant users is presumably necessary for the optimum implementation of the algorithms in the development of speech processors to control overall loudness and the loudness at the level of each electrode. There have been suggestions that the compression algorithm used can affect speech intelligibility [6]. It is also necessary to understand how loudness is encoded and may affect the relative performance in speech intelligibility. Acoustical sound needs to be compressed into the electrical range of hearing in a cochlear implant processor due to the large differences in dynamic range, going from the lowest sound one can hear to the loudest sound one can tolerate, with the electrical dynamic range being much smaller than the acoustic dynamic range (approximately 10:1 for the range of current level in electric hearing, as compared to approximately 1,000,000:1 for the range of sound pressure levels in acoustic hearing). Compression is implemented in the cochlear implant speech processor to fit a larger dynamic range of sounds into the electrical range of hearing and as an attempt to match the logarithmic compression that is missing in the bypassed damaged cochlea. This is based on the current assumption that loudness growth is an exponential function of electrical amplitude and can be predicted exclusively by the threshold and most comfortable levels of loudness, [11, 31]. Having an appropriate loudness growth model is important to cochlear implant users because of the much smaller dynamic range of hearing and distorted loudness growth functions compared to normal hearing listeners [12, 28, 29]. To compensate for this, appropriate models of loudness are necessary for the design of cochlear implant speech processors. Consequently, commercial cochlear implant speech processors currently use a logarithmic function to map acoustic sound pressure to electrical current. Mapping loudness incorrectly from the acoustic sound pressure to electric current may distort speech sounds and reduce speech recognition abilities for cochlear implant users [23, 20, 9, 10, 7]. Findings have shown that speech recognition scores in quiet can drop by a small but clinically significant 8% to up to 15% when distorting the compressive function in the speech processor and the best speech recognition performance occurred when the normal loudness function was restored [1, 2, 6, 12, 30]. Changing the loudness exponent up to a factor of 2 has also been found to have a small effect on speech recognition performance at least in quiet listening conditions, but may have a much greater impact on speech intelligibility performance in noisy listening conditions [12]. In addition, speech intelligibility performance in noisy environments may be detrimentally affected, presumably because environmental noise and background sounds may occupy too much of the already limited dynamic operating range in a cochlear implant.
The major objectives of this study were to examine psychophysical loudness growth measures in twelve adult cochlear implant listeners and to determine whether the Absolute Magnitude Estimation (AME) and Absolute Magnitude Production (AMP) psychophysical methods as previously discussed in Zwislocki and Goodman 1980, can be successfully applied for describing and measuring the sensation of loudness produced by cochlear implants as a function of current level. These methods avoid the presumed bias related to a priori assignment of standards [37].
Currently, there is no consensus on the actual shape of the loudness growth function, with several “expansive” functions being suggested, as described above. The shape of the loudness growth functions measured in cochlear implant users were reexamined based on some a priori models. Since choosing the bestfit models in an objective way can be difficult, Akaike’s Information Criterion (AIC, [17]) approach was used as a quantitative way to find the bestfit loudness functions in cochlear implant listeners, and to compare various functions with different numbers of parameters mentioned in previous studies by evaluating goodness of fit. There is a tradeoff between goodnessoffit and parsimony in model selection, but given the limited a priori knowledge, it would seem more appropriate to use a simple model approach. Hence, the AIC method of model selection was used in this study by choosing the bestfit model to the data with the smallest number of parameters.
Some additional goals were to investigate the variation in loudness growth across the population of listeners, for different electrode pairs representing different places along the electrode array, and to investigate the hypothesis that a listener’s developmental history may have an effect on the shape of the loudness function. The relationships between electrical dynamic range and threshold values and the shapes of the loudnessgrowth functions were also compared.
Subject 
D.O.B. 
Sex 
Implant Type 
Cause of Hearing Impairment 
Age of Hearing Loss Onset 
Age of Diagnosed Profound Hearing Loss 
Age of CI activation (yrs.) 
Pre, or Postlingual Deafness 
S1a 
3/1/1983 
M 
Nucleus 22 
Unknown/ possible neuropathy 
Birth 
Diagnosed at age 2 yrs  but severe to profound from birth 
7 
Prelingual  first learned to speak at age 7 yrs Wore hearing aids starting at age 2 yrs 
S1b 
3/1/1983 
M 
Nucleus 24 
Unknown/ possible neuropathy 
Birth 
Diagnosed at age 2 yrs  but severe to profound from birth 
17 
Prelingual  first learned to speak at age 7 yrs Wore hearing aids starting at age 2 yrs 
S2 
10/3/1984 
F 
Nucleus 22 
Usher's Syndrome 
Birth 
4 months 
6 
Prelingual 
S3 
12/9/1978 
M 
Nucleus 22 
Unknownprogressive bilateral hearing loss 
Birthdiagnosed at 18 months of age 
? *Subject to confirmation* 
12 
Postlingual Learned to speak at age 2 yrs 
S4 
10/3/1951 
F 
Nucleus 24 Contour 
Hereditary hair cell loss progressive bilateral hearing loss 
37 yrs. 
46 
50 
Postlingual 
S5 
1/6/1957 
M 
Nucleus 22 
Trauma 
35 yrs. 
35 
35 
Postlingual 
S6 
9/19/1968 
F 
Nucleus 24 Contour 
Illness  progressive bilateral hearing loss 
4 to 5 yrs. 
early 30's 
34 
Postlingual 
S7 
9/21/1931 
M 
Nucleus 24 Contour 
Unknown  progressive bilateral hearing loss 
61 
66 
70 
Postlingual 
S8 
10/2/1964 
F 
Nucleus 24 Contour 
Rubella Epidemic 
Birth 
Birth 
37 
Prelingual  wore hearing aids starting at age 2 yrs 
S9 
8/9/1966 
F 
Nucleus 24 
UnknownGenetic/progressive bilateral hearing loss 
4 
32 
36 
Postlingual 
S10 
10/20/1984 
F 
Nucleus 22 
Unknown 
Birth 
1 
4 
Prelingual  wore hearing aids since age 2 yrs 
S11 
3/14/1951 
F 
Nucleus 24 Contour 
loss 
15 
26 
50 
Postlingual 
S12 
3/3/1960 
M 
Nucleus 24 Contour 
Unknown  Progressive 
10 
late 30's 
44 
Postlingual 
Signals consisted of trains of 200μsec/phase biphasic pulses with a 20 μsec silent interval between pulse phases. The biphasic pulse trains were 100mseclong at a fixed stimulation rate of 500 pps. The stimuli were presented in bipolar (BP + N) mode with varying separation of electrodes. Electrode pair (A, B) therefore indicates current flow between electrode A and electrode B = A+1+N where the lower numbered electrode, A, serves as the source for the initially negative current pulse and the return is via electrode B. N represents the number of electrodes between the current source and reference electrode. Electrode pairs were chosen to represent different lengths along the implant arrays, different dynamic ranges, and different
absolute thresholds. Experiments involved presenting stimuli on electrode pairs including; 4,7; 10,13; 16,19; 10,12; 13,16; 12,13; and 10,16. The most typical bipolar pairs measured were 4,7; 10,13; 16,19; 10,16. For precautionary reasons, measurements were not made on electrodes that were flagged or turned off by the subjects’ audiologists as indicated by their most recent maps. Effects of varying other stimulus parameters such as stimulus frequency, rate of stimulation, interphase gap, phase duration, and stimulation mode (bipolar, monopolar) were not investigated.
Signal thresholds were approximated either by asking the subject to adjust the current to a just audible level and more precisely by using a 3interval forcedchoice paradigm, with feedback, converging at the 79.4% correct point [13]. Maximum Allowable Levels (MALs) were determined by asking subjects to adjust the current to the maximum level that they felt was tolerable for experimentation with the particular electrode configuration. Although loudness appeared to play a key role in the choice of MAL, subjects reported that their MAL was also determined by a variety of unacceptable sensations and physiological responses such as facial twitches that would be produced by higher current stimulation. Thresholds, MALs, dynamic ranges (the difference between the MAL and threshold), and loudness growth functions were measured on the abovementioned bipolar electrode pairs to assess amplitude cues representing various locations across the cochlea. Currents of adjustable magnitude were presented under dedicated computer control, with current changing in steps of one clinical unit when controlled by the subject.
Loudness was measured as a function of current level using the methods of Absolute Magnitude Estimation (AME) and Absolute Magnitude Production (AMP). These methods, based on a wide range of psychophysical studies [36, 25], assume that subjects utilize an internal loudness scale in order to provide numbers or line lengths whose subjective magnitudes match the loudness of the corresponding sounds. The subject’s numerical responses are considered a “direct” measurement of their perception of sensation magnitudes. Hence subjective magnitude is assumed to have a single common scale or dimension that is shared by all of these modalities: loudness, numbers, and indeed other sensory magnitudes such as brightness. Furthermore, it is often assumed that loudness is directly proportional to the matching numbers and line lengths, which in turn are approximately proportional to one another. This method has the potential for measuring and comparing loudness under various conditions without utilizing constraints in which particular numbers or line lengths are a priori assigned by the experimenter to particular magnitudes or sensations, hence the term “absolute” is used as opposed to relative which involves a predetermined standard. Standards have been used in the previously cited studies and presuppose, for example, that the experimenter can assign the appropriate number for different electrode pairs, i.e. 100 is equated to the MAL. The “absolute” approach attempts to minimize potential biases created by the number range chosen by the experimenter, since such numbers may not necessary correspond to the numbers that the subject would normally use. These biases have been shown to change the shape of the loudness growth curve, for example, increasing the steepness or flattening the curve [37]. In the method of Absolute Magnitude Estimation, the subject is asked to produce a numeric value corresponding to a loudness magnitude he or she perceives based on his or her mental scale, without being given a predetermined range of numbers to use.
Before beginning the loudness experiments, the subjects were given an initial training session of AME and AMP of line lengths and numbers. Subjects were first asked to match numbers to line lengths, and vice versa, avoiding any explicit reference to upper or lower bounds of numbers. This training or practice session between line lengths and numbers were used to allow the subjects to become more familiar with using their own number scaling system, similar to methods in Collins and Gescheider [5]. Lines of adjustable length were projected on the wall of a dimly lit room against a featureless background. The subjects sat 265 cm from the wall and observed a thin line projected on the wall at a height of 231 cm. First lines of random length were presented on the wall and the subjects were instructed to assign numbers to match the magnitude of the line length. The subjects were then presented cards with numbers in the range of numbers that were used in the initial training session. The subject could adjust the line length in small steps of approximately 0.2% of the maximum length of 143 cm by turning an unmarked Griffin Technology PowerMate® USB Multimedia controller knob connected to our computercontrolled research interface. Results from the practice session revealed that all subjects in this study were able to use their own number scaling to match their perceptual concept of line length effectively and results were generally close to a linear relationship, as found in previous studies [25].
Listeners were then instructed to match loudness of current amplitude stimulation of the cochlear implant to numbers, again using their own natural scales in the AME and AMP experiments. In the AME experiments, the biphasic pulse train signals on a single bipolar electrode pair were presented at different “quasirandom” electrical amplitudes between and including the subject’s threshold and MAL for each electrode pair. At least 10 different amplitude levels were presented in a run and each run was repeated three times, with the amplitude levels rerandomized and presented in all three runs. The first run was considered to be a “calibration period” and the geometric mean was taken of only the last two magnitude estimations for each current level and was included in the data analysis, similar to methods in Stevens, 1975.
In the AMP experiments, the subjects were given cards with numbers on them within the range of numbers that they derived in the AME experiments. At least 10 different numbers were quasirandomly presented in a run and each run was repeated three times, similar to the AME experimental methods. The current level of the biphasic pulses were manipulated by the subject by turning an unmarked Griffin Technology PowerMate® USB Multimedia controller knob connected to our computercontrolled research interface.
In an additional series of experiments, not included in this paper, subjects were asked to match line length to loudness and the results appear to closely resemble the experimental results using numbers [21]. Before each of the experiments, the subjects were given a sheet of paper with instructions. Some examples follow:
Absolute Magnitude Estimation: In this experiment, I will present a sound to you. From one presentation to another, the sound may seem different in loudness, at random. Some sounds may seem loud, and others may seem quiet. I am interested in your impression of how loud each sound seems to you. I would like you to assign a number to every sound so that your subjective impression of how large the number is matches your impression of how loud the sound is. You may use any positive numbers that seem appropriate to you – whole numbers, decimals, or fractions. When you think of a number, it may appear subjectively large or small or in between. Obey your intuition and remember that numbers have no limits. There can always be a larger number than the largest you assign and a smaller one than the smallest you assign.
After every presentation of a sound, I would like you to write the number you have assigned on a piece of paper and place it face down in a neat pile on the table next to you. Say “ok” when you are ready for the next presentation. Any questions?
Absolute Magnitude Production: In this experiment, I will give you cards, one at a time, with a number on each. During every card selection, I would like you to adjust the loudness of the sound using the unmarked knob, to match your subjective impression of the magnitude if the number on the card. After you have adjusted the loudness of the sound to match your subjective impression of the magnitude of the number, just say “ok,” and I will move on to another card. Any questions?
KaleidaGraph™ graphical software was used to produce all of the graphical presentation of the figures in this paper and GraphPad Prism version 4 statistical trial software package, San Diego, C.A., was used for curvefitting and the AIC analyses. SPSS® version 15 (SPSS, Inc.) was used specifically for statistical analysis. A variety of models were used, as per those used in earlier studies, these include some of the following and a straight line was also used based on visual inspection.
1. Linear Function Model (LFM)
L = ax + k
2. Power Function Model (PFM)
L = a(x)^{b} + k
3. Exponential Function Model (EFM)
L = a^{*}e^{bx} + k
4. Fourthorder polynomial model
L = a_{0} + a_{1}x^{1} + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4}
Correlation coefficients for models or functions may have close R2 values (i.e. 0.90 vs. 0.92) and there is no way to quantify differences in the correlation coefficients for different “bestfit” functions because R2 has no units and only reveals the amount of variability in the data that can be explained by the model [17]. Functions with the highest number of parameters may automatically have higher R^{2} values because more parameters can better fit fluctuations or noise in the data curves. The least squares method using Ftest or Pvalues for significance can tell you what is significant or not significant or that the models may be statistically different, but cannot tell you which model to use by a direct comparison between two or more models. F tests should only be used for nested models (meaning one function is a simpler case of the other as in multiple polynomial functions) and the Ftest cannot be extended to compare three of more models because multiple P values cannot be directly compared [17]. We used a modified AIC method to compare bestfit growth of loudness functions, called AICc, which is a variation of the AIC equation that utilizes a second term that takes into account the number of parameters [17]. The AICc formula is recommended to use when the number of data points is small relative to the number of parameters. The AICc can be computed as follows;
where N equals the number of data points, SS is the sum of squares of the vertical distances of the points from the curve or the residuals, K is the number of parameters fit by the regression plus 1 [17]. If the N/K ratio is larger than 40, the second term becomes negligible. The AICc method of bestfit model comparison has been used previously to investigate nonlinear psychophysical modeling [19] and is commonly used for analysis of biological system model comparisons. Comparing AIC values is a good method for comparing nonnested nonlinear models and for evaluating the strength of evidence of the probability that one model is better at approximating the loudness data than the other comparison models or growth functions. The AIC method estimates which function or model most likely best approximates the data and quantifies how much more likely based on the function with the smallest number of parameters without overfitting or underfitting the data [17]. The AIC method can also be used for goodnessoffit approximation by estimating the relative distance between the fitted model and the data, using a combination of a maximum likelihood method and information theory approach to minimize the residuals [17].
The bestfit growth of loudness function can be determined by comparing the AICc values once fit to the data points and choosing the model with the lowest AICc value. The differences between the AICc values between two models can also help explain how the bestfit model can be selected. The differences between the AICc values were also calculated and quantified by approximating the strength of evidence with the evidence ratio of probabilities that one model is better at approximating the data than the other models in comparison. AICc was used to find the model that correctly best fits the loudness growth function data, and was used to quantify how much more likely it fit the data than another comparison models by comparing the output range of probabilities of one model fit compared to the others. The AICc values were computed for all of the possible model comparisons, (i.e. linear vs. power; power vs. exponential; linear vs. exponential, etc.), for each data set or loudness growth curve and the model with the lowest AICc value was selected as the “bestfit” model between the model comparison cases.
Subjects were given the freedom to use any range of numbers corresponding to loudness with the AME procedure,
In contrast, the loudness functions in figure 2 illustrate that for this second set of individuals (Subjects S1, S2, S8, S10), both AME and AMP yield “linear” functions for the electrode pairs utilized. Results from Subject 1 demonstrated a linear relationship between stimulating current and loudness with both the subject’s Nucleus 22 and Nucleus 24 implants, represented by Subject 1a and Subject 1b data (Figure 2).
Visual inspection of loudness growth curves as in figures 1 and 2 appeared to reveal at least two classes of subjects; those that produced the previously seen “expansive” loudness growth functions and those that produced “linear” functions when plotted as a linear function of current level in microamperes. These particular loudness growth functions were wellfit by either exponential (Fig. 1), linear functions (Fig. 2) as evidenced by their correlation coefficients.
Tables 2 and 3 some results of AICc bestfit functions analysis of loudness growth functions measured by Absolute Magnitude Estimation on different electrode pairs for the postlinguallydeafened subjects. Many of these functions were included in the plots in figure 1. It should be noted that highdegree polynomials were unstable and diverged outside of the data range, making it difficult to fit the data, so that data was not included here. The bestfit function was chosen by selecting the model with the lowest AICc values between the linear, power, and exponential fits. The accuracy of the fits for each loudness growth data set can be obtained by pairing two models and finding the AICc difference and comparing the probability between each of the three models based on the difference in AICc values, resulting in the percent likelihood that each model is the correct fit. The percentages, or Akaike’s weights, were calculated using the relative likelihood method described by Motulsky & Christopoulos, [17]. The last column of Table 2 lists the power fit function exponent for each data set and the results are discussed in further detail below. Note that in some individuals, both power and exponential best
Table 3 shows examples of the AICc bestfit function results measured on various electrode pairs for the prelinguallydeafened subjects [Table 3].
According to the results from the AICc analysis, the shape of the loudness growth functions were consistent with the classifications based on observation and indicated that there were two classes of growth functions; linear and expansive (power and exponential best fit functions). Subjects S1a, and S1b, S2, S8, and S10 seem to demonstrate bestfit linear loudness functions, whereas subjects S3, S4, S5, S6, S7, S9, S11, and S12 mostly demonstrate bestfit expansive functions (power, exponential). Subjects were classified as having “linear” or “expansive” based on the modal score or the most frequent bestfit functions measured on multiple electrode pairs for that individual. Some subjects with expansive growth of loudness demonstrate both power and exponential bestfit functions on different electrode pairs, as shown by Subject S3 in Table 2.
Figure 4 (left) demonstrates linear functions for all electrode pairs and electrode separations with an increasing slope value for wider bipolar configuration for Subject S1a. Results in figure 4 (right) are shown for another subject (S3) demonstrating more “expansive” growth of loudness functions for different electrode pairs and electrode separations. Similar to Subject S1a, this subject demonstrates an increasing exponent value and a decrease in dynamic range as for wider bipolar stimulation, similar to results found in Chatterjee [4]. These preliminary data reveal that the general shape of the growth of loudness function is independent of stimulus configuration, but the exponent or slopes of the loudness growth functions are dependent on electrode separation, with wider bipolar separation producing steeper curves. The steeper curves are presumably produced because more nerve fibers are recruited as electrode separation increases but this effect was not studied in detail.
The loudness growth functions obtained via AMP methods generally yielded steeper slopes or higher exponents than for results obtained with AME measured on the same electrode pair, demonstrating the “regression effect,” that describes the phenomenon in which subjects typically reduce the range of the variable that they control. This phenomenon has been commonly observed in previous studies using these methods [26, 14].
The mean powerfunction exponent for loudness functions bestfit by power functions, based on the AICc analysis was approximately 2.47 for AME experiments only, similar to the exponent 2.72 reported by Fu and Shannon [12]. The mean powerfunction exponent obtained for only the classified bestfit power functions, based on the AICc analysis for AMP experiments, was approximately 3.05. The histograms in figure 5 (AME results shown on the left, AMP shown on the right) reveal that when the AICc obtained bestfit functions were not powerfunctions, there were larger distributions of exponents, as shown by the histograms in Fig. 5, the exponents were generally higher for exponential bestfit functions (mean AME exponent =4.74; mean AMP exponent = 5.34) and lower for linear bestfit functions closest to an exponent of around 1, (mean AME exponent = 1.266; mean AMP exponent = 1.16). The mean exponents (+/ 1 s.d.) for powerfit functions fit to AICc –classified linear, power, and exponential bestfit functions and standard deviations were plotted in figure 6 for the AME (left) and AMP (right experiments). There was some overlap in the distributions between the
Table 2: AICc Analysis Results for Postlinguallydeafened Subjects 

Subject 
Electrode Pair 
AICc values 
Models 
AICc Difference 
Probability Correct in Percent 
Best Fit Model 
Power Function Fit Exponent 

Linear 
Power 
Exponential 

S3 
10, 13 
36.32 
5.91 
2.55 
lin/pow 
30.4 
0.01 / 99.9 
Power 
6.427 
pow/exp 
3.36 
15.71/84.29 
Exponential 

lin/exp 
33.77 
0.01 / 99.9 
Exponential 

S3 
16, 19 
109.61 
88.62 
95.61 
lin/pow 
20.99 
0.01 / 99.9 
Power 
3.62 
pow/exp 
6.99 
97.07 / 2.93 
Power 

lin/exp 
13.99 
0.09 / 99.91 
Exponential 

S4 
10, 13 
36.32 
5.91 
2.55 
lin/pow 
30.4 
0.01 / 99.99 
Power 
5.207 
pow/exp 
3.36 
15.71/84.29 
Exponential 

lin/exp 
33.77 
0.01 /99.99 
Exponential 

S4 
4,7 
27.56 
2.57 
0.2 
lin/pow 
30.13 
0.01 / 99.99 
Power 
3.617 
pow/exp 
2.37 
76.08/23.92 
Power 

lin/exp 
27.76 
0.01 /99.99 
Exponential 

S5 
12, 14 
80.53 
61.94 
64.73 
lin/pow 
18.59 
0.01 / 99.99 
Power 
2.655 
pow/exp 
2.79 
80.11/19.89 
Power 

lin/exp 
15.8 
0.04/99.96 
Exponential 

S6 
4, 7 
44.21 
29.38 
25.59 
lin/pow 
14.83 
0.06 / 99.94 
Power 
4.046 
pow/exp 
3.79 
13.06/86.94 
Exponential 

lin/exp 
18.62 
0.01 /99.99 
Exponential 

S6 
10, 13 
126.47 
117.9 
116.12 
lin/pow 
8.57 
1.36 / 98.64 
Power 
5.552 
pow/exp 
1.78 
29.18/70.82 
Exponential 

lin/exp 
10.35 
.56 / 99.44 
Exponential 

S7 
10, 13 
34 
15.89 
13.78 
lin/pow 
18.11 
0.01 / 99.9 
Power 
4.23 
pow/exp 
2.11 
25.88/74.12 
Exponential 

lin/exp 
20.22 
0.01 / 99.9 
Exponential 

S7 
4, 7 
46.9 
38.44 
40.99 
lin/pow 
8.46 
1.43 / 98.57 
Power 
3.225 
pow/exp 
2.55 
78.19/21.81 
Power 

lin/exp 
5.91 
4.95 / 95.05 
Exponential 

S9 
10, 13 
36.32 
5.91 
2.55 
lin/pow 
30.4 
0.01 / 99.99 
Power 
6.91 
pow/exp 
3.36 
15.71/84.29 
Exponential 

lin/exp 
33.77 
0.01 /99.99 
Exponential 

S11 
4,7 
0.14 
8.36 
13.7 
lin/pow 
8.5 
2.35 / 97.65 
Power 
2.441 
pow/exp 
5.34 
93.92 / 6.08 
Exponential 

lin/exp 
13.84 
.16 /99.84 
Exponential 

S12 
16,19 
1.29 
24.73 
6.04 
lin/pow 
23.44 
0.01 / 99.99 
Power 
2.3 
pow/exp 
18.69 
93.30 / 6.70 
Power 

lin/exp 
4.75 
0.01 / 99.99 
Exponential 

S12 
10,13 
10.7 
27.67 
4.35 
lin/pow 
16.97 
0.02 / 99.98 
Power 
2.017 
pow/exp 
23.32 
99.99 / 0.01 
Power 

lin/exp 
6.35 
96.00 / 4.0 
Linear 



Figure 6 shows the mean powerfit exponent for all linear, power, and exponential bestfit functions as determined by the AICc analysis, with +/ standard deviation error bars. Single factor ANOVA analysis revealed a significant difference in the powerfit function exponents between all groups (F(2,42) = 34.24, p< 0.05) for the AME experiments, as well as for the AMP experiments (f(2, 28) = 39.85, p< 0.05). Significant differences were confirmed by the Bonferroni and Fischer’s Exact test.
The mean powerfunction exponents obtained using the AME method, shown in figure 7 (left), for prelingual subjects (M=1.68, SD=1.08, N=5) and figure 7 (right)  postlingual subjects (M=4.22, SD=1.61, N=7), were significantly different and lower for the prelingual group of subjects, as revealed by twosample ttest assuming unequal variances (t(10) = 3.27, p< 0.05). The mean powerfunction exponents obtained using the AMP method, as shown in the right panel of figure 8 for prelingual subjects (M=1.24, SD=0.95, N=5) and postlingual subjects (M=3.069, SD=1.62, N=7) were also significantly different (t (10) = 2.45, p< 0.05), as shown by a two sample ttest assuming unequal variances. Results were confirmed by the Tukey HSD posthoc analysis. The power function exponents were significantly lower for the prelingual group. The means and standard deviations are plotted for the prelinguallydeafened versus the postlinguallydeafened subjects in Figure 8 for the AME experiments (left) and AMP experiments (right). Hence, when the power function is used to fit all loudness growth curves, loudness growth was significantly shallower for the prelingually deafened group, compared to the postlingually deafened group (Figure 8).
The relationship between electrical dynamic range in current level (Amps) and the shapes of the loudness growth functions is shown in figure 9. The mean dynamic range is shown for each AICc bestfit function category (linear, power, and exponential) for the loudness growth function results obtained for the BP+2 electrode pair configurations for all subjects and their dynamic ranges measured on those electrode pairs. The mean dynamic ranges were calculated for each electrode pair on which the loudness growth functions were measured, and were plotted in the bar graph, including the standard deviations as the error
Table 3: AICc Analysis Results for Prelinguallydeafened Subjects (AME Experiments only) 

Subject 
Electrode Pair 
AICc values 
Models 
AICc Difference 
Probability Correct in Percent 
Best Fit Model 
Power Function Fit Exponent 

Linear 
Power 
Exponential 

S1a 
24.92 
38.86 
39.48 
lin/pow 
13.94 
99.91/ .09 
Linear 

pow/exp 
0.62 
57.63 /42.37 
Power 
0.951 

lin/exp 
14.56 
99.93 / 0.07 
Linear 

S1a 
4,7 
32.81 
44.17 
34.6 
lin/pow 
11.36 
99.66/ .34 
Linear 

pow/exp 
9.57 
.82/99.18 
Exponential 
1.853 

lin/exp 
1.79 
70.85 /29.15 
Linear 

S1b 
10,13 
78.91 
80.33 
106.78 
lin/pow 
1.42 
67.06 /32.94 
Linear 

pow/exp 
26.45 
99.99 / 0.01 
Power 
0.779 

lin/exp 
27.87 
99.99 / 0.01 
Linear 

S2 
10,13 
41.95 
45.87 
44.37 
lin/pow 
4.22 
87.67 /12.33 
Linear 

pow/exp 
1.5 
32.15 /67.85 
Exponential 
1.243 

lin/exp 
2.42 
77.11 / 22.89 
Linear 

S2 
10,13 
22.94 
26.11 
38.02 
lin/pow 
3.16 
83.03 /16.97 
Linear 

pow/exp 
11.91 
99.74/ .26 
Power 
0.8506 

lin/exp 
15.08 
99.95 / 0.05 
Linear 

S8 
4,7 
13.29 
14.86 
33.03 
lin/pow 
1.57 
68.74/31.26 
Linear 

pow/exp 
18.17 
99.99/0.01 
Power 
1.331 

lin/exp 
19.74 
99.99/0.01 
Linear 

S8 
4,7 
33.82 
37.61 
47.65 
lin/pow 
3.791 
86.94 /13.06 
Linear 

pow/exp 
10.04 
99.34 / 0.66 
Power 
1.067 

lin/exp 
13.83 
99.90 / 0.10 
Linear 

S10 
10,13 
13.29 
14.86 
33.03 
lin/pow 
1.57 
68.74/31.26 
Linear 

pow/exp 
18.17 
99.99/0.01 
Power 
2.025 

lin/exp 
19.74 
99.99/0.01 
Linear 

S10 
16,19 
15.85 
18.55 
22.95 
lin/pow 
2.7 
79.45 /20.55 
Linear 

pow/exp 
4.4 
90.19/9.81 
Power 
1.661 

10,13 
lin/exp 
7.1 
97.27/2.73 
Linear 
It has been suggested that the sensation of threshold and loudness could be related to the physiological status at the electrodeneuron interface, i.e. the local neuron survival or the location of the electrode relative to the nerve [22], and possible morphological and physiological differences may exist between prelingual and postlingual patients [27]. Mean thresholds for all of the BP+2 electrode pairs were also compared between the AICc classified loudness growth function groups (linear, power, exponential) and between prelingual and postlingual subject groups. Figure 10 shows the relationship between the AICc bestfit linear, power, and exponential functions for AME obtained results for the BP+2 electrode pair configurations for all subjects and their threshold values measured on the respective electrode pairs (Figure 10).
The mean thresholds that were measured on each of the BP+2 electrode pairs on which the loudness bestfit functions (linear, power, and exponential) were obtained, and were plotted in the bar graph including the standard deviations as error bars. Mean thresholds measured in current μAmps),level (were lower for the AICcclassified exponential bestfit functions than for the classified bestfit linear or power functions, but a single factor ANOVA analysis revealed that there was no significant difference between the threshold means (F(2,79) =1.70, p=0.1877). Directly comparing the dynamic ranges (MAL – threshold) measured in current level (μAmps), for the prelingual versus postlingual groups revealed the mean dynamic ranges for prelingual subjects (M=545.18, SD = 285.36, N=5) were not significantly different from the dynamic ranges for postlingual subjects (M=387.17, SD
The mean thresholds for prelingual subjects (M=191.75, SD = 27.01, N=5) were not significantly different from the mean thresholds of postlingual subjects (M=149.96, SD = 89.29, N=7), (t(8)=1.07, p=0.31), as shown by twosample ttest assuming unequal variances.
There were no significant differences of dynamic range or thresholds accross all subjects’ growth of loudness functions using the same electrode configuration.
The above results can be seen by visually observing and comparing the measured growth of loudness functions. In addition, the AICc method of modeling is effective for objectively analyzing and comparing growth of loudness functions and agrees with the perhaps more subjective observations. Bestfit loudness growth functions as determined by AIC appear to show a difference in the loudness growth functions across subjects and across electrode pairs within individual subjects. The range of functions observed is greater than previously reported and goes from linear to expansive, suggesting that individual variations in loudness growth should be incorporated in the design of cochlear implant sound processors. Our results reveal that some subjects may show bestfit linear, power, and exponential functions on different electrode pairs and different individuals can show a range of these bestfit functions. By accurately identifying the shape of the loudness growth functions for different subjects, it should be possible to modify the shape of the mapping functions to restore or attempt to optimize normal growth of loudness for individual cochlear implant users.
Results of the powerfit function exponents also revealed that there is a relationship between the exponents of the loudness functions and the two subject populations (prelingual vs. postlingual), showing that prelingual subjects may generally show significantly lower growth of loudness function exponents when fit with powerfunctions.
Loudness growth functions that were previously measured using magnitude estimation have shown that loudness growth may have been best fit by a power function with an approximate exponent of 2.72 [12]. Our results reveal that the exponent of the powerfit function, when proven as the bestfit function, agrees with previously obtained approximate exponents. However, when a powerfit function is fit to a bestfit linear or exponentialfunction, the exponents reveal a much wider distribution of values, approximating a value near 1 for linear functions and a value close to 5 for exponential functions. In Shannon, et al. 2004, [23] and Fu and Shannon 1998,[12] it was predicted that a crossmodal match between the bestapproximate loudness function exponents of normal hearing listeners (0.6) and the approximated loudness function exponent of the cochlear implant listeners (2.72), by taking the ratio of the two exponents, revealed that the exponent resulting from the ratio of the two agreed with the peak in the speech intelligibility performance of cochlear implant listeners. All of the subjects in their experiments were postlinguallydeafened and our data agrees with the bestfit loudness growth function exponent range obtained on the postlinguallydeafened subjects in this study. However, the few prelinguallydeafened subjects in our study show bestfit approximate exponents around 1, demonstrating a much more linear loudness growth function. These results may suggest that an appropriate crossmodal match between acoustic and electric amplitude to obtain more normal loudness growth may be different for the prelinguallydeafened population of subjects.
An exponential loudness growth model proposed by Zeng and Shannon [28] assumes that the loudness growth function can be predicted by determining threshold and uncomfortable loudness levels, but our preliminary results suggest that the shape of the loudness growth function is independent of electrical dynamic range and thresholds for the BP+2 electrode pairs.
The results suggested that there were no significant differences in thresholds or dynamic ranges between prelinguallyand postlinguallydeafened subjects. These results may be due to the large variability found in thresholds and dynamic ranges measured on several BP+2 electrode pairs within individual subjects as well as between subjects. In this study, results suggest that there may be two populations of cochlear implant listeners: those who have “developed” the expansive nonlinearities in the central auditory system and those, mainly earlydeafened subjects implanted at four or more years after birth, who have not “developed” the expansive loudness growth in the central auditory system. It should be noted, however, that prelinguallydeafened subjects with early implantation – before 3 years of age, were not included in this study. Since these results show that the shapes of the loudness functions vary both across subjects, as shown by the “linear” and “expansive” bestfit classifications, and within an individual subject on each electrode pairs, as shown by individuals displaying both power and exponential bestfit functions. Since current cochlear implant speech processors assume a uniform loudness growth function for all electrodes and subjects, it may be indicated that speech processors may need to be redesigned to take into account individual differences in loudness growth to maximize listening performance for cochlear implant listeners.
There are limited previous studies performed on loudness growth functions and speech recognition in prelinguallydeafened subjects. Earlydeafened patients may have limited auditory experience and due to longterm sound deprivation prior to cochlear implantation, these patients may have poorer abilities of speech recognition as well as differences in psychophysical function [3, 27]. Although speech recognition abilities for the subjects in this study were not measured, there may be a possible relationship between speech recognition abilities for prelinguallydeafened subjects who have a lack of early normal speech listening experience, and improper amplitude transformation between acoustic and electrical stimulation. These variables will be addressed further in future investigations.
 Boex CS, Eddington DK, Noel VA, Rabinowitz WM, Tierney J and Whearty WE. Restoration of normal loudness growth for CIS sound coding strategies. Abstracts of 1997 Conference on Implantable Auditory Prostheses. 1997;26.
 Boex CS, Pelizzone M, Piloux V and Montandon P. Use of loudness scaling measurements to determine compressive mapping in speech processing for cochlear implants. Abstracts of 1995 Conference on Implantable Auditory Prostheses. 1995;57.
 Busby PA, Tong YC, and Clark GM. Psychophysical studies using a multipleelectrode cochlear implant in patients who were deafened early in life. Audiology. 1992;31(2):95111.
 Chatterjee M. Effects of stimulation mode on threshold and loudness growth in multi electrode cochlear implants. J Acoust Soc Am. 1999;105(2):850860.
 Collins AA and Gescheider GA. The measurement of loudness in individual children and adults by absolute magnitude estimation and crossmodality matching. J Acoust Soc Am. 1989;85(5):20122021.
 Cosendai G and Pelizzone M. Effects of the acoustical dynamic range on speech recognition with cochlear implants. Audiology. 2001;40(5):272281.
 Drullman, R. Speech intelligibility in noise: relative contribution of speech elements above and below the speech level. J Acoust Soc Am. 1995;98(3):17961798.
 Eddington DK, Boex SpanoC, Tierney J, Noel V, Whearty M. Speech processors for auditory prostheses. Seventh Quarterly Progress Report, NIH Contract N01DC62100, 1997.
 Freyman RL and Nerbonne GP. The importance of consonantvowel intensity ratio in the intelligibility of voiceless consonants. J Speech Hear Res. 1989;32(3):524535.
 Freyman RL and Nerbonne GP and Cote HC. Effect of consonantvowel ratio modification on amplitude envelope cues for consonant recognition. J Speech Hear Res. 1991;34(2):415426.
 Fu QJ. Loudness growth in cochlear implants: effect of stimulation rate and electrode configuration. Hear Res. 2005;202(12):5562. doi:10.1016/j.heares.2004.10.004
 Fu QJ and Shannon RV. Effects of amplitude nonlinearity on speech recognition by cochlear implant users and normalhearing listeners. J Acoust Soc Am. 1998;104(5):25702577.
 Levitt H. Transformed updown methods in psychoacoustics. J Acoust Soc Am. 1971;49(2):467477.
 Marks LE. Sensory Processes: The New Psychophysics. Academic Press, New York. 1974.
 McKay CM. “Psychophysics and Electrical Stimulation in Cochlear Implants,” In: F.G. Zeng, A.N. Popper, and R.R. Fay (Eds.). Auditory Prostheses and Electric Hearing, Springer Verlag, NY; 2004: 286333.
 McKay CM, Remine MD and McDermott HJ. Loudness summation for pulsatile electrical stimuli of the cochlea: Effects of rate, electrode separation, level, and mode of stimulation. J Acoust Soc Am. 2001; 110(3):1514 1524.
 Motulsky HJ and Christopoulos A. Fitting models to biological data using linear and nonlinear regression. A practical guide to curve fitting. GraphPad Software Inc, San Diego CA. 2003.
 Müller CG. Survey of cochlear implant work. J Acoust Soc Am Suppl. 1981;70(1):S52. doi: 10.1121/1.2018916
 Pitt MA, Myung IJ and Zhang S. Toward a method of selecting among computational models of cognition. Psychological Review. 2002;109(3):472491.
 Plomp R. The negative effect of amplitude compression in multichannel hearing aids in the light of the modulationtransfer function. J Acoust Soc Am. 1988;83(6):23222327.
 Sanpetrino NM and Smith RL. The growth of loudness functions measured in cochlear implant listeners using absolute magnitude estimation and compared using Akaike’s Information Criterion. Paper in: Engineering in Medicine and Biology Society, EMBS 28th Annual International Conference of the IEEE, New York. 2006;1:16421644. doi:10.1109/IEMBS.2006.259984
 Shannon RV. Threshold and loudness functions for pulsatile stimulation of cochlear implants. Hearing Research. 1985;18(2):135143.
 Shannon RV, Fu QJ, Galvin JJ III and Friesen L. Speech perception with cochlear implants, Chapter 8 in Auditory Prostheses, Zeng FG, Popper AN and Fay RR (Eds.), Springer Handbook of Auditory Research, Vol. X, SpringerVerlag, New York, 2004;20:334376.
 Shannon RV, Zeng FG, Wygonski J, Kamath V and Ekelid M. Speech recognition using only temporal cues. In: the Auditory Processing of Speech: from Sounds to Words, Schouten MEH (Ed.), Mouton de Gruyter, Berlin. 1995;270(5234):303304.
 Stevens SS. Psychophysics: Introduction to its Perceptual, Neural, and Social Prospects, edited by G Stevens. Wiley, New York. 1975;99133.
 Stevens SS and Greenbaum HB. Regression effect in psychophysical judgment. Perception & Psychophysics. 1966;1(5):439446. doi:10.3758/BF03207424
 Tong YC, Busby PA, Clark GM. Perceptual studies on cochlear implant patients with early onset of profound hearing impairment prior to normal development of auditory, speech, and language skills. J Acoust Soc Am 1988;84(3):951962.
 Zeng FG and Galvin JJ 3rd. Amplitude mapping and phoneme recognition in cochlear implant listeners. Ear & Hearing. 1999;20(1):6074.
 Zeng FG, Grant G, Niparko J, Galvin J, Shannon R, Opie J, et al. Speech dynamic range and its effect on cochlear implant performance. J. Acoust. Soc. Am. 2002;111(1):377386.
 Zeng FG and Shannon RV. Loudness balance between acoustically and electrically stimulated ears. Hear Res 1992;60:231235.
 Zeng FG and Shannon RV. Loudness coding mechanisms inferred from electric stimulation of the human auditory system. Science. 1994;264(5158):564566. doi: 10.1126/science.8160013
 Zeng FG & Shannon RV. Loudness of simple and complex stimuli in electric hearing. Annals of Otology Rhinology & Laryngology, Suppl. 1995;166:235238.
 Zeng FG, Shannon RV and Hellman WS. Physiological processes underlying psychophysical laws,” In: A.R. Palmer, A. Rees, A.Q. Summerfield, R. Meddis (Eds.), Psychophysical and Physiological Advances in Hearing, Proceedings of the 11th International Symposium on Hearing. 1997;473481.
 Zeng FG and Shannon RV. Psychophysical laws revealed by electric hearing. NeuroReport. 1999;10(9):19311935.
 Zwicker E and Scharf B. A model of loudness summation. Psychology Review. 1965;72:326.
 Zwislocki JJ. Group and individual relations between sensation magnitudes and their numerical estimates. Perception & Psychophysics. 1983;33(5):460468.
 Zwislocki JJ and Goodman DA. Absolute scaling of sensory magnitudes: A validation. Perception & Psychophysics. 1980;28(1):2838.