Dispersion in models of categorical perception

JOURNAL

OF MATHEMATICAL

Dispersion

PSYCHOLOGY

29, 271-288

in Models

A. J. ROZSYPAL,

(1985)

of Categorical

D. C. STEVENSON, AND J. T.

Perception* HOGAN

University of Alberta

A signal detection theory model of auditory discrimination with a nonlinear mapping from stimulus continuum to perceptual continuum can account for the enhanced discrimination at the category boundary found in categorical perception. Properties of this transformation are specified by a unimodal “dispersion function.” Furthermore, it is shown that a system consisting of two acoustic feature detectors with an associated decision function is also a dispersive system, which models categorical perception of a stimulus continuum as well as boundary shifts under adaptation. The effect of detector adaptation on discrimination is discussed in view of three different types of decision variable and different types of detector noise. I( ’ 1985 Academic

Press, Inc.

Categorical perception in speech was originally characterized by Liberman, Harris, Hoffman, & Griffith (1957) in a classic experiment in which it was shown that the discriminability of a series of synthetic /b/-/d/-/g/ stimuli was poorer within phonetic categories than between categories. Their study established a methodology for investigating categorical perception. They constructed a model of ABX discrimination based on the assumption that the ability to discriminate between speech stimuli was strictly a result of covert phonetic classification of stimuli. This model became the standard test for categorical perception: If the discrimination results could be adequately predicted by the model, then the continuum was “categorically perceived” (Studdert-Kennedy, Liberman, Harris, & Cooper, 1970). According to their criteria for the demonstration of “categorical perception,” the discrimination results were required to show an enhanced peak at the phoneme boundary and this peak had to be predicted on the basis of the labelling probabilities. Categorical perception of sound stimuli is not unique to speech. It is suggested that it may be a general property of biological sensory communication systems (Miller, Wier, Pastore, Kelly, & Dooling, 1976). For a detailed review of the issues related to categorical perception, including speech perception by infants and animals, selective adaptation, and cross-linguistic phenomena, the reader is referred to Repp (1983). * This paper is based on parts of the Ph. D. dissertation of D. C. Stevenson submitted to the Faculty of Graduate Studies and Research of the University of Alberta; his present address is: MacDonald, Dettwiler. and Associates, Vancouver, B. C., Canada.

271 0022-2496/85

$3.00

Copyright 8 1985 by Academic Press, Inc. All rights 01 reproduction in any form reserved.

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AND

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Although categorical perception can be fairly easily demonstrated for various speech and nonspeech continua, it has yet to be explained in any psychophysical terms. Several mechanisms for categorical perception have been proposed, such as rapid decay of auditory information with slow decay of phonetic information (Fujisaki & Kawashima, 1969, 1970; Pisoni, 1975), comparison of a single variable stimulus component against a fixed stimulus context (Miller et al., 1976), perceptual limits independent of phonetic processing (Wood, 19761, or a stable dichotomy forming naturally a category boundary (Pastore, Ahroon, Baffuto, Friedman, Puleo, & Fink, 1977). Although these mechanisms are all plausible, they have not been explicitly formalized.

DISPERSION

IN SENSORY

MAPPING

To characterize the possible nonlinearity of transformation between the physical stimulus dimension x and the perceptual dimension y, a dispersion function D(x) is introduced. This function is defined as the rate at which the perceptual variable y changes with respect to the physical variable X, D(x) = dy/dx.

(1)

For the case of direct mapping y = x the dispersion is constant, D(x) = 1. Assuming constant sensory noise amplitude, the discrimination along the x continuum must also be constant. However, the existence of a peak in the discrimination curve in categorical perception studies indicates that the dispersion cannot be constant: Within-category stimuli are mapped onto the perceptual dimension y such that the subjective distance between them is small, whereas stimuli near the boundary are separated by larger perceptual distances. In other words, the discrimination is better and hence the dispersion must be greater in the vicinity of the category boundary. For a given dispersion function D(x), the mapping of a stimulus x on the perceptual dimension y is

where the dummy variable < represents distance along the physical dimension x, and x0 is some convenient reference point. Cast in these terms, the enhancement of discriminability at the perceptual boundary xk is indicated by a peak in an underlying dispersion function. Such a peak positioned at xk will tend to map all values of x< xk onto one end of the y scale, and all values of x> xk onto the other. This results in a spreading of the y dimension with respect to the x continuum in the vicinity of the category boundary, with a more-or-less step-like transition from one y “state” to the other at the boundary. The steepness of the transition is determined by the width of the underlying dispersion function and by the distribution of the perceptual noise.

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DISPERSION IN MODELS OF PERCEPTION

As the width of the dispersion function peak narrows to zero, the dispersion function approaches a delta function and y(x) becomes a unit step function. This case represents perfect categorization, since y can only take on values of either 0 or 1. The other extreme, corresponding to continuous perception, occurs when the dispersion is constant. For intermediate cases, depending on the width and shape of the dispersion function, various degrees of “categorical perception” are possible. Consequently, instead of making dichotomous distinctions between “categorical” perception and “continuous” perception, it is probably more suitable to view some continua as “more categorical” or “less categorical” than others. Attributing dispersion to the receiver system captures conceptually and mathematically what has been observed in discrimination studies all along: Signals spaced equally along a physical continuum need not be spaced equally on any phychological one (Macmillan, Kaplan, & Creelman, 1977). The nonlinear mapping between x and y evidently appears to be either the “phoneme boundary effect” of Wood (1976) or the varying “combined response” of two feature detectors discussed by Elman (1979) or the “stable dichotomy” to which Pastore et al. (1977) refer, or the “Type 1” effect of Ades (1977), or the “perceptual discontinuity” of Hary & Massaro (1982). The dispersion function cannot normally be directly observed in order to be a priori specified. In these cases some explicit form of the dispersion function with appropriate attributes can be chosen and the estimates of its parameters extracted by fitting the model to the observed data. This would result in a curve-fitting model, but one which at least will allow parameterization of the perceptual continuum. A Gaussian curve may serve as a suitably flexible choice for a first approximation, even though there is no theoretical motivation for this function. One example, where the shape of D(x) can be a priori estimated, is offered here: Although the auditory range of the Greater Horseshoe bat is at least 9 to 96 kHz, the numbers of inner and outer hair cells and spiral ganglion neurons (Bruns & Schmieszek, 1980) as well as the number of inferior colliculus neurons (Schuller & Pollak, 1979), is disproportionately distributed. They are concentrated into an “acoustic fovea,” a relatively narrow frequency range between about 78 and 88 kHz, which spans the range of the bat’s orientation calls. Frequency resolution in hearing appears to follow closely the number of sensory cells per a given frequency interval (Zwislocki, 1965, p. 55). Thus a reasonable approach to modelling frequency resolution in this particular case would be to postulate a priori the dispersion function based on the neurophysiological data.

SIGNAL DETECTION

THEORY AND CATEGORICAL

PERCEPTION

Signal detection theory (Green & Swets, 1966) has been applied to the specific problem of categorical perception by Macmillan et al. (1977) and by Rosner (1984). In this section, a model of the AX discrimination process will be developed utilizing the familiar concepts of the theory of signal detection. A nonlinear transformation, 480/29/3-3

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AND

HOGAN

characterized by the dispersion function, will be incorporated into this model in order to simulate categorical perception. This model assumes that in an identification task the stimulus from a physical continuum x is mapped onto a perceptual continuum y. Due to perceptual noise a normally distributed noise component ~(0, a) of zero mean and variance a2 is added to the perceptual variable y, resulting in an internal continuous variable Y, Y=y+&(O,

r3).

(3)

Thus y is the expected value of Y. Initially, the mapping from x to y will be assumed to be y =x. If the perceptual continuum y is divided into two category regions by a boundary criterion y,, then the probability of an arbitrary stimulus being identified as the second category, for instance, is mYk)=~=

1’1

0(5,Y, cl &,

(4)

where 4 is a dummy variable of integration. Function 4 is the probability distribution of Y. For a Gaussian noise component E, the peak of 4 is located at y. The resulting identilication function would be an ogive. In the AX discrimination tasks, the two presented stimuli are mapped into two perceptual random variables Y, and Y,. As shown by Sorkin (1962), Zinnes & Kurtz (1968), and Zinnes & Wolfe (1977), the probability P, that these variables will be discriminated, i.e., separated by some subjective criterion dy,, is

(5) The function Qj is the cumulative normal distribution of 4. To incorporate dispersion into this signal detection theory model of discrimination, Eq. (5), the dispersion function D(x) must be specified. The mappings y, and y, for any stimulus pair x, and x2 are found by means of Eq. (2). Then the corresponding AX discrimination model can be computed from Eq. (5). In order to simulate categorical perception, any unimodal function would serve well for the dispersion function, such as the Gaussian function 1

D(x)=-e

- (x ~ -Y F/Z?72

(6)

J&

with the peak located at the category boundary xk and the width given by deviation v. Using Eqs. (6), (2), and (5), the AX discrimination probability be calculated for all stimulus pairs x, and x2. Computer solutions for values ranging from 0 to 1, various observer criteria Ay,, and disperson are shown in Fig. 1. As dispersion is increased, by decreasing the value peak of D(x) becomes higher and narrower, eventually appraoching

standard Pd can stimulus widths q of q, the a delta

DISPERSION

IN MODELS

27s

OF PERCEPTION

1.0 1.0 x2

0.5

?I

10

1.0 1.0

x2

0.5

?I

0 0

0.5 Xl

1.0

0

0.5

I.0

Xl

FIG. 1. Probability of discrimination P&C,, .x2) predicted by the signal detection theory model of AX discrimination assuming an underlying Gaussian dispersion function of standard deviation 1. Two values of the subjective criteria dy,. Perceptual noise is the same for all six cases, D = 0.1: (a) dy, = 0,5, 4 = 0.05; (b) Ayt = 0.5, r~= 0.2; (c) dyk = 0.1, q = 0.05; (d) dy, = 0.1, q = 0.2; (e) small 1: perfectly categorical discrimination; (f) large q: continuous discrimination.

function. In such case, the discrimination becomes perfectly categorical, as shown in Fig. le. While perfect categorization may not be demonstrated eperimentally, it is certainly the desired limiting behavior of the model. With decreasing dispersion, the width 7 of the dispersion function increases and eventually D(x) becomes approximately constant over the range of X. In this case the probability of discrimination, for a constant step size dx, i.e., along any line x2 =x, + dx, remains constant, never forming a peak, as shown in Fig. If. Constant dispersion function is thus characteristic for “continuous” perception. As the criterion dy, is changed, the entire level of “different” responses increases or decreases by the same amount anywhere along this line.

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STEVENSON,

AND

HOGAN

Discrimination predicted by this model, for several step sizes Ax, is illustrated in Fig. 2. For Ax = 0 the discrimination function has a constant value, derived from Eq. (5), dependent only on the observer criterion Ay,, P,=2{1-B[$]}.

(7)

This is true even when in the limit the dispersion function becomes a delta function, in which case the discrimination function has a singularity at x1 =x2 = 0.5, as illustrated in Fig. le. As the step size Ax is increased, a peak in the discrimination function emerges and gradually broadens.

1.0 -

0.5 -

0

0

I 1.0

0.5 X

lb)

FIG. 2. Calculated AX discrimination functions sizes Ax. Perceptual noise (U = 0.1) and the spread both cases: (a) Ay, = 0.5; (b) dy, = 0.05.

P,(x,, x2) for the dispersion model for various step of the dispersion function (q = 0.05) are the same in

DISPERSION IN MODELSOF PERCEPTION

A DETECTOR MODEL OF CATEGORICAL

277

PERCEPTION

Since the experiments of Eimas & Corbit (1973) it has been assumed in several speech perception studies that decoding of the speech signal is mediated by phonetic or, more recently, acoustic “feature detectors.” Two such detectors presumably span different segments of a stimulus continuum and are characterized by response functions representing the sensitivity of the detectors to stimuli along this continuum. The major support for this theory comes from shifts in the labelling curves in selective adaptation studies. It is proposed that, under repeated presentation of an adapting stimulus, one of the detectors becomes desensitized due to fatigue, with the result that the stimulus value for which the two detector outputs are equal, that is, the unbiased category boundary, shifts in the direction of the adapting stimulus (e.g., Eimas & Corbit, 1973; Miller, 1975; Ainsworth, 1977; Brady & Darwin, 1978; Cooper, 1979; Landahl & Blumstein, 1982; Miller, Connine, Schermer, & Kluender, 1983). This view is not universally accepted (Simon & Studdert-Kennedy, 1978; Elman, 1979; Remez, 1979, 1980). Diehl (1981), in reviewing the adaptation literature, proposes an alternative explanation in terms of contrast effects. To aid in the interpretation of experimental results (e.g., Miller, 1975; Ainsworth, 1977), the detectors are usually presented as graphical constructs. The following mathematical analysis shows that a system of two feature detectors and an associated decision function will have the dispersion properties necessary for categorical perception and, from these properties, the corresponding dispersion function can be derived. Consider a 2-detector system with properties similar to those assumed by Elman ( 1979 ). Response functions of these detectors are ui(x) and Q(X), assuming again the trivial relation y = x between the perceptual dimension y and the physical stimulus continuum x. The detector outputs Ui will be assumed to be normally distributed random variables, as in Eq. (3), centered around the expected values ui, ui = 24,+ go, a),

i= 1, 2.

(8)

In the interests of mathematical tractability, the width of both detectors will be equal, specified by the standard deviation 6 of the Gaussian response functions ui(x),

where z?, and i2, are the locations of maximum sensitivity of the response functions ui(x) and u,(x), respectively, as shown in Fig. (3). If the perceptual noise ~(0, a) is considered as a random additive component of the output of each detector, the uncorrelated random detector output variables U, and U2, similar to Y in Eq. (3), would be distributed as a circular normal

278

ROZSYPAL, S’I’EVENSON, AND HOGAN

X

FIG. 3. Two Gaussian detector response functions u,(x) with 6 = 0.2, spanning the stimulus continuum x, and positioned at 2, = 0.3 and & = 0.7.

probability density function in a a1 - uz sensory plane. The expected value points (u,, u2) are the centroids and (r the standard deviation of this density function. As the stimulus takes on various values on the x continuum, the points (ur , UJ trace out a line, referred to as the stimulus trajectory. The solid curve in Fig. (4) shows such stimulus trajectory for the Gaussian detector functions ui(x) from Fig. (3). The open circles indicate the mapping of equal steps dx. The decision line is represented by the dash-dotted line at 45”, corresponding to U, = u2. The probability that a given stimulus x will be classified as a member of the second category, u2 > ul, is then

“2

“I FIG. 4. Solid line: stimulus trajectory created by the locus of points (u,, ur) of two detectors with output functions u,(x) and IQ(X). Dashed line: effect of adaptation on a two-detector system. Only the first detector is adapted so that only the output u,(x) of the first detector is affected. The arrows connect identical values of x. Note that the category boundary, i.e., the point where the stimulus trajectory crosses the dash-dotted decision line, shifts towards the category of the adaptor.

DISPERSION IN MODELSOFPERCEPTION

279

where (z+ - ~,)/a is the normal distance from the point (ui, ZQ) to the decision line. The identification function for the stimulus trajectory shown by the solid line in Fig. (4) will be ogival in shape since this trajectory crosses the decision line only once. For more complicated detector response functions uJx), the stimulus trajectory may cross the decision axis several times, which will in general lead to multimodal identification functions. DISPERSION FUNCTIONS

In the 2-detector model of discrimination a direct mapping y = x is assumed, which cannot introduce dispersion. The dispersion function can be specified at a higher stage, in terms of the output functions ui(x) of the two detectors. In order to derive an expression for the dispersion function D(x), it is also necessary to define a single decision variable W(X) from the 2-detector outputs. From the previous definition of the l-dimensional dispersion function, Eq. (2), the corresponding 2dimensional vector valued function can be stated as

w(x) =jxD(T). br=ixD(?). (uidxi+ uidxj).

(11)

x0

x0

Here r = ui(x) i + U*(X) j, aF = du,i + L&J, and U; = du,/dx. The path of integration runs from an arbitrary point x,, to the presented stimulus x. In order for this line integral to be path independent, which is equivalent to stating that the similarity between two stimuli x, and x2 depends only on their respective positions on the stimulus trajectory, D(i;) must be related to W(X) as D(r) = grad w(x).

(12)

The above integral thus becomes w(x)=j-(gu;+$u;)dx 1

(13) 2

and it follows that the dispersion function can be specified as a function of x as

D(x)+;+&%;. I

(14) 2

Normal Distance Decision Variable

One obvious choice for the decision variable w(x) is the distance normal to the decision line, w(x)

U2--Ul = -. 3

(15)

280

ROZSYPAL,

STEVENSON,

AND

HOGAN

Its partial derivatives substituted into Eq. (14) yield the dispersion function

(16)

which is evidently the form suggested by Elman (1979), who, however, does not use the term “dispersion,” but states that the discriminability of two adjacent stimuli depends on the differences between the slopes of the two detector functions. For the Gaussian detector functions defined by Eqs. (9), this dispersion function is shown as the dashed line in Fig. (5). The point of maximum dispersion is located at the category boundary U, = u2. Angular Distance Decision Variable Another decision variable W(X) satisfying Eq. (12) can be defined as the angle between the position vector (u,, u2) and the decision line, u2-u1

tan w(x) = u2+u,’

(17)

In this case the dispersion function is

D(x)=

ulu;-uu;u~ u2+u2 1

t

(18)

2

shown as the solid line in Fig. (5).

FIG. 5. Dispersion line: normal distance, length, Eq. (20).

functions Eq. (16).

for two Gaussian detectos for three different decision variables. Dashed Solid line: angular, Eq. (18). Dash-dotted line: stimulus trajectory path

DISPERSION IN MODELS OF PERCEPTION

281

Path Length Decision Variable

A third possible measure of similarity of signals in the sensory plane is the path length along the stimulus trajectory separating the sensory mappings of the stimulus x and the boundary stimulus xk (19) where < is again the dummy

variable. The corresponding D(x) = dm

dispersion function (20)

is shown as the dashclotted line in Fig. (5). Next, the relative merits of the three possible dispersion functions proposed above will be evaluated. As seen in Fig. (5), all show a peak at the category boundary and decrease monotonically in the immediate vicinity of that peak. The dispersion function for the angular metric, Eq. (18), has mathematically the most desirable properties inasmuch as it is unimodal. The dispersion function based on

FIG. 6. Calculated AX discrimination response surfaces Pd(x,, x2) for two Gaussian detector response functions positioned at ?, = 0.3 and & = 0.7, as in Fig. (3). The angular distance dispersion function is given by Eq. (18). The decision criterion in (a) is dy, = 0.5 and in (b) dy, = 0.05. In both cases the perceptual noise is the same, 0 = 0.1.

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ROZSYPAL,STEVENSON,

AND HOGAN

the normal distance decision variable, Eq. (16), inverts the perceptual scale y with respect to the stimulus scale x in the range of the negative values of the dispersion function. Another property of this function is indicated by its integral, Eq. (2): It maps the extremely distant points of the x continuum, both to the left and to the right of the boundary, and the boundary point itself, into the same sensory value y. The dispersion function derived from the stimulus trajectory path length, Eq. (20), has local minima at the detector peaks and secondary maxima of enhanced sensitivity further away from the boundary, at the far side of the detectors. Choosing, for instance, the angular distance dispersion function D(x) defined by Eq. (18) for the case of two Gaussian detectors, the AX discrimination function P, can be calculated from Eqs. (5) and (2). The corresponding response surfaces are shown in Fig. (6).

SELECTIVE ADAPTATION

IN THE DETECTOR

MODEL

The model of categorical perception as mediated by a 2-detector system described above will now be investigated with respect to selective adaptation. According to the detector model, the effect of adaptation is to desensitize one of the detectors. This can be modelled by incorporating scaling factors a, into the detector response functions defined by Eqs. (9) u,(x) = &

1

e - (.‘(- .t,)2/2@,

i=l,2,

(21)

with a, = 1 for unadapted detector and 0
In this equation, the first term on the right side stands either for the unadapted boundary, or for a boundary produced by two detectors adapted by the same degree. In both cases a, = u2. The second term represents the boundary shift due to adaptation. For instance, for the detector configuration illustrated in Fig. (3), desensitization of the second detector results in reduction of a2 with respect to a,, the second term becomes larger and the adapted boundary shifts in the direction of the second, adapted, detector. This formula has four implications. First, desensitization of one of the detectors will cause a boundary shift in the direction towards the adapted detector. Second, desensitization of both detectors simultaneously by the same degree will cause no change in the boundary. The first result has been verified many times in the selective adaptation literature (e.g., Eimas & Corbit, 1973). The second result has also been demonstrated (Miller, Eimas, & Root, 1977; Sawush & Pisoni, 1976). The third implication is that, with other

283

DISPERSION IN MODELSOF PERCEPTION

things equal, larger boundary shifts will occur for less strongly categorized continua characterized by detectors with larger 6. Lastly, larger shifts should be expected for smaller detector separations (a, - 9,).

THE EFFECT OF ADAPTATION

ON THE STIMULUS

TRAJECTORY

The effect of adaptation on the stimulus trajectory is illustrated by the dashed line in Fig. (4), in which the detector for the first category is being desensitized. Desensitizing one detector is equivalent to scaling down the corresponding axis of the sensory plane. This causes a distortion of the stimulus trajectory such that the point of equality of the detector outputs, ui = u2, now corresponds to a different value of stimulus x from the preadapted state. An interesting result following from this is that the model predicts the experimentally measured boundary shift due to desensitization of one detector to be equivalent to the shift due to a change in response bias. In an adapted state the unbiased decision line u2 = ui may intersect the adapted stimulus trajectory at the same stimulus as a biased decision line u2 = flui in an unadapted state. For tan /? = al/al the boundary shifts, either due to bias or adaptation alone, are thus equal. This suggests that by simply measuring the boundary shift the effects of response bias and adaptation are experimentally inseparable, although it is evident that in the model this boundary stimulus for these two cases is mapped into two distinct points in the u1 - u2 sensory plane. This poses a difficult experimental challenge. To distinguish between these two cases, the researcher would have to design an experimental method allowing comparison of subjects’ internal sensory responses in unadapted, adapted, and biased state.

DISCRIMINATION

UNDER CONDITIONS

OF ADAPTATION

Notwithstanding whether in an adapted or unadapted state, discrimination is dependent on the shape of the dispersion function. As shown above, this function depends on the choice of the decision variable. In the following, the model behavior will be examined for the angular, normal, and path length variables. The following discussion applies only for the case of additive sensory noise, that is, noise whose standard deviation is independent of the value of the detector output. This assumption will be questioned later. Angular Distance Decision Variable

Consider the dispersion function defined by Eq. (18) for the angular decision variable. For the special case of the Gaussian detector functions given by Eq. (21) the dispersion function has the following form D(x)J~*-evz P(u:

+

u:,

.

(23)

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ROZSYPAL,

STEVENSON,

AND

HOGAN

The peak of the dispersion functon occurs for x such that dD(x)/dx = 0. Differentiating Eq. (23) and setting it to zero produces (u, - u2)(u1+ U?) D(x) = 0,

(24)

provided that ui and u2 do not go to zero simultaneously. Since the dispersion function D(x) for Gaussian detectors is nonzero for finite values of x, it follows from Eq. (24) that u1 = u2. Thus, this detector model of categorical perception using the angular distance variable has the property that for Gaussian detectors the peak of the dispersion function, and hence also of the discrimination function, will coincide with the category boundary, U, = u,, even under adaptation. This property of the angular distance version of the model agrees with findings of Eimas & Corbit (1973) and Cooper (1974). Their results show that the shift in the peak of the discrimination function is in the same direction and of approximately the same magnitude as the shift in the boundary of the corresponding identification function, suggesting an intimate relationship between the location of the identification boundary and the peak of the discrimination function, or, in the light of the previous discussion, the peak of the dispersion function. It can be shown that adaptation has no effect on the peak magnitude of the discrimination function, suggesting that discriminability at the boundary is not changed with the degree of adaptation. Normal Distance

For Gaussian Eq.( 16) will be

Decision

Variable

response functions

the normal

D(x) = 0% - xl

distance

- (A, - .x1 UI J&s

u2

dispersion

function,

(25)

The search for extrema reveals a maximum at the unadapted boundary stimulus value and two minima, one at (2, - 6) and the other at (2, + 6). Thus the position of the dispersion peak is independent of adaptation whereas the boundary shifts with adaptation. Dispersion peak magnitude is proportional to (a, + a2) and thus discrimination will be reduced due to adaptation. Path Length Decision

Variable

Finally, for the path length decision variable the corresponding dispersion function, Eq. (20), after substituting Gaussian detector functions, will have the form D(x) =

(2, - x)~ u; + (2, - x)~ ~$43~.

Its analysis yields five extrema: a value, two minima at the detector (a, - 6) and (A, + 6). The position case, is independent of adaptation.

(26)

maximum at the unadapted boundary stimulus peaks 2, and A2, and two secondary maxima at of the dispersion peak, as in the normal distance The position of the maxima for the path length

DISPERSION

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case corresponds to that of the minima in the normal distance case. The magnitude of the dispersion peak is proportional to ,/m and therefore discrimination will be reduced by adaptation.

DISCUSSION

One question still to be resolved is the question of the sensory noise. In all signal detection models, including Elman’s (1979) and the one presented in this paper, the noise is assumed to be additive, that is, its parameters are kept constant, independent of y, as in Eq. (3). In such case the sensitivity index d’ of signal detection theory is proportional to the dispersion function D(x), &!LD(x)~y c-i CJ

,.

(27)

One may rightly question the validity of the assumption of equal noise distribution. For instance, if the noise is interpreted as random fluctuation in detector sensitivity, its value will be necessarily proportional to the expected response value of the detector and thus be multiplicative in nature. In case the noise is considered as a constant Gaussian error in mapping from stimulus variable x to sensory variable y, the noise again has a multiplicative character. Its value is greatest at the inflection points of the detector response function, where the noise is bipolar, preserving approximately its Gaussian character. For a stimulus coinciding with the detector peak, the sensory noise is skewed, assuming only negative polarity. It has been shown that the choice of decision variable W(X) determines the shape of the dispersion function. However, as Eq. (27) indicates, discrimination depends also on the magnitude of the sensory noise 0. For noise of the additive type, discrimination along the stimulus continuum follows the dispersion function. The effect of adaptation on the position of the category boundary and on the position and magnitude of the dispersion peak, and consequently the discrimination peak, were discussed for the three decision variables above: For the angular variable the position of the dispersion peak under adaptation follows the shift of the category boundary. Its magnitude is not affected by adaptation. For the normal distance and path length variables the position of the dispersion peak is independent of adaptation, while the category boundary is shifted. The magnitude of the dispersion peak for both of these variables decreases with adaptation. For noise of the multiplicative type, these relationships can be different. Although the dispersion functions are identical as in the additive noise case, discrimination is inversely proportional to the magnitude of the sensory noise which decreases with adaptation. Thus, for the normal distance and path length variables, discrimination

ROZSYPAL,STEVENSON,

286

AND HOGAN

in the boundary region can be expected not to depend on adaptation. The same is true also for the angular variable, since as adaptation brings the stimulus trajectory closer to the origin of the ui - u2 sensory plane, the noise at any particular point is proportionaly reduced. But in this metric the effectiveness of the noise depends on the angle under which it is viewed from the origin and that angle is not changed.

CONCLUSIONS

In summary, the design of the dispersion model of auditory discrimination can be approached two ways: starting with an a priori postulated dispersion function D(x), or with assuming two detector response functions along with a decision variable. Both approaches lead to a model of categorical perception provided that the dispersion function displays a peak at the category boundary. In the second case, when D(x) is not known, the analysis presented above shows that a 2-detector configuration results in a nonlinear mapping from the stimulus to the sensory continuum. The shape of the underlying disperson function D(x) can be computed from the detector output functions ui(x) and the chosen decision variable w(x). Two detectors which have opposite slopes at the boundary will always lead to enhanced dispersion in the vicinity of the category boundary, which is the point where the detector outputs are equal. The degree of dispersion, and hence discrimination, is highly dependent on the slopes of the detector response functions, the steeper the slope near the boundary, the larger the dispersion. In discrimination, the subjects’ task is one of perceiving differences in stimuli. These differences may be small or large, depending on the physical differences between the signals, and the extent to which the auditory system is sensitive to these differences.

REFERENCES ADES,

A. E. (1977). Vowels, consonants, speech and nonspeech. Psychological Review, 84, 524530. W. A. (1977). Mechanisms of selective feature adaptation. Perception & Psychophysics, 21,

AINSWORTH,

365-370.

C. J. (1978). Range effect in the perception of voicing. Journal of fhe 63, 15561558. BRUNS.V., & SCHMIESZEK,E. (1980). Cochlear innervation in the greater horseshoe bat: Demonstration of an acoustic fovea. Hearing Research, 3, 27-43. COOPER,W. E. (1974). Adaptation of phonetic feature analyzers for place of articulation. Journal of the BRADY,

S. A., &

DARWIN,

Acoustical

Society

of America,

Acoustical

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of America,

56, 617-627.

COOPER,W. E. (1979). Speech perception and production: Studies in selective adaptation. Norwood, N J: Ablex. DIEHL, L. R. (1981). Feature detectors in speech: A critical reappraisal. Psychological Bulletin, 89, l-18. EIMAS, P. D., & COREIT, J. D. (1973). Selective adaptation of linguistic feature detectors. Cognitive Psychology,

4, 99-109.

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