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

LETTER Communicated by Carlos Brody

Deviation from Weber’s Law in the Non-Pacinian I TactileChannel: A Psychophysical and Simulation Study of IntensityDiscrimination

Burak [email protected] Engineering Institute, Bogazici University, Istanbul 34342, Turkey

This study involves psychophysical experiments and computer simula-tions to investigate intensity discrimination in the non-Pacinian I (NP I)tactile channel. The simulations were based on an established popu-lation model for rapidly adapting mechanoreceptive fibers (Guclu &Bolanowski, 2004a). Several intensity codes were tested as decision cri-teria: number of active neurons, total spike count, maximal spike count,distribution of spike counts among the afferent population, and syn-chronization of spike times. Simulations that used the number of activefibers as the intensity code gave the most accurate results. However, theWeber fractions obtained from simulations are smaller than psychophys-ical Weber fractions, which suggests that only a subset of the afferent pop-ulation is recruited for intensity discrimination during psychophysicalexperiments. Simulations could also capture the deviation from Weber’slaw, that is, the decrease of the Weber fraction as a function of the stim-ulus level, which was present in the psychophysical data. Since the psy-chophysical task selectively activated the NP I channel, the deviationeffect is probably not due to the contribution of another tactile channelbut rather is explicitly produced by the NP I channel. Moreover, becausesimulations with all tested intensity codes resulted in the same effect, theactivity of the afferent population is sufficient to explain the deviation,without the need for a higher-order network. Depending on the intensitycode used, the mechanical spread of the stimulus, rate-intensity functionsof the tactile fibers, and the decreasing spike-phase jitter contribute tothe deviation from Weber’s law.

1 Introduction

Intensity discrimination is the detection of small changes in the stimulusintensity and is of paramount importance for encoding the physical stimu-lus by a sensory system. The ecological significance of this cannot be under-estimated, since survival often depends on judging the magnitude variationof a sensory input and taking action according to the increase or decrease

Neural Computation 19, 2638–2664 (2007) C© 2007 Massachusetts Institute of Technology

Deviation from Weber’s Law in the Non-Pacinian I Tactile Channel 2639

of the input level. The sense of touch is acute at our fingertips, where manytactile fibers are concentrated. However, most of the studies on intensitydiscrimination in the human tactile system have focused on the thenar emi-nence of the hand (e.g., Gescheider, Bolanowski, Verrillo, Arpajian, & Ryan,1990; Gescheider, Bolanowski, Zwislocki, Hall, & Mascia, 1994; Gescheider,Thorpe, Goodarz, & Bolanowski, 1997), because it can be mechanicallystimulated more easily by contactor probes of various sizes. One goal of thestudy reported here is to compare its experimental results, which were ob-tained by stimulating the fingertip, with previous work that stimulated thethenar eminence. Psychophysical detection thresholds have been measuredat the fingertip (Guclu & Bolanowski, 2005a) and were found to be similarto the results of previous studies that applied 40 Hz vibratory stimuli tothe thenar eminence. Therefore, it would be expected that the results ofintensity-discrimination experiments on the fingertip should turn out to beconsistent with the literature as well.

The primary importance of this letter is that it attempts to predict apsychophysical phenomenon, the deviation from Weber’s law, by using aphysiological population model (Guclu & Bolanowski, 2002, 2004a, 2005b).This model mimics the properties of rapidly adapting (or fast adaptingtype I) nerve fibers, and has recently been improved by incorporating time-dependent activity (Guclu & Bolanowski, 2004b). Rapidly adapting fibersare associated with Meissner’s corpuscles in the skin and constitute one offour classes of mechanoreceptive fibers (Greenspan & Bolanowski, 1996). Itis assumed that the input from a population of rapidly adapting fibers isprocessed in the non-Pacinian I (NP I) psychophysical channel (Bolanowski,Gescheider, Verrillo, & Checkosky, 1988; Gescheider, Bolanowski, & Verrillo,2004). In order to compare the performance of the population model withthe psychophysical response of the NP I channel, the NP I channel wasexperimentally isolated by forward masking (Guclu & Bolanowski, 2005a,2005b). This approach avoids interactions between channels. Although anarrower range of intensities can be tested with this method (cf. Gescheideret al., 1990), the psychophysical properties of the NP I channel can beobtained selectively.

Weber’s law states that a just-noticeable change in intensity (or ampli-tude) is approximately proportional to stimulus intensity (or amplitude).This is an almost universal law for all sensory systems, but deviations arepossible in certain conditions. For example, a near miss to Weber’s lawwas observed when sinusoidal stimuli were used in hearing (Moore &Raab, 1974) and in touch (Knudsen, 1928; Gescheider et al., 1990). Thatis, some improved discrimination was found at high intensities, althoughWeber’s law asserts a worsening of discrimination. The psychophysical ba-sis for this deviation is unclear, but it may be due to recruitment of othersensory-input channels as the intensity is increased (Gescheider et al., 1997).This does not exclude purely cortical properties (e.g., see Dykes, 1983), but

2640 B. Guclu

in the simplest sense, more recruited channels may mean more input tofacilitate discrimination. Therefore, it may be hypothesized that by isolat-ing the NP I tactile channel, the deviation from Weber’s law should disap-pear. On the other hand, if deviation is still found during the activation ofa single psychophysical channel, the deviation may be attributed to eitherphysiological properties of afferents or a certain intensity code representedin the activity of higher-order neurons. For this letter, several hypotheticalcodes were tested. These codes specify either the number of activated tactileneurons or the spike activity of the neurons. Note that either approach maybe a modified view of the same representation, since many active neuronswould increase the activity of a higher-order unit by synaptic convergence.In addition, the distribution of activity among the population of neuronsand synchronization of spikes was considered in the modeling analysespresented.

2 Methods

2.1 Population Model. The population model is similar to the one usedby Guclu and Bolanowski (2002, 2004a, 2005b); it simulated the rapidlyadapting afferent output that mediates the NP I channel.

2.1.1 Anatomical Organization. Since the contactor location does not havean effect on the detection thresholds measured across the finger (Guclu &Bolanowski, 2005b), a uniformly random distribution was used to model thereceptive field locations of the afferent fibers. The innervation density of therapidly adapting fibers was set to 0.75 fibers per mm2 (Johansson & Vallbo,1979). Therefore, the rectangular terminal-phalanx skin model (3 cm × 2cm) contained 450 rapidly adapting fibers. The x- and y-coordinates of thereceptive field centers were randomly determined by using the followingindependent probability density functions:

fX(x) ={

1/30; 0 ≤ x ≤ 30 mm

0; else.fY(y) =

{1/20; 0 ≤ x ≤ 20 mm

0; else(2.1)

2.1.2 Time-Independent Physiological Properties. The rate intensity func-tions of the rapidly adapting fibers were determined by, among others,Guclu and Bolanowski (2003, 2005b). Since the statistical firing propertiesof human tactile fibers were not adequately described at this time, the modelwas based on monkey data (Johnson, 1974). The rate intensity functions can


be approximated by

r =

0; 0 < A < a0

fs

a1 − a0(A− a0); a0 < A < a1

fs; a1 < A < a2

fs

(1 + A− a2

a3 − a2

); a2 < A < a3

2 fs; a3 < A

for a0 < a1 < a2 < a3,

(2.2)

where r is the average firing rate, A is the effective stimulus amplitude atthe fiber’s receptive field center, fs is the stimulus frequency, and (a0, a1,a2, a3) are the parameters, which are distributed randomly among fibers.These parameters are not independent, but Johnson (1974) showed inde-pendence between the random variables a0/a1 and a1 and between a2/a3

and a3. Therefore, the following probability density functions were used toconstruct the rate intensity function of each rapidly adapting fiber:

p(a0/a1) = 3.093 exp

{−30.046

[a0

a1− 0.32

]2}

p(a1) = 0.456a1

exp

{−3.458

[log

(a1

35.94

)]2}

p(a2/a3) = 3.093 exp

{−30.046

[a2

a3− 0.62

]2}

p(a3) = 0.625a3

exp

{−6.516

[log

(a3

278.26

)]2}

. (2.3)

Note that in equation 2.3, the probability densities of parameter ratios arenormal, but the probability densities of the given parameters are log normal.

2.1.3 Spike Generation. The average firing rate of each nerve fiber speci-fied the activity regime that affects the transition probabilities in a Markovchain used for spike generation (Guclu & Bolanowski, 2004b). The rapidlyadapting fiber’s output can occupy one of three states during each cycle ofthe stimulus waveform: no spike at a given stimulus cycle (ε0), one spikeat the positive phase of the stimulus cycle (ε1), and two spikes (one in thepositive phase and the other in the negative phase of the stimulus cycle(ε2)). It was assumed that the fiber changed its state randomly at each cycle

2642 B. Guclu

c = 1, 2, 3, . . . Starting from an initial state, the simulated process proceededin a chain such as ξ (1) → ξ (2) → ξ (3) → · · ·, where ξ (c) is the state of therapidly adapting fiber at cycle c. It can be shown (Guclu & Bolanowski,2004b) that

Prob{ξ (c + 1) = ε j } =∑

i

p ji Prob{ξ (c) = εi }, i, j = 0, 1, 2, (2.4)

where p ji is the probability that the fiber will go into state ε j at the nextcycle given that it is in state εi at some stimulus cycle c. This probability isindependent of the fiber’s trajectory before the cycle c. p ji ’s were obtainedas a function of the average firing rate (r ) from Guclu and Bolanowski(2004b). The timing of the first spike phase relative to the stimulus cycle wasmodeled using the Laplace distribution. The probability density functionof this distribution is:

f (x) = 1

σ√

2exp

(−

√2

σ|x − µ|

). (2.5)

It was reported that the phase advances to smaller values, and the distribu-tion becomes narrower as the stimulus amplitude is increased in monkeyfibers (Talbot, Darian-Smith, Kornhuber, & Mountcastle, 1968). Therefore,the mean (µ) and the standard deviation (σ ) varied as a function of theeffective stimulus amplitude at the receptive field center in the simulations:

µ = ma e−mb A + mc

σ = sa e−sb A + sc . (2.6)

The parameters (ma , mb , mc , sa , sb , sc) were obtained from Guclu andBolanowski (2004b). When there were two spikes per stimulus cycle, theearlier spike was modeled by equation 2.5, but the phase difference betweenthe first and second spike was modeled by a normal distribution with theparameters given in Guclu and Bolanowski (2004b). In order to simulate thepropagation of action potentials from the periphery to the central nervoussystem, a randomly gaussian variation was added to the simulated spiketimes based on the estimate of conduction speed reported by Darian-Smithand Kenins (1980; average distance: 1 m; mean conduction speed: 50 m/s,standard deviation of conduction speed: 10 m/s).

2.1.4 Mechanical Stimulus. The stimulus used in the simulations was a40 Hz sinusoidal wave, which had duration of 0.5 s and varying intensity.The skin model had the anatomical properties given in section 2.1.1 and wasstimulated by a circular contactor probe (radius: 2 mm, area: 0.126 cm2). Thestimulation site was the center of the skin model. The effects of the stimulus


spread radially, but attenuated as a function of distance from the contactoraccording to (Johnson, 1974)

A(x, y) =

As; (x − xs)2 + (y − ys)2 ≤ r2s

As

(rs√

(x − xs)2 + (y − ys)2

)1.9

; else, (2.7)

where (xs , ys) are the coordinates of the stimulus contactor, rs the radiusof the contactor, As the stimulus amplitude, (x, y) the coordinates of thereceptive field center, and Athe effective stimulus amplitude at the receptivefield center of the rapidly adapting fiber.

It is important to note that the responses of the nerve fibers are corre-lated in the model because fibers close to each other are more likely to havesimilar firing rates due to the mechanical attenuation (see equation 2.7).On the other hand, the intensity rate functions that actually determine theaverage firing rate of each fiber were generated from independent proba-bility distributions, and the spike generation for each fiber was performedindependently as described above. This is because the tactile afferents donot interact with each other in the periphery. The model presented above isbiologically accurate and mimics the population response of rapidly adapt-ing fibers, but it is not strictly independent because of mechanical coupling.The entire model has numerous levels where stochastic variability is incor-porated. Additionally, the model is noisy due to the spike generation mech-anism employed. Physiological and anatomical parameters are altered insuccessive simulation experiments to mimic experimental variability (e.g.,testing a different subject).

2.2 Simulations and Decision Criteria. The intensity discriminationexperiment was simulated in Matlab Version 6.5 by using the two-intervalforced-choice paradigm, which has two temporal observation intervals.Each observation interval included a stimulus, but the amplitude of thestimulus in one interval was set greater than the amplitude of the stimu-lus (the comparison stimulus) in the other interval. The interval with thestronger stimulus (the test stimulus) was determined randomly. The in-tensity of the comparison stimulus was kept constant during a simulatedexperiment.

The simulated observer’s task was to select the interval that presumablycontained the test stimulus. Therefore, the decision criterion was to selectthe interval that had the higher neural variable (n1 or n2) according to theassumed intensity code (see below). The decision criterion was tested bythe simulated observer for each simulated trial. If n1 > n2, the stimulusin the first interval was judged to be of higher amplitude by the simu-lated observer. If n1 < n2, the second interval was selected by the simulated

2644 B. Guclu

observer. Note that the simulated observer could make an error by selectingthe interval with the higher neural output, although that interval containedthe weaker stimulus. This is because of the random factors incorporated inthe population model.

The frequency of correct decisions in 100 trials was found, and thisvalue was considered to be the probability of discriminating among stimuliwith different amplitudes. Half of the trials that had equal neural vari-ables (n1 = n2) were assumed to yield correct decisions. Probabilities ofdiscriminations were found between a stimulus with a given amplitude(the comparison stimulus) and stimuli with slightly higher amplitudes.The amplitude change detected with 0.75 probability was set to be the dis-crimination threshold of the simulated observer, that is, the just-noticeabledifference.

When presenting the results, the just-noticeable difference in amplitude(δA) was converted to the just-noticeable difference in intensity (differencelimen, DL) by using

DL = 20 log10As + δA

As, (2.8)

where As is the amplitude of the comparison stimulus and As + δA isthe amplitude of the stronger stimulus. DL was given in dB units. TheWeber fraction was calculated as the ratio of δA to As , (δA/As); therefore,it shows the discrimination capacity relative to the stimulus amplitude. Itis customary to plot the Weber fraction as a function of the sensation level(SL), which is defined as

SL = 20 log10As

Ath, (2.9)

where Ath is the detection threshold in amplitude units. Thus, SL was givenin dB units referenced to the detection threshold—the weakest stimulusdetected by the observer. Note that Weber’s law implies that the Weberfraction is approximately constant as a function of the intensity of the com-parison stimulus. Here, deviation refers to a decrease in the Weber fractionas a function of comparison stimulus intensity.

The simulations were repeated for five hypothetical observers who weregenerated by using the population model described. The criterion for de-tecting the stimulus was at least 10 active fibers, which has been shownto give accurate simulation results when compared to psychophysical data(Guclu & Bolanowski, 2005b). That is, at least 10 active rapidly adaptingfibers should be present in the population for detecting the 40 Hz sinusoidalstimulus. Detection thresholds of the simulated observers were found andSLs could be calculated by using this criterion.


2.3 Intensity Codes. The following hypothetical intensity codes weretested in the simulations. In the descriptions below, {ok} refers to the outputof the population, that is, a set of spike times for each kth fiber. i specifiesthe interval and is either 1 or 2. n is the neural variable that contains theintensity information.

2.3.1 Number of Active Fibers. In this code, the intensity information isrepresented by the number of active fibers in the population. Therefore,the temporal interval that produces the highest number of active fibers isselected by the simulated observer. This code implies that there is a vastconvergence of the afferent output within the central nervous system. Bysynaptic convergence, the information encoded in the number of fibers maybe converted to an increase of activity in a higher-order neuron. The codecan be mathematically expressed as

ni =∑

k

I[1,∞) (|{ok}|). (2.10)

In equation 2.10, I[...](·) is the indicator function, which has a value of 1 ifits argument belongs to the set indicated by the subscript; otherwise, thefunction is 0. Here, the cardinality (the number of elements shown by barnotation) of the spike time set is evaluated. Note that cardinality is 1 orgreater if there is at least one spike in the set.

2.3.2 Total Activity. This code summates all the spike counts in the pop-ulation. It implies that the intensity information is represented by the en-tire spike activity in the population. Therefore, the temporal interval thatproduces the highest number of total spikes is selected by the simulatedobserver. This code suggests that the activity of a higher-order decisionnetwork is modulated by all afferent spikes. The mathematical expressionfor the code is

ni =∑

k

|{ok}|. (2.11)

In equation 2.11, the cardinalities of the ouput sets (i.e., the number ofspikes) are summed.

2.3.3 Maximal Spike Count. This code assumes that the intensity informa-tion is represented by the activity of the afferent fiber that has the maximalspike count. Therefore, the temporal interval that produces the highest spikecount in an individual fiber is selected by the simulated observer. This codeimplies that there is a competition between afferent fibers in a higher-orderdecision network (e.g., winner-takes-all type). The mathematical expression

2646 B. Guclu

for the code is

ni = maxk

|{ok}| . (2.12)

In equation 2.12, the maximum cardinality found in the population outputis set as the neural variable.

2.3.4 Distribution of Activity. This code considers the distribution of spikecounts (q ) among the fiber population. The simulated observer decides on atemporal interval according to the signal detection theory (Green & Swets,1966; Gescheider, 1997). In this approach, the maximum likelihood ruleyields the optimum decision (Srinath, Rajasekaran, & Viswanathan, 1996).If I1 and I2 are the stimulus intensities in two intervals, the maximumlikelihood rule can be given as

p (q |I2 )p (q |I1 )

I2

>

<

I1

1. (2.13)

In equation 2.13, p(q |Ii ) is the conditional probability density of spike countsgiven the stimulus with intensity Ii . The simulated observer selects theinterval according to the rule in equation 2.13. In other words, if the ratiois greater than one, the second interval is selected; if it is less than one, thefirst interval is selected. The density functions in equation 2.13 were foundby repeating the simulation for 100 trials.

2.3.5 Synchronized Activity. Although temporal information in the spiketrain may help discrimination, the central decision unit probably does notknow about stimulus parameters a priori. Therefore, synchronization withrespect to the stimulus waveform may be a less probable code for intensity.With no a priori information, the interspike interval is the simplest temporalparameter that may signal the intensity of the sinusoidal stimulus. If thespikes are regular, the standard deviation of the interspike intervals is low,and the afferent fibers are synchronized. The neural variable tested in thesimulations is based on the entire population output:

si = std⋃

k

{ok}. (2.14)

In equation 2.14, the union (∪ symbol) set of all the spike times in the pop-ulation is first obtained within the stimulus duration. The neural variable(s) is equal to the sample standard variation (std) of the union set. Notethat the decision criterion based on s is reversed as compared to n in the


simulations. That is, s1 > s2 implies that the second temporal interval con-tains the stronger stimulus, because the synchronization is higher.

2.4 Psychophysical Experiments. The psychophysical experimentswere similar to the experiments by Guclu and Bolanowski (2005a) exceptthe final procedure, which measured intensity discrimination. Again, un-masked detection thresholds were measured first. The apparatus was iden-tical. All psychophysical experiments used the two-interval forced-choicetask. The test stimulus frequency was chosen as 40 Hz because the Pacinian(P) channel is not very sensitive at this frequency (see Bolanowski et al.,1988). Thus, the NP I channel can more easily be isolated. It is also impor-tant to note that the population model has been validated specifically by40 Hz data in the literature.

2.4.1 Subjects. Two female and three male human subjects in good healthwere recruited. The experiments adhered to National Institutes of Healthethical guidelines for testing human subjects and were approved by theEthics Committee for Human Subjects of Bogazici University. The subjectscame from a narrow age group (22 to 27 years). Note that the differentialsensitivity to changes in stimulus intensity is not affected much by age(Gescheider, Edwards, Lackner, Bolanowski, & Verrillo, 1996). The terminalphalanx of the left middle finger was stimulated on the center of the volarsurface. All subjects declared that they were right-handed. The averagesurface temperature of the second finger in the left hand was 29◦C to 34◦Cduring the experiments. The temperature was not controlled, because theNP I channel is not affected by temperature (Greenspan & Bolanowski,1996).

2.4.2 Stimuli. The stimuli were sine wave bursts of mechanical displace-ments superimposed on a static indentation of 0.5 mm (see Figure 1). Theywere applied by a 0.126 cm2 circular contactor probe (radius, 2 mm). Theburst stimuli started and ended as cosine-squared ramps with 50 ms riseand fall times. The stimulus duration was determined with reference to thehalf-power points of the stimulus burst. In the intensity discrimination mea-surements, 40 Hz test stimuli were preceded by 250 Hz stimuli to mask theP channel (Guclu & Bolanowski, 2005a). This forward-masking paradigmwas to ensure that the 40 Hz stimulus is detected by the NP I channel butnot by the P channel. For one set of experiments, 250 Hz sine waves wereused as test stimuli to determine the masking functions of each subject. Thetest stimulus levels were changed during the experiment according to anup-down rule, which tracked 0.75 correct decision probability (Zwislocki& Relkin, 2001). In this rule, the stimulus level is increased one step foreach incorrect answer; and it is decreased one step for three, not necessarilyconsecutive, correct answers. The step size was 1 dB for detection experi-ments and 0.4 dB for intensity discrimination experiments. An experiment

2648 B. Guclu

a

b

c


ended when the stimulus level was within a small range (±1 dB for detec-tion, ±0.4 dB for intensity discrimination) for the previous 20 trials. A 0.75probability level is referred to as the threshold.

2.4.3 Procedures. To obtain a complete set of experiments for each subject,four serial procedures were required. Each measurement was repeated fivetimes to enable statistical analyses of the data. The procedures were basedon the four-channel theory of tactile psychophysics (Bolanowski et al., 1988).In this theory, tactile channels are assumed to be independent, and by mask-ing (e.g., forward masking) a channel at a special frequency, the thresholdof another channel can be measured (Verrillo & Gescheider, 1977; Hamer,Verrillo, & Zwislocki, 1983; Bolanowski et al., 1988; Guclu & Bolanowski,2005a; Hollins, Lorenz, & Harper, 2006). Figure 2 illustrates the main princi-ples of the theory. In Figure 2a, the threshold curve (solid line) of a channelis shown. X1 to X3 are three stimuli with different frequencies. Note thatX1 is below the threshold, so it is not detected. X2 and X3 are detected.However, X3 is above the threshold and induces masking of the channel.The masked channel has elevated thresholds (dotted line) shifted parallelto the original threshold curve in Figure 2a. Figure 2b depicts the detection

Figure 1: Stimulus timing diagrams for psychophysical experiments. All stim-uli are sinusoidal bursts of displacements superimposed on a static indentation.Stimuli occur in two temporal intervals signaled to the subject by red and greenlights. The subject gives a response when the yellow light is on. Therefore,all psychophysical tasks conform to the two-interval forced-choice paradigm.(a) In a simple detection task, the test stimulus with a given frequency andintensity occurs either in the first (shown as the gray burst) or in the second in-terval (shown as the dotted burst) randomly. In the experiments presented here,thresholds were measured for 40 Hz and 250 Hz test stimuli. (b) In a detectiontask with forward masking, a high-level 250 Hz masking stimulus is applied atthe beginning of each interval. The test stimulus with a given frequency and in-tensity occurs in either the first (shown as the gray burst) or the second interval(shown as the dotted burst) randomly. For finding the masking function of thePacinian channel, a 250 Hz test stimulus was used. For finding the thresholdof the NP I channel, a 40 Hz test stimulus was used. Note that the thresholdof the NP I channel cannot be measured with certainty if the threshold of thePacinian (P) channel is not elevated by forward masking. (c) In an intensitydiscrimination task regarding the NP I channel, the P channel is first maskedby high-level 250 Hz stimuli at the beginning of the intervals. Then the 40 Hzstimuli are applied in both intervals. However, the 40 Hz stimulus in one ofthe randomly determined intervals has a higher intensity (test stimulus) thanthe 40 Hz stimulus in the other interval (comparison stimulus with constantintensity). In this diagram, the comparison stimulus is in the first and the teststimulus is in the second interval.

2650 B. Guclu

Figure 2: Diagrams explaining the principles of the four-channel theory(Bolanowski et al., 1988) of tactile psychophysics. (a) The threshold curve (solidline) of a hypothetical tactile channel is plotted as a function of frequency. X1to X3 are stimuli with different frequencies. Because X1 is below the thresholdcurve, it cannot be detected by the channel. X2 and X3 are detected by the chan-nel. X3 is above the threshold curve and causes some masking of the channel.The threholds of the masked channel (dotted line) are elevated. It is impor-tant to note that masking at any given frequency (e.g., frequency of X3) shiftsthe thresholds up, parallel to the original curve (solid line). (b) The detectionmechanism that is valid in the psychophysical experiments presented in thisletter is shown. A weak 40 Hz stimulus is detected by the P channel because thethreshold of this channel is lower than the threshold of the NP I channel at 40Hz. However, if the P channel is masked, the threshold of the NP I channel canbe measured at 40 Hz.

mechanism based on previous experimental results (Guclu & Bolanowski,2005a). A weak 40 Hz stimulus is typically detected by the P channel, be-cause this channel has a lower threshold than the NP I channel at 40 Hz.However, if the P channel is sufficiently masked, the threshold of the NP Ichannel can be measured at 40 Hz. Note that the inputs of the NP I chan-nel (rapidly adapting fibers) and the P channel (Pacinian fibers) are pro-cessed separately in this theory. Masking a channel does not entirely deleteit; a sufficiently intense stimulus at a particular frequency (e.g., a stimu-lus with an intensity that reaches the masked P curve in Figure 2b) canreactivate it.

Unmasked detection thresholds. Without applying a masking stimulus,unmasked detection thresholds were measured for each subject. In thisprocedure, only one of the two intervals contained the test stimulus (seeFigure 1a). The interval that contained the test stimulus was determinedrandomly, and the subject’s task was to find this interval. The intensity ofthe test stimulus was varied during each experiment, and the unmaskedthreshold was measured according to the up-down rule. The threshold wasfound separately for a 40 Hz test stimulus and a 250 Hz test stimulus.


Masking functions. A 250 Hz high-level masking stimulus precededa 250 Hz test stimulus, which varied in intensity during the experiment.By using this procedure, a shift in sensitivity (masked threshold minusunmasked threshold for 250 Hz stimulus) was obtained in the P channelas a function of the level of the masking stimulus. Again, only one of thetwo intervals contained the test stimulus (see Figure 1b). The interval thatcontained the test stimulus was determined randomly, and the subject’s taskwas to find this interval without paying attention to the masking stimulus.A masking level that produced sufficient masking effect (10–20 dB shift)was selected for further experiments.

Detection thresholds of the NP I channel. By using the forward-maskingprocedure, the sensitivity of the P channel was decreased, and the detectionthreshold of the NP I channel could be measured at 40 Hz. In this procedure,the 250 Hz masking stimulus with the predetermined level was applied justbefore the 40 Hz test stimulus (see Figure 1b). Only one of the two intervalscontained the test stimulus. The interval that contained the test stimuluswas determined randomly, and the subject’s task was to find this intervalwithout paying attention to the masking stimulus. The intensity of thetest stimulus was varied during each experiment, and the threshold wasmeasured according to the up-down rule.

Intensity discrimination in the NP I channel. The previous procedureswere necessary to ensure detection of the 40 Hz test stimulus by the NP Ichannel. Once this was verified for each subject, intensity discriminationexperiments could be performed. In order to mask the P channel, a 250 Hzmasking stimulus with the predetermined level preceded the 40 Hz stim-uli. Both of the intervals contained 40 Hz stimuli (see Figure 1c), but oneof the 40 Hz stimuli (the test stimulus) had greater amplitude than thatof the other stimulus (the comparison stimulus). The interval that con-tained the test stimulus was determined randomly, and the subject tried tofind that interval. The level of the comparison stimulus was kept constant,but the intensity of the test stimulus was varied during each experiment.The discrimination threshold was measured according to the up-downrule.

3 Results

3.1 Simulations

3.1.1 Detection Thresholds. Five observers were simulated by generat-ing five populations of afferents that were modeled with slightly differentanatomical and physiological properties. Model skin regions were stimu-lated by 40 Hz stimuli. The psychometric function of each simulated ob-server is given in Figure 3. The detection thresholds (stimulus intensity at

2652 B. Guclu

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25Stimulus intensity [dB re 1 µm peak]

Pro

bab

ility

of

det

ecti

on

O1O2O3O4O5

Figure 3: Psychometric functions of simulated observers (O1–5). The 0.75 prob-ability of detection is defined as the detection threshold. The probabilities werecomputed from 100 simulations.

0.75 detection probability) are 20.2, 15.8, 15.1, 20.0, and 17.0 dB (referencedto 1 µm amplitude) for simulated observers O1 to O5, respectively. There isno significant difference (two-sample t-test; p = 0.072) between these simu-lated detection thresholds and the thresholds of the NP I channel measuredin humans (Guclu and Bolanowski, 2005a).

3.1.2 Intensity Discrimination by Number of Fibers. The probabilities ofdiscrimination (probabilities of correct decisions) were found for each sim-ulated observer by using comparison stimuli with various intensities. TheDLs of O5 are 0.4, 1.2, 1.2, 0.8, 0.5, 0.3, 0.7, 0.3, and 0.2 dB, respectivelyfor the comparison stimulus intensities of 17, 18, 19, 20, 22, 24, 26, 31, and36 dB. For this particular intensity code (the number of active fibers), thediscrimination is very good. The DL values are significantly lower thanpsychophysical results (approximately 1.5 dB, t-test; p < 0.001) reported byGescheider et al. (1990). Note, however, that Gescheider et al. (1990) used a25 Hz test stimulus and did not isolate the NP I channel by forward masking(see below).

The Weber fraction is plotted as a function of SL in Figure 4a. The datapoints in Figure 4a are averages of five simulated observers. There is alarge variation among observers near the threshold, but the variation isdecreased as the SL is increased. On average, the Weber fraction is 0.1near the threshold and decreases to 0.02 at 20 dB SL. Therefore, there is adeviation from Weber’s law. The data points in the 10 to 20 dB SL range


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Figure 4: Weber fractions (in terms of displacement amplitude: δAs/A) fromsimulations with various intensity codes. Intensity discrimination was basedon the number of active fibers (a), the total spike count (b), the maximal spikecount (c), and the standard deviation of interspike intervals (d). The data pointsare averages of five simulated observers, and the error bars are the standarddeviations.

are significantly lower than the data points in the 0 to 10 dB SL range(two-sample t-test; p < 0.001).

3.1.3 Intensity Discrimination by Total Activity. The probabilities of dis-crimination were found for each simulated observer by using comparisonstimuli with various intensities. The intensity of the comparison stimuluswas set at 17, 18, 19, 20, 22, 24, 26, 31, and 36 dB. The DLs of O5 are 0.2, 0.2, 0.2,0.3, 0.3, 0.2, 0.2, 0.2, and 0.2 dB, respectively, for the given list of comparisonintensities. Therefore, the discrimination is very good. The DL values aresignificantly lower than the psychophysical results (approximately 1.5 dB,t-test; p < 0.001) by Gescheider et al. (1990).

The Weber fraction is plotted as a function of SL in Figure 4b. The datapoints in Figure 4b are averages of five simulated observers. On average,the Weber fraction is 0.03 for all tested SLs. There is a slight decrease of theWeber fraction up to 0.02 in the higher half-range of the SLs; this change isstatistically significant (two-sample t-test; p = 0.007).

3.1.4 Intensity Discrimination by Maximal Spike Count. For this intensitycode, the DLs of O5 are 0.6, 0.4, 0.4, 7.2, 5.5, 7.9, 8.8, 9.5, and 4.7 dB,

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respectively, for the comparison intensities 17, 18, 19, 20, 22, 24, 26, 31,and 36 dB. These DLs are significantly higher than psychophysical results(t-test; p = 0.011) by Gescheider et al. (1990).

The Weber fraction is plotted as a function of SL in Figure 4c. Again,there is a large variance near the threshold. Nevertheless, there is also aconsiderable decrease of the Weber fraction from 1.66 in the lower half-range of the SLs to 0.91 in the higher half-range of the SLs; this change isstatistically significant (two-sample t-test; p = 0.022).

3.1.5 Intensity Discrimination by Distribution of Activity. The distributionof spike counts in the afferent population of O5 is given in Figure 5a forvarious stimulus inputs. If the stimulus intensity is below the detectionthreshold (e.g., 15 dB), almost all fibers are inactive. If the stimulus intensityis increased, some fibers are activated, and eventually they are entrainedto the stimulus frequency. That is, they fire one spike per cycle of stimulus,and thus, the distribution has a peak at 20 spikes. (Note that stimulusfrequency is 40 Hz, and duration is 0.5 s.) As a result of this phenomenon,the distribution of activity is typically bimodal: one mode at zero activityand the second mode at 20 spikes. If the stimulus intensity is very high, theremay be two spikes generated per cycle. Then the distribution of activity istrimodal, although the peaks at 0 and 40 spikes are lower than the peak at20 spikes. For this intensity code, the DLs of O5 are 35.6, 34.6, 33.6, 32.6,30.6, 28.6, 26.6, 21.6, and 18.4 dB, respectively, for the comparison intensities17, 18, 19, 20, 22, 24, 26, 31, and 36 dB. These DLs are much higher (t-test;p < 0.001) than the psychophysical results reported by Gescheider et al.(1990). Weber fractions are also very high (mean: 33.93; see Figure 5b),but they decrease as a function of the stimulus level similar to the resultsobtained from the other intensity codes. An intensity code based on thedistribution of spike counts in the population yielded a very unfavorablediscrimination capacity.

3.1.6 Intensity Discrimination by Synchronized Activity. As the stimulusintensity is increased, more afferent fibers within the population are en-trained, and they generate one spike per cycle of the stimulus. Thus, thestandard deviation of the interspike intervals would be expected to de-crease. For the simulated observer O5, the DLs are 2.7, 1.5, 1.1, 1.0, 8.4, 16.7,14.2, 9.9, and 0 dB, respectively, for the comparison intensities 17, 18, 19,20, 22, 24, 26, 31, and 36 dB. These DLs are significantly higher than thepsychophysical results (t-test; p = 0.029) of Gescheider et al. (1990).

Because of the high variance in the discrimination thresholds, the vari-ances of the Weber fractions are also high, especially near the detectionthreshold (see Figure 3d). Weber fractions are very high (mean: 5.72), butthere is a considerable decrease from 7.87 in the lower half-range of the SLsto 1.44 in the higher half-range of the SLs, and this change is statisticallysignificant (two-sample t-test; p = 0.017).


Figure 5: Simulations based on the distribution of activity as the intensity code.(a) The probabilities of spike counts among the population are given for O5obtained from 100 simulations. Different graphs are presented for indicatedstimulus intensities (dB re 1 µm peak). (b) Weber fractions are plotted as afunction of the sensation level. The data points are averages of five simulatedobservers, and the error bars are the standard deviations.

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3.2 Psychophysical Experiments

3.2.1 Detection Threshold of the NP I Channel. The results from five subjectsshow that the NP I channel could be isolated (see Figure 6a). This conclusioncan be clarified as follows. The white bars in Figure 6a are the detectionthresholds measured for the 40 Hz stimulus when there was no 250 Hzmasking. The gray bars are the detection thresholds measured for the 40 Hzstimulus when there was forward masking by the 250 Hz stimulus. Since thegray bars are higher than the white bars (two-sample t-test; all p’s < 0.015),detection was probably mediated by different channels. That is, the whitebars must represent the threshold of the P channel at 40 Hz, and the graybars must represent the threshold of the NP I channel while the P channelwas masked by the 250 Hz stimulus (see section 2.4.3). These results areentirely consistent with previous data obtained by the same procedure(Guclu & Bolanowski, 2005a). This explanation can be validated further byadding threshold shifts obtained from masking functions (not shown) to thewhite bars to obtain the black bars. The black bars represent the elevatedthresholds of the P channel in each subject at 40 Hz and should be higherthan the gray bars. The data in Figure 6a show that this explanation is valid(two-sample t-test; all p’s < 0.001).

The NP I thresholds of S1 to S5 are 17.1, 17.7, 20.4, 18.4, and 20.8 dB,respectively, at 40 Hz. These data were compared to data previously ob-tained from five different subjects (Guclu & Bolanowski, 2005a). No sta-tistical difference could be found (two-sample t-test; p = 0.226). The cur-rent psychophysical data also were compared to the thresholds obtainedfrom simulations. No statistical difference could be found between exper-imental and simulation results (two-sample t-test; p = 0.401). Therefore,the rapidly adapting population model used in the simulations mimics thepsychophysical detection process of the NP I channel very well.

3.2.2 Intensity Discrimination by the NP I Channel. Since SLs were ad-justed for each subject’s NP I channel characteristics, the measurements of

Figure 6: Results of psychophysical experiments from five subjects (S1–5). Thedata points are averages of five measurements, and the error bars are the stan-dard deviations. (a) Thresholds measured for a 40 Hz stimulus without (whitebars) and with forward masking (gray bars). The white bars are considered tobe the thresholds of the Pacinian channel, and the gray bars are thought to bethe thresholds of the NP I channel at 40 Hz. The black bars refer to the elevatedthresholds of the Pacinian channel when forward masking is used. (b) Weberfractions are plotted separately for each human subject. (c) Weber fractionsfrom psychophysical experiments (filled circles) are plotted with simulated We-ber fractions (open circles) based on the number of active fibers as the intensitycode.


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intensity discrimination were performed at SLs spaced irregularly apart.The DLs obtained from all subjects are a little higher on average (mean:2.3 dB) than the average DL (1.5 dB) reported by Gescheider et al. (1990).This difference is statistically significant (two-sample t-test; p = 0.002).

The Weber fraction is plotted as a function of SL in Figure 6b. The averageWeber fraction is 0.32 for all SLs tested. The Weber fraction decreases from0.38 in the lower half-range of the SL to 0.18 in the higher half-range. Thischange is highly significant (two-sample t-test; p < 0.001).

The Weber fraction found in psychophysical experiments (0.32) is closestto the Weber fraction found in the simulations (0.07), which were based onthe number of active fibers as the intensity code. Psychophysical Weberfractions and simulated Weber fractions, based on the number of activefibers, are plotted on the same graph in Figure 6c. The psychophysicalresults are significantly greater than the simulation results (two-samplet-test; p < 0.001). Similarly, the psychophysical Weber fractions are signif-icantly higher than the simulated Weber fractions (average: 0.03) whenthe intensity code was the total activity (two-sample t-test; p < 0.001).On the other hand, the psychophysical Weber fractions are considerablylower than the results obtained from the other intensity codes tested (two-sample t-test; p’s < 0.005). The mean Weber fraction based on the max-imal spike count is 1.41. It is 33.93 when the intensity code is based onthe distribution of activity and 5.72 when the intensity code is based onsynchronization.

4 Discussion

The population model used in the simulations produced simulated detec-tion thresholds (average: 17.9 dB) very close to experimental thresholds(average: 19.0 dB). Similar to previous studies that used the same decisioncriterion of at least 10 active rapidly adapting fibers, the population modelmimics the detection process very well.

In addition, the results of the population model exhibited a deviationfrom Weber’s law, just like the current psychophysical results on intensitydiscrimination. However, the average Weber fraction of the psychophysi-cal data (0.32) obtained from a 20 dB SL range was higher than the Weberfraction of the nearest simulation data (0.07), which were based on thenumber of active fibers as the intensity code. An intensity code based ontotal activity gave better discrimination (Weber fraction: 0.03), whereas theother codes tested (i.e., maximal spike count, distribution of activity, syn-chronization) gave much worse results. Intensity discrimination based onthe distribution of spike counts was especially unfavorable (Weber fraction:33.93).

The experimental and simulation results are discussed below in relationto the hypotheses put forth in section 1.


4.1 Previous Literature. The Weber fractions from psychophysical ex-periments (average: 0.32) are slightly higher that the those reported byGescheider et al. (1990). The average Weber fraction in their study is 0.19.Therefore, the subjects in our study could not detect small changes in am-plitude as well as the subjects in Gescheider et al. (1990). There may beseveral explanations for this inconsistency. In addition to some differencesin temporal stimulation parameters, the current study used a forward-masking paradigm to isolate the NP I channel. Therefore, the results pre-sented here are related only to the NP I channel. On the other hand,Gescheider et al. (1990) tested the overall tactile discrimination capacity.More than one psychophysical channel could have contributed to discrim-ination and improved the Weber fraction in their work. This idea is alsosupported by frequency insensitivity in their results. That is, the intensitydiscrimination capacity was probably not altered by frequency because thechannels were not selectively activated.

4.2 Deviation from Weber’s Law. It was implied in section 1 that thedeviation from Weber’s law may be because of the recruitment of additionaltactile channels, and it was hypothesized that the NP I channel shouldnot display this phenomenon. However, the current psychophysical resultsindicate that the deviation is present even if a single channel (i.e., NP Ichannel) is activated. Then a second question arises: Is the deviation fromWeber’s law a result of some specific higher-order processing or in theinherent activity of the afferent population? No matter which intensity codewas tested, Weber fraction decreased as a function of the stimulus level. Theactivity of the afferent population seems sufficient to generate the deviation.In order to investigate the mathematical basis for that, it may be assumedthat there exists a response measure (i.e., intensity code) R defined over thepopulation activity, and it is equal to a function of the stimulus level (S):R = f (S). Additionally, it is assumed that S > 0 and d R

d S = f ′(S) > 0, that is,the response measure is monotone increasing. For a given criterion changedR* > 0 in the response measure, the Weber fraction (W) is W = d S

S = d R∗S· f ′(S) .

Then the change in the Weber fraction over the change in the stimulus levelis

dWd S

= −d R∗( f ′(S) + S · f ′′(S))S2 · f ′2(S)

. (4.15)

Since the denominator in equation 4.1 is positive, W is decreasing if andonly if f ′(S) + S · f ′′(S) > 0. Therefore, all concave up or constant slope(i.e., f ′′(S) ≥ 0) response measures yield a decrease in the Weber fractionas seen in the experimental and simulation data. However, that is not anecessary condition; for example, power functions (R = β · Sα ; β > 0 )commonly used in neurophysiology and psychophysics also yield de-creasing Weber fractions: f ′(S) + S · f ′′(S) = α2β · Sα−1 > 0. Note that for

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decreasing response measures (i.e., d Rd S = f ′(S) < 0) like the synchonized

activity code presented above, dR* < 0 and a decreasing Weber fractionrequires f ′(S) + S · f ′′(S) < 0. (The nontrivial solution of f ′(S) + S · f ′′(S) =0 is R = f (S) = c1 ln S + c2, which produces a constant Weber fraction, thatis, exact Weber’s law.)

The values of the intensity codes obtained from the simulations pre-sented in this letter are plotted as a function of the stimulus amplitudein Figure 7. Note that all graphs in Figure 7 are generally consistent withthe mathematical explanation given above. Although the graphs in section3 were plotted in dB units, the same mathematical conclusions are validthere as well, because logarithm function is monotonic. There is no singleresponse measure for the distribution of activity, but a similar explana-tion can be extended by considering the number of nerve fibers for eachspike count used to construct the distribution. The mathematical analyses,however, do not explain the physical causes of the deviation effect.

The major physical cause for the deviation from Weber’s law is themechanical spread of the stimulus. Given a criterion amplitude A* < As , theskin area that is activated as the effective stimulus amplitude is attenuated to


the criterion amplitude is π · r2s

( AsA∗

)2/1.9according to equation 2.7. Note that

this area increases almost linearly as the stimulus amplitude is increasedand hence causes an increase in the number of activated nerve fibers. Thismechanical factor affects all the intensity codes tested in this letter, andbecause of equation 4.1, it generates a decreasing Weber’s fraction. Thesecond cause for the deviation is physiological: the monotonic rate intensityfunctions (see equation 2.2). This factor also affects all the tested codes. Thethreshold parameters (a0) of the rate intensity functions determine whichfibers are activated. As the stimulus intensity is increased, the total spikecount and the maximum spike count increase according to the rate intensityfunctions. Additionally, the distribution of activity and the synchronicity ofthe spikes change. There is another physiological factor that is relevant onlyif the intensity code is based on the timing of the spikes. That is, accordingto equation 2.6, the phase jitter of the spikes decreases as the stimulus levelis increased, and that causes a decrease in the Weber fraction.

In summary, the peripheral population response contains sufficient in-formation to generate the deviation from Weber’s law in the NP I tactilesystem. Similarly in hearing, the excitation pattern of neurons is believedto be the cause of the near-miss to Weber’s law (Moore & Raab, 1974).However, severe departure from Weber’s law can sometimes be observedin hearing at high frequencies (e.g., 6500 Hz: Carlyon & Moore, 1986). Thetactile sense, however, is very insensitive to high frequencies (Bolanowskiet al, 1988).

4.3 Intensity Codes. The population model can be improved by select-ing a better intensity code. It is important to note that the codes tested inthis study are not optimal. More information regarding the stimulus can beobtained if the small fluctuations in the spike rates and spike timing of indi-vidual afferent fibers are considered. Nevertheless, the intensity code basedon the number of active fibers has yielded better discrimination than dataobtained from psychophysical measurements. This means that the nervoussystem is operating at a suboptimal level. What then is a more proba-ble code? Something more stringent than the number of active fibers canproduce better results. For example, higher-order decision networks maychoose to analyze only a subset of the afferent population. Furthermore,this subset should not be concentrated close to the stimulus-input location;otherwise, intensity discrimination would be better at lower stimulus lev-els, which is opposite the deviation effect. Perhaps a uniform distributionof afferents, but fewer in number than the entire population, may be thebasis for an accurate intensity code. This and other points discussed in thenext section will be considered in future modeling studies.

4.4 Future Work. Gescheider, Zwislocki, and Rasmussen (1996) foundthat intensity discrimination was not affected by stimulus duration when

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the gated-pedestal method was used (i.e., the psychophysical method usedin the current study). This was attributed to the absence of a major temporalsummation effect. Temporal summation is the improvement of the threholdas the stimulus duration is increased. Since the population model and thedecision rules that were used in this letter did not explicitly include temporalsummation, duration effects cannot be easily predicted by the populationmodel. However, stimuli with longer durations would generate more spikesand shift the detection criteria. Therefore, it would be worthwhile to testwhether some duration effects may be produced by the population model.It is interesting to note that, in hearing, stimulus duration has only a smalleffect on discrimination (Green, Nachmias, Kearney, & Jeffress, 1979).

The population model of intensity coding may also be tested using thepsychophysical method of absolute magnitude estimation. Sensation mag-nitude is typically represented as a power law function of the stimulusamplitude with growth of sensation magnitude faster at lower stimulusintensities (Verrillo & Chamberlain, 1972; Gescheider et al., 1994). It is prob-able that the nervous system uses the same intensity code for intensitydiscrimination and magnitude estimation. This hypothesis may be testedby using the population model described in this letter. If the same intensitycode gives accurate results in both discrimination and magnitude estima-tion tasks, the hypothesis would be supported.

Acknowledgments

This work was supported by Bogazici University Research Fund grant04HX101 and TUBITAK grant 104S228. I thank Ronald T. Verrillo and DianeOzbal for helpful comments and editing the manuscript. I am grateful toIbrahim Mutlu for the mechanical construction of the psychophysical setup.I also thank Ozge Kalkancı for her help in the psychophysical detectionexperiments.

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Received March 31, 2006; accepted October 29, 2006.

Documents

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