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The Journal of Neuroscience, October 1, 2001, 21(19):7751-7763
Tactile Discrimination of Edge Shape: Limits on Spatial
Resolution Imposed by Parameters of the Peripheral Neural
Population
Heather E.
Wheat and
Antony W.
Goodwin
Department of Anatomy and Cell Biology, University of Melbourne,
Victoria 3010, Australia
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ABSTRACT |
When the flat faces of a coin are grasped between thumb and index
finger, a "curved edge" is felt. Analogous curved edges were
generated by our stimuli, which comprised the flat face of segments of
annuli applied passively to immobilized fingers. Humans could scale the
curvature of the annulus and could discriminate changes in curvature of
~20 m 1. The responses of single slowly adapting
type I afferents (SAIs) recorded in anesthetized monkeys could be
quantified by the product of two factors: their sensitivity and a
spatial profile dependent only on the radius of the annulus. This
allowed us to reconstruct realistic SAI population responses that
included noise, variation in fiber sensitivity, and varying innervation
patterns. The critical question was how relatively small populations
(~70 active fibers) can encode edge curvature with such precision. A
template-matching approach was used to establish the accuracy of edge
representation in the population. The known large interfiber
variability in sensitivity had no effect on curvature resolution.
Neural resolution was superior to human performance until large levels
of central noise were present showing that, unlike simple detection,
spatial processing is limited centrally. In contrast to the behavior of
mean response codes, neural resolution improved with increasing
covariance in noise. Surprisingly, resolution for any single population
varied considerably with small changes in the position of the stimulus relative to the SAI matrix. Overall innervation density was not as
critical as the spacing of receptive fields at right angles to the edge.
Key words:
tactile resolution; mechanoreceptive afferents; somatosensory; form processing; spatial coding; innervation density; curvature; edges
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INTRODUCTION |
It has been recognized for a long
time that edges are an important feature of tactile stimuli.
Information about these features is relayed to the CNS principally by
the slowly adapting type I afferents (SAIs) that are highly sensitive
to edges (Vierck, 1979 ; Phillips and Johnson, 1981a; Johansson et al.,
1982). The reason for this sensitivity is that SAIs respond to strain
energy density or its equivalent (Phillips and Johnson, 1981b;
Srinivasan and Dandekar, 1996); therefore they are also very responsive
to rectilinear corners (Blake et al., 1997b) and to circular punctate stimuli with small diameters (Mountcastle et al., 1966). Studies to
date have characterized human psychophysical performance and SAI
responses only for edges that are straight and relatively long
(Phillips and Johnson, 1981a). However, the edges of many important
tactile stimuli are not straight, but are curved. This situation is
exemplified by tasks such as grasping the flat surfaces of a coin
between the thumb and index finger. The stimulus is the curved edge of
a surface that is flat in the plane of the skin; this is in marked
contrast to grasping a sphere, where the curvature is at right angles
to the skin surface. When identifying or manipulating objects such as
the grasped coin, it is not sufficient to know that an edge is present;
the shape (curvature) of the edge and its position on the skin must be
signaled to the CNS. There is no information available about the
precision with which humans can discriminate such curved edges.
Currently it is not known how the edge sensitivity of cutaneous
mechanoreceptors changes with the curvature of an edge, but it is
obvious that enhanced mechanoreceptor responses cannot, in themselves,
provide information about the shape of an edge. Such information can
only be conveyed by distributed spatial signals within a population of
afferents (Doetsch, 2000). Tactile spatial coding has been analyzed for
a variety of stimuli, including three-dimensional objects of various
shapes (LaMotte and Srinivasan, 1987, 1996; Goodwin et al., 1995;
Dodson et al., 1998; LaMotte et al., 1998) and patterns of raised dots
and letters (Phillips et al., 1990; Connor and Johnson, 1992; Johnson
et al., 1995). Understanding how curved edges could be encoded presents
a particular challenge because the number of afferents activated will
be relatively small compared with the spatial precision required
(Johansson and Vallbo, 1979; Darian-Smith and Kenins, 1980). Thus, it
might be expected that parameters of the afferent population, such as
innervation density, could have a profound effect on resolution.
Because it is not currently possible to record from the entire
population simultaneously, such issues can only be addressed by
simulating population responses.
In this study we quantified the human capacity to scale and
discriminate the curved edges of flat stimuli. We then recorded from
single primary afferent fibers and used the data to reconstruct realistic SAI population responses, which were then compared with the
human performance. By varying the population parameters (individual afferent sensitivities, pattern and density of innervation, neural noise, and correlation) we elucidated the neural mechanisms underlying this type of form processing in the tactile system.
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MATERIALS AND METHODS |
The stimuli consisted of a series of annular segments of Delrin,
1.5-mm-wide, with curvatures ranging from 107 m1 (radius of curvature 9.3 mm) to a
straight segment, curvature 0 m1 (radius
of curvature  ) (Fig.
1A,B). For all but the
largest curvatures, 107 and 84.7 m1
(smallest radii), the length of the chord b through the end
points of each segment was 25 mm. This ensured that the ends of the
segments did not contact the fingerpad. The stimuli were fixed at
a to a hub attached to a gravity operated, balanced-beam
stimulator described previously (Goodwin et al., 1991) that lowered a
selected stimulus onto the fingerpad (dark line c highlights
the surface that made skin contact). Contact force was set by a
counterbalance weight on the beam and calibrated to a resolution of 0.1 gram force (gf). A rotary damper controlled vertical motion of the beam
so that the surface of the stimulus contacted the skin at a velocity of
~20 mm/sec. The beam was mounted on an x-y stage fitted with micrometers and dial indicators; this enabled stimuli to be
positioned with a resolution of 0.01 mm. The stimuli were applied
passively to the fingerpad and were orientated orthogonal to the axis
of the finger with the concave side distal (Fig. 1C).
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Figure 1.
Stimulus dimensions and conventions.
A, The stimuli, annular segments of Delrin 1.5-mm-wide,
were fixed by post a to a hub attached to a
gravity-operated, balanced-beam stimulator. The superimposed dark
line c indicates the stimulus surface that made contact with the
skin. The chord b was 25-mm-long for all stimuli, except
curvatures 107 and 84.7 m1 where it was 20 mm.
B, The six stimuli used in the human scaling and the
neural experiments. Roman numerals I and
II identify the two standard stimuli used in the human
discrimination experiments, 25.6 and 61.7 m1,
respectively. C, In monkey neural experiments the origin
of the x-y coordinate system was located
at the receptive field center (rfc), and the center of
the surface (e) was positioned at different
points on the skin. The center of the circle of curvature is shown by
o, and d is the shortest distance from
the rfc to the circle. The circle forms the midline of
the annular segment, i.e., 0.75 mm from both edges. D,
In population reconstructions the origin of the
x'-y' coordinate system was located at
the center of the annular segment (e), and
different fibers in the population had receptive field centers
(rfc) located at different points on the skin.
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Human psychophysics
Human capacity to perceive the curvature of an annular segment
was measured in two ways; first in a series of scaling experiments and
then in a series of discrimination experiments. The same six subjects
(five females, one male) ranging in age from 20 to 25 years, took part
in both of these experiments. The subject was seated comfortably with
forearm supinated and the index finger of the dominant hand secured in
a plasticine finger mold to prevent lateral movement between stimulus
and finger; a curtain prevented the subject from seeing either the
stimulator or their finger. The stimulus in each experiment was applied
to the distal portion of the fingerpad at a contact force of 0.49 N (50 gf). Between trials, the position of the stimulus was varied randomly
along the long axis of the finger to ensure that subjects were using information about the shape of the stimuli and not any spurious cues.
The range of random position variation was ±1 mm from the initial
contact point.
Each subject underwent an extensive training period, during which
performance improved, so that by the time data were collected, the
subject's performance had stabilized at the optimum level.
Curvature scaling. Six stimuli with different
curvatures, 0-107 m1 (Fig.
1B), were presented in random order in blocks of 30 trials; each trial consisted of the presentation of a single stimulus for 1 sec with 3 sec between trials and a 2 min rest break between blocks. A reference stimulus, which had a mid-range curvature of 61.7 m1, was presented a number of times at
the commencement of each block and after every 10 trials within each
block. To establish some consistency in scale, subjects were asked to
assign this reference stimulus an arbitrary value of 50 and to estimate
the magnitude of the curvature of all stimuli relative to this, e.g., if they perceived a test stimulus as being twice as curved as the
reference stimulus, they were instructed to assign it a value double
that of the reference. Eight such blocks were presented so that
n = 40 for each stimulus for each subject. The order of stimulus presentation was varied randomly within and between blocks.
Curvature discrimination. A two-alternative forced-choice
paradigm was used to measure the subject's ability to discriminate small differences in the curvature of annular segments. Stimuli were
presented in pairs: the standard, for 1 sec, followed by the
comparison, for 1 sec, with an interval of ~2 sec between them. Two
stimulus conditions were used: in one
(Ss) the curvature of the comparison
stimulus was the same as that of the standard, and in the other
condition (Sd), the curvature of the
comparison stimulus was greater (smaller radius) than that of the
standard. Subjects were required to judge whether the two stimuli in
the pair were the same (Rs) or
different (Rd). Two standard stimuli were used: curvatures 25.6 and 61.7 m1
(Fig. 1B). The curvatures of the comparison stimuli
were 31, 41.2, and 61.7 m1 for the less
curved standard and 70.2, 84.7, and 107 m1 for the more curved standard. The
range of comparison curvatures was chosen to provide values below and
above the likely difference thresholds. In each experimental session,
six blocks of trials were presented, three for each standard (using the
corresponding three comparison stimuli); each block consisted of 20 trials, 10 of which were Ss and 10 Sd, presented in random order. The order of presentation within and across blocks of trials was varied randomly from session to session for all subjects. After an initial training period, data were collected over five sessions
(n = 300 per standard per subject). From the
conditional probabilities p(Rd/Sd)
and
p(Rd/Ss),
the bias-free measure of discrimination d' was calculated (Johnson,
1980), and difference limens were determined by linear interpolation of
those values for each subject.
Neural recording
Single mechanoreceptive fiber recordings were performed on two
Macaca nemestrina monkeys weighing 2.5 and 4.5 kg,
respectively. All experimental procedures were approved by the
University of Melbourne Ethics Committee and conformed to the National
Health and Medical Research Council of Australia's Code of Practice
for nonhuman primate research. An intramuscular dose of ketamine
hydrochloride (15 mg/kg) plus atropine sulfate (60 µg/kg) was given
before the induction of surgical anesthesia by intravenous
administration of sodium pentobarbitone (15 mg/kg). An endotracheal
tube was inserted (after the application of topical xylocaine spray) to maintain a patent airway and to enable continuous measurement of
end-tidal carbon dioxide levels. Anesthesia was monitored throughout the experiment and maintained with titrated doses of sodium
pentabarbitone. This was delivered in isotonic saline (dilution 12 mg/ml) via an intraperitoneal catheter together with additional saline
to maintain hydration. Respiration rate, end-tidal carbon dioxide level, blood pressure, core temperature, heart rate, and oxygen saturation levels were monitored throughout the experiment. Body temperature was maintained at 37°C by a heat pad and insulating blankets. Antibiotic cover was provided throughout the experiment by
intramuscular administration of amoxicillin (18 mg/kg) and at the end
of the experiment by a single dose of procaine penicillin (60 mg/kg).
Using aseptic surgical techniques, single fibers were isolated by
microdissection after exposing the median nerve first in the upper arm
and then in the lower arm. The process was repeated in the other arm,
making a total of four experiments for each monkey. Each experiment
lasted a maximum of 18 hr, and there was a rest period of at least 2 weeks between each experiment. During this rest period the monkeys were
housed in large cages, together with or adjacent to other monkeys, with
access to an outdoor exercise area. They were observed closely and
regularly by trained personnel and at all times were found to be in
good health with no evident signs of pain or distress. Buprenorphine
hydrochloride (8 µg/kg) was available for pain relief but was judged
to be unnecessary. At the end of the series, the monkeys were in prime
condition and showed no signs of sensory or motor deficits. They were
returned to the breeding colony.
The afferents principally activated by our stimuli were the SAIs. This
afferent class was therefore selected for study. Fibers were classified
by established response criteria (Talbot et al., 1968; Vallbo and
Johansson, 1984). The most sensitive spot in the receptive field of
each fiber was located using calibrated von Frey filaments; this is
subsequently referred to as the receptive field center. Only those
fibers with receptive field centers close to the central, relatively
flat region of the fingerpad were used. This ensured that any effects
caused by the curvature on the sides or end of the fingerpad or
attributable to changes in skin mechanics close to the interphalangeal
joint were minimized. Once the most sensitive spot had been identified,
the finger was immobilized (with the most sensitive spot uppermost) in
a customized mold with the fingernail glued to the mold; this was then
secured to a hand holder. The finger was positioned so that when the
stimulus made contact with the skin, the surface of the stimulus was
tangential to the fingerpad, and the line of force was normal to that
plane; contact force was set at 0.196 N (20 gf). The contact force used in the neural recording experiments is approximately equivalent to that
used in the human psychophysics experiments scaled to take account of
the difference in size between human and monkey fingerpads (for
rationale, see Goodwin et al., 1997).
Six stimuli were used for all fibers: one stimulus was a straight
segment (curvature 0 m1, radius ),
and the others had curvatures of 25.6, 34.2, 61.7, 84.7, and 107 m1 (radii of curvature 39.1, 29.2, 16.2, 11.8, and 9.3 mm, respectively) (Fig. 1B). Responses
were recorded from a total of 14 fibers using the following protocol.
For each fiber, the origin of the x-y coordinate system was
located at the receptive field center. First, the center of the segment
(Fig. 1C, e) was presented at positions separated by 0.5 mm
along the y-axis, starting with the most proximal position
used. At each position the stimulus was applied for 1.5 sec (a single
trial), and the time between successive presentations was 3 sec. This
presentation sequence was performed for all six stimuli, and then the
entire sequence was repeated twice (n = 3 per stimulus
for each fiber); n = 3 was deemed sufficient because we
and others have shown that variation in the responses of peripheral fibers is low (Edin et al., 1995; Wheat et al., 1995; Vega-Bermudez and
Johnson, 1999). This entire procedure was repeated along lines parallel
to the long axis of the finger and separated by 1 mm in ±x
directions. The order of the lines depended on the location of the
receptive field. The order of presentation of the different curvatures
was varied randomly between fibers, but for individual fibers it was
maintained across all data collection lines, i.e., at all values of
x. At the commencement of each traverse of the receptive
field, a lead stimulus with the same time sequence, 1.5 sec on and 3 sec off, was presented to minimize differential interaction effects.
Data from this trial were not included in the analyses with the result
that for each trial analyzed, the intertrial time period was constant.
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RESULTS |
Human psychophysics
Curvature scaling
Subjects scaled the curvature of six stimuli ranging from a
straight segment (curvature 0 m1) to one
with a curvature of 107 m1 (radius of
curvature, 9.3 mm). Stimuli were presented passively to an immobilized
finger, therefore the only source of information about the stimuli
available to the subjects derived from cutaneous sources. An additional
feature of the protocol we adopted, i.e., randomly varying the position
of successive stimuli on the fingerpad, ensured that estimations about
stimulus magnitude were based on information about the comparative
shape of the stimuli and not on any spurious positional cues.
As stimulus curvature increased, the subjects' perception of the
magnitude of curvature also increased. This is illustrated in Figure
2A, which shows mean
estimates for each stimulus for each subject; the thick line represents
the mean across all six subjects.
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Figure 2.
Human performance. A, Scaling of
segment curvature. Fine lines show perceived curvature
of the six stimuli for each of the six subjects (mean curvature
estimates, n = 40 for each stimulus for each
subject). The thick solid line is the mean across all
six subjects. B, Curvature discrimination for the 25.6 m1 standard. Fine lines show
d' values for each of the six subjects, and the
thick solid line shows mean d' values.
Filled circles show difference limens (d' = 1.35) for each subject. C, Comparison of
d' values (averaged across subjects;
n = 6) for the two standards. Error bars show
unidirectional SE.
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Curvature discrimination
All subjects could scale stimulus curvature over the range 0-107
m1. To determine the smallest difference
in curvature that could be discriminated, the same six subjects took
part in a series of discrimination experiments using subsets of stimuli
from within this range. Two standard stimuli were used (curvature 25.6 and 61.7 m1) to determine whether
performance depended on the magnitude of stimulus curvature. Indices of
discrimination d', calculated from the conditional
probabilities
p(Rd/Sd)
and
p(Rd/Ss),
are shown for all six subjects in Figure 2B for the
25.6 m1 standard. The solid line shows
the mean d' values across the six subjects. The relationship
between the curvature of the annular segment and discrimination
performance was approximately linear. For each subject, the difference
limen was estimated by linear interpolation of the two data points on
either side of d' = 1.35 (Fig. 2B,
filled circles).
Performance with the more curved (61.7 m1) standard was similar to that with
the less curved standard. Difference limens for each subject are shown
in Table 1 for both standards. The mean d' values across the six subjects are compared for the two
standards in Figure 2C. The mean difference limens, 23.0 and
20.8 m1 for standards of 25.6 and 61.7 m1, respectively (Table 1) were not
significantly different (p = 0.59; two-tailed
paired t test; n = 6). Neither were there
significant differences in performance between standards when all 18 pairs of data points (six subjects × three curvatures) were
considered: there were no significant differences
(p > 0.1) in either slope or elevation between
performances for the two standards (Zar, 1984).
Neural responses
To generate receptive field profiles for single SAI fibers,
stimuli were presented for 1.5 sec successively at a matrix of points
on the skin; points were separated by 0.5 mm in the y
direction and 1 mm in the x direction. The x and
y coordinates refer to the position of the center
e of the annular segment with respect to the receptive field
center (Fig. 1C). Care was taken to avoid values of
x that were large enough for the end points of the stimulus to contact the skin. The response measure used is the mean response (n = 3) evoked in the first second of stimulus contact.
Figure 3A compares responses
of a typical single fiber to all six stimuli of different curvature at
different positions along the y-axis. These profiles show
three key features. First, all six profiles superimpose, indicating
that the response of the afferent depended only on the distance of the
receptive field center from the center of the stimulus and not on the
curvature of the stimulus as such. Second, there are two response peaks corresponding to the two edges of the annular segment. Third, the
skirts of the profiles are Gaussian in shape. In Figure 3B, profiles along a line parallel to the axis of the finger at a distance
of 6 mm from the receptive field center are illustrated for four of the
stimuli. As the curvature of the segment increases, the profiles shift
to the left of the figure. Also, with increasing curvature the profiles
become broader and increasingly asymmetric; it is clear that at a
curvature of 107 m1 (thick broken line)
the skirt on the right side (increasing y) is steeper than
the skirt on the left side (decreasing y). In Figure
3C, profiles for one stimulus, curvature 84.7 m1, are shown along a number of lines,
parallel to the finger axis, at increasing distances from the receptive
field center. With increasing values of x, the profiles
shift to the left and become broader and increasingly asymmetric with a
steeper skirt on the right; compare profile at x = 0 (solid line) with that at x = 6 mm (thick broken line).
For all afferents, responses were highly consistent among the three
repetitions in keeping with previous observations on the low
variability of peripheral neural responses. A second order effect can
be seen in Figure 3A; the asymmetry of the two peaks
increases with increasing curvature.
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Figure 3.
Responses of a single fiber to stimuli of
different curvature at different positions in the receptive field. The
x and y coordinates define the position
of the center of the annular segment with respect to the afferent's
receptive field center. Responses are the number of impulses occurring
in the first second of stimulus contact. A, The six
profiles show the mean responses (n = 3) to each of
the six stimuli along the y-axis (x = 0). B, Mean responses elicited along a line (parallel
to the y-axis) at x = 6 mm for four
of the stimuli. For clarity, only one representative bidirectional SD
is shown for each of two profiles (n = 3).
C, Mean responses for one stimulus, curvature 84.7 m1, along lines parallel to the
y-axis.
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All fibers responded with similarly shaped profiles, but the magnitudes
of the responses varied widely, indicating a wide spread in fiber
sensitivity. To reveal the underlying response characteristics common
to all fibers in our sample (n = 14) in the absence of
sensitivity differences, we normalized the responses of each fiber.
This was done by dividing the responses of each fiber by the average
response of that fiber to the straight stimulus over seven data points
along the y-axis spanning the center of the receptive field;
this region corresponded across all fibers. Consistent with our
previous studies, the normalizing factors were normally distributed
with a coefficient of variation of 0.35 (Goodwin et al., 1995; Wheat
and Goodwin, 2000). When the normalized responses were pooled, it was
clear that all fibers in the sample produced similarly shaped profiles.
This is exemplified in Figure 4A, which shows mean
normalized responses of the 14 SAI fibers to the straight stimulus
(curvature 0 m1) along the
y-axis. Figure 4B shows that, as with the
single fiber data illustrated in Figure 3A, responses of all
fibers along the y-axis were invariant with stimulus
curvature; data shown in this figure are mean normalized responses of
14 fibers to each of the six stimulus curvatures (for clarity SE are
shown for one stimulus only). All stimuli produced response peaks in
the receptive field profiles which corresponded to the edges of the
stimuli. The consistency of the profiles between fibers across the
extent of the receptive field is shown in Figure 4C. These
normalized profiles reflect the shape of the annular segments as is
clearly evident in a representative contour plot of the two-dimensional
profiles (Fig. 4D). The normalized data verify an
important principle that is essential for the population reconstructions that follow. All 14 SAI responses to an annular segment
can be quantified by the product of two factors. The first factor is
the sensitivity of the fiber, and the second factor is the normalized
profile, which is independent of the fiber sensitivity.
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Figure 4.
Normalized responses for the 14 SAIs.
A, Normalized responses of each of the 14 fibers to the
straight stimulus (curvature 0 m1) at
x = 0. B, Mean normalized responses
(n = 14) to each of the six stimuli at
x = 0; ±SE for one representative profile only.
C, Mean normalized responses (n = 14) to one of the most curved stimuli (curvature 84.7 m1) along lines at four locations along the
x-axis; ±SE at two representative points.
D, Contour plot of mean normalized responses
(n = 14) for curvature 84.7 m1; isometric lines are separated by increments of
0.12 (black = 0; white = 1.2).
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Mathematical characterization of response profiles
The normalized receptive field profiles were similar for all
fibers in the sample and could therefore be described by a single mathematical function. The data in Figures 3 and 4 suggest that the
profiles conform well to the sum of two offset Gaussians (in effect one
for each edge of the segment) and that the only positional variable is
the radial distance from the receptive field center to the segment
(Fig. 1C, d). Thus, the function used was:
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(1)
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where NR is the normalized response, and a,
b, and c are the constants of the Gaussians. For
an annular segment of radius r the distance d is
determined by the x and y coordinates of the stimulus from the relationship, d =   r. Thus:
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(2)
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The constants a, b, and c
were determined by nonlinear regression using the data from all 14 afferents, for all six stimuli, at all tested locations in the
receptive field: 5239 data points. These data points are plotted in
Figure 5A as normalized
response versus distance d, together with the regression
function. The six regression constants are given in Table
2. The close correspondence between the
mean normalized data and the fitted function is illustrated for a
representative range of curvatures and positions in Figure 5B-D. The correspondence along the y-axis is
shown for one of the most curved stimuli, 84.7 m1 (Fig. 5B) and for the
straight stimulus 0 m1 (Fig.
5C). Figure 5D shows the fit for the stimulus
with curvature 84.7 m1 along a line 6 mm
from the receptive field center. Naturally, the function matches the
changes in profile position, width, and asymmetry seen in the data.
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Figure 5.
Comparison between normalized data and the best
fitting function from Equations 1 and 2. A, Data points
(n = 5239) show normalized responses from 14 SAIs,
for all six stimuli, at all x and y
coordinates tested, plotted as a function of the distance
d in Equation 1. The solid line is the
regression function. Note that because n is so large,
many data points are hidden because they overlap, particularly close to
the regression function. B, Responses to one of the most
curved stimuli (curvature 84.7 m1) at
x = 0. C, Responses to the straight
stimulus (0 m1) at x = 0. D, Responses to the stimulus with curvature 84.7 m1 along a line 6 mm from the center of the
receptive field. B-D, Solid lines show mean normalized
responses (n = 14; ±SE at representative
locations), and dashed lines show the fitted
function.
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Simulated SAI populations
The critical issue in this series of experiments is the precision
with which the neural population can reflect the essential features of
the stimuli. In the following sections we explore this by simulating
realistic SAI population responses. These reconstructions take into
account population parameters that are not evident in single fiber
responses. Fundamental parameters which degrade the representation of
the stimulus in the population response are (1) the variation in
sensitivity between fibers, (2) random variation in the responses of
individual afferents, and (3) the sampling density and geometry.
The conventions for the simulation are shown in Figure
1D. The receptive field center (rfc) of
afferents in the population are specified by their x' and
y' coordinates with the origin located at the center of the
annular segment. Initially the matrix of receptive field centers
(x'ij,
y'ij) is uniform with a spacing of 1.2 mm, which corresponds to the estimated innervation density of 0.7 mm2 for human fingerpad skin (Johansson
and Vallbo, 1979). The total area spanned by the fibers is 12 × 12 mm, which is equivalent to the size of the central part of the
average human fingertip.
The normalized response of an afferent with a receptive field center at
(x'ij,
y'ij) obeys Equation 2. Note that
x'ij = xij and
y'ij = yij; for example if the coordinates
of e in Figure 1C are (4, 4) then the coordinates
of rfc in Figure 1D are (4, 4), and
these two figures are equivalent. As a result, population responses are
mirror images of receptive field profiles. If the afferent at position
(x'ij,
y'ij) has a sensitivity
sij then its response
REij (impulses elicited in the first
second of contact) is given by:
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(3)
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where r is the radius of curvature of the annular
segment, and a, b, and c are the
constants given in Table 2. The factor  ij
allows for random neural noise in the processing pathway; initially
ij is set to zero, and in later sections we
will vary it systematically. The sensitivity sij
of each fiber in the matrix varies randomly from fiber to fiber with a
Gaussian distribution with a coefficient of variation of 0.387; the
distribution of sensitivities was derived from our current and previous
experimental data (Goodwin and Wheat, 1999).
Figure 6A (shaded
area) shows a slice, parallel to the y' axis at
x' = 4.8 mm, through a simulated ideal population response to a stimulus with a curvature of 61.7 m1. The fibers within this ideal
population had uniform sensitivity and a high innervation density. No
noise was superimposed on these responses so that
ij = 0. Corresponding response profiles
for a selection of four more realistic populations are shown by the thin lines and symbols in Figure 6A. The constituent
fibers had receptive fields that were arranged in a uniform
configuration with a density of 0.7 mm2
(spacing 1.2 mm). Fiber sensitivities were randomly distributed following a Gaussian with a coefficient of variation of 0.387, and the
pattern of sensitivities was different for each population. Comparison
between the realistic and ideal profiles makes it abundantly clear that
varying sensitivity has a dramatic distorting effect on the shape of
the response profiles, even in the absence of response noise.
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Figure 6.
SAI population simulation.
A, Slices through simulated response profiles for four
populations of SAIs (thin lines and
symbols), each with a different sensitivity profile.
Receptive field centers were uniformly spaced with a separation of 1.2 mm (density 0.7 mm2). For comparison, a
corresponding response profile for an ideal population (shaded
area) is also shown. Stimulus curvature was 61.7 m1. Slices are taken parallel to the long axis of
the finger at x' = 4.8 mm. B, Curvature
estimates  from each of the five populations illustrated in
A for the stimuli used in the human scaling experiments.
Curvature () was extracted from the population responses by template
matching. The close correspondence between stimulus and estimated
curvature is indicated by r = 0.99 for all
populations. C, Distribution of curvature estimates
() for one representative population with a stimulus curvature of
61.7 m1. Response variation was the sum of two
random variables, one for proportional noise (variance of
ij = 1.5 REij),
and one for additive noise (SD of ij = 6 impulses/sec). Mean was 60.0 (± 5.44 SD; n = 500), and the distribution was Gaussian (shown by the thick
line). The Kolmogorov-Smirnov test found no significant
difference between the distribution of and that of the normal
distribution (p = 0.65). Fiber configuration
for this simulation was uniform with a spacing of 1.2 mm (density 0.7 mm2).
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Representation of curvature in the population responses
To determine the precision with which the curvature of the
stimulus was represented in the simulated population responses, we
adopted a template-matching approach. We determined the curvature of
the annular segment that would minimize the mean square deviation of
the population response from the template response by regression of the
equation:
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(4)
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The regression results in a value for the two constants,  the
radius of curvature of the matching template and  the mean sensitivity of the afferents in the population. We will express the
result in terms of the curvature of the match, given by 1/. In
the case of an ideal population response (Fig. 6A, shaded area), will of course be identical to the curvature of the
stimulus (Fig. 6B, thick line). In realistic
populations with distortions, noise and finite innervation density, will differ from the stimulus curvature. The advantage of this approach
is that it provides a quantitative measure of the precision of
representation in realistic populations without having to make
assumptions about candidate neural codes. Moreover, this is likely to
measure the optimal representation (see Discussion). The nonlinear
regression was performed using the Levenberg-Marquardt method (Press
et al., 1986).
In spite of the severe distortions in the four population responses
indicated in Figure 6A, the curvatures extracted
from the responses of each of those populations were similar (Fig. 6B). More striking is the almost perfect relationship
between stimulus curvature and curvature estimated from the population responses; r = 0.99 for all four populations. The
stimulus curvatures used in these simulations were those used in the
human scaling experiments.
Effect of response variability on resolution in the population
Although the variability of peripheral afferents is low, there is
considerable noise in the ascending pathway that will reduce resolution. In our model we account for such noise by adding a random
component, ij, to the response of each fiber
in the population. Two broad categories of lumped noise are used. For
one category, termed "proportional noise", the level of noise is
proportional to the magnitude of the response of each fiber; thus
ij in Equation 3 is a normally distributed
random variable with a mean of zero and a variance proportional to
REij. For the second category, "additive noise", the noise is independent of the magnitude of the
afferent's response; thus, ij is a normally
distributed random variable with a mean of zero and a SD that is
the same for all i and j. These choices are
justified in Discussion.
With a constant stimulus, variability in each fiber's response leads
to variability in the whole population response and therefore to
variability in estimates of curvature from the population response.
Figure 6C shows the distribution of estimated 500 times
with a combination of proportional noise (variance of
ij = 1.5 REij) plus additive noise (SD of
ij = 6 impulses/sec). The distribution
is Gaussian. The mean value of is 60.0, which is close to the
stimulus curvature of 61.7 m1, but the
distribution is broad so that most estimates of curvature from the
population response will differ from the stimulus curvature, leading to
errors in perception. This type of distribution was found under all
stimulus and noise conditions.
To quantify the effect on discrimination of the variations in curvature
estimates illustrated in Figure 6C, we used a signal detection theory approach analogous to that used in our human psychophysics experiments. Stimuli were presented to the model in
pairs. For 500 pairs the first and second stimuli were both the
standard, and for an additional 500 pairs the first stimulus was the
standard, and the second stimulus was the comparison, which had a
larger curvature (smaller radius). Curvature estimates were
extracted from the population response to each stimulus. For each pair,
the second stimulus was judged to be different if
2 1  the
decision boundary (defined as half the difference between the mean
value of for the comparison stimulus and the mean value for the
standard stimulus), otherwise the two stimuli were judged to be the
same. From the resultant conditional probabilities, the index of
discrimination d' was calculated, and discrimination thresholds were determined by linear regression.
In Figure 7A the ability of
the model to discriminate small differences in curvature, using the
more curved standard from our psychophysics experiments (61.7 m1), is illustrated for two populations
with varying sensitivity profiles and one population with uniform
sensitivity. Various magnitudes of proportional and additive noise were
used. As expected, increasing either component of noise decreased
resolution. Doubling additive noise had a greater impact on performance
than doubling proportional noise. The difference limens for populations
1 and 2 are similar and, surprisingly, so are the difference limens for
the population with uniform fiber sensitivity. In other words, even if
the CNS compensates for varying sensitivity, there would be no
improvement in performance as demonstrated by the white bars in Figure
7A.
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Figure 7.
Effect of different noise components on
discrimination performance of the model. A, Standard
curvature 61.7 m1. Performance of three
populations: the gray and hatched columns
show performance of two populations with different sensitivity profiles
(Gaussian distribution with a coefficient of variation of 0.387), and
the white columns show the performance of a population
with uniform sensitivity. B, Comparison of performance
between the two standards used in the psychophysics experiments, 61.7 and 25.6 m1 (same population as
population 1 in A). For all populations,
fibers were uniformly arranged with 1.2 mm spacing (density 0.7 mm2). Dotted lines in
A and B indicate human difference limens
for standards of 61.7 and 25.6 m1, respectively.
Noise levels in both A and B were (from
left to right) as follows: proportional
noise alone with variance = 0.75 REij
and 1.5 REij, additive noise alone
with SD = 4 and 8 impulses/sec, and for A only,
combined proportional noise (variance = 3.5 REij) plus additive noise (SD = 12 impulses/sec).
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For all populations tested, performance of the model was superior to
human performance (indicated by the dotted line) at the moderate noise levels represented by the four leftmost sets of bars.
The right column in Figure 7A shows that high levels of noise needed to be introduced before human and model performances were
on a par; in this illustration proportional noise with variance = 3.5 REij plus additive noise with
SD = 12 impulses/sec. Figure 7B confirms that
performance of the model with the less curved standard used in the
psychophysics experiments is approximately equivalent to that with the
more curved standard when tested under the conditions illustrated in
Figure 7A; the population used to illustrate this is
population 1. The effects of the two standards on the discriminatory
capacity of the model parallel their effects on psychophysical performance.
If the only noise present were that in the primary afferent fibers,
where the coefficient of variation is a few percent (Wheat et al.,
1995), then the precision of curvature representation in the SAI
population would far exceed the performance of our subjects.
Response covariance
The analyses so far have assumed that the noise attributed in the
model to individual afferents is independent of the noise on the other
afferents. However, such independence is unlikely, and correlation
among the variation of responses may have a significant effect on the
resolution of the population. Although covariance among lumped noise
represents a simplification of the real situation, it serves to
highlight the major effects that covariance has on the resolution of
stimulus features represented in the population response. Values for
the range of covariances occurring in the tactile pathway have not been
established, so we illustrate effects over a large range to demonstrate
the trends (see Discussion).
In the following section, the noise components
ij have been generated such that the
correlation coefficient between each i, j pair is
0, 0.4, or 0.8. Five representative levels of noise are illustrated in
Figure 8: proportional noise alone with
variance = 0.75 REij and 1.5 REij, additive noise alone with
SD = 4 and 8 impulses/sec, combined proportional noise
(variance = 1.5 REij) plus
additive noise (SD = 4 impulses/sec).
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Figure 8.
Effect of correlation on resolution in
the population response for two standards: 61.7 and 25.6 m1. Correlation coefficients for noise components
ij are 0, 0.4, or 0.8 for all i,
j pairs. For a realistic population with varying
sensitivity (population 1), five representative levels of noise are
illustrated: proportional noise alone with variance = 0.75 REij (line 1) and 1.5 REij, (line 2),
additive noise alone with SD = 4 impulses/sec (line
4) and 8 impulses/sec (line 5),
and a combination of proportional noise with variance = 1.5 REij plus additive noise with SD = 4 impulses/sec (line 7). For comparison,
performance of a population with uniform sensitivity
(U) is shown for proportional noise with
variance = 1.5 REij,
(line 3) and for additive noise with SD = 8 impulses/sec (line 6).
Lines with symbols indicate
performance of population 1, with varying sensitivity;
lines with no symbols indicate the
performance of a population with uniform sensitivity.
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Figure 8 illustrates that for both standards used in the psychophysics
experiments, 61.7 and 25.6 m1,
resolution improved as correlation increased from 0 to 0.8. In this
illustration a realistic population with varying sensitivities (population 1 described previously) was used. For the less curved standard (Fig. 8B), the trend illustrated was similar
for all noise levels tested and for both types of noise (proportional and additive). For the more curved standard (Fig.
8A), however, the functions for additive noise alone
(lines 4, 5, and 6) were flatter than those for proportional noise
alone (lines 1, 2, and 3). Maintaining uniform sensitivity across the
population (lines 3 and 6) resulted in similar performance (illustrated
for two levels of noise) to that of the population with varying
sensitivity between fibers. Thus, even if the CNS were able to
compensate for varying afferent sensitivity, little advantage in
performance would be gained, as was the case when noise was uncorrelated.
Innervation density and geometry
So far we have assumed that afferents innervate the skin uniformly
with a separation between receptive field centers of 1.2 mm, which
corresponds to the estimated innervation density of 0.7 mm2. However, this density is only an
estimate (Johansson and Vallbo, 1979), and the real value may be
considerably different. Moreover, the innervation is unlikely to be
completely uniform. Given the relatively sparse innervation compared
with the spatial detail that must be resolved to discriminate our
stimuli, the details of innervation geometry are likely to have a
considerable impact, the nature of which is not obvious a
priori. In the following section we first explore the effect of
varying density with a uniform pattern of innervation, and then we
explore the effect of nonuniform innervation geometry.
Three uniform sampling densities were compared: the nominal density of
0.7 mm2 (fiber spacing 1.2 mm), a higher
density of 1.78 mm2 (fiber spacing 0.75 mm), and a reduced density of 0.25 mm2
(fiber spacing 2 mm). Even for this simple comparison, there are two
confounding factors. First, for more realistic populations with varying
sensitivity, changing the density in a fixed area of skin also, of
necessity, changes the distribution of sensitivities within the
population and the distribution relative to the stimulus. To avoid this
complication, we used populations with uniform sensitivity so that the
effects of innervation density could be isolated. The second factor is
the position of the sampling matrix relative to the stimulus, which
also changes with changing density. This is demonstrated in Figure
9B, which shows a slice along
the y' axis through a population response. The thin dotted
lines and open circles show sampling at 3 mm intervals for one position of the fiber matrix, and the thicker broken lines and solid circles show sampling at the same density with a different position of the
fiber matrix. It is obvious that, at least for this slice, the images
of the stimulus "seen" by the two populations are different, although the only difference between the populations is a shift in
origin. It is not possible to vary innervation density without changing
the relative position of some of the afferents in the population with
respect to the stimulus.
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Figure 9.
Resolution as a function of innervation density
and geometry. Resolution is measured by   , the SD of
(the estimate of curvature in the population response). The
standard stimulus had a curvature of 61.7 m1.
A, Distributions of for various fiber
populations. In each case, the position of the population was varied
randomly in both the x' and y' directions
within a range of ±0.5 times the fiber spacing; n = 500 per population. Fiber spacing was the same in the
x' and y' directions for distribution 1 (spacing 0.75 mm, density 1.78 mm2), distribution
2 (spacing 1.2 mm, density 0.7 mm2), and
distribution 3 (spacing 2 mm, density 0.25 mm2).
For distributions 4 and 5, density 0.28 mm2,
spacings were 3 mm x', 1.2 mm y', and 1.2 mm x', 3 mm y', respectively.
B, Indication of the effect of shifting a population
relative to the stimulus. The response function being sampled is a
slice, at x' = 0, through the profile of an ideal
population with infinite sampling density. The function is sampled
every 3 mm by two populations with different relative locations
(thin dotted lines and thick broken
lines, respectively). C, Resolution indicated by
the median value of . The solid black
bars (marked 1-5), show the median values of
the distributions (marked 1-5) in A and
thus indicate the resolution of five different populations with
regularly spaced receptive fields. White bars show
resolution when the five populations have some scattering imposed on
the receptive fields, assuming the CNS "knows" the exact positions
of the fields. Hatched bars show resolution if the CNS
does not know those exact positions. D, Geometric
arrangement of fibers at two densities. Curved lines
superimposed over the matrices represent a stimulus with a curvature of
61.7 m1. For density 0.25 mm2 (top
two panels), fiber configuration is either uniform and
symmetrical (left) or scattered (right),
in which case each fiber has been randomly shifted from the uniform
configuration in the x' and y' directions
within ±1 mm. The bottom two panels (density, 0.28 mm2) show two alternative fiber configurations:
x' spacing 1.2 mm, y' spacing 3 mm
(left) and x' spacing 3 mm,
y' spacing 1.2 mm (right). Responses for
the analyses depicted in A and C varied
randomly with a combination of proportional noise (variance of
ij = 1.5 REij)
plus additive noise (SD of ij = 6 impulses/sec).
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To address this second factor, we varied the origin of the fiber matrix
randomly in the x' and y' directions within the
range ±0.5 times the fiber spacing. Using this approach, we created 500 populations that all had the same density, but each had a different
offset with respect to the stimulus. For each of these populations, the
model estimated the resolution of the representation of stimulus
curvature in the population. We illustrate the results for a stimulus
with a curvature of 61.7 m1 with
response noise ij represented by a combination
of proportional noise (variance = 1.5 REij) and additive noise (SD = 6 impulses/sec). For each of the 500 populations, the value of was
estimated 500 times, and its SD was
calculated (Fig. 6C). According to signal detection theory,
if both the standard and comparison stimuli had the same SD of ,
then the difference limen calculated by our model would be 1.35 ×  2 × (Macmillan and Creelman, 1991). Note that in Figure 6C, 1.35 × 2 × = 10.4, which is a close approximation to
the difference limen of 10.7 calculated more rigorously without the
assumption of invariant SD. Because of the complexity of the
computations involved in the following analysis, we will use
as a measure of resolution rather than the
more precise, but less tractable, difference limen.
The solid black distribution (marked 2) in Figure 9A shows
the distribution of across all 500 populations with a uniformly distributed density of 0.7 mm2. Populations with different
positions relative to the stimulus have different values of
and hence different resolutions; for
purposes of comparison, the resolution at this density is characterized
by the median value of the distribution
(5.08) shown by the solid black column marked 2 in Figure
9C. With an increase in innervation density to 1.78 mm2, the distribution of
(marked 1) is narrower and shifted to the
left, with a median value (marked 1 in Fig. 9C), clearly showing better resolution than at the density of 0.7 mm2. The converse is true when
innervation density is reduced to 0.25 mm2 (marked 3). The degradation in
resolution with decreasing innervation density was expected, but it was
not initially obvious that resolution, with fixed innervation density,
would depend on the positioning of the fiber matrix.
To determine how variations in the uniformity of innervation impact on
resolution, we first examined the effects of changing fiber spacing in
the x' direction to 3 mm while maintaining the spacing in
the y' direction at 1.2 mm (marked 4 in Fig.
9A,C) and changing fiber spacing in the y'
direction to 3 mm while maintaining the spacing in the x'
direction at 1.2 mm (marked 5). Although the density was the same in
both scenarios, 0.28 mm2, the resolution
was highly dependent on the direction of increased spacing. When the
spacing was increased in the x' direction (marked 4) the
resolution at 0.28 mm2 was better than
at the comparable uniform density of 0.25 mm2 (marked 3) and indeed was close to
that at the uniform density of 0.7 mm2
(marked 2), which had the same spacing of 1.2 mm in the y'
direction. Note that in the population underlying performance marked 2, 72 afferents were active, whereas in the population underlying
performance marked 4, only 33 fibers were active, and yet their
resolutions were similar. Conversely, when the spacing in the
y' direction was increased to 3 mm, and spacing in the
x' direction was maintained at 1.2 mm (marked 5), the
resolution was inferior to a uniform population of density 0.25 mm2. The critical factor here is the
spacing in the y' direction. It is clear why this is the
case from the slice in Figure 9B and from the bottom two
panels of Figure 9D, which show the stimulus (curvature 61.7 m1) positioned over two differently
configured populations, but both with a density of 0.28 mm2. The difference limens will be
smaller when innervation density is higher in the y'
direction than when it is higher in the x' direction. This
clearly demonstrates that overall density per se is not the crucial
factor for this type of stimulus.
The solid black columns in Figure 9C show the resolution for
populations in which the spacing is regular in both the x'
and y' directions (even if the spacing magnitude is
different in the two directions). But real populations of SAI fibers
are likely to have some scattering rather than be regularly spaced. We
simulated such a scenario by adding a random component (±0.5 times the
fiber spacing) to the position of each fiber in both x' and
y' directions. Scattering the fibers in this way, albeit by
small perturbations, showed that if the central processors have
"knowledge" of the actual locations of the scattered receptive
field centers, then resolution is similar to that obtained for
populations with regular spacing; compare white and black columns in
Figure 9C. However, if the actual locations are unknown, and
the CNS assumes a uniform and symmetrical array, then resolution is
degraded as shown by the hatched columns in Figure 9C.
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DISCUSSION |
Our stimuli were applied passively to the fingerpad with a low
contact velocity, so that only SAIs were activated appreciably. From
the responses of single fibers to these stimuli, realistic SAI
population responses were reconstructed to examine how the required
spatial information could be relayed by the afferents and to elucidate
how the inherent properties of afferent populations affect the spatial
representations. For the stimuli used in this study, high-resolution
form processing is effected by the SAIs, which have already been shown
to underlie form processing of ellipsoidal objects contacting and
scanned across the skin (Goodwin et al., 1995; LaMotte et al., 1998)
and of patterns of raised dots and squares scanned across the skin
(Connor and Johnson, 1992; Blake et al., 1997a).
Psychophysics
There is an extensive body of literature on perception of the
shape of three-dimensional objects; in most studies the objects were
explored with active touch, but in a few experiments passive touch was
used (for review, see Appelle, 1991; Vogels et al., 1999). Of more
direct relevance to the experiments reported here are a number of
studies of planar tactile drawings (Kennedy et al., 1991; Millar, 1991;
Lakatos and Marks, 1998). These suggest that humans are able to
perceive changes in curvature in the plane of the skin, but they
provide no information about the nature or resolution of this capacity.
All subjects could scale the curvature of our stimuli, which ranged
from a straight edge to a curvature of 107 m1. The model (Fig.
6B) accounts for the near linearity of the scaling functions and the minor differences among five of the six subjects. For
one subject there is an additional gain factor and offset, which could
easily occur in the stage at which perception is translated into the
subject's notion of an appropriate number scale. It is interesting
that for all subjects, although the straight edge was perceived as
having lower curvature than any other stimulus, it was not rated as
having zero perceived curvature. We cannot say whet |
TOP
ABSTRACT
INTRODUCTION
