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INNOVATIVE METHODOLOGY
Department of Neurobiology, Weizmann Institute of Science, Rehovot, Israel
Submitted 13 July 2004; accepted in final form 16 November 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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| TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Fee et al.
1997; Jog et al.
2002; Kralik et al.
2001; O'Connor
et al. 2002; Prigg et al. 2002). However, during
unobstructed free ranging, events that produce sensory responses,
such as self-generated whisking and object touch ( Fanselow et al.
2001;
Krupa et al. 2004; Prigg et al. 2002; Szwed et al. 2003), need to be
accurately monitored.
Existing methods for monitoring movements of whiskers in unrestrained
rodents employ techniques that involve considerable manual data
collection, occasionally with the aid of customized software (
Berg and Kleinfeld 2003a; Brecht et al. 2004; Carvell and
Simons 1990, 1995; Hartmann et al. 2003; Sachdev et al. 2003; Sachdev et
al. 2002; Welker 1964; Wineski 1983). Automation
has been limited to conditions in which whiskers were tagged
with glass-beads or other reflective materials. However, since
the mass of whiskers is small, any material chosen to tag them
is likely to influence their dynamical properties, such as resonant
frequencies or center-of-mass position ( Hartmann et al. 2003; Neimark et
al. 2003). With a less
intrusive method ( Bermejo et al. 1998), an array of light
sensitive elements is illuminated by a collimated light source, and
the shadow of a whisker is detected as a drop in voltage at a
location along the array. However, this method is limited to
head-restrained animals and requires additional tagging when tracking
individual whiskers in an intact whisker field. Recordings of EMGs of
whisker pad muscles have been used in unrestrained rats ( Berg and
Kleinfeld 2003a, b; Carvell et al. 1991). EMG is a good predictor
of whisking timing, but cannot provide information about the exact
trajectory of the whisker in space or its elastic behavior. Thus
methods available for accurate measurements of whisker movements
are either too labor intensive or too restrictive to be of
general use in freely moving paradigms. Furthermore, none of these
methods allow efficient tracking of the entire whisker length and
may thus potentially miss important kinematic parameters.
For accurate measurement of whisker movements, we chose high-speed video, since it allows a large area to be covered simultaneously at high spatial (>5 pixels/mm) and temporal (>500 frames/s) resolution. High spatial resolution allows accurate determination of whisker shape and location. High temporal resolution allows visible changes in whisker shape and location to be correlated with neuronal events. Furthermore, correlation between successive images will be high since head and whiskers will not move much from frame to frame. Here, we take advantage of this fact to track the movements of the whiskers. Although high-speed video is easy to use, it generates enormous amounts of data (typically hundreds of megabytes for a few seconds of raw, uncompressed bitmaps) and therefore requires efficient tools for analysis.
We present an improved videographic technique for measuring
whisker movements, efficient and general enough to be used in
most freely moving paradigms involving rodents (rats, mice,
hamsters) that require accurate determination of whisker movements.
Using high-speed video, we acquired movies of rodents whisking
at high spatial and temporal resolution. With relatively simple
and fast image processing techniques, we completely automated
tracking of head movements and partially automated tracking of
individual whiskers without applying markers of any sort to the head
or whiskers. Since we tracked the entire length of whiskers, we were
able to approximate the true angle of a whisker at its base, its
curvature, and the distance of any point along the whisker shaft from
fixed environmental features at 1- to 2-ms resolution. This method
and some results have been presented in abstract form ( Knutsen et
al. 2004).
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METHODS |
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| TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Our method for tracking whisker and head movements was applied to
video movies of freely moving rats, mice restrained by the means of a
nylon bag, and videos of artificially whisking anesthetized rats.
Three adult male albino rats were trained in a freely moving object
localization task (see Fig. 1 for
experimental setup). Briefly, water-deprived rats were trained to
poke their head through a hole in the wall of an enclosure and to
approach a nosepoke about 8 cm from the wall. Two vertical objects
were positioned 23 cm from either side of the face at randomly
selected anterior-posterior locations. Rats were rewarded with
fruit juice for correctly judging the relative locations of the
objects. Although task specifics and performance are not reported on
here, whisking was present in both naïve and successfully trained
rats. One rat, in addition to the three above, was trained (with
nosepoke and objects removed) to poke its head through the hole in
the wall and wait for a water reward presented through a sipper
approaching from the left or right side. This task typically evoked
more exploratory whisking type behavior (data generated for Fig. 5) than
did the localization task (data generated for Figs. 610). One
mouse was restrained in a nylon bag, allowing it to move its head
only. The mouse was placed inside a small plastic tube for added
support and placed beneath the camera at an 
30° upright angle. Whisker
movements occurred spontaneously or were elicited by occasional
tactile stimulation of the face or whiskers. In addition, whisker
movements were acquired of a urethane anesthetized rat induced
to move its whiskers by electrical stimulation of the facial
motor nerve (see Szwed et al. 2003). Animal maintenance
and experimental procedures were conducted in accordance with
National Institutes of Health and Institute guidelines.
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The imaging system consisted of an integrated high-speed video
system (MotionScope PCI, Redlake, San Diego, CA) and personal
computer (Pentium 4, 2 GHz). Whisker movements of behaving rats
and mice were captured at 500 frames/s at 320 x 280 pixels, and those of anesthetized
rats ( Szwed et al. 2003) at 1,000
frames/s at 320 x 156 pixels. The camera
was equipped with a high-speed lens (DO-1795, Navitar, Rochester,
NY). The MotionScope system is by standard fitted with an IR blocking
filter. This filter was removed, yielding a sensitivity of 3 V/΅J/cm2 at the
peak illumination wavelength (940 nm) of the LEDs. Exposure time at
500 fps was limited one-half or one-third duty-cycle, corresponding
to 1 or 0.665 ms exposure per frame, respectively. The lens had a
focal length of 17 mm, with a focal ratio between f/3 and f/5. The
camera was placed at 0.5-m viewing distance and had a depth of field
suitable for tracking whiskers of 10 cm.
Whisker visibility was enhanced by illuminating from below and
placing the camera above the head of the animal (see Fig. 1).
With anesthetized rats, we used a fiber optic light source emitting
visible light (FL6000, Micro-Lite, Three Rivers, MA) and a
fiber-optic backlight (PANELite, Schott Fostec, Auburn, NY) covered
with one layer of diffusing glass (F02149
[GenBank]
, Edmund Industrial Optics, Barrington, NJ). For
the behavioral task experiments, we used a custom-made 10 x 10 array of infrared (940 nm) light-emitting
diodes (L940-04AU, Epitex, Kyoto, Japan), each outputting 20 mW/sr, covered with two
layers of diffusing glass.
Synchronization with neurophysiological data was achieved in several ways. Run-time commands to the camera were logged with timestamps on the controlling PC. Second, two black-and-white squares in the top left corner of the video frame were toggled during recording to mark specific events, such as the interruption of an infrared beam, which indicated entry of the rat into the frame, and a signal from a touch sensor, indicating contact with the nosepoke. Third, the signal controlling the camera shutter itself was digitized in real-time.
Implementation of tracking algorithms
Tracking of head and whisker movements was implemented by code written in MATLAB (v6.5, MathWorks, Natick, MA) and the C programming language. The algorithms are described below and supplied as supplementary material.
Tracking of head movements
Head movements were tracked by following the eye reflections made by two overhead infrared (880 nm) LED spotlights (F54845, Edmund Industrial Optics) 1015 cm above the rats (Figs. 1 and 3). The reflections in both eyes were manually located in the first frame of every movie. In the next and subsequent frames, a small region (usually 2 x 2 mm) centered on the location of the eye in the previous frame was smoothed with a low-pass filter constructed from the difference of two Gaussians. Since movement of each eye from frame to frame was small, the location of the peak luminance indicated the new position of the eye. This procedure was repeated for all frames of the same trial until either eye disappeared out of the camera's view. If the eye reflection disappeared out of sight momentarily for a few frames (e.g., during an eye-blink or when the head was tilting), this procedure generally managed to relocate the eye. When this was not the case, manual intervention was required. Location of the nose was estimated as the vertex of an isosceles triangle with the base between the eyes. The base-nose angle was kept straight, and its distance constant, for all frames in each movie.
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Movements of individual whiskers were tracked frame-by-frame as follows: 1) stationary features were removed, 2) a region of interest parallel to the face was extracted and rotated, 3) the image was filtered according to the anticipated whisker position, 4) the region of interest was filtered according to the local angle of the whisker in the previous frame, and 5) piecewise polynomials (splines) were fitted to whisker-like features in the transformed image. Briefly, a whisker was located manually in the first frame of every trial by selecting three or more points along the shaft and interpolating the shape of the whisker with a piecewise polynomial function. The algorithm determined the shape of the whisker in subsequent frames by moving these points to new locations (within a predetermined range) and interpolating a new spline for all their possible permutations. The interpolated spline overlaying the largest cumulative sum of pixel values was selected as the new location and shape of the whisker. The various steps involved in the algorithm are shown in Fig. 2. Pseudo-code outlining the algorithm is available as supplementary material1 .
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A rectangular region adjacent and parallel to a line intersecting the ipsilateral eye and nose was extracted and rotated. The cutout was rotated such that the axis parallel to the whisker pad was vertical. Thus anterior/posterior whisker movements are represented by up and down movements in the cutout. A rotated segment without and with the background removed from the image is depicted in Fig. 2, A and B, respectively.
New locations of each point along the whisker shaft were linearly extrapolated from the trajectory of locations in preceding frames. A Gaussian-profile filter (Fig. 2C), aligned along the expected location of the whisker, reduced the visibility of other nearby whiskers, while enhancing the whisker of interest (Fig. 2D). The width of the Gaussian was always broad (approximately one-half of the anterior-posterior width of the entire whisker field), such as to just introduce a slight bias in the estimation of whisker location.
Discriminability of the whisker was enhanced by convolving the image in the radial direction with a set of oriented edge-filters co-aligned to the local angle of the whisker in the preceding frame. The local angle of the whisker was computed for each 1-mm segment of the whisker shaft. The cross-section of the filter was a vector of ones in the center, flanked by zeros on both sides (Fig. 2E). The width of the ON center should be approximately the same as the typical width of whiskers in the image. In general, it is sufficient to configure the imaging system such that the thinnest whisker of interest can be subjectively separated from the background. In our experience, a whisker width of one or two pixels is sufficient for reliable tracking with a static background. The edge filter was 12 mm long and convolved with the image in 1 x 1-mm blocks along the dimension of the whisker shaft. This operation attenuated other whiskers in the image with local orientations differing from that of the whisker of interest.
After applying the filters above, whiskers appear bright against a
dark background. Thus the cumulative sum of pixel values of a spline
that best overlaps the whisker shaft would be higher than those of
partially overlapping splines. Coordinates that defined the
interpolated spline in the preceding frame were repositioned within a
predetermined range, and a set of new splines was interpolated from
all possible permutations of new coordinates. The spline with the
largest cumulative sum of pixel values was selected as the new
location of the whisker. Points were repositioned only to whole
pixels and within a range that corresponded to the maximal
anticipated frame-to-frame whisk displacement. Assuming whisker
lengths upward to 7 cm ( Brecht et al. 1997) and a maximal whisker
displacement upwards to 3,000 deg/s (Fig. 6), the
maximal anticipated velocity of the distal tip of the whisker is 3.5 mm/ms. Typically, however,
whiskers are shorter and movements are slower, thus bringing down
the anticipated range of whisker displacement and velocity.
Since the algorithm needs to evaluate fewer possible whisker
locations when the range is reduced, working with the smallest
possible range effectively makes the algorithm run faster. The
algorithm works with a separate range for each of the spline points,
since the speed of the tip is higher than that of the base of
the whisker. We solved this in part by adding to the individual
range of each spline point the sum of ranges of all proximal
points. For example, if the first, second, and third spline
points were assigned ranges of one, two, and three pixels,
respectively, the net range (and thus its maximal displacement) of
the third point was 1 + 2 + 3 = 6 pixels (see Fig.
2F). These ranges are typically hand-tuned once per movie,
although parameters are generally transferable to similar cases such
as movies of the same rat across different trials, assuming the
position of the camera has not changed significantly. The
spline-fitting step can accept splines with more than three
interpolation points, although this reduces the speed of the
algorithm (see pseudo-code in supplementary material).
Data analysis
Whisker angle was computed at base (the most proximal point of the
tracked whisker) from the coefficients of the most proximal
polynomial of the spline representation of the whisker
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(1) |
; Hartmann et al.
2003) but differs
from that of Bermejo et al. (1998) who computed whisker
angle between a fixed reference point and a location about 1 cm out
on the whisker shaft. We maintain that this difference is crucial,
since the pivot point of the whisker moves during whisking ( Berg and
Kleinfeld 2003a), and thus
assuming a fixed reference would distort the angle between the
whisker and the face. Furthermore, whiskers often continue to
protract when obstacles are encountered. If whisker angle is
calculated on the flexible portion of the whisker, the angle may
appear to decrease (due to bending), whereas the angle of the whisker
at base is actually increasing.
A full-length spline representation of whiskers allows direct
measurement of the bending of the whisker both during whisking
and on object touch. We computed bending (curvature) as the
inverse of the maximal radius of the whisker. Whisker curvature
can also be estimated at its base from the coefficients of the
first polynomial of the whisker spline
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(2) |
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RESULTS |
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We applied our method to tracking of whisker movements in unrestrained
rats, restrained mice, and artificially whisking anesthetized
rats (Fig. 3). In
unrestrained animals, head movements were also tracked so that the
effect of head rotation and translation on whisker movements could be
isolated. This allowed us to express whisker location both in a
head-centered and world-centered frame of reference. We tracked head
movements by following the reflections of two overhead spotlights in
the eyes of the animal. Such tracking was completely automated and
allowed accurate localization of both eyes. Tracked head movement and
orientation in one trial is depicted in Fig. 4. In
anesthetized rats, where whisking can be induced by stimulation of
the facial motor nerve ( Szwed et al. 2003), tracking of whisker
movements does not require tracking of the head as the head is
immobilized.
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. Furthermore, a clear
correlation was seen between whisk phase and curvature of the
whisker. Curvature was most positive (convex anterior) at whisk onset
and most negative (concave anterior) during retraction (Fig. 5, B
and D). This observation is clearly seen within single
whisks (Fig. 5,
B and C), as well as in the average trace of
curvature in all whisks (Fig.
5D). A movie demonstration of simultaneous tracking of
head movements and four whiskers is provided as supplementary
material.
Performance of tracking algorithm
The performance of tracking was dependent on illumination conditions and spatial resolution. Optimal visibility of the large whiskers (macrovibrissae) during rapid movement was achieved using strong and evenly distributed backlight illumination to enhance whisker-background contrast and an appropriate choice of a bright, high-speed lens. Tracking of individual whiskers within an intact whisker array was semiautomatic, requiring occasional intervention by an operator. The frequency of interventions depended mainly on video quality and the discriminability of individual whiskers. Manual intervention was only required if whiskers completely overlapped when video quality was optimal. Automated tracking was resumed after adjustments to compensate for errors were made manually. Most errors, however, were small, restricted to a few frames and were corrected only after tracking of the entire movie was complete.
In addition to testing our algorithm on whiskers within an intact whisker field, we also tracked whisker movements after clipping all but a single whisker (C2) an arc of whiskers (arc 2) or a row of whiskers (row C). Under such simplified conditions, tracking was almost completely automatic. Of the few manual interventions required, the most typical was to change the maximal range the spline was allowed to traverse from frame to frame (see METHODS). During large-amplitude, high-velocity whisking, this range typically had to be increased since the range was deliberately kept as low as possible for the algorithm to run as fast as possible.
We found that the method described here significantly increases
the efficiency of tracking planar movements of head and whiskers.
The method is optimal when high temporal resolution is called
for, as in electrophysiological experiments and detailed studies
of whisker kinematics, where a large number of frames need to
be tracked. The efficiency of the method compared with manual
data collection is easily shown numerically. Tracking a single
whisker for 1 s at 1,000 fps involves collecting a total of
3,000 data points (with 3 spline points). Assuming each data
point takes 1 s to acquire manually, 50 min are needed to track
1 s of movement of a single whisker. In comparison, Table 1
shows the typical durations involved in tracking with the
semiautomatic method. Depending on the experimental condition (e.g.,
how many whiskers are left intact) and the number of whiskers that
can be tracked simultaneously, the method improves the
efficiency of data point acquisition by 
20 times (Table 1).
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The high yield achieved with our high-speed video and simple image
processing analysis, as well as the reproducibility of the method, is
shown in Fig.
6. After tracking the head and whisker movements of three rats,
we identified 2,521 individual whisks with amplitudes >5° of angle
collected from 800 experimental trials (14 s each). The correlation
between whisk amplitude and peak velocity, particularly during
retraction, was high, as previously reported by others ( Bermejo et
al. 1998; Carvell and Simons
1990).
Whisker oscillations
Recent work by Hartmann et al. (2003) has suggested that
resonant oscillations of whiskers may be important during behavior.
Our method of tracking enables tracking of both first- and
higher-order oscillations with one or more radii. Figure 7 shows
tracking of the C2 whisker with a single radius (3 spline
interpolation points) as it hits and passes a vertical bar at three
separate instances. The whisker oscillates for two to three cycles
at 66 Hz.
Higher-order vibrations can also be detected using splines with two
or more radii.
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The method we describe accurately estimates the position and shape
of the entire visible portion of the whisker shaft. In contrast,
tracking of a single point along the whisker shaft yields no
information on whisker shape and probably inaccurate information on
important kinematic parameters, such as whisker angle at its base
(base angle). Estimation of base angle from a single tracked point on
the whisker assumes a static, nonmoving reference point (or pivot)
and rigidity of the whisker shaft. That the shape of the whisker
changes during free-air whisking is shown in Fig. 5.
Actions of the muscles that drive whisking may also move the pivot
point ( Berg and Kleinfeld 2003a).
To estimate the errors produced by tracking a single point along
the whisker, we compared whisker angle computed using the full
whisker-representation (Eq. 1;
Fig.
8A) with an angle computed between a fixed reference and a
particular radial location on the whisker shaft (Fig.
8B). The first estimate, which we will refer to as the
moving-reference estimate, is influenced both by movements of the
whisker shaft base and by whisker bending. For the fixed-reference
estimate, we used as fixed reference the intersection of two lines
drawn through the base of the whisker at onset of protraction and at
peak protraction; the distal end of the line was varied between 10
and 30 mm from base. The two estimates of angle calculation were
compared separately for whisks in free-air and those that brought the
whisker in contact with an object between protraction onset and
half-way to maximal protraction (0 to 
/2). The averaged whisk trajectories
of angle and angular velocity are depicted in Fig. 9. For
nontouch whisks, the moving-reference estimate of whisk amplitude
(maximum angle -minimum angle; Fig. 9, A
and B) was significantly different from the
fixed-reference estimate only when the latter was computed 30 mm out
on the whisker shaft (z = 11.0518, P < 0.001; Wilcoxon
rank-sum test). However, estimates of whisk amplitudes during
touch were significantly different even when the fixed-reference
estimate was computed just 10 mm from the whisker base (z =
6.59, P < 0.001). Estimations of angular velocity (Fig. 9, C
and D) were more sensitive to the calculation method
used and differed significantly even at 10 mm from base both
during nontouch (z = 12.50, P < 0.001) and touch
(z = 13.12, P < 0.001) whisks. Thus when an angle is
calculated between a particular location on the whisker shaft and a
fixed reference point, the estimate can deviate significantly from a
direct measure of the whisker angle at a moving base, both during
nontouch whisking and touch whisking (see absolute magnitudes of
deviations in Fig. 10).
Deviations in angle and angular velocity estimates are larger during
touch than nontouch, but can also be significant during nontouching
whisking.
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DISCUSSION |
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Here, we describe how head and whisker movements can be tracked
semiautomatically in both immobilized and unrestrained rodents,
using high-speed video and image processing, at high spatial
(>5 pixels/mm) and temporal (1251,000 frames/s) resolution.
The method allows tracking under different experimental conditions
and illumination settings (Fig. 3). When
spatial resolution is such that individual whiskers can be resolved,
multiple whiskers can be tracked separately in the same movie. Thus
the entire whisker array can be left intact throughout an experiment.
Manual intervention is occasionally called for when tracking
individual whiskers in a crowd of whiskers or when a whisker of
interest overlaps completely with other whiskers. With good spatial
resolution, however, or under simplified conditions (e.g., by
trimming whiskers not of interest) most tracking is automatic. Our
algorithm achieves accurate and fast determination of the updated
whisker position by matching a limited number of splines to
whisker-like features in the image. The method significantly
increases the efficiency of data collection compared with previously
described methods. When we applied our method to freely moving rats
engaged in a tactile discrimination task, as well as an unstructured
task, we achieved high yield of data with kinematic properties
similar to those observed by others (e.g., Bermejo et al. 1998; Carvell and
Simons 1980).
Whereas other reported methods track a single parameter (such as
location of a particular point on the whisker shaft or time-varying
EMG of muscles that move the whiskers), our method tracks both
the shape and location of a whisker over time. We found significant
differences between whisker angles measured directly at the
whisker base from its piecewise polynomial representation and
those measured between a fixed reference and a particular location
on the whisker shaft, both when whiskers moved freely or against
obstacles (Figs. 9 and 10). Previous
studies have indicated that bending affects whisker angle at base and
is important during exploratory behavior ( Carvell and Simons 1990; Hartmann et
al. 2003; Krupa et al.
2001). It is
therefore necessary, as we show here quantitatively, to track both
the base of the whisker as well as its full length. The importance of
the latter point has long been recognized, and we introduce here a
flexible and efficient method of tracking whiskers that provides
such information, previously unavailable using other automated
techniques.
Full-length whisker tracking also allows characterization of the mechanical effect of obstacles on the whisker shaft, such as whisker bending and small oscillations (Fig. 7), probably necessary for extrapolating receptor responses within the follicle. Locations of whiskers, relative to objects in the environment, is readily obtained by coordinate transformations between relative (head-centered) and absolute (world-centered) coordinates. Thus environmental features can be marked and moments of touch automatically detected and cross-referenced with neuronal signals. Since results of full-length tracking are unambiguous, this method can efficiently and accurately characterize kinematic parameters of whisking, especially when whiskers come into contact with stationary or moving objects and textures.
Tracking head and whisker movements with a single viewpoint
precludes a three-dimensional description of head and whisker
orientation. Thus, the top-down viewpoint used here does not
capture vertical excursions of the whisker trajectories ( Bermejo
et al. 2002). Our observation that
curvature changes throughout protraction may be due to such
excursions, such as may be caused by rotation of the follicle.
Furthermore, anterior-posterior movements can only be estimated
correctly when it is assumed that the head neither has pitch nor
roll. Our whisker tracking method does not assume a particular
viewpoint, and a two-viewpoint solution (using extra cameras or
mirrors) may therefore in principle allow a more accurate description
of whisker movements and curvature changes in all three
dimensions.
Our tracking method requires video movies of good quality and
assumes that whisker displacement is small from frame to frame.
The second criterion is achieved using a fast video acquisition
system; we observed maximal efficiency with frame rates >100
fps. As fast video integrates over very short periods of time,
a strong source of illumination is required. We used an array
of infrared light-emitting diodes or a fiber-optic light source,
combined with a bright lens, to achieve strong lighting without
excessive heating. We also found that backlighting gave better
results than illumination of the whiskers from the top or sides.
With our imaging system, we achieved stable tracking even when
the diameter of a whisker (140 ΅m for the C2 whisker; Neimark et al. 2003) was less than the width
of a pixel (250 ΅m at
the horizontal plane of the whisker).
We chose high-speed video to track whisker movements because it
does not require additional invasive procedures, and the underlying
technology is scalable as the resolution of high-speed cameras
continues to improve. Our method of automated analysis of high-speed
video allows effective, direct, and accurate measurements of head and
whisker kinematics. Furthermore, the method readily allows the
experimenter to identify and characterize whisker interactions with
environmental features at a resolution sufficient for analyzing their
correlation with fast neuronal signaling. Thus the method is well
suited to investigate mechanisms related to active sensing of the
whisking sensorimotor loop ( Ahissar and Kleinfeld 2003), and somatosensory
processes in general.
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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1 The Supplementary Material for this
article (a movie and text) is available online at http://jn.physiology.org/cgi/content/full/00718.2004/DC1.
Address for reprint requests and other correspondence: P. M. Knutsen, Dept. of Neurobiology, The Weizmann Inst. of Science, 76100 Rehovot, Israel (E-mail: per.knutsen{at}weizmann.ac.il )
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) was estimated as the tangent of the curvature at base
(see 