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The Journal of Neuroscience, February 1, 1999, 19(3):1122-1141
Computation of Object Approach by a Wide-Field,
Motion-Sensitive Neuron
Fabrizio
Gabbiani,
Holger G.
Krapp, and
Gilles
Laurent
Computation and Neural Systems Program, Division of Biology,
California Institute of Technology, Pasadena, California 91125
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ABSTRACT |
The lobula giant motion detector (LGMD) in the locust visual system
is a wide-field, motion-sensitive neuron that responds vigorously to
objects approaching the animal on a collision course. We investigated
the computation performed by LGMD when it responds to approaching
objects by recording the activity of its postsynaptic target, the
descending contralateral motion detector (DCMD). In each animal, peak
DCMD activity occurred a fixed delay  (15  35 msec)
after the approaching object had reached a specific angular threshold
 thres on the retina (15° thres 40°). thres was independent of the size or velocity of
the approaching object. This angular threshold computation was quite
accurate: the error of LGMD and DCMD in estimating thres
(3.1-11.9°) corresponds to the angular separation between two and
six ommatidia at each edge of the expanding object on the locust
retina. It was also resistant to large amplitude changes in background
luminosity, contrast, and body temperature. Using several
experimentally derived assumptions, the firing rate of LGMD and DCMD
could be shown to depend on the product  (t  )
· e   (t ), where (t)
is the angular size subtended by the object during approach,
(t) is the angular edge velocity of the object and the
constant, and  is related to the angular threshold size [ = 1/tan(thres/2)]. Because LGMD appears to receive
distinct input projections, respectively motion- and size-sensitive,
this result suggests that a multiplication operation is implemented by
LGMD. Thus, LGMD might be an ideal model to investigate the biophysical implementation of a multiplication operation by single neurons.
Key words:
looming; multiplication; locust; LGMD; DCMD; lobula; collision-avoidance
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INTRODUCTION |
The processing of sensory
information by neural circuits is known to depend critically on the
implementation of nonlinear operations. Multiplication, for example, is
thought to be the elementary building block underlying the detection of
visual motion in insects (Reichardt, 1987 ; Borst and Egelhaaf, 1989) or
the generation of gain fields in posterior parietal neurons of the primate neocortex (Andersen et al., 1985). Despite years of efforts, the precise biophysical and network mechanisms underlying these nonlinear operations remain elusive (Koch and Poggio, 1992). Here, we
study two identified locust visual neurons, which may prove well suited
to investigate the biophysical implementation of a multiplication operation.
The lobula giant motion detector (LGMD) belongs to a class of neurons
sometimes called "jittery movement detectors" (Glantz, 1974;
Frantsevich and Mokrushov, 1977, for review, see Wehner, 1981). Its
dendritic arborizations ramify in the third neuropil (lobula) of the
locust optic lobes and consist of three dendritic subfields (O'Shea
and Williams, 1974). The main subfield is thought to receive in part an
excitatory retinotopic projection, which is sensitive to motion,
whereas the remaining two dendritic subfields receive massive
feed-forward inhibitory inputs, which are size-dependent (Palka, 1967;
Rowell et al., 1977). In contrast to directionally selective neurons
involved in optomotor response behaviors and gaze stabilization, such
as those extensively studied in the fly visual system (Hausen and
Egelhaaf, 1989), LGMD is inhibited by whole field motion (Pinter, 1977;
Zaretsky and Rowell, 1979). It responds to movement of small objects,
irrespective of their location in its receptive field, in a
nondirectionally selective way. It is also vigourously excited by
objects approaching on a collision course with the animal (Schlotterer,
1977; Rind and Simmons, 1992). In the locust brain, both left and right
LGMD neurons each synapse onto one descending contralateral motion detector (DCMD) neuron, whose fast-conducting axon, the largest in the
contralateral nerve cord, projects to thoracic motor centers involved
in the generation of jump and flight maneuvers (Burrows and Rowell,
1973; Pearson et al., 1980; Simmons, 1980; Robertson and Pearson,
1983). The connection between LGMD and DCMD is very strong and reliable
such that each action potential in LGMD elicits an action potential in
DCMD. Conversely, under visual stimulation, each action potential in
DCMD is caused by an action potential in LGMD (O'Shea and Williams,
1974; Rind, 1984). These anatomical and physiological properties
suggest that LGMD and DCMD are part of an early warning system aimed at
eliciting escape or avoidance behaviors in face of an imminent danger.
Although the preferential response of LGMD and DCMD to looming objects
on a collision course with the animal has been recognized for some time
(Schlotterer, 1977; Rind and Simmons, 1992), the neural computation
performed by LGMD during approach remained unclear. The experiments and
theoretical analysis presented here were aimed at understanding this
computation and the algorithm used to perform it (Marr, 1982). A
part of our results has been published recently (Hatsopoulos et al.,
1995) but has been criticized on the basis of the low refresh rate of
the monitor used in those experiments (Rind and Bramwell, 1996, note
added in proof; Judge and Rind, 1997; Rind and Simmons, 1997). In
addition to confirming our original findings, the present work shows
that the peak in DCMD activity depends solely on the retinal image size
of the approaching object. It also provides a better theoretical
foundation for our results by deriving from a few plausible assumptions
a model describing the firing rate of LGMD and DCMD in response to
approaching objects. This model generalizes that of Hatsopoulos et al.
(1995); its free parameters are determined experimentally.
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MATERIALS AND METHODS |
Preparation. Experiments were performed on adult
locusts (mostly female) taken from the laboratory colony 3-4 weeks
after their final molt. Locusts were mounted dorsal side-up on a
plastic holder, which was then fixed vertically to a clamp located
directly under a dissection microscope. The head of the locust was
placed between the jaws of an alligator clip mounted on a
micromanipulator and was carefully aligned with reference points marked
on a reticular grid inserted in one of the microscope eyepieces. After
alignment, the head was fixed in place with a few drops of beeswax.
This, together with the calibration procedure described below, allowed us to reliably align the center of the locust's right eye with the
center of our stimulation screen. A small piece of rigid plastic paper
(transparency film; Eastman Kodak, Rochester, NY) cut to fit the
dimensions of the outer edge of the locust's eye was glued in place as
close as possible to the anterior rim of the right eye with fast epoxy
glue. A waterproof wax cup terminating on this plastic sheet was then
built around the locust's head. In some experiments no plastic sheet
was used, and the wax cup was built directly up to the rim of the eye.
The entire dorsal half of the head was bathed in locust saline (Laurent
and Davidowitz, 1994), except for the right eye, which remained outside
the cup formed by the plastic sheet and the wax, with an unobstructed field of view of at least 100° measured from the center of the eye.
The brain (supraoesophageal ganglion) and the optic lobes were exposed
by opening a rectangular window in the frontal head cuticle. The gut
was removed to minimize coupling of abdominal respiratory movements to
the brain and to expose the connectives. The suboesophageal ganglion
was grabbed with a pair of fine forceps, and both connectives were
sectioned as close as possible to the suboesophageal ganglion. In a few
of the experiments reported here, locusts were also prepared for
simultaneous intracellular recordings from the dendrites or axon of
LGMD by desheating the right optic lobe with fine forceps after
softening the protective sheet surrounding the brain with protease
(XIV; Sigma, St. Louis, MO). The results of these experiments will be
reported elsewhere.
Electrophysiology and data acquisition. The locust was fixed
to a clamp with its longitudinal body axis parallel to the stimulation screen, and the cut end of the proximal, left connective (contralateral to the stimulated eye) was placed in a suction electrode. Extracellular signals were amplified with a differential AC amplifier (A-M Systems, Everett, WA) and recorded using a digital tape recorder (Micro Data
Instruments, Woodhaven, NY; sampling rate, 11.5 kHz) together with the
transistor-transistor logic (TTL) synchronization pulses generated by
the computer controlling the stimulation screen. DCMD typically
produced the largest action potentials in the nerve cord and was easily
identified from its characteristic responses to small objects moving in
the visual field of the animal or to looming stimuli (see Fig. 2). The
preparation was sometimes allowed to recover from the dissection for
15-30 min before starting an experiment, and stable recordings from
DCMD could be maintained up to 4 hr, depending on the robustness of the
animal and the quality of the dissection.
Stimulus generation. Visual stimuli were generated on a
monochrome monitor coated with an ultrafast P46 phosphor (10% decay time, 1 µsec; Vision Research Graphics, Durham, NH) emitting in the
green-yellow range (normalized intensity reaching >95% between 520 and 565 nm and >50% between 505 and 610 nm), close to the peak in the
absorption spectrum of locust photoreceptors (Lillywhite, 1978; Osorio,
1986a). The monitor image was refreshed at a rate of 200 Hz, well above
the temporal cutoff frequency of locust photoreceptors (<100 Hz;
Miall, 1978; Howard, 1981; Howard et al., 1984). The monitor was placed
at a distance of 120 mm from the center of the locust's eye. Alignment
of the center of the locust's eye with the center of the monitor and
distance adjustment were achieved by (1) leveling horizontally and
vertically the monitor and the air table supporting the clamp for the
preparation, and (2) placing a locust with the head fixed to its holder
(see Preparation above) in the clamp and adjusting the distance and eye
position with respect to the monitor with the help of an optical bench
consisting of two 10 mW helium-neon lasers and two reflection mirrors
(Uniphase, Manteca, CA).
The dimensions of the image were 300 × 209 mm (736 × 500 pixels) corresponding to a spatial resolution of 5 pixel/° at the center of the locust's eye [spatial resolution of the locust
ommatidial array: interommatidial angle, 1.25°, (Horridge, 1978);
photoreceptor acceptance angle, 1.5° in light- and 2.5° in
dark-adapted animals (Wilson, 1975)]. An unobstructed field of view
from the locust's eye to the monitor was achieved by opening a
rectangular window in the Faraday cage surrounding the preparation. To
minimize the electromagnetic noise generated by the monitor, a glass
shield coated with indium tin oxide was inserted between the monitor and the preparation (front surface reflectance <0.5% in the visible range; Thin Film Devices, Anaheim, CA) and connected to the chassis ground of the instrumental rack. The luminance of the screen was calibrated linearly between 0 cd/m2 and a maximal
luminance Imax of 95 cd/m2,
as measured at a distance of 46 cm from the center of the screen with a
photometer (PR-504 and PR-502; Photometer Research, Chatsworth, CA).
The stimulation screen was controlled by a personal computer equipped
with a high-resolution graphic controller (Cambridge Research Systems,
Rochester, England). Stimulation programs were written in C using the
VisionWorks library of function calls (Vision Research Graphics,
Durham, NH).
Kinematics of object approach. The stimuli used simulated
dark squares of various sizes approaching with constant velocity and on
a collision course with the animal (Fig.
1A). The response of
LGMD and DCMD to bright objects on a dark background is qualitatively similar to the responses of dark objects on a bright background (Rind
and Simmons, 1992, their Fig. 9). Because they were presented monocularly, the time course of the angular size subtended by the
objects on the locust's retina is the variable characterizing the
approach. Let x denote the position of the object with
respect to the eye of the animal; i.e., x > 0 before
collision, and x = 0 at collision. If we define
t = 0 as the time of expected collision and
t < 0 before collision, an approach at constant
velocity v is described by the equation:
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(1)
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where the velocity v is negative (reflecting the fact
that the object is approaching) according to these conventions. The angular size of the object at the retina is given by trigonometry as
twice the inverse tangent of l/vt:
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(2)
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where l denotes the object's half-size. The
half-size lscreen(t) of the simulated
object on the screen is then similarly determined from:
where xscreen-eye denotes the distance
between the screen and the eye (120 mm). Both the angular size,
(t), and angular edge velocity, (t), of the
object,
|
(3)
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are nonlinear functions of time (Fig. 1B). As
can be seen from Equations 2 and 3, both functions depend only on the
ratio l/|v| between the object's half-size,
l, and its approach speed, |v|. In other
words, an object of half-size l approaching with speed
|v| will generate the same visual stimulation on the
locust's retina as an object twice as large approaching twice as
fast.
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Figure 1.
Diagram of experiments and temporal
characteristics of the stimulus. A, The right eye (see
drawing of half locust head on right) was stimulated from the side by
presenting squares of half-size l approaching at a constant
velocity v toward the center of the eye at 90° relative to
the animal's body axis. Because the stimulus is monocular, the
variable of importance is the time course of the angular size of the
object subtended at the retina, (t). B, Time
course of (t) and of the angular velocity
(t) as a function of time before collision for a slow
velocity of approach (or a large object). Both functions are nonlinear
in time; see Equations 2 and 3. Final angular size at collision:
180°; i.e., the object covers the entire visual field.
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Because l/|v| is the relevant parameter for the stimuli
considered here, it is it that we will use in the following sections. For the range of object sizes (l = 6-14 cm) and speeds
(|v| = 2-10 m/sec) simulated in the present
experiments, l/|v| ranged from 5 msec (for small or fast
moving objects) up to 50 msec (for large or slow-moving objects; Table
1).
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Table 1.
Correspondence between the half-size of the object
(l), the speed of approach |v|, and the
parameter l/|v| characterizing the time
course of the angular size on the retina
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The following modifications have been introduced in the notation
introduced above compared with Hatsopoulos et al. (1995): (1) the angle
(t) defined in Equation 2 was previously denoted by
2 · (t); (2) the slope of the linear regression
line defined by Equation 5 (see Results) is now called instead of
/2 (compare with Eq. 2 of Hatsopoulos et al., 1995); and (3) the
angular velocity of edge motion (t) defined in Equation 3
corresponds to  (t) in Hatsopoulos et al. (1995).
These changes make the present notation (see Fig. 1A)
identical to the one used by Sun and Frost (1998).
Stimulation protocols. The Faraday cage surrounding the
experimental setup was covered with a sheet of black felt, and all experiments were performed in the dark, except for the brightly lit
stimulation screen. This allowed avoidance of any reflections of
experimental objects (manipulators, microscope, etc.) on the computer
screen and its glass shield. The luminance range to which the
stimulated eye was constantly exposed during the experiments (24-95
cd/m2; see below) covered the middle to upper range
of intensity values encountered in interior lightening situations.
Therefore, the eyes of our experimental animals were in a light-adapted state.
The first series of experiments consisted of presenting 10 repetitions
of looming squares approaching at various values of l/|v| (ranging from 5 to 50 msec in steps of 5 msec, 10 protocols) pseudo-randomly interleaved. This protocol was applied to
N = 6 animals. The initial size subtended by the
objects at the beginning of approach was <1° in visual angle
(n = 4 pixels), and the full final angle subtended at
the end of approach was always equal to 80°, independent of the value
of l/|v|. Thus, in each case, exactly the same portion
of the ommatidial array was stimulated, and only the time course of the
visual stimulation for each individual ommatidium differed as
l/|v| changed. The luminance of the object IO was equal to 0 cd/m2, and
the constant luminance of the background was set to
Imax = 95 cd/m2. To minimize
the effects of habituation (which can be pronounced in some animals;
Rowell, 1971b), the intertrial interval was 40 sec. An experiment (100 trials) therefore lasted ~1 h 15 min. This 40 sec interval was not
always sufficient to completely avoid some degree of habituation (e.g.,
Fig. 3) but represented an acceptable compromise between minimizing
such effects and keeping the experiment duration within reasonable
limits. In a variant of this series of experiments, we studied
more closely the range of validity of our results for small values of
l/|v| (corresponding to small objects or high speeds of
approach); the value of l/|v| was varied from 10/2 = 5 msec, 10/1.8 = 5.6 msec, ... , up to 10/0.2 = 50 msec (i.e., l/|v| = 10/y msec, with y decreasing
from 2 to 0.2 in steps of 0.2; 10 protocols × 10 repetitions = 100 trials; N = 9 animals).
In a second series of experiments, we studied the effect of monitor
refresh rate by pseudo-randomly changing the image refresh from its
default value (200 Hz) to 100 and 67 Hz. This was achieved by updating
the image only every second frame (i.e., every 10 msec) or third frame
(i.e., every 15 msec), respectively. Eight values of
l/|v| were tested (5, 7.5, 10, 12.5, 15, 25, 35, and 45 msec) with five repetitions for a total of 3 × 8 × 5 = 120 trials (N = 5 animals).
In a third series of experiments, we tested the effects of background
luminance and contrast on the time course of the LGMD and DCMD
response. We used the following protocols. The background luminance
IB = B · Imax was set to four
different values (B = 100, 75, 50, and 25%) of the
maximal luminance (Imax = 95 cd/m2). Object luminance IO = O · Imax was varied with values of O from O = B 0.25 to O = 0 (in
steps of 25%), giving a total of 10 different combinations of object
and background luminance. The corresponding contrast C = (IO IB)/IB of the
object's edges sweeping through the visual field of the animal thus
ranged between 1.00 and 0.25. The stimuli were interleaved
pseudo-randomly and presented at intervals of 40 sec. Between two
stimuli the blank screen was set to the background luminance value used
for the next protocol, thus allowing for 40 sec adaptation time to the
new background. For each combination of IB and
IO three values of l/|v| were
tested (10, 25, and 40 msec) with four repetitions (total:
10 × 3 × 4 = 120 trials; N = 6 animals).
In a fourth series of experiments, we studied the effect of body
temperature on the time-course of the DCMD response by heating the
animal with a small electric fan from room temperature (21-24°C) up
to 30-33°C. The temperature of the head capsule was monitored at
regular intervals with a calibrated thermocouple probe
(Sensortek, Clifton, NJ) placed in the saline bath. A heating or
cooling time of 20 min was allowed before starting an experiment. Five
values of l/|v| were used (10, 20, 30, 40, and 50 msec)
with eight repetitions. Both transitions from low to high and from high
to low temperature were studied (N = 8 animals).
Data analysis. The extracellular recordings and the TTL
synchronization pulses were acquired at a sampling rate of 10 kHz using
an analog-to-digital board (NB-MIO-16X; National Instruments, Austin,
TX) and transferred to a Unix workstation for further processing. Each
trace was examined separately, and the DCMD spike occurrence times were
extracted using two variable thresholds adjusted to select the unit
generating the largest positive and negative voltage deflections in the
extracellular recordings (Fig. 2). Each individual raster trial was
then smoothed with a 20 msec Gaussian window (Fig. 2) and an estimate
of the instantaneous firing rate was obtained by normalizing the
resulting waveform f(t) so that:
where n is the number of spikes emitted
during the trial. This procedure circumvents the artifacts caused by
temporal binning of the responses usually used to compute peristimulus
time histograms (Richmond et al., 1990). The peak of the firing rate
was localized in each individual trace, and the mean value and SD were
computed from the n (n = 4-10) repetitions of each
trial (Fig. 3). Similar results were obtained for smoothing windows of
15 and 10 msec width (Fig. 2), although the SD in the peak time
estimates was usually larger with the shorter smoothing windows.
Linear regressions and estimates for the slope and intercept parameters
and their confidence intervals (Fig. 4) were obtained as described by
Press et al. (1992, Chap 15).  2 values per degrees of
freedom (2/F) for linear fits such as the one
illustrated in Figure 4A ranged from 0.07 to 1.09 (mean ± SD, 0.07 ± 0.27; N = 15 neurons).
The assumption that the residuals of the linear fits were normally distributed was tested using a Kolmogorov-Smirnov test (K.-S. test;
Press et al., 1992). In all cases, the distribution of errors was
consistent with the Gaussian assumption (significance levels ranging
from p  0.31 to p 0.99; N = 15 neurons). Computation of the error in the estimate of the
angular threshold (see Fig. 6B) given by Equation 6
(see Results) was obtained by error propagation (Bevington and
Robinson, 1992, Sec 3.2), i.e., using the formula  f = |df/dx| · x for f = f(x).
The linear fit of the peak occurrence time SD,
 tpeak, as a function of
l/|v| (see Eq. 8 in Results) was compared to fits with
the following nonlinear functions:  (l/|v|)2,
(l/|v|)3, el/|v|, and
e l/|v|. Estimates for the goodness
of fit (2) and its SE were obtained by a bootstrap
analysis on the fit residuals (Efron and Tibshirani, 1993, Chap 9). In
all but one case analyzed, linear fits were better or as good as those
obtained using the nonlinear functions described above (N = 15 neurons).
For the 15 neurons tested in the first series of experiments, we
compared the experimental times of peak firing rate relative to
collision to the following model: for each value of l/|v|, |tpeak| is distributed normally with a mean value
|tpeak| = · l/|v| and an SD tpeak = 1/2(1 + 2) · thres · l/|v| (see
Results, Eqs. 5 and 7). For each experiment, the 100 peak times
collected were separated in two groups: 50 peak times (10 values of
l/|v| × 5 repetitions) were used to estimate the
parameters , , and thres (according to
Eqs. 5, 8, 9 in Results). The remaining 50 peak times were used to test
the model by computing the 2 value per degrees of
freedom (2/F) of the fit and the F
statistics:
In this equation, the mean peak times and their SDs are
determined by the model as:
where (l/|v|)i is the ith
value of l/|v|, i = 1, ... 10 (see description
of the first series of experiments above),
|tij| is the peak time of the jth
repetition (j = 1, ... 5) for the ith
stimulus presentation, and | i exp|
is the experimental mean peak time value: 1/5
 j=15|tij|. Under the hypothesis
that the linear model is correct, F follows the statistics
of an F(10,40) random variable (Lindgren, 1976;
Secs 7.1.5, 12.1.1, 12.2.4). Furthermore, the standardized residuals of
the fit are expected to follow a standard normal distribution. This
assumption was tested by applying a Kolmogorov-Smirnov test.
The static nonlinearity g(z) described in Appendix 3 and
Results was fitted to the following sigmoid function:
|
(4)
|
where x = log(z) is the natural
logarithm of the kinematic parameter z (see Eq. 12). The fit
procedure was as follows: fits were first performed for the part of
g corresponding to the rising phase of the firing rate (see
Figs. 12B, 13A) with a
constrained to zero. The parameter b characterizing the
asymptotic value of the firing rate for large values of z
was determined by fitting the static nonlinearity obtained for the
smallest l/|v| (=5 msec) and was fixed to that value
afterward. The slope and half-activation parameters c and
x1/2 (as well as b for
l/|v| = 5 msec) were obtained by iterating a nonlinear
least square algorithm. The part of g corresponding to the
falling phase of the firing rate was subsequently fitted with a,
c, and x1/2 as free parameters and the
constraint that it should match the value obtained for the rising phase
at the largest value of z. Representative values of these
parameters are given in Table 4 for four different experiments.
All the data analysis software was written using Matlab 5.0 and its
graphical interface (The MathWorks, Natick, MA). The data of one
experiment used in Figures 2-4, 7, and 12-14, as well as a Matlab M
file generating Figure 3, are available on the World Wide Web (http://www. klab.caltech.edu/~gabbiani/abstracts/Gabbiani_etal1998.html).
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RESULTS |
The results presented here are based on recordings and complete
analysis of the activity of 34 DCMD neurons in 34 different animals.
Peak response of DCMD and LGMD to simulated looming objects
The spontaneous activity of LGMD and DCMD was typically low (<1
Hz). LGMD and DCMD responded vigorously to the movement of small
objects presented anywhere in the visual field of the stimulated eye
and to simulated approach of looming objects on a collision course with
the animal, as reported previously (Rowell, 1971a; Schlotterer, 1977;
Pinter et al. 1982; Rind and Simmons, 1992). The response of LGMD and
DCMD to looming objects was quite characteristic: it started early
during the approach phase, usually before the object reached 10° in
visual angle (Fig. 2). The firing rate
gradually increased as the object grew larger, as if the cell were
"tracking" the object over the approach. The firing rate then
peaked and eventually decreased. On some occasions, the response was
bimodal, with a brief interruption of spiking during object approach
(Fig. 2, arrow), as reported by others (Schlotterer, 1977;
Pinter et al., 1982). During the course of an experiment (which usually lasted between 1 h 15 min and 1 h 30 min and in which trials
with different values of l/|v| were interleaved
pseudo-randomly; see Materials and Methods), the response to repeated
presentations of the same stimulus was quite variable from trial to
trial. In some of the neurons studied, part of this variability was
attributable to habituation of the response to the stimulus, as
illustrated in Figure 3, left
panels (see in particular the trials for l/|v| = 50, 45, and 35 msec). This habituation was reflected by the high SD in
the number of spikes elicited per trial, relative to the mean. Clear
effects of habituation could be seen by visual inspection of the raster
plots in 2 of 15 cells studied. Even when the SD in the number of
spikes elicited was low (as illustrated in Fig. 3, right
panels), repeated presentations were characterized by a
substantial jitter in the spike occurrence times.
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Figure 2.
Extracellular recording from DCMD in response to
object approach (l/|v| = 45 msec). Top panel,
Time course of size of the approaching object on the stimulation
screen, as monitored by TTL synchronization pulses. Bottom
panel, Extracellular recording from the contralateral connective,
showing DCMD as the largest unit. Note the short interruption in
spiking during approach (curved arrow). Thick,
thin solid lines, dashed line, Estimate of the
instantaneous firing frequency obtained by smoothing the response with
20, 15, and 10 msec Gaussian windows, respectively. *Peak firing rate.
Note that in this example, peak firing time is not determined with
great certainty because of the two local maxima separated by the short
interruption (arrow). For this experiment, l/|v| = 45 msec was the least favorable stimulus parameter to estimate the
peak, as can be seen from the error bars for the point l/|v| = 45 msec in Figure 4A. Left inset,
20 msec Gaussian window used to smooth the spike rasters (same
horizontal time scale as the extracellular trace, not drawn to scale in
the vertical direction). spk, Spike.
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Figure 3.
Full data set for a single DCMD experiment. Each
panel shows, for a given value of l/|v|, 10 spike
rasters (bottom) obtained in response to the stimulus shown
on top. The smooth trace in the middle
is the estimated instantaneous firing rate (averaged over 10 trials,
mean ± SD; solid and dotted lines,
respectively). The number, n, of elicited spikes per trial
(mean ± SD) is given on the left. The trace in Figure
2 corresponds to the second raster of the panel l/|v| = 45 msec. Dashed lines, Mean time of the peak firing
rate; f.f., firing frequency.
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To study the time-course of the firing rate of DCMD and LGMD during
object approach, each individual spike raster was convolved with a 20 msec Gaussian window to obtain an estimate of the instantaneous firing
frequency and its SD (Figs. 2, 3, solid and dotted
lines; also see Materials and Methods). Visual inspection of the
instantaneous firing rate as a function of l/|v|
revealed that the peak in DCMD firing rate consistently shifted toward
collision as l/|v| decreased (Fig. 3, read from
top to bottom and from left to
right) to eventually occur at, or even after, predicted
collision for small values of l/|v| (corresponding to
small and/or fast-moving objects). This observation could also be made
directly from inspection of the instantaneous frequency plots computed
from single trials (see Fig. 2) and was independent of the variability
described in the previous paragraph.
A plot of the time of peak firing, |tpeak|,
versus l/|v| revealed that these two variables were
linearly related (Fig.
4A): correlation
coefficients between |tpeak| and
l/|v| ranged from 0.98 to 1.0 (mean ± SD,
0.99 ± 0.01; N = 15 neurons). Similar results
were observed for all 34 neurons building our database. We
therefore performed, for each experiment, a linear regression of the
peak time versus l/|v|,
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(5)
|
and estimated the slope (called ) and the intercept (called
) characterizing these linear fits (Fig. 4A,
inset) as well as the errors in the estimates of and . Note
that although is dimensionless, is in units of time. This
analysis revealed that the estimates of and were not
independent of each other, as might be expected a priori,
but rather were tightly correlated (range of observed correlation
coefficients, 0.73-0.96; mean ± SD, 0.85 ± 0.09;
N = 15 neurons). In other words, a higher (lower) value
of is more likely to coincide with a higher (lower) value of ,
respectively. This is illustrated in Figure 4B, where
the 68.3% confidence region for the values of and are
depicted. The tilted ellipsoidal shape of the confidence region
reflects the correlation between the estimates of the two variables
(uncorrelated estimates would yield ellipses with principal axes
parallel to the x- and y-coordinate axes).
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Figure 4.
Relation between the time of peak firing rate and
l/|v| is linear. A, Plot of the time of peak
firing rate, |tpeak|, obtained from Figure 3
as a function of l/|v| (mean ± SD). Note the
increase in SD as l/|v| increases (visual inspection of
the rasters and firing rate estimates in Fig. 3 shows a clear
tightening of the responses for small values of l/|v|).
This increase in SD with l/|v| was observed in all
preparations. Dashed line, Best least square fit of the data
to Equation 5 ( = 4.7 ± 0.3; = 27 ± 3 msec).
Inset, Schematics illustrating the geometric significance of
(intercept of the y-axis and the dashed line)
and (slope). B, Two-dimensional plot of the estimated
value of and together with the 68.3% confidence region for
these parameters. This confidence region is an ellipse
tilted from the horizontal, reflecting the fact that the estimates in
these parameters are well correlated (correlation coefficient, 0.76).
In other words, if were overestimated, then would also be
expected to be overestimated and vice versa. Conversely, if were
underestimated, then would also be expected to be underestimated
and vice versa. Significance level of the Kolmogorov-Smirnov test:
p 0.80.
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|
Significance of the linear relation between peak firing time and
l/|v|
The significance of the experimentally determined linear relation
between the peak DCMD firing rate relative to collision and the
parameter l/|v| as well as the correlation between the estimates of and can be explained by the following observation.
For a given DCMD neuron, let be the slope and the intercept of
the linear regression between peak time and l/|v| (see Fig. 4A, Eq. 5). Consider the angular size, call it
thres, subtended by the approaching object msec before the peak (see Fig. 1A for the definition
of ). Equation 5 states that this angular size is independent of
l/|v|, i.e., independent of the particular size or speed
of the approaching object. To see this, note that by trigonometry,
Furthermore, this calculation shows that the angular threshold
size characterizing the peak response of a given DCMD neuron (to occur
msec later) is completely determined by the slope of the linear
regression line as:
|
(6)
|
For the DCMD neuron depicted in Figures 2-4, this means that the
peak firing rate always occurred 27 msec (±3 msec) after that the object had reached a full angular size
(thres) of 24° (±1.5°) on the
locust's retina, for all values of l/|v|, as
illustrated in Figure 5.
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Figure 5.
Diagram illustrating the significance of Equation 5 for the time course of the DCMD response to looming objects.
Top panel, Angle of an approaching square for three
different values of l/|v| (10, 30, and 50 msec; Eq. 2).
Bottom three panels, Time course of DCMD firing rate in
response to these stimuli. The time of peak firing follows the same
linear relation (Eq. 5) observed in Figs. 3 and 4. In each case, the
time of peak firing is indicated by the dotted vertical
lines. The three vertical dashed lines are placed = 27 msec before the peak. At this moment the angle subtended by the
object is (t) = 24° for all three experiments. [Each
vertical dashed line intersects its angular size stimulus
curve (×) at a constant y = thres = 24°.]
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|
Equation 6 also offers a simple explanation for the experimental
correlation observed between the parameter and characterizing the linear relation between the peak time and l/|v|
(Fig. 4B). If the threshold angle
thres is overestimated (corresponding to an
underestimation of , according to Eq. 6), then we expect to
underestimate the delay between the angular threshold size and the time
of the peak firing rate. Conversely, if the angular threshold size is
estimated as being smaller than its real value, we should overestimate
the time interval separating the instant when the object reaches the
threshold angle and the peak firing rate.
Estimates of the slope, delay, and angular threshold size are plotted
for 15 animals in Figure 6. The values of
these parameters varied from animal to animal but were consistently
found to range between 15 and 35 msec delays for angular threshold
sizes between 15 and 40°. Part of this variability might arise from
interindividual differences in the geometric characteristics and size
of the eye.
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Figure 6.
Experimentally determined values of , , and
thres for 15 neurons. A, Plot of the slope
of the best linear fit between l/|v| and
|tpeak| versus the intercept (see Fig.
4A, inset). Error bars represent 1 SD from
the estimated value. B, Plot of the corresponding angular
size thres (computed using Eq. 6) subtended by the
object msec before the peak. Error bars for thres
were obtained from those on by error propagation (see Materials and
Methods).
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|
Accuracy of the angular threshold computation
If the angular threshold computation implemented by LGMD and DCMD
is performed with a fixed angular accuracy,
thres, one expects the variability in peak
occurrence time, tthres, to increase with
l/|v|. This point is illustrated in Figure
7A. The dotted horizontal
lines represent a constant error thres for the
angular threshold size thres. In the time domain, this
constant error translates into increasingly large tolerance windows
tthres for the peak occurrence time
(delimited by pairs of dotted vertical lines) because as
l/|v| increases, the rate of angular change (d/dt) decreases at the threshold angle
thres (see Fig. 7A, Eq. A8 in Appendix
2).
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Figure 7.
The time jitter in peak firing rate of LGMD and
DCMD corresponds to a fixed angular error in the encoding of
thres. A, The two horizontal dotted
lines show a fixed error (independent of l/|v|) in
the encoding of thres (indicated with
crosses) by the peak firing rate of LGMD and DCMD. This
translates in the time domain to increasingly wide time windows
tthres for the jitter in peak firing rate as
l/|v| increases. The value of thres
depicted in this example, 6.2°, corresponds to ±1 SD in Figure
4A (i.e., thres = 2 · thres according to the linear fit depicted in
B below). B, The SD of the peak firing time shown
in Figure 4A has been replotted as a function of
l/|v| and fitted with Equation 8. The correlation
coefficient between l/|v| and
tpeak is equal to 0.92, consistent with
the prediction made by Equation 7. Corresponding angular error obtained
from Equation 9: thres = 3.1°.
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|
As derived in Appendix 2, the accuracy of peak timing is predicted (to
first order) to increase linearly with l/|v| for
thres fixed,
|
(7)
|
where is the slope of the regression line defined by Equation 5. The SD, tpeak, of
|tpeak| characterizes the accuracy of peak
timing. To assess the validity of Equation 7, we therefore performed a
linear regression of tpeak as a function
of l/|v|,
|
(8)
|
(Fig. 7B). Correlation coefficients between
tpeak and l/|v| ranged
from 0.56 to 1.0 (mean ± SD, 0.87 ± 0.15; N = 15 neurons), consistent with this linear assumption.
Next we tested whether the peak time in DCMD response and its
variability could be described by a simple model depending only on
three free parameters: (1) the threshold angle thres
encoded by the peak firing rate of LGMD and DCMD; (2) the delay between the angular threshold size and peak time; and (3) the accuracy thres with which the threshold angle is encoded
by a given neuron. The first two parameters can be obtained from the
experimental data by linear regression of peak time versus
l/|v| (see Eqs. 5 and 6), whereas
thres can be obtained by combining Equations 7
and 8,
|
(9)
|
where  is the slope of the linear regression between
tpeak and l/|v| (see
Bevington and Robinson, 1992, Sec 3.2).
Ten of 15 neurons tested had a distribution of peak firing times as a
function of l/|v| in close agreement with the model (see
Materials and Methods; 2/F range, 0.19-4.1;
mean ± SD, 1.88 ± 1.52; F statistics range, p 0.05-0.96; mean ± SD, 0.38 ± 0.29;
K.-S. test range, p 0.01-0.96; mean ± SD,
0.58 ± 0.32). No neuron failed to pass more than one of these
three tests: two neurons had high values of
2/F (5.0 and 11.5, respectively); one failed
to pass the F test (p < 0.05); and
the remaining two failed the K.-S. test (p < 0.01). In three of these five cases, failure was attributable to a
small number of outliers (one to three) among the 50 test peak time values, causing a significant distorsion in the distribution of fit
residuals. The remaining two cases appeared to represent genuine failure of the model to represent the data. For the 10 neurons best
described by the model we also verified that the experimental values of
and , their SDs, and correlation coefficient could be reproduced
by the model. To this end, we generated synthetic data sets consisting
of 100 peak times and recomputed these parameters. For the neuron
depicted in Figures 4 and 7B, for example, 25 synthetic sets
yielded a range of values ( = 4.47-4.77; SD = 0.18-0.30; = 24.1-28.7 msec; SD = 1.7-4.5 msec; correlation
coefficient = 0.71-0.79) in close agreement with the experimental
ones ( = 4.68 ± 0.29; = 27 ± 3.2 msec; correlation
coefficient = 0.76).
Among the 10 neurons that successfully passed all three criteria
described above, the angular errors computed using Equation 9 ranged
from 3.1 to 11.9°. These angular errors correspond to the angle
defined by two to six ommatidia on each side of the expanding object.
Test of possible artifacts attributable to video refresh
Because images of approaching objects were updated every 5 msec on
the monitor screen (200 Hz refresh; see Materials and Methods), the
resulting visual stimulation was only an approximation of the
continuous motion of real objects. A possible artifact caused by such
discontinuous image updates would be a progressive failure to stimulate
the neuronal motion circuitry presynaptic to LGMD when the jump in
image size exceeds 3° (i.e., twice the locust's photoreceptor
acceptance angle; Osorio, 1986b). In this case, the early decrease in
DCMD activity could be attributable to a lack of appropriate
stimulation after the time at which angular increments exceed 3°
(Rind and Bramwell, 1996, note added in proof; Rind and Simmons, 1997;
Judge and Rind, 1997). It is therefore important to rule out the
possibility of such a stimulation artifact. The following two
observations show that no such artifact occurred in our experiments.
First, for large objects or low velocities of approach (l/|v| 25 msec) the peak firing rate occurred well before the angular increment reached 3°. This is illustrated in Figure 2 for one experiment (l/|v| = 45 msec) in which only the last
image update, 112 msec after the peak firing rate, led to an
angular increment >3°, and in Table 2
for 15 other experiments (l/|v| = 25 msec). The average
angular step at threshold was 0.60 ± 0.33°, ranging from 0.2 to
1.2° (Table 2). Second, assuming an artifact caused by stimulation
failure is equivalent to stating that the peak in DCMD firing rate
should be correlated with an angular velocity threshold
thres = 600°/sec (3°/refresh × 1 refresh/5
msec) rather than an angular size threshold
thres (between 15 and 40°), as reported here. By
setting thres = 600°/sec and solving for
|tpeak| as a function of
l/|v| in Equation 3 (see Appendix 1), one obtains a
relation between peak time and l/|v| that is plotted in
Figure 8A (dashed
line), together with the experimental results of one DCMD
experiment. As can be seen, the relation between peak time and
l/|v| predicted by an angular velocity threshold model (Fig. 8 A, dashed line) provides a very poor fit
of the data compared with the excellent fit obtained with Equation 5
(Fig. 8A, solid line). For low values of
l/|v|, for example, large angular jumps occur soon
during the stimulation, and the peak time is predicted by the angular
velocity threshold model to actually occur earlier than
observed experimentally. This prediction could be confirmed in none of
N = 9 animals specifically tested for this using a protocol containing five values of l/|v| < 10 msec
(Fig. 8A; variant of first protocol, see Materials
and Methods). Introducing a constant delay between the curve
thres = 600°/sec and the peak time produced a better
fit of the experimental data for l/|v|  5 msec but
much poorer fits for all other values of l/|v| (Fig.
8A, dotted line).
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Figure 8.
No indication of angular velocity threshold
artifacts was seen both at high and at low image refresh rates with the
video stimulation system used in the present experiments. A,
The experimental dependence of peak firing time as a function of
l/|v| (filled dots, mean ± SD) is
well fitted by the linear relationship corresponding to an angular
threshold thres = 26° and a delay = 27 msec
between the time of threshold angle and peak time (solid line
a; see Eq. 5). In contrast, the prediction that the peak time
should occur immediately after an angular jump of 3° in the image
(dashed line b) only poorly fits the data. The same is true
when a fixed delay is inserted between the time of 3° image jump and
the peak, thus shifting the curve downward (dotted line c;
same delay = 27 msec as for the angular threshold line
thres = 26°). See Appendix 1 for the derivation of the
angular threshold curve (Eq. A7). B, Dependence of peak time
as a function of l/|v| for three different refresh rates
of the video monitor (squares, 200 Hz; circles,
100 Hz; triangles, 67 Hz). The best linear fits to Equation 5 (dotted line, short dashed line, long
dashed line, respectively) virtually overlap in all three cases.
For the sake of clarity only the largest SD of the mean peak time
estimate in one direction is shown at each value of
l/|v|. Data in A and B are from
two different experiments.
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|
Because our earlier experiments were performed with a refresh rate of
1/13.9 msec (72 Hz; Hatsopoulos et al., 1995), we also verified that
low refresh rates did not affect the validity of our results. To this
end, we used three different refresh rates (pseudorandomly interleaved
trials) in five animals (200, 100, and 67 Hz; second protocol,
Materials and Methods). The results of one such experiment and the best
linear fits to the data are plotted in Figure 8B. No
significant change in the time of peak firing rate as a function of
l/|v| was observed in three of five experiments. In one
preparation, lower refresh rates led to slightly earlier peak times. In
the fifth preparation, lower refresh led to slightly later peak times
(Table 3). In accordance with these results, and in contrast to those reported by Rind and Simmons (1997),
little or no phase locking of the DCMD response to the image refresh
could be observed at 67 or 200 Hz. This could be seen from the spike
rasters obtained in response to individual trials, as illustrated in
Figure 9 (compare with Rind and Simmons, 1997, their Fig. 4). Rind and Simmons (1997) also reported that a
failure to properly stimulate DCMD results in a significant decrease in
the number of action potentials elicited by a stimulus. We therefore
compared the number of action potentials elicited at 200 Hz
(n200) to that obtained at 67 Hz
(n67). The mean difference,  n200 n67 = 0.74 ± 6.1 action potentials, was not significantly different from
zero (mean ± SD averaged over five experiments and eight
protocols). DCMD was therefore equally well excited by stimuli
presented at 67, 100, and 200 Hz, and none of the artefacts reported by
Rind and Simmons (1997) could be observed in the present experiments.
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Figure 9.
No phase locking of the DCMD response could be
observed at either 200 or 67 Hz refresh. Top panel,
Half-angle of the object presented on the screen as a function of time
for a value of l/|v| = 5 msec and a refresh rate of 200 Hz (angular image increments are determined by differences in
successive values of /2). The five rasters below show the DCMD
response to this stimulus (the dotted lines are placed 5 msec apart and aligned with the time of image refresh). The next five
rasters report the response of the same DCMD neuron to the same
stimulus but with the monitor image refreshed only every 15 msec (67 Hz), as illustrated in the bottom panel (the
dotted lines are placed 15 msec apart and aligned with the
time of image refresh).
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|
Effect of background luminance and stimulus contrast on peak
firing time
Next, we investigated whether the mean light level input to
photoreceptors or the contrast of the edges of the object sweeping across the receptive field during visual stimulation affected the
timing of the peak response and/or its relation to the angular threshold size characteristic of a given DCMD neuron. To this end, a
series of experiments was performed where the relative background
luminance, B, and object luminance, O, were
systematically varied. Data was collected at three values of
l/|v| (third protocol, Materials and Methods). These
three values were selected among those used in the previous two
protocols to yield the most reliable estimates of the slope and delay
parameters ( and , respectively; see Eq. 5). In this way, a large
range of background and contrast combinations could be explored in a
single DCMD neuron while retaining the ability to detect possible
deviations from linearity in the peak time versus l/|v|
relation as well as possible changes in and/or . As illustrated
in Figure 10, A and
B, in one experiment, the linear relation between peak time
relative to collision and l/|v| remained valid for a
wide range of background and contrast values. This result and the
following ones were observed in all neurons tested (N = 6). No statistically significant changes in the angular threshold
size (thres) or the delay () were observed for
decreases in background luminance up to one-fourth of the maximal value
(Fig. 10A). Similarly, the number of action
potentials elicited per trial did not change in a statistically
significant way as the background luminance decreased (see one example
in Fig. 10B). Note that for the experimental results
depicted in Figure 10, B and D, there was a
statistically significant decrease in the number of action potentials
elicited at l/|v| = 40 msec (35 per trial) compared
with l/|v| = 10 msec (20 per trial). This observation
could be made in only two of six preparations (the experiment
illustrated in Fig. 3, for example, showed no significant decrease in
the number of action potentials with l/|v|). On the other hand, the number of action potentials elicited per trial was
always significantly dependent on the contrast of the edge of the
stimulus in the range tested (from 1.00 to 0.25), as illustrated in
one example in Figure 10D. To characterize this dependence, we pooled for each experiment data obtained at different background luminances and performed a linear regression of the mean
number of action potentials elicited as a function of contrast separately for each value of l/|v|. The slope of these
regression lines ranged from 6.3 ± 2.6 to 35.3 ± 4.7 spikes/unit of contrast and was significantly correlated with
l/|v| in four of six cases (larger values of
l/|v| leading to larger drops in the number of action
potentials elicited per trial as contrast varied from 1.00 to
0.25). Remarkably, this decrease in mean stimulus-evoked activity
affected neither the timing of the peak nor the values of the angular
threshold size and delay (Fig. 10C). No statistically significant increase in the SD of peak time estimates could be observed. Therefore, in the present experiments the angular threshold computation characterizing the response of DCMD to looming stimuli remained valid over a wide range of background luminances and contrasts.
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Figure 10.
The linear relation between l/|v|
and peak firing time is independent of background luminance or stimulus
contrast, despite decreased responses at lower contrasts. A,
Peak firing time relative to collision as a function of
l/|v| for four background luminances (B = 100%; 75, 50, and 25% of Imax = 95 cd/m2). Dashed line, Best linear fit for
B = 75% ( = 4.20 ± 0.34; = 11.9 ± 6.6 msec). For clarity, only one linear fit is shown, and only the
largest error for each value of l/|v| is illustrated in
one direction. B, Mean number of spikes ±SD elicited per
trial for the same four background luminances as in A
(symbols as in A). C, Peak firing time as a
function of l/|v| at four contrasts (C = 1.00, 0.75, 0.50, and 0.25). Dashed line, Best
linear fit for C = 0.25 ( = 4.47 ± 1.56; = 17.9 ± 16.6 msec). D, Mean number of spikes ± SD elicited per trial for the same four contrasts as in C
(symbols as in C). Data in A-D are from a single
experiment.
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Effect of body temperature on peak firing time
Because locusts exert only a limited control over their body
temperature, it is expected to fluctuate over a large range of values
depending on external conditions and on metabolic rate. In particular,
in flying insects, body temperature is known to rise significantly
during intense periods of muscular activity such as flight episodes
(Weis-Fogh, 1956, 1964; Neville and Weis-Fogh, 1963; Stavenga et al.,
1993). We studied the effect of body temperature on the timing of peak
firing rate, the angular threshold, and delay parameters by comparing
the responses to looming stimuli elicited at room temperature
(21-24°C) with those obtained at 30-33°C (fourth protocol,
Materials and Methods). In all neurons tested (N = 8),
raising body temperature resulted in a significant shortening of DCMD
action potentials (see example in Fig.
11, top inset; mean
decrease, 35.5 ± 4.5%, measured in N = 4 animals
with best recording signal-to-noise ratio) but no consistent changes in
the number of action potential elicited during each trial was observed.
The timing of the peak response was also unaffected (N = 8 neurons), as illustrated in Figure 11. In this experiment, the
response to looming stimuli was recorded successively at room temperature (21°C, squares), 30°C (circles),
and back at room temperature (triangles). The values of the
parameters and obtained by fitting straight lines to the data
were accordingly not significantly affected by temperature (Fig. 11,
bottom inset). Therefore, our characterization of the DCMD
response remained valid over a 12°C range of ecologically relevant
temperatures.
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Figure 11.
Effects of body temperature on the DCMD response
to looming stimuli. Top inset, Two extracellular action
potentials recorded at room temperature (21°C, trace i)
and at 30°C (trace ii). Note the significant decrease
(43%) in the width of trace ii compared with i
(measured as the time difference between the arrows shown on
the inset). Main panel, Time of peak firing
relative to collision as a function of l/|v| at room
temperature (squares), 30°C (circles), and
during a subsequent control experiment back at room temperature
(triangles). For clarity, only the largest error bar is
shown at each value of l/|v|. Note the increased
variability in the peak response compared with Figure 7 or 9, for
example. This was presumably because of the longer time span of the
experiment (2 h 30 min in this case) caused by heating and cooling of
the animal. Solid lines, Best linear fits to the data using
Equation 5. Bottom inset, Values of the parameters and
for each of the three fits ± SD showing no significant
differences in the mean values. Data are from a single
experiment.
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|
Time course of firing rate as a function of retinal object size and
edge angular velocity
We next studied the dependence of the angular threshold
computation implemented by LGMD and its presynaptic circuitry on the physical variables characterizing object approach. Two such kinematic variables that can be measured at the retina are the angular size and
edge velocity (see Fig. 1). We therefore looked for a class of
functions f of these variables that could describe the time course of LGMD and DCMD firing rate over the range of values of l/|v| used in the present experiments (see Fig. 3).
Remarkably, only three assumptions derived from experimental data
tightly constrain the functional dependence of the firing rate on
angular size and velocity of approach, as we now explain.
From earlier anatomical and electrophysiological characterizations of
LGMD responses (Palka, 1967; O'Shea and Rowell, 1976; Rowell et al.,
1977) and from the results presented here, it is plausible that the
class of functions f describing the firing rate of LGMD and
DCMD neurons should satisfy the following three assumptions:
(1) f should depend only on the angular size,
(t), and angular edge velocity, (t), of the
approaching object. These two variables are thought to be represented
in the form of inhibitory (size-dependent) and excitatory
(motion-dependent) inputs to the various dendritic subfields of LGMD
(Palka, 1967; O'Shea and Rowell, 1976; Rowell et al., 1977; also see
Discussion). Another kinematic variable of the approach that might play
a role in describing the firing rate of LGMD and DCMD (Rind and Simmons
1990, 1992), the angular acceleration of the object, was ruled out as
sole explaining parameter in an earlier series of experiments
(Hatsopoulos et al., 1995).
(2) The firing rate at time t depends only on the value of
and at time t . In other words, if we set
x(t) = (t) and y(t) = (t), then:
|
(10)
|
(see Eq. 5 for the definition of and ). A delay between
firing rate and stimulus is expected because of the lags introduced by
synaptic and cellular elements along the neuronal pathways converging
onto LGMD. Here, this delay is , the delay observed between the peak
response and the threshold angle preceding it (Fig. 5).
(3) The peak of f(t) should satisfy the linear relation
observed experimentally between peak firing rate time and
l/|v| (see Figs. 4, 10, 11), regardless of the
particular values of thres and implemented by the
LGMD neuron of a given animal. In other words, we seek to characterize
a class of functions that can take into account the different angular
threshold sizes and delays (see Fig. 6) as well as the variability in
the time course of the firing rate observed across animals (data not
shown, but see Table 4).
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Table 4.
Numerical values of the parameters a,
b, c, and x1/2 (see Eq. 4) characterizing the static nonlinearity g in four
experiments (experiment 10797 is illustrated in Figs. 2-4, 11-13)
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|
As derived in Appendix 3, it follows from assumptions 1-3 above that
f has to be of the form,
|
(11)
|
where g is a static nonlinearity that characterizes the
transformation between the kinematic variable z = · e and the firing rate (see next
section; Fig.
12A).
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Figure 12.
Determination of the static nonlinearity
g from experimental data. A, The diagram on the
left illustrates the time course of the firing rate
f(t) and of the kinematic variable z(t) during
approach. Because f = g(z), one obtains g by
disregarding the time parameter t and plotting for each
t the pairs of values [z(t), f(t)] as
illustrated on the right. According to Equation 11,
g should be independent of l/|v| and two time
values t1, t2 on either side
of the peak (rising and decay phase) for which
z(t1) = z(t2) should
yield f(t1) = f(t2).
B, Static nonlinearity g obtained as in
A for 10 values of l/|v| (same experiment as
in Figs. 2-4). The part of g corresponding to the rising
phase is a steep nonlinearity well approximated by a saturating
exponential (sigmoid function). For l/|v| 25 msec,
the firing rate f during the decay phase shows a hysteresis
with larger values than during the rising phase.
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|
Thus each function belonging to the class given in Equation 11 depends
on three parameters: , , and the static nonlinearity g. To understand why this class of functions can describe
the time course of LGMD firing rate during object approach, note that because g is a static (time-independent) function, the time
course of the firing rate is entirely determined by:
|
(12)
|
Both and are nonlinear increasing functions of time during
the approach (Eqs. 2, 3; Fig. 1B). The combination of
and in Equation 12 is such that acts as an excitatory term
(an increase in leads to an increase in z and thus in
the firing rate), whereas acts as an inhibitory term (an increase
in leads to an increase in e
and thus a decrease in z and in firing rate; see Eq. 12).
Therefore, the apparent excitatory and inhibitory effect of motion and
size (respectively) on the LGMD response (Palka, 1967; O'Shea and
Rowell, 1976; Rowell et al., 1977) can be predicted on the basis of
assumptions 1-3 alone under our experimental conditions. At the onset
of approach, the motion-dependent excitation represented by increases faster than e, leading to
an increase in firing rate. Some time later, size-dependent inhibition
gains importance because of its exponential dependence on . It
eventually overcomes the excitatory term, leading to a decrease in the
response. The time at which inhibition overcomes excitation always
occurs a fixed delay after that the object has reached an angular
threshold 2 · tan11/, independent of
l/|v| (Eq. 12), as observed experimentally. In addition,
under assumptions 1-3 above, this is the only functional combination
of and with this property.
Experimental determination of the static
nonlinearity g
It follows from Equation 11 that the static nonlinearity
g characterizes the transformation between the kinematic
variable z = · e
and the firing rate during approach (Fig. 12A,
left). To determine the function g(z) that best
fits the experimental data, the values of and were first
obtained for each DCMD neuron by a linear fit of the peak firing time
relative to collision versus l/|v|, as explained earlier
(Fig. 4, Eq. 5). We then computed z(t) as a function of time
during object approach (see Eq. 12) and plotted for each value of
t the experimental value of the firing rate versus
z (nonparametric plot; Fig. 12A,
right). This procedure was repeated for each value of
l/|v| tested experimentally. The dependence of
g on z obtained for 10 different values of
l/|v| for a single DCMD neuron (first protocol,
Materials and Methods) is shown in Figure 12B (note
the logarithmic scale of the horizontal axis in this figure and in
Figs. 13A,
14A,
inset). In contrast to the prediction of Equation 8,
g was not independent of l/|v| (see Discussion): the dependence of g on z was
typically steeper for large values of l/|v|. The
function g also exhibited hysteresis at small values of
l/|v| (20 msec). This meant that, for a fixed value of
z, the firing rate was typically higher after than before the peak (Fig. 12A, B), reflecting a
slower "shutdown" than "buildup" of excitation during object
approach.
View larger version (22K):
[in this window]
[in a new window]
|
Figure 13.
Least square fits of g by the sigmoid
function of Equation 4 (same preparation as in Fig.
12B). A, Static nonlinearity g
at four values of l/|v| determined following the
procedure outlined in Fig. 12A. For each panel the
solid line is the mean value of g, and the
dotted line represents 1 SD from the mean. The dashed
line is the best least square fit with the sigmoid function of
Equation 4 (Materials and Methods). B, Fits of the time
course of DCMD firing rate during object approach using Equation 11 and
the function g determined in A. In each panel the
solid line is the time course of the firing rate during
object approach (mean over 10 trials), the dotted line is
the time course of the kinematic variable z(t) (see Eq. 12),
and the dashed line is the fit with g, , and
determined as explained in Results. Note that in the
bottom panel the fit is good up to the time point indicated
by the arrow: for larger values of t, z(t) is
identically equal to zero, and the time course of the firing rate falls
outside the range of validity of Equation 11. A failure to fit the
entire time course of LGMD and DCMD firing rate was observed in all of
our 15 neurons at values of l/|v| = 5 and 10 msec (see
Discussion).
|
|
View larger version (13K):
[in this window]
[in a new window]
|
Figure 14.
Dependence of g on
l/|v| (same preparation as in Figs. 12, 13).
A, Value of the half-activation parameter
x1/2 and the slope a for 10 different
values of l/|v|. The points taken from left
to right correspond to values of l/|v| from
50 to 5 msec, respectively. Note the decrease in the slope parameter
and the shift in half-activation as l/|v| decreases.
This is illustrated in the inset, which shows the 10 different sigmoids corresponding to the parameters shown in the
main panel (a = 0; b = 209 Hz; see Eq. 4). B, Hysteresis parameter as a function of l/|v|
(a is the asymptotic value of the firing rate as z
becomes small; see Eq. 4).
|
|
To characterize quantitatively these two observations, we fitted
separately the parts of g corresponding to the rising and falling phase of the firing rate at each value of l/|v|
with a sigmoid function (see Eq. 4, Materials and Methods; Table 4). This led to satisfactory fits of g and of the time course of
the firing rate during approach in 13 of 15 neurons (Fig.
13A, B). The parameters
x1/2 (half-activation) and c (slope)
characterizing g for the rising phase of the firing rate
(Eq. 4) were positively and negatively correlated with
l/|v| (Fig. 14A), reflecting the shift
in the dependence of g on z with
l/|v| and its shallower slope (Fig.
14A, inset), respectively. The dependence
of the parameter a characterizing hysteresis as a function
of l/|v| for the neuron of Figure 12B
is illustrated in Figure 14B.
| |
DISCUSSION |
This series of experiments and theoretical analysis were aimed at
characterizing the computation performed by LGMD and DCMD in response
to objects approaching on a collision course with the animal. Because
the peak firing rate always occurred a fixed delay after the object had
reached a constant angular size on the animal's retina, this suggests
that LGMD computes the time at which this threshold angle is reached
during object approach. For each animal, thres is
constant over one order of magnitude of the kinematic approach
parameter l/|v|. The algorithm underlying this
computation requires a multiplication of two nonlinear functions of
time that characterize the stimulus projection on the retina: these
functions are the angular size, (t), and the angular
velocity, (t), of the approaching object.
The visual stimulus
In these experiments, the locust's visual system was stimulated
with a monitor screen refreshed at 200 Hz. This allowed for a good
control of the stimulus time course but represented only an
approximation of true motion. Several arguments rule out the possibility of stimulation artifacts.
First, the temporal cutoff frequency of locust photoreceptors is less
than half the monitor refresh rate, ruling out a locking of the
photoreceptor response to video refresh (Miall, 1978; Howard, 1981;
Howard et al., 1984). Osorio (1986b) reported a decrease in the
response of medullary motion-detecting neurons for temporally offset
flashes spaced more than 3° apart. His results do not apply to our
experimental situation, because all photoreceptors on the looming
motion trajectory were stimulated.
Second, a stimulation artifact predicts that the peak firing rate
should follow a very different relation with l/|v| than that observed experimentally. Because our earlier experiments (Hatsopoulos et al., 1995) were performed at a lower refresh rate (72 Hz) we verified that none of our results depended on refresh rate and
that DCMD was equally well stimulated at lower refresh frequencies.
Thus, the statement that "The peak in DCMD activity measured by
Hatsopoulos et al. (1995) occurred when the jump size on the monitor
first exceeded approximately 3°" (Judge and Rind, 1997, p 2210) is
incorrect (Fig. 2, Table 2) and the statement that the "DCMD response
reported by Hatsopoulos et al. (1995) is an artifact" (Rind and
Simmons, 1997, p 1032) (also see Rind and Bramwell, 1996, note added in
proof) is erroneous.
Finally, the occurrence of peak DCMD activity before collision at high
values of l/|v|, as observed here, is supported by earlier reports (Schlotterer, 1977, their Fig. 3; Pinter et al., 1982,
their Figs. 1A, 2A(i)),
including Rind and Simmons' own recordings (Rind and Simmons, 1992,
their Fig. 4). As demonstrated here, however, the timing of the peak
depends on l/|v|. Therefore, if sufficiently small
values of l/|v| are selected, a substantial portion of
the spikes, including the peak, will occur after collision (for example, see Fig. 3, l/|v| = 5 msec; also see Judge
and Rind, 1997, their Fig. 4, top, which corresponds to
l/|v| = 6 msec).
Behavioral relevance of the angular
threshold computation
Peak DCMD activity always occurred a fixed time after that the
size of the object reached an angular threshold. This information might
thus be used by the animal to initiate an escape behavior in face of an
imminent danger. In our earlier experiments (Hatsopoulos et al., 1995,
note 15), the time of peak DCMD activity was correlated with the
subsequent timing of a femoro-tibial flexion for presumed jump
preparation (for a similar observation, also see Wallace, 1958; Wehner,
1981, p 479). This flexion occurred a fixed delay (20 msec) after the
peak, and after that the object had reached a fixed angular threshold
size on the retina. Robertson and Johnson (1993a,b) recently
characterized the cues used by locusts in visually triggered collision
avoidance maneuvers during flight. They concluded that the variable
most closely related with the avoidance maneuver was the angular size
of the object 65 msec before its initiation. The threshold angles
reported by them, 10° (Robertson and Johnson 1993a, their Fig. 3, b,
their Table 2), are smaller than the typical threshold angles reported
here. Because in their study, objects approached on a frontal rather
than side collision course, this difference might be related to the
better acuity in the frontal visual field (Horridge, 1978).
These two lines of evidence as well as anatomical cues suggest that
LGMD and DCMD are part of a fast neuronal pathway initiating escape
responses. In pigeons, neurons with response properties very similar to
those described here have been recently reported in nucleus rotundus,
the homolog of the mammalian inferior caudal pulvinar (Sun and Frost,
1998). Interestingly, this nucleus also contains neurons sensitive to
angular velocity and others yet sensitive to time to contact. The
nucleus rotundus is thought to play a important role in the
"tracking" of approaching objects. Clearly, angular threshold size
has disadvantages when compared with time to contact (Lee and Reddish,
1981; Wagner, 1982; Wang and Frost, 1992): given a fixed angular
threshold size, the timing of the response will occur later for objects
moving faster. However, this might prove sufficient for a restricted
range of object sizes and velocities as encountered in natural
environments (Robertson and Johnson, 1993a,b). In this context, it is
also important to assess the dependence of this computation on the
nature of the looming stimuli. Our results demonstrate stability over a
wide range of backgrounds and contrasts despite a decrease in spike rate with contrast (for a similar dependence of spike activity on
contrast, see Palka and Pinter, 1975; Judge and Rind, 1997). Similarly,
we showed invariance to physiological body temperature changes.
Whether the peak activity or some physiologically related variable is
extracted by the motor circuitry postsynaptic to DCMD remains an open
question. Formally, peak detection is equivalent to the
detection of a zero crossing in the time derivative of the firing rate,
a problem extensively investigated in mammalian vision (Marr, 1982).
Alternatively, it is possible that an integration mechanism similar to
the one proposed for the triggering of the landing response in flies
(Borst and Bahde, 1988; also see Sun and Frost, 1998, their
Fig. 3a) might underlie the processing of DCMD firing rate
by postsynaptic neurons. Isolating this mechanism is complicated by the
divergent connections made by DCMD and by the existence of >10 lobula
neurons with response properties similar to DCMDs (Gewecke and Hou,
1993).
Representation and algorithm
We have shown that a particular combination of the object angular
size (t) and of its angular edge velocity
(t) can describe phenomenologically the time dependence
of the peak firing rate on l/|v|. Furthermore, this
combination of (t) and (t) is unique, as
seen from Equation 11. Assumptions 1-3 (see Results) used in Appendix
3 to derive this equation follow either directly from our experimental
observations (Figs. 3, 6) or from earlier electrophysiological and
anatomical investigations (Palka, 1967; O'Shea and Rowell, 1976;
Rowell et al., 1977). At the algorithmic level, this allows to
breakdown the computation performed by LGMD into four distinct steps:
(1) a calculation of the angular velocity of the edges of the expanding
object on the retina; (2) a calculation of the size subtended by the
object at the retina; (3) a multiplication of these two positive
variables; and, finally, (4) a transformation of the result into a
firing rate through the static nonlinearity g.
A closer examination of the time course of the firing rate of LGMD and
DCMD to various values of l/|v| reveals that the simple model of Equation 11 is not entirely satisfied by experimental data
(Figs. 12-14). Experimental results differ from Equation 11 in two
respects: (1) the part of g corresponding to the rising
phase of the firing rate depends on l/|v|, with a
steeper and more abrupt activation at high than low values of
l/|v|; and (2) the part of g corresponding to
the falling phase of the firing rate shows significant hysteresis at
low values of l/|v| (Figs. 12B,
13A, 14B). For l/|v| = 5 or
10 msec, the model of Equation 11 does not fit the firing rate over the
entire time course of the stimulation (Fig. 13B,
arrow).
The first difference between theoretical prediction and experimental
data may be entirely attributable to the biophysical properties of the
conductances shaping the response of LGMD and, possibly, of its
presynaptic elements (see next subsection). These properties were not
taken into account in the derivation of Equation 11 but are expected to
influence the response of the neuron. Intracellular recordings suggest
that several active conductances are present in the membrane of LGMD in
addition to those presumably responsible for action potential
generation (Gabbiani et al., 1997). The change in the time course of
synaptic input activation across the dendritic tree of LGMD with
l/|v| might result in a different recruitment of active
membrane conductances and be sufficient to explain the change in
g described in Figure 14A. Biophysical
modeling of LGMD (Gabbiani et al., 1997) and further experiments to
characterize its membrane properties along the lines of Haag et al.
(1997) will be useful to address these questions.
The second difference is also probably attributable in part to the
biophysical properties of LGMD: at small values of
l/|v|, the time constant of the membrane (in the range
of 5-10 msec; Gabbiani et al., 1997) might for example be the limiting
factor in determining the termination of excitation by feed-forward
inhibition. In addition, part of the hysteresis is explained by our
choice of a 20 msec Gaussian window to estimate the firing rate of LGMD and DCMD. The same analysis with a 10 msec window or using the spike
rasters showed that the slower termination of the response compared
with its activation was reduced, although present. Thus, we conclude
that the hysteresis described in Figures 12B,
13A, and 14B is real, but that its
apparent magnitude is affected by uncertainties in the estimation of
the firing rate of LGMD and DCMD.
In our earlier report (Hatsopoulos et al., 1995) the time course of the
firing rate was fitted directly with Equation 12 instead of 11 (i.e.,
setting g = identity in Eq. 11). This led to reasonable fits for values of l/|v| < 25 msec as used in those
experiments but was found to work poorly at values of l/|v| > 25 msec in the current experiments (for example, see Fig.
13B). In this respect, our present results have thus
extended our original ones.
Biophysical implementation
How could the algorithmic steps characterizing the proposed
computation be implemented biophysically? A simple possibility, suggested by Equation 11, is that local motion computation is performed by neurons presynaptic to LGMD and represented by an excitatory input
logarithmic in angular velocity [log(t)] (for the solid objects used here). The inhibitory input to LGMD would be expected to
be proportional to the angular size of the object (), and the
multiplication operation between (t) and
e could be implemented by linear
summation followed by an approximate transform from logarithmic to
exponential coordinates. This corresponds to rewriting Equation 11 as
follows:
where (g exp(·)) = g(exp(·)) represents
the composition of g and the exponential function.
Alternatively, as recently suggested (Rind and Bramwell, 1996; Rind and
Simmons, 1998), part of the nonlinear interaction between
motion-dependent excitation and size-dependent inhibition could occur
presynaptically, via a lateral inhibitory network, which is already
known to protect excitatory synapses onto LGMD from habituation to
whole-field motion (O'Shea and Rowell, 1975).
Because LGMD can be reliably identified from one preparation to the
next, and its excitatory and inhibitory inputs can be studied in
isolation (Rowell et al., 1977), these hypotheses are amenable to
experimental testing. Thus, LGMD offers a unique opportunity to
investigate how a nonlinear neuronal computation involving a
multiplication operation is implemented biophysically.
| |
FOOTNOTES |
Received July 8, 1998; revised Nov. 4, 1998; accepted Nov. 19, 1998.
This work was supported by grants from the National Science Foundation
(NSF), the NSF Presidential Faculty Fellow Program, the Sloan Center
for Theoretical Neuroscience, the Center for Neuromorphic Systems
Engineering as part of the NSF Engineering Research Center program, and
a travel grant from the Deutsche Forschungsgemeinschaft.
We thank J. Burns, J. Douglass, S. Panish, and M. Wicklein for
help in setting up the stimulation system and Martin Egelhaaf for
careful reading and valuable comments on this manuscript.
Correspondence should be addressed to Dr. Fabrizio Gabbiani or Dr.
Gilles Laurent, Division of Biology, 139-74 Caltech, Pasadena, CA 91125.
Dr. Krapp's present address: Lehrstuhl fuer Neurobiologie,
Universitaet Bielefeld, Postfach 1001 31, D-33501 Bielefeld, Germany.
| |
APPENDIX 1 |
This appendix provides additional details on the derivation of
Equation 5 and on the equation characterizing the relation between peak
time and l/|v| plotted in Figure 8A
for an angular velocity threshold = 600°/sec.
As shown in Results, the tangent of the threshold angle is determined
from trigonometry by:
|
(A1)
|
The denominator in this equation represents the position of the
object msec before the peak, because the object approaches at
constant velocity v. We start by multiplying out the two
terms in the denominator,
|
(A2)
|
Because according to our conventions, v < 0, tpeak < 0 and > 0, we have
vtpeak = |v| · |tpeak| and v = |v| · . Plugging this in Equation A2 gives:
At this point we may use Equation 5 to simplify the right hand
side,
Using this last result in Equation A1 gives Equation 6.
To derive the equation describing the value of
|tpeak| as a function of
l/|v| for an angular velocity threshold of 600°/sec (plotted in Fig. 8A), we start by rewriting Equation 3,
|
(A3)
|
with = l/|v| (note that v < 0
so that = l/v). Assume that the peak occurs a fixed
delay after that has reached a fixed angular velocity threshold
thres = 600°/sec. In angular units of radians, for
which Equation A3 holds, thres = 3/5 ·  /180 rad/msec = 0.010472 rad/msec. The peak time is then determined by:
|
(A4)
|
This equation gives an implicit relation between
|tpeak| and l/|v| = ,
which we now solve. First note that Equation A4 is equivalent to:
Thus,
Taking the square root on both sides gives:
|
(A5)
|
Because tpeak < 0 and > 0,
|
(A6)
|
From A5 and A6 we obtain:
|
(A7)
|
The value of |tpeak| as a function of
is plotted in Figure 8A for = l/|v| = 5-50 msec and for two values of ( = 0 msec and = 27 msec, dashed and dotted lines, respectively).
| |
APPENDIX 2 |
To derive Equation 7 we first note that, by combining Equations 2
and 6, the time at which the threshold angle is reached during approach
is given by tthres = · l/|v|. According to Equation 3, the angular expansion of the
object at tthres is given by:
|
(A8)
|
If we denote by tthres a small change in
time around threshold and by thres the corresponding
change in angle, we obtain by expanding Equation A8 to first order in
tthres,
where o(tthres) denotes a remainder term
tending to zero faster than tthres:
Rearranging this last equation yields Equation 7.
| |
APPENDIX 3 |
We start by proving the following result.
Lemma. For a square of half-size l approaching at
constant velocity v (corresponding to a kinematic parameter
= l/|v|) the linear relation characterizing
tpeak,
|
(A9)
|
is equivalent to the statement that tpeak
is determined by the equation:
|
(A10)
|
independent of the kinematic approach parameter .
Proof. First note that from Equation 3,
|
(A11)
|
and
we derive that
|
(A12)
|
Assume that Equation A9 is true. Then from A12,
But from A9, tpeak = so
that:
which is Equation A10. Conversely, assume that Equation A10 holds
for t = tpeak. By combining Equations A10
and A12,
Because 2(t)  0 for all
t, we obtain Equation A9. This completes the proof.
We now state our main result.
Theorem. The class of functions f,
with x(t) = (t) = (t)/2 and y(t) = (t), whose peak value follows Equation A9 independent of the
value of thres, is determined by:
|
(A13)
|
Proof. At peak time we have,
where xpeak = x(tpeak ), and ypeak = y(tpeak ). Using Equation A10 we obtain:
Because 0 (see Eq. A11), we obtain:
|
(A14)
|
at (x, y) = (xpeak,
ypeak). Equation A14 should hold for all values of
= l/|v|, that is, for x > 0 (using
Eqs. A11 and A9). Furthermore, we note that for t = tpeak, ypeak = (tpeak ) = thres.
Because we require the peak of f to follow Equation A9
independent of thres this means that Equation A14 should
hold over the range of values ypeak = 2tan11/ that we observe experimentally
[peak  (3.5; 3.9) for (3; 7.5);
see Fig. 6A]. Therefore f should satisfy
the partial differential equation:
|
(A15)
|
over the range x > 0, ypeak (3.5;
3.9). Equation A15 is a linear partial differential equation whose
solution is given by:
|
(A16)
|
(see Courant and Hilbert, 1968, Chap I). Plugging in the
definition of x and y yields Equation A13. The
arbitrary function g in the solution A16 must clearly be
monotonic increasing if Equation A9 is to be the only peak in the response.
| |
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Abstract
Introduction
