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The Journal of Neuroscience, January 1, 2001, 21(1):314-329
Invariance of Angular Threshold Computation in a Wide-Field
Looming-Sensitive Neuron
Fabrizio
Gabbiani1, 2,
Chunhui
Mo1, and
Gilles
Laurent1
1 Computation and Neural Systems Program, Division of
Biology, California Institute of Technology, Pasadena, California
91125, and 2 Division of Neuroscience, Baylor College of
Medicine, Houston, Texas 77030
 |
ABSTRACT |
The lobula giant motion detector (LGMD) is a wide-field
bilateral visual interneuron in North American locusts that acts as an
angular threshold detector during the approach of a solid square along
a trajectory perpendicular to the long axis of the animal (Gabbiani et al., 1999a
). We investigated the dependence
of this angular threshold computation on several stimulus parameters
that alter the spatial and temporal activation patterns of inputs onto the dendritic tree of the LGMD, across three locust species. The same angular threshold computation was implemented by LGMD in all three
species. The angular threshold computation was invariant to changes in
target shape (from solid squares to solid discs) and to changes in
target texture (checkerboard and concentric patterns). Finally, the
angular threshold computation did not depend on object approach angle,
over at least 135° in the horizontal plane. A two-dimensional model
of the responses of the LGMD based on linear summation of
motion-related excitatory and size-dependent inhibitory inputs
successfully reproduced the experimental results for squares and discs
approaching perpendicular to the long axis of the animal. Linear
summation, however, was unable to account for invariance to object
texture or approach angle. These results indicate that LGMD is a
reliable neuron with which to study the biophysical mechanisms
underlying the generation of complex but invariant visual responses by
dendritic integration. They also suggest that invariance arises in part
from non-linear integration of excitatory inputs within the dendritic
tree of the LGMD.
Key words:
invariant responses; looming; locust; LGMD; DCMD; collision avoidance
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INTRODUCTION |
Invariance of neuronal responses is
a key aspect of sensory processing. It characterizes the extraction of
specific features in a stimulus, independent of their context
(Cavanagh, 1978). Some invariant responses, such as
those underlying the detection of edges in an image regardless of
contrast level, rely on gain control mechanisms that are relatively
well understood at the cellular and network levels (Galarreta
and Hestrin, 1998; Sanchez-Vives et al.,
2000a,b).
In the inferotemporal visual area (IT) of the monkey, evidence
suggests that the responses of many neurons are invariant to
translation and scale transformations (Schwartz et al.,
1983; Rolls and Baylis, 1986), a possible
prerequisite for object identification (Sary et al.,
1993). Although little is known about how this type of
invariance arises, some authors have recognized the need for
mechanistic investigations and proposed models (Olshausen et
al., 1993; Salinas and Abbott, 1997;
Riesenhuber and Poggio, 1999). The neural basis of
response invariance, however, remains poorly understood. It is not
known whether it results from network properties, from dendritic
processing within single cells, or from combinations of both. We study
two monosynaptically connected neurons in the visual system of the
locust and demonstrate that they exhibit invariance properties to many
attributes of the stimuli that best excite them. Because the neuronal
processing leading to these invariant responses is thought to occur
within the dendritic tree of one of these neurons, it might prove
appropriate to study the biophysical mechanisms underlying the
generation of invariant visual responses in single neurons.
The lobula giant motion detector (LGMD) is located in the
third visual neuropil of the locust optic lobe (Strausfeld and
Nässel, 1981). Its dendritic tree consists of three
subfields that arborize in the lobula (O'Shea and Williams,
1974). The largest subfield receives excitatory inputs from
afferents that are sensitive to local motion over the whole visual
hemifield, whereas the remaining two subfields are thought to receive
feedforward inhibitory projections that are size dependent
(Palka, 1967; Rowell et al., 1977). LGMD synapses onto the descending contralateral motion detector (DCMD) (Killmann et al., 1999), a neuron the large axon of
which projects to thoracic motor centers responsible for the generation
of jump and flight steering maneuvers (Pearson et al.,
1980; Simmons, 1980; Robertson and
Pearson, 1983). The synaptic connection between LGMD and DCMD
is so strong as to cause a one-to-one correspondence between
presynaptic and postsynaptic spiking activity under visual stimulation
(O'Shea and Williams, 1974; Rind, 1984).
LGMD and DCMD are vigorously excited by objects looming toward the
animal (Schlotterer, 1977; Rind and Simmons,
1992). The peak firing rate during approach of such
looming objects signals the moment when the object reaches a fixed
angular threshold size (Gabbiani et al., 1999a). This
stimulus variable has in turn been related to the generation of escape
and collision avoidance behaviors (Robertson and Johnson,
1993; Hatsopoulos et al., 1995).
Invariance of the peak firing time of LGMD and DCMD to changes of body
temperature and stimulus luminance/contrast have already been
demonstrated (Gabbiani et al., 1999a). In the present
work, we investigate the extent to which the peak firing time of
LGMD/DCMD
and thus the angular threshold detection
computationchanges as we vary parameters of the approaching object
that are expected to directly affect the time course of the
spatially distributed excitatory and inhibitory inputs impinging
onto the dendrites of the LGMD.
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MATERIALS AND METHODS |
The experimental materials and methods used were similar to
those of Gabbiani et al. (1999a). The following brief
account mainly emphasizes the differences with that work.
Preparation. Experiments were performed on three different
species of North American and African locusts. Male and female specimens of the species Schistocerca americana (American
Bird Grasshopper) were taken from the laboratory colony 3-4 weeks
after their final molt. Adult male specimens from the African species Schistocerca gregaria (Desert Locust) and Locusta
migratoria (African Migratory Locust) were imported (USDA-APHIS
permits 36933 and 42816, respectively) from the International Center
for Insect Physiology and Ecology (ICIPE, Nairobi, Kenya) and the
University of Bielefeld (Germany). After the animals were fixed to a
plastic holder, the head was aligned under a microscope with reference points marked on a reticular grid inserted in one of the eyepieces. This procedure and the calibration protocol described in
Gabbiani et al. (1999a) allowed us to reliably align the
center of the locust eye with the center of the video monitor used for
visual stimulation. Animals were prepared for electrophysiological
recordings in one of the following three manners: (1) mounted dorsal
side up on the plastic holder, frontal dissection of the head capsule, performed as in Gabbiani et al. (1999a); (2) mounted
ventral side up, connectives exposed by an incision of the neck cuticle
(Hatsopoulos et al., 1995); or (3) mounted dorsal
side up, rectangular incision of the pronotum. The gut was grabbed with
a pair of fine forceps, cut as close as possible to the mouth, and
removed through an abdominal incision, thus exposing the connectives.
In all cases, the preparation was bathed in locust saline
(Laurent and Davidowitz, 1994).
Electrophysiology and data acquisition. Locusts were fixed
to a clamp with their longitudinal body axis parallel to the
stimulation screen, except for the turntable experiments described
below. For preparation 1, the connective contralateral to the
stimulated eye was placed in a suction electrode (Gabbiani et
al., 1999a). Alternatively, in preparations 2 and 3, two hook
electrodes made of 50 µm (0.02 inch) stainless steel wire isolated up
to the tip with H-Formvar (California Fine Wire Co., Grover City, CA)
were placed around the connective. The two electrodes were electrically isolated from each other with Vaseline (Hatsopoulos et al.,
1995). Extracellular signals were amplified with a
differential AC amplifier (A-M Systems, Everett, WA) and a Brownlee
amplifier (model 210A; Brownlee, San Jose, CA). They were acquired
together with transistor transistor logic (TTL) pulses
synchronizing the visual stimuli with the recordings (Gabbiani
et al., 1999a) using a 12 bit A/D board (win30; United
Electronic Industries, Watertown, MA) connected to a personal computer
running the QNX real-time operating system (QSSL, Kanata,
Ontario, Canada). Each recorded spike train was inspected visually;
DCMD action potentials (typically the largest in the nerve cord) (Fig.
1A) were selected on-line
using custom software written in C.
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Figure 1.
Responses of DCMD to simulated approaches of
looming squares (L. migratoria). A, The time
course of the angular size,  (t), subtended by the object
on the retina (inset) is illustrated on top (l/|v| = 50 msec). Each jump in angular size corresponds to a video
screen refresh (see Materials and Methods). Bottom panel,
Extracellular recording obtained from the connective contralateral to
the stimulated eye. B, Ten repetitions of each stimulus were
presented, and spike occurrence times were obtained by thresholding the
recorded extracellular signals. The corresponding spike rasters are
illustrated on the bottom (top raster corresponds
to extracellular trace in A). C, The time of peak firing
rate ( ) shifted consistently toward collision as the stimulation
parameter l/|v| decreased. The mean peak firing times
(obtained from similar graphs) and their SDs (obtained from the
repetitions of each stimulus) are plotted as a function of
l/|v| in Figures 3-7.
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Stimulation. Stimuli were generated using a fast monochrome
video monitor refreshed at a rate of 200 Hz, as described in
Gabbiani et al. (1999a). The sequences of video images
simulated circular and square objects with various textures approaching
on a collision course with the animal. Each image was computed by
central projection from the center of the stimulated eye onto the
screen plane (Gabbiani et al., 1999a, their Fig. 2A and
Eq. 2). The distance eye-screen was 120 mm. The parameters
characterizing each object approach were the radius (half-size),
l, of the approaching disk or square (in centimeters), and
the simulated constant speed of approach, v (in meters per
second) (see Fig. 1A, inset). If we set
t = 0 as the time of expected collision and adopt
the convention that t < 0 before collision, then
the velocity v is also negative when the object is
approaching (Gabbiani et al., 1999a, their Eq. 1). Because the objects approached at a constant speed, elementary trigonometry shows that the image sequence is characterized by the
ratio of these two parameters l/|v| (in milliseconds)
(Gabbiani et al., 1999a, their Fig. 1A). The
range of values used for l/|v| was from 5 to 50 msec
(see species, shape, texture, and approach angle protocols below),
corresponding to approach speeds between 2 and 10 m/sec for half-sizes
between 6 and 14 cm (Gabbiani et al., 1999a, their Table 1).
Species protocols. In the first series of experiments, we
compared the responses across different locust species by presenting to
each animal 10 repetitions of looming squares approaching at 5 values
of l/|v| pseudorandomly interleaved (from
l/|v| = 10-50 msec in steps of 10 msec; 50 protocols
total). An interval of 40 sec was inserted between successive stimulus
presentations to minimize habituation (Gabbiani et al.,
1999a). The squares were black (0 cd/m2) on
a bright background (95 cd/m2). This protocol was
applied to n = 5 specimens of the species S. gregaria, 3 L. migratoria, and 64 S. americana. An additional eight L. migratoria were
stimulated 10 times with 10 different values of l/|v|
(from 5 to 50 msec in steps of 5 msec) interleaved pseudorandomly (100 protocols total; an experiment lasted ~1.25 hr). This protocol was
identical to the one used in Gabbiani et al. (1999a) and
allowed for a more stringent test of the linear relationship between
peak firing time and l/|v|, because an additional 50 trials at 5 supplementary l/|v| values were gathered to
test the linear model.
Shape protocol. In the second series of experiments we
compared the responses to black disks and black squares looming toward the animal by presenting 10 times these two objects at 5 values of
l/|v| (l/|v| = 10-50 msec in steps of 10 msec; two
shapes × 5 l/|v| values × 10 repetitions = 100 protocols; n = 5 animals, S. americana). Both the shape of the presented object and
the value of l/|v| were interleaved pseudorandomly, with
a 40 sec interstimulus interval.
Texture protocols. In the third series of experiments, we
studied the effect of target texture by comparing the responses to a
black looming square and a square textured with a 3 × 3 checkerboard pattern as illustrated in Figure 5A
(inset) (l/|v| = 10-50 msec in steps of 10 msec; 100 pseudorandomly interleaved protocols; 40 sec interstimulus
interval; n = 12 animals, 5 S. gregaria, 7 S. americana). The background was bright with a
luminance IB = B·Imax (B = 100%,
Imax = 95 cd/m2), and the
checkerboard pattern had five squares of luminance IO = O·Imax, and
four squares of luminance IP = P·Imax with O = 0% and
P = 30%. In two experiments, the contrast between the squares of the checkerboard pattern was increased by setting
P = 70%.
The fourth series of experiments was a modified version of the previous
series, with a textured pattern consisting of four concentric squares
as illustrated in Figure 6A (inset). The luminance of
the background, IB, and object,
IO, IP, were identical to
those of the previous series (B = 100%, O = 0%,
P = 30%; 100 pseudorandomly interleaved protocols; 40 sec
interstimulus interval; n = 12 animals, 9 S. americana, 3 L. migratoria).
Approach angle protocols. The last three series of
experiments investigated the effect of target approach angle. Animals
were fixed to a clamp attached to a turntable that allowed rotation of
both the animal and the micromanipulators holding the recording electrodes around a virtual vertical axis passing through the center of
the eye. The turntable was equipped with a graduated dial allowing
angles to be read with an accuracy of ±1°. We denote by 0° the
position for which the longitudinal body axis is perpendicular to the
stimulation screen (i.e., the animal faces the screen; see Fig.
7A, inset). Positive angles denote counterclockwise
rotation, and negative angles denote clockwise rotations. The turntable could be positioned from 135° (the object approaches from the same
side as the stimulated eye, almost from the back) to 
45° (the
object approaches from the side contralateral to the stimulated eye).
These experiments were performed using only the third dissection described above (see Preparation).
In the fifth series of experiments, the effect of approaching angle was
tested at two positions: 0° and 90°. During the first six
experiments, the angles were alternated between 0° and 90°, and the
value of l/|v| (l/|v| = 10-50 msec in steps of 10 msec) were varied pseudorandomly from trial to trial (2 positions × 5 l/|v| values × 10 repetitions = 100 protocols). During the next seven experiments, the angle was fixed at
0° for the first 50 trials (pseudorandomly interleaved values of
l/|v|) and then at 90° for the remaining 50 trials
(n = 13 animals total, S. americana; 40 sec
interstimulus interval).
In the sixth series of experiments, the effect of approaching angle was
tested at four positions on the same side as the stimulated eye (0°,
45°, 90°, and 135°, from front to back) and 3 values of
l/|v| (l/|v| = 10, 30, and 50 msec). At the end of a
trial the animal was moved from one position to the next, and a new value of l/|v| was chosen pseudorandomly. Each
combination of position and l/|v| value was repeated 10 times (4 positions × 3 l/|v| values × 10 repetitions = 120 protocols; n = 5 animals, S. americana; 40 sec interstimulus interval).
In the seventh series of experiments, five positions close to 0°
(animal facing the screen) were investigated (45°, 22.5°, 0°,
22.5°, and 45°) at 3 values of l/|v| (l/|v| = 10, 30, and 50 msec). At the end of a trial, the animal was moved
from one position to the next, and a new value of l/|v|
was chosen pseudorandomly. Each combination of position and
l/|v| value was repeated eight times (5 positions × 3 l/|v| values × 8 repetitions = 120 protocols; n = 11 animals, S. americana; 40 sec interstimulus interval). In this experiment, recordings were
obtained from both connectives by placing one hook electrode around
each connective and amplifying the voltage signal of each channel with
respect to a far reference electrode implanted in the body.
Data analysis. Data analysis methods followed closely those
of Gabbiani et al. (1999a). The extracellular traces
obtained in response to the various stimuli described above were
thresholded to extract the spike occurrence times of the DCMD (Fig.
1A). Each spike train was smoothed with a 20 msec Gaussian
window to obtain an estimate of the instantaneous firing rate (Fig.
1B) (Gabbiani et al., 1999a). Estimates of
the mean time of peak firing rate and its SD were obtained from the
repetitions (from 8 to 10) of each trial (Fig. 1B). The
timing of the peak consistently shifted closer to collision time as the
parameter l/|v| characterizing the stimulus was
decreased (Fig. 1C). This dependence was studied by plotting
the peak time as a function of l/|v| (see Figs. 3-7). Linear regressions of peak firing time as a function of
l/|v| were performed for each experiment separately (see
Figs. 3-7) to obtain the slope, 
, and the intercept parameter, 
,
of the best linear fit together with their SD (Press et al.,
1992). The angular threshold angles (see Figs. 3-7) and their
SD were computed from using Equation 6 of Gabbiani et al.
(1999a):
|
(1)
|
and by error propagation, respectively (i.e., using the formula
f = |df/dx|·x, for f = f(x)) (Bevington and Robinson, 1992, sect. 3.2).
The delay between the time when the angular threshold size is reached
and the peak firing rate, (see Figs. 3-7), is equal to the
intercept parameter obtained by the linear fit described above. Both
mean threshold angles and mean delay values are illustrated in Figures
4-7 with their SDs. The SDs correspond to 68.3% confidence intervals
on the mean values of these parameters, whereas two SDs correspond to
95.4% confidence intervals (Press et al., 1992, sect.
15.6). All data analysis was performed using Matlab 5.2 (The MathWorks,
Natick, MA).
Two-dimensional modeling of the responses of the LGMD. The
time course of LGMD/DCMD firing rate in response to solid squares or
discs approaching toward the center of the eye along a trajectory perpendicular to the body axis may be described by the following one-dimensional model:
|
(2)
|
In this equation, (g 
exp)(·) = g(exp(·)) represents the composition of the exponential
function and a static nonlinearity g(·) described in
Gabbiani et al. (1999a; their Eq. 11 and Discussion). The term log 
(t ), where (t) = 1/2
(t) is the angular velocity of the edges
of the objects on the retina, represents motion-dependent excitation,
whereas (t ) is an inhibitory term
proportional to the angular size of the object. Equation 2 describes
the approach of a solid square or circle based on the angle that it
subtends with respect to the center of the eye and its temporal
derivative. Hence, it cannot predict the responses to more complex
(e.g., textured) stimuli or to the different approach angles explored in the present study. We therefore generalized Equation 2 to take explicitly into account the two-dimensional projection of the object on
the retina. In the following, we will set the delay parameter = 0 for simplicity. This does not affect the generality of the
arguments presented below because only causes a global translation
of the firing rate along the time axis and thus a translation along the
vertical axis of the linear relationship between the peak firing time
relative to collision and l/|v| (see Figs. 3-7).
Similarly, the static nonlinearity (g exp)(·) does not
influence the prediction of the peak firing rate time by the model
because it is a monotonic function. It was therefore not taken further
into consideration.
The eye was modeled as a hemisphere. Because the eye radius did not
affect the output of the model, it was normalized to R = 1 in the following description (see Appendix 1 for complete equations, including explicitly R). The object was projected
onto the eye surface from the center of the hemisphere (Fig.
2A, left diagram).
The motion-dependent excitatory input was obtained by integrating along
each luminosity edge on the hemisphere the logarithm of the optic flow
generated by the expanding object. Biophysically plausible mechanisms
to compute this quantity have been described in insects
(Hildreth and Koch, 1987; Reichardt et al.,
1988; Zanker et al., 1999). Let
[p0; p1] be a closed
interval and 
(t, p), p 
[p0;
p1] denote a parametrization of the edge. Each value of the parameter p is required to describe the same physical
point on the edge, independent of time. Let
t0(p) = 
/t
(t, p)|t=t0 denote the instantaneous
velocity of expansion of the point (t, p) at time
t = t0 and let
nt0(p) be the outward pointing normal unit vector (Fig. 2A, right diagram). The
excitation, Ex(t0, l/|v|), caused by
the motion of the edge on the retina was obtained by integrating the
logarithm of the optic flow weighted by a factor w((t0, p)) along the luminance
edge:
![<UP>Ex</UP><FENCE>t<SUB>0</SUB>, <FR><NU>l</NU><DE>‖v‖</DE></FR></FENCE>=<FR><NU>1</NU><DE>2&pgr;</DE></FR><LIM><OP>∫</OP><LL>[<UP>p<SUB>0</SUB>;p<SUB>1</SUB></UP>]</LL></LIM>dl<SUB>&ggr;</SUB>(p)·w(&ggr;(t<SUB>0</SUB>, p))<SUP><UP>−1</UP></SUP>](/content/vol21/issue1/fulltext/314/img004.gif)
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(3)
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In this equation dl
(·) denotes the
line element associated with the curve
(t0, ·) on the sphere
(Dubrovin et al., 1991), and 
·; · 
is the Euclidian scalar product. In Equation 3, the optic flow term has
been renormalized by a constant Ccutoff:
|
(4)
|
The constant Ccutoff was chosen so that
log+t0(p);
nt0(p) was equal to zero when
the object subtended a half-angle 1/2cutoff = 1° (corresponding to
the optical resolution of the eye). During the remainder of the
approach trajectory log+t0 (p); nt0(p) was
positive. The weight factor w((t0, p)) depended on the distance of the local motion stimulus to the central axis of the eye. The weight factor mimicks the experimental decrease in
LGMD sensitivity observed as a local motion stimulus is shifted from
the eye center toward the periphery. Experimental evidence suggests
that this sensitivity decreases faster in the vertical than in the
horizontal plane. Iso-contour lines of equal weight were therefore
chosen as elongated ellipsoids (Fig. 2B, right diagram). Let (
) denote the coordinates of
(t0, p) with respect to the coordinate
system illustrated in Figure 2B, (left diagram). Then:
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(5)
|
with a = 2 and
1/2cutoff = 1°. The choice
a = 2 yields the ellipsoidal contour lines illustrated
in Figure 2B, and
1/2cutoff = 1° causes the weight
factor to saturate beyond the optical resolution of the eye. The
dependence of w(, ) on the angular distance to the
eye center in the vertical and horizontal planes is illustrated in
Figure 2C.
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Figure 2.
Two-dimensional modeling of LGMD/DCMD responses.
A, The eye was described as a hemisphere; the image of a
looming square on the eye was obtained by central projection
(left diagram). The excitatory term of the response was
calculated by projecting at each boundary point the instantaneous
expansion vector () onto the unit normal (n) to the
boundary (optic flow vector; right diagram). The logarithm
of the optic flow vector, multiplied by the inverse of a weight factor,
was then integrated along the whole boundary (see Eq. 3). B,
A point on the hemisphere (represented by a cross on the
left diagram) was described by a pair of angles (, ).
is the angle made with the y-axis by the projection of
the point onto the (y, z) plane. is the angle between
the point and the x-axis. This angle is measured in the
plane defined by the x-axis and the projection of the point
onto the (y, z) plane. The right diagram
illustrates ellipsoids of constant weight a = 2 (Eq. 5). C, Value of the weight (a = 2 in Eq. 5)
as a function of the angular distance to the center of the eye [i.e.,
the point with coordinates (1, 0, 0)] in the vertical,
(x, z)-plane [dashed line, (0,  /2), = /2] and the horizontal, (x, y)-plane
[solid line, (0; /2), = 0].
D, Dependence of the inhibitory term in on
the surface subtended by the object on the retina (Eq. 6).
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The size dependent inhibitory term In(t, l/|v|) was
obtained by converting the surface, S(t), covered by the
object on the retina into an equivalent angle,
in(t):
|
(6)
|
Note that in this equation S(t) is normalized by the
surface of the eye (=2 = surface of a hemisphere of radius 1).
The value of in as a function of S is plotted
in Figure 2D. For comparison with experimental data, the
parameter was set equal to 5, a value that lies in the middle of
the typical range observed for a large sample of LGMD neurons
(Gabbiani et al., 1999a, their Fig. 6A).
When simulating approaches from angles other than 90°, the optic flow
was integrated only across that portion of the object that effectively
projected onto the eye. Similarly, only the surface of the object
covering the retina was taken into account in Equation 6.
Appendix 1 gives a precise mathematical analysis of this model for an
approaching circle. Appendix 2 describes the parametrization used for
the edges of approaching squares, as well as the conditions describing
the portion of the approaching square projecting onto the retina for
approach angles different from 90°.
For the concentric square pattern, the excitatory term was obtained by
linear summation of the excitatory terms corresponding to each of the
concentric squares composing the pattern. If we denote by
Exsquare(t, l/|v|) the excitation at time
t attributable to a square of half-size l
approaching at speed |v|, then:
Similarly, the excitation term for the checkerboard pattern,
Excheckerboard(t, l/|v|), was obtained by
taking into account the two additional expanding vertical and
horizontal inner edges. If the corresponding term is denoted by
Exinner(t, l/3·|v|), then:
Appendix 2 explains how Exinner(t,
l/3·|v|) is computed.
The symbolic algebra package Maple (Maple Inc., Waterloo, Ontario,
Canada) was used to obtain numerical values for Ex(t,
l/|v|) and In(t, l/|v|) under the tested
experimental conditions. Although In(t, l/|v|) could be
computed in closed form, the indefinite integral corresponding to
Equation 3 could not be obtained in closed form for approaching
squares. The integrand of Equation 3 was therefore computed
symbolically and then integrated numerically using an 8th order
Newton-Côtes quadrature method (Forsythe et al.,
1977). Numerical calculation of Ex(t, l/|v|) and
In(t, l/|v|) for all condition tested (l/|v| = 10, 20, 30, 40, 50 msec for solid discs, squares, concentric
square patterns, and checkerboard patterns approaching at 90°; same
values of l/|v| for solid squares approaching at angles
from 90° down to 30° in 15° steps), and all combinations of
model parameters tested (a = 1, 2, and
1/2cutoff = 0°, 1° in Eq. 5) took
approximately 1 week on a computer equipped with a 450 MHz Pentium III
processor (Intel, Santa Clara, CA). None of the results depended
qualitatively on the choice of a or
1/2cutoff in Equation 5. Maple simulation
files and Matlab functions reproducing Figures 8 and 9 may be obtained
on the World Wide Web at http://glab.bcm.tmc.edu by following the
"Publications" link and selecting the page corresponding to the
title of this article.
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RESULTS |
The results described below are based on recordings and complete
data analysis from 58 different animals. Part of the data used in Fig.
3D was collected in 35 additional animals (S. americana) for different
purposes.
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Figure 3.
The same linear relationship between time of peak
firing rate and l/|v| is found across locust species.
A-C, Plot of the time of peak firing rate as a function of
l/|v| (mean ± SD) in three animals belonging to
three different locust species (A, L. migratoria,
= 2.7 ± 0.5, = 8.7 ± 7.9 msec, same
animal as in Fig. 1; B, S. americana, = 3.5 ± 0.8, = 6.2 ± 13.1 msec; C, S.
gregaria, = 3.3 ± 0.9, = 9.3 ± 16.7 msec). D, Values of the mean angular threshold size and the
mean delay between angular threshold and peak firing rate in 64 animals
of the species S. americana, 5 animals of the species
S. gregaria, and 11 animals of the species L. migratoria.
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Is the angular threshold computation species dependent?
The response of LGMD/DCMD in L. migratoria to the
simulated approach of a solid square on a collision course with the
animal is illustrated in Fig. 1A. The time course of visual
stimulation on the retina is characterized by the ratio of the
half-size of the square, l, and its approach velocity,
|v|, (Fig. 1A, inset) [see also
Materials and Methods and Gabbiani et al. (1999a)]. The
response starts early during the approach, when the object reaches
approximately 10° in visual angle. It gradually increases and reaches
its peak value before collision for large values of l/|v| (
10 msec) (Fig. 1A-C). Each
stimulation was repeated 8-10 times to obtain estimates of the mean
time of peak firing rate and its SD (Fig. 1B). As may be
seen in Figure 1C, the time of peak firing rate relative to
collision () decreases systematically as the stimulation parameter
l/|v| decreases (i.e., as the approach velocity,
|v|, of the moving object increases for a fixed object half-size, l). Figure 3A replots the time of peak
firing rate, tpeak, obtained from Figure
1C as a function of the kinematic parameter
l/|v| characterizing the approach [see also
Gabbiani et al. (1999a), their Figs. 3, 4]. To a very
good approximation, tpeak follows a linear
relationship with l/|v|. Similar results have been
reported for S. americana (Gabbiani et al.,
1999a) and are illustrated in Figure 3B for one
specimen of this species. Furthermore, as reported in Gabbiani
et al. (1999a), this linear relationship between
tpeak and l/|v| implies that the
peak in firing rate occurs a fixed delay (5-40 msec on average
in the present experiments) after the object has reached a fixed
angular threshold size (between 15° and 40° on average) on the
locust's retina, independent of the value of l/|v|
(Gabbiani et al., 1999a, their Fig. 5). To test further
the linear dependence of peak firing time on l/|v|, an
additional eight animals were presented looming stimuli at 10 values of
l/|v| with a protocol identical to the one used in
Gabbiani et al. (1999a, their Fig. 4). The results of
these experiments were also similar to those reported in S. americana. Thus, in L. migratoria as in S. americana, the peak firing rate can be seen as the output of an
angular threshold detector and may contribute to trigger visually
evoked escape behaviors [Hatsopoulos et al.
(1995); see Gabbiani et al. (1999a) for a
detailed discussion]. To ascertain whether this is also the case in a
second African locust species, we performed similar experiments on
S. gregaria. Figure 3C illustrate the results of experiments performed on one animal of this species. In all experiments analyzed, the same linear relationship was found in both the American and African species. Figure 3D plots the mean angular
threshold sizes and delays observed for S. gregaria (n = 5, ) and L. migratoria (n = 11, 
).
These values are well into the range observed in a large sample of
animals from the American locust S. americana (n = 65, 
).
Angular threshold computation and target shape
Shape differences between solid objects looming toward the
locust's eye translate into temporal and spatial differences in the
stimulation of ommatidia on the retina. In the case of a disc and
square having the same radius (or half-size), l, the
temporal sequence of light intensities during approach is the same only along the two horizontal and vertical symmetry axes passing through the
center of the objects. For a disc, the angular velocity of expansion on
the retina at any given time points out from the center with a
magnitude that is constant along the whole disc boundary. Because this
vector is always perpendicular to the disc boundary, it coincides with
the optic flow vector (i.e., the projection of the expansion velocity
vector onto the unit normal to the boundary) that can be measured
locally at the retina (Appendix 1). By contrast, the angular
velocity of expansion of a square is not always perpendicular to the
square edges and its magnitude is greatest at its four corners (Fig.
4A, inset). After
projection on the unit normal to the boundary, the optic flow vector
magnitude ison the contrarygreatest at the middle of an edge rather
than at its corners. Both quantities may be computed from the equations given in Appendix 2. The ratio of the optic flow magnitude at the
corner of a square to that of a disc decreases monotonically during
approach from a peak value of 1 when both objects are far away to a
minimum value of 0.5 at collision time. Thus, the motion stimulus
generated by an expanding square is always less strong than the one
generated by a disc, whereas the surface that a square of half-size
l subtends at the retina is always larger than the one
subtended by a disc of radius l.
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Figure 4.
The linear relationship between time of peak
firing rate and l/|v| is the same whether the target is
a disc or a square. A, Plot of peak firing rate as a
function of l/|v| (mean ± SD) for a square (target
1:  , = 5.1 ± 0.8, = 11.8 ± 12.8 msec)
and a disc (target 2: , = 4.7 ± 0.7, = 11.9 ± 12.1 msec) measured in the same preparation. For clarity,
only the largest SD is shown in one direction for each values of
l/|v|. Bottom inset, The rate of angular expansion
of a square target early during approach is largest at its corners
where it equals 1.35 times the rate of expansion of a disc (for
l/|v| = 50 msec and t = 225 msec,
approximate peak time). At this time, the optic flow vector at the
corner (dotted line) is still almost identical, however, to
the one of a disc. B, Plot of the delay versus the angular
threshold (mean ± SD) for approaching squares and discs
(illustrated by and , respectively) measured in five different
preparations (S. americana). Experiments performed on the
same animal are connected by dashed lines. Solid lines
illustrate SDs in one direction only for clarity. The extensive overlap
of SDs for experiments performed on the same animal indicate no
significant differences between the two stimulation conditions.
C, Plot of the delay (mean ± SD) for target 2 versus
target 1 for the same five preparations as in B. The
dashed diagonal line represents identity. Delays cluster
within 1 SD of the diagonal, indicating no significant differences
between the two conditions. D, Plot of the threshold angle
(mean ± SD) for target 2 versus target 1 for the same five
preparations as in B. In three of five experiments,
threshold angles cluster within 1 SD from the diagonal, indicating no
significant differences between the two conditions. The remaining two
cases were not statistically different at the 95.4% confidence level.
Note that mean threshold angles for square targets are consistently
smaller than those for discs (i.e., all points lie above the
diagonal).
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We investigated whether such shape differences affected the timing of
peak firing by presenting to the same animal square () and circular
() targets interleaved pseudo-randomly over the course of an
experiment. Figure 4A illustrates the outcome in one
preparation: the differences in mean times of peak firing rate measured
with both objects could not be distinguished from the intertrial
variability (SD) (Fig. 4A, solid bars). Figure 4B summarizes the outcome of five such experiments in
different preparations by plotting mean angular thresholds versus mean
delays obtained for square () and circular () targets and their
SDs (solid lines parallel to the axes). For each mean angle
and delay pair, the SDs outline the boundary of the two-dimensional
68.3% confidence region on their value (Gabbiani et al.,
1999a, their Fig. 4B; Press et al., 1992, sect.
15.6). As can be seen from the plot, the two-dimensional
confidence regions strongly overlap, and therefore mean angular
thresholds and delays were not statistically different between the
responses to these two stimulus conditions. Figure 4C plots
the mean delays for square versus circular targets and their SDs along
the delay axis. No significant trend could be observed, and the data
always lay within 1 SD of the diagonal characterizing equal delays
under both conditions. By contrast, the mean angular thresholds were
consistently above the diagonal (Fig. 4D), corresponding to
smaller average threshold angles, or equivalently earlier peak times
for square targets. Differences in mean threshold angles were not
significant in single preparations. In two cases, the SDs of the
threshold angles did not intercept the diagonal, but these two points
were located within two SDs of the diagonal and were thus not
significantly different at the 95.4% level. The difference in mean
threshold angles averaged over all preparations was small: 3.2 ± 2.1° (mean ± SD), but statistically significant (paired
t test, p < 0.005). Thus, these results
point to a small (i.e., not statistically significant in single
preparations) but consistent trend across preparations of higher
threshold angles and later peak firing times for disc targets.
Angular threshold computation and target texture
Variations in target texture will also result in different
temporal and spatial activation of ommatidia on the locust's retina. We investigated whether such changes affected the time of peak firing
rate during approach by comparing the responses obtained for solid
square targets to two different textured targets: a checkerboard
pattern (CBP) (Fig. 5A,
inset) and a concentric squares pattern (CSP) (Fig.
6A, inset). For the
CBP, the inner edges move in opposite directions as the object grows on
the retina, whereas for the CSP, inner edges consistently follow
the leading edges of the square although at a different
instantaneous angular velocity. In both cases, the luminance of the
pattern object was set at 30% (dark gray) and 0% (black) of the
background luminance (95 cd/m2); similar results
were obtained in two animals with CBP relative luminances of 70%
(light gray) and 0%. Figures 5A and 6A
illustrate the outcome of one experiment for the CBP (Fig.
5A, 
) and the CSP (Fig. 6A, ),
respectively. In both cases, the time of peak firing rate was
statistically indistinguishable from the controls obtained with black
square targets (Figs. 5A, 6A, ). Figures 5B
and 6B plot mean angular threshold versus delay obtained
with the CBP (, n = 10 animals) and the CSP (,
n = 12 animals), respectively, and compares them with
those obtained in the same preparations with black square targets
(). Experiments performed on the same preparation are connected with
a dashed line. In both cases, no consistent changes were
observed with respect to the control condition (). This was
confirmed by first plotting the mean delays and SDs obtained in the
control condition versus those of the two test conditions (CBP and
CSP). Figures 5C and 6C illustrate the results
for the CBP and CSP, respectively: the mean delays in the test versus
control conditions lie within 1 SD of the diagonal and therefore are
not significantly different from one another. Figures 5D and
6D illustrate that the mean threshold angles were also not
statistically different from those obtained in control experiments. Two
mean threshold angles of 12 preparations lay >1 SD away from the
diagonal, but <2 (Fig. 6D). Therefore the difference was
not statistically significant at the 95.4% confidence level.
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Figure 5.
The linear relationship between time of peak
firing rate and l/|v| does not change with changes in
target texture. A, Plot of peak firing rate as a function of
l/|v| (mean ± SD) for a square target (target 1:
, = 6.1 ± 0.8, = 26.1 ± 10.7 msec) and
a checkerboard textured target (target 3: , = 6.1 ± 0.8, = 30.6 ± 13.2 msec) measured in the same
preparation (S. americana). For clarity only the largest SD
is shown in one direction for each value of l/|v|. B,
Plot of the mean delay for target 1 () and target 3 () versus
mean angular threshold in 10 different preparations (5 S. americana and 5 S. gregaria). Experiments performed on
the same preparation are connected by a dashed line.
C, Plot of the delay (mean ± SD) for target 3 versus
the delay for target 1 in the same 10 preparations as in B.
For clarity, SDs are illustrated in one direction only. The
dashed diagonal line represents identity. Delays cluster
within 1 SD of the diagonal, indicating no significant differences
between the two conditions. D, Plot of the threshold angle
(mean ± SD) for target 3 versus target 1 in the same 10 preparations as in B. Same illustration conventions as in
C. Threshold angles also cluster within 1 SD of the
diagonal, indicating no significant differences between the two
conditions.
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Figure 6.
The linear relationship between time of peak
firing rate and l/|v| remains unchanged when several
edges are added to the looming target. A, Plot of the peak
firing rate as a function of l/|v| for a square target
(target 1: , = 2.7 ± 0.5, = 8.7 ± 7.9 msec) and a textured target consisting of four concentric squares
(target 4: , = 3.5 ± 0.8, = 18.5 ± 10.2 msec) measured in the same preparation (L. migratoria).
For clarity, only the largest SD is shown in one direction for each
value of l/|v|. B, Plot of the mean delay for target 1 () and target 4 () versus mean angular threshold in 12 different
preparations (9 S. americana and 3 L. migratoria). Experiments performed on the same preparation are
connected by a dashed line. C, Plot of the delay
(mean ± SD) for target 4 versus the delay for target 1 in the
same 12 preparations as in B. For clarity, SDs are
illustrated in one direction only. The dashed diagonal line
represents identity. Delays cluster within 1 SD of the diagonal,
indicating no significant differences between the two conditions.
D, Plot of the threshold angle (mean ± SD) for target
4 versus target 1 in the same 12 preparations as in B. Same
illustration conventions as in C. Threshold angles cluster
within 1 SD of the diagonal, indicating no significant differences
between the two conditions.
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Angular threshold computation and target approaching angle
We investigated the effect of the angle of approach on LGMD/DCMD
firing by rotating the animal around a virtual vertical axis passing through the center of the stimulated eye. This allowed us
to present looming targets from a wide range of approaching angles.
Let 0° be the position at which the animal's front faces the
stimulation screen and positive angles denote counterclockwise rotation. If we consider the right eye, the angular range covered target approach angles from the back (135°) on the same side as the
stimulated eye to 45° on the contralateral side (Fig.
7A, inset).
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Figure 7.
The linear relationship between time of peak
firing rate and l/|v| is invariant over a wide range of
object approach angles, whereas the number of spikes decreases with
presentation angle. A, Plot of the peak firing rate as a
function of l/|v| (mean ± SD) for a target
approaching from the front (0°: , = 3.6 ± 0.9, = 10.7 ± 17.8 msec) and for a target approaching from
the side (90°: , = 3.7 ± 0.6, = 8.9 ± 11.2 msec). For clarity, only the largest SD in one direction is
shown at each value of l/|v|. Inset, Schematic
diagram illustrating the definition of the approach angle with respect
to the screen and the animal. Counterclockwise rotation from 0°
defines positive angles. B, Plot of the delay versus angular
threshold (mean ± SD) for the same two approaching directions as
in A, measured in five preparations (S. americana). Experiments performed on the same animal are connected
by a dotted line. Solid lines illustrate SD in
one direction only for clarity. The extensive overlap of SDs (with one
exception) for experiments performed on the same animal indicate no
significant differences between the two stimulation conditions.
Arrow, Angle that lies at >1, but <2 SD of its control.
C, Number of spikes (mean ± SD) elicited per trial as
a function of presentation angle at three different values of
l/|v| (50 msec: ; 30 msec: ; 10 msec: )
measured in the same preparation. D, Number of spikes
(mean ± SD) elicited as a function of presentation angle recorded
simultaneously from both DCMDs, the axons of which run in the
right () and the left () connectives
(l/|v| = 30 msec). The sum of both mean spike counts
() is approximately independent of the presentation angle.
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Figure 7A illustrates an experiment in which the animal was
stimulated at 0° (, facing the screen) and at 90° (,
perpendicular to the screen). No statistical differences were observed
in the timing of the peak firing rate under these two conditions. The variability in the time of peak firing rate, however, usually increased
from 90° to 0°. In Figure 7A for example, the largest SDs in the peak firing time at each value of l/|v| were
in all but one case (l/|v| = 20 msec) attributable to
stimulation at 0°. Figure 7B summarizes the results of 5 similar experiments of 13 that had the most reliable responses at 0°.
The mean angular thresholds and delays under these two experimental
conditions are plotted together with their SDs, and experiments
performed on the same animal are connected by a dotted line.
There were no statistical differences between the mean delay values in
all cases. In one experiment (see , top left), the mean
threshold angles were different at the 95.4% level, whereas the
remaining threshold angles showed no statistically significant
differences (one mean threshold angle, indicated by an
arrow, lay within 2 SDs of its control experiment and was
therefore not statistically different at the 95.4% confidence level).
Here also, in all but one case, variability increased from 90° to
0°. This increase in variability was correlated with the number of
action potentials elicited per stimulus presentation at various
approach angles. Figure 7C plots for one experiment the
mean number of action potentials (±SD) elicited at three values of
l/|v| for looming black squares with approach angles
135°, 90°, 45°, and 0°. As in Figure 7, A and
B, no statistically significant differences were observed in
the timing of the peak firing rate in all cases analyzed (n = 5). Furthermore, the number of action potentials elicited per trial was remarkably constant over a large portion of the visual field
ranging from 135° to 45°. Between 45° and 0°, however, the response declined abruptly, by >50% in this experiment; similar results were obtained in the remaining four experiments. In the experiment illustrated in Figure 7C, there was a clear
decrease in the number of action potentials elicited per trial as
l/|v| decreased from 50 to 10 msec. However, this
observation could not always be made so clearly on other preparations
(Gabbiani et al., 1999a).
This decrease in the number of spikes per trial produced at 0° was
compensated for by the contralateral DCMD. Figure 7D
illustrates the number of action potentials (±SD) recorded in each of
the two DCMDs during an experiment in which the target approaching angle was varied between 5 values in the frontal visual field: 45°,
22.5°, 0°, 22.5°, and 45°. The response recorded in the left connective () decreased from 45° to 0°, as in Figure
7C, and essentially vanished at 45°. Concomitantly, the
number of action potentials recorded from the right connective ()
increased over the same range, resulting in the total number of action
potentials elicited from both DCMDs per trial () being constant over
the entire range (45°-45°). Similar results were obtained in
n = 11 additional animals.
Two-dimensional linear summation model for motion-dependent
excitation and size-dependent inhibition
The experimental results described above were compared with those
obtained in a two-dimensional model of the responses of the LGMD to
looming stimuli. The motion-dependent excitatory input of the model was
obtained by integrating the logarithm of the optic flow along the
luminance edges of the expanding object weighted by a factor depending
on the distance of the local motion stimulus to the central axis of the
eye (Figs. 2A-C and Eqs. 4 and 5). When the approaching
object contained additional luminance edges, such as the CBP and the
CSP described above, excitatory inputs were summed linearly for all
luminance edges. The size-dependent inhibitory input was obtained at
each time point by computing the surface covered by the object on the
retina and transforming it by using the nonlinear function plotted in
Figure 2D. Figure 8,
A and B, illustrates the time course of
excitation and inhibition for a disc approaching perpendicular to the
body axis toward the center of the eye. The excitatory and inhibitory
terms were subtracted from each other (Fig. 8C), and the
times of peak firing rate were computed and plotted as a function
l/|v| (Fig. 8D, ). As may be seen from
this Figure, the prediction of the model is a linear function of
l/|v|, although the slope of the linear relation is slightly smaller (4.69) than the prediction of the one-dimensional model [dashed line; slope identical to = 5) of
Gabbiani et al. (1999a)]. This, in turn, corresponds to
a slightly larger threshold angle (24° vs 22°). The difference,
however, is well within the uncertainty of the experimental estimates
for these parameters (Fig. 4B, D). As proven in Appendix 1,
the two-dimensional model reduces exactly to the one-dimensional model
for a looming disc if a less realistic choice of parameters is made,
corresponding to circular iso-contour lines for the weight factor
(a = 1 in Eq. 5) and no weight saturation beyond the
optical resolution of the eye
(1/2cutoff = 0° in Eq. 5).
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Figure 8.
Model response to a disc looming toward the animal
along a trajectory perpendicular to the body axis. A, Time
course of excitation before collision for the five values of
l/|v| used experimentally. B, Time course of
inhibition before collision. C, Model output obtained by
linear combination of excitation and inhibition. D, Peak
time (obtained from model, C) as a function of
l/|v|. The dashed line is the prediction of
the one-dimensional model that matches experimental data.
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Figure 9 summarizes the results for
the remaining stimuli used in our experiments. As may be seen by
comparing Figure 9A with Figure 8D, the time of
peak firing rate for a square is closer to the prediction of the
one-dimensional model, and accordingly the slope is higher (4.86)
corresponding to a smaller threshold angle (23°). These results are
consistent with those observed experimentally, although the difference
in threshold angles in the model (~1°) is smaller than the mean
value derived from Figure 4D (~3°). In the model, the
earlier peak firing time for a square stimulus versus a disc is not
surprising: as explained in Results, the optic flow associated with a
square is smaller than that of the corresponding disc and its surface
is larger. Consequently, inhibition will start to dominate the
excitatory term earlier, leading to earlier peak times.
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Figure 9.
Peak time (model) as a function of
l/|v| for the various experimental targets.
A, Square target. B, CBP. C, CSP.
D, Solid square approaching at three different angles
(75°, 45°, and 0°) measured from the eye center in the horizontal
plane. In A-D, the dashed line is the prediction
of the one-dimensional model that matches experimental data.
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Figure 9, B and C, illustrates the results for
the CBP and the CSP patterns: the time of peak firing still varies
linearly with l/|v|, but as the strength of the
excitatory term is increased by the additional moving edges in the
stimulus, the timing of the peak moves closer to collision time. The
values of the slopes in these two cases (1.56 and 0.92) correspond to
predicted threshold angles of 65° and 95°, respectively, that are
incompatible with the experimental observations of Figures 5 and 6.
Similarly, presentation of square targets approaching at different
angles disrupted significantly the predictions of the model for
deviations as small as 15° with respect to the 90° approach
direction (Fig. 9D) (the slope equals 3.73 and the value of
the threshold angle is 30°). Approach angles of 45° and 0°, such
as those used experimentally, could not be explained by the linear
model (slopes of 1 and 0 corresponding to threshold angles of 90° and
180°, respectively).
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DISCUSSION |
TOP
ABSTRACT
INTRODUCTION
