Invariance of angular threshold computation in a wide-field looming-sensitive neuron

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 degrees 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.

Fig. 1.

Responses of DCMD to simulated approaches of looming squares (L. migratoria). A, The time course of the angular size, θ(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.

Fig. 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 (γ˙) 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 theleft diagram) was described by a pair of angles (ϑ, ϕ). ϕ is the angle made with the y -axis by the projection of the point onto the (y, z) plane. ϑ 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 diagramillustrates 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, ϑ ∈ (0, π/2), ϕ = π/2] and the horizontal, (x, y) -plane [solid line, ϑ ∈ (0; π/2), ϕ = 0]. D, Dependence of the inhibitory term θ_in on the surface subtended by the object on the retina (Eq. 6).

Fig. 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, α = 2.7 ± 0.5, δ = 8.7 ± 7.9 msec, same animal as in Fig. 1; B, S. americana, α = 3.5 ± 0.8, δ = 6.2 ± 13.1 msec; C, S. gregaria, α = 3.3 ± 0.9, δ = 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 speciesS. gregaria, and 11 animals of the species L. migratoria.

Fig. 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: ■, α = 5.1 ± 0.8, δ = 11.8 ± 12.8 msec) and a disc (target 2: ●, α = 4.7 ± 0.7, δ = 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 linesillustrate 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. Thedashed 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).

Fig. 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: ■, α = 6.1 ± 0.8, δ = 26.1 ± 10.7 msec) and a checkerboard textured target (target 3: ▵, α = 6.1 ± 0.8, δ = 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. Thedashed 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.

Fig. 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: ■, α = 2.7 ± 0.5, δ = 8.7 ± 7.9 msec) and a textured target consisting of four concentric squares (target 4: ▵, α = 3.5 ± 0.8, δ = 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 linerepresents 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.

Fig. 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°: ●, α = 3.6 ± 0.9, δ = 10.7 ± 17.8 msec) and for a target approaching from the side (90°: ▵, α = 3.7 ± 0.6, δ = 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 theright (●) and the left (★) connectives (l/‖v‖ = 30 msec). The sum of both mean spike counts (▵) is approximately independent of the presentation angle.

Fig. 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.

Fig. 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.