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FIGURE 7.1-4 Another Zorkoczy system that uniquely senses unit dark spots moving with v = +1 across the receptive field. See text for analysis.

moving objects in such a manner as to keep the image on the same part of their compound eyes, much the same way humans track a moving object with their eyes and head to keep an image on their foveas.

One early method used to measure the optomotor behavior of beetles used the Y-globe maze. A beetle was suspended inside a vertically striped drum by gluing its back to a fixed, vertical support. The beetle reflexively grasped a Y-globe maze and began to walk. (See Figure 7.1-5 for a description of this setup.) If the drum were stationary, there is about a 50% probability that the beetle will turn right vs. left at each Y junction. When the striped drum is given a uniform velocity, e.g., clockwise looking down on the beetle, the beetle tends to select more right turns than left, trying to follow the stripes. As a measure of the beetle's directional preference, workers have defined a "reaction" parameter, R = (W - A)/(W + A), where in an experimental run with the drum turning at fixed velocity, W is the number of times the beetle turns with the direction of the drum, and A is the number of times it turns opposite to the drum. A + W = N, the total number of times the beetle turns at a Y. It has been found experimentally when plotting R(0) vs. log(0) that R makes a bell-shaped curve with a peak whose value depends on stripe period and contrast. Figure 7.1-6 illustrates the optomotor reaction for the milkweed bug, Oncopeltus fasciatus,

FIGURE 7.1-5 Schematic drawing of a Y globe optomotor testing apparatus. A restrained beetle holds a Y globe while viewing a moving object (in this case, vertical stripes moved at constant velocity toward the beetle's right). As the beetle "walks," it rotates the Y globe toward it. When the beetle comes to a Y junction, the beetle generally chooses the right-hand path if the stripes are moving to the right. That is, it tries to follow the stripes.

FIGURE 7.1-5 Schematic drawing of a Y globe optomotor testing apparatus. A restrained beetle holds a Y globe while viewing a moving object (in this case, vertical stripes moved at constant velocity toward the beetle's right). As the beetle "walks," it rotates the Y globe toward it. When the beetle comes to a Y junction, the beetle generally chooses the right-hand path if the stripes are moving to the right. That is, it tries to follow the stripes.

responding with a Y maze to different rotation velocities of a surrounding drum with black/white stripes with a 20° period (Bliss et al., 1964). Figure 7.1-7 illustrates a typical Chlorophanus beetle's, Y- maze, optomotor R-response to continuously moving, sinusoidal stripes of fixed spatial wavelength (Reichardt, 1964). Note that the curve has a single broad peak and falls off for high drum velocity.

The heads of most insects are so articulated that they can swing from side to side in the horizontal plane (yaw), rotate (roll) around the body axis, and nod up and down (pitch). John Thorson (1966a, b) examined the head optomotor response in locusts by measuring the head roll torque in response to sinusoidal striped drum rotation. The drum rotational axis was aligned with the animal's long body axis, and the drum was centered on and enclosed the head. The animal's body was fixed down, but the head was free to turn. A torque-measuring sensor was attached to the front of the locust's head. In one set of data, the stripe period was 8°. The peak-to-peak amplitude of the striped drum oscillation was a tiny 0.03°. Even at this very low input amplitude, the reflex head roll torque was measured reliably over a drum frequency range of 0.01 to 6 Hz. In most cases, the ratio of peak-to-peak, sinusoidal neck torque to peak-to-peak sinusoidal drum amplitude peaked at ~0.5 Hz. The phase was nonminimum. Thorson (1966b) gave several Bode plots of 20 log (peak-to-peak neck torque/reference torque) showing this peak response. The Bode plot

FIGURE 7.1-6 Record of Y globe turning reaction index, R, for the milkweed bug, Onco-peltus fasciatus, as a function of stripe velocity. The restrained bug and its Y globe were in the center of a rotating, striped drum. Note that the striped drum speed that elicits a maximum turning reaction is between 10° and 20°/s. The stripe period was 18°. R = (C - I)/(C + I), where C is the total number of correct turns the beetle makes at a "Y" (i.e., to the right if the stripes are rotating to the right, as seen by the beetle) and I is the total number of incorrect turns at Ys (to the left). (From Bliss, J.C. et al. Final Report for Contract AF49C638)-1112, Stanford Research Institute, Menlo Park, CA, 1964.)

DRUM SPEED — degrees/sec

FIGURE 7.1-6 Record of Y globe turning reaction index, R, for the milkweed bug, Onco-peltus fasciatus, as a function of stripe velocity. The restrained bug and its Y globe were in the center of a rotating, striped drum. Note that the striped drum speed that elicits a maximum turning reaction is between 10° and 20°/s. The stripe period was 18°. R = (C - I)/(C + I), where C is the total number of correct turns the beetle makes at a "Y" (i.e., to the right if the stripes are rotating to the right, as seen by the beetle) and I is the total number of incorrect turns at Ys (to the left). (From Bliss, J.C. et al. Final Report for Contract AF49C638)-1112, Stanford Research Institute, Menlo Park, CA, 1964.)

Reaction index, R

Reaction index, R

FIGURE 7.1-7 Representative graph of the average Y globe turning index, R, of the beetle Chlorophanus, as a function of drum speed. (Based on a graph from Reichardt, 1964.) A vertical, sinusoidal intensity pattern with a 4.7° period was used.

FIGURE 7.1-7 Representative graph of the average Y globe turning index, R, of the beetle Chlorophanus, as a function of drum speed. (Based on a graph from Reichardt, 1964.) A vertical, sinusoidal intensity pattern with a 4.7° period was used.

sloped up to the peak at about +20 dB/decade at low frequencies, and the high-frequency attenuation slope was between 30 and 40 dB/decade. Response was down about 30 dB from the peak at 5 Hz.

Thorson (1966a) examined the stripe period that gave maximum neck torque response at a given oscillation frequency (0.25 Hz) and amplitude (0.03°) for both roll and yaw optomotor responses. For yaw, the peak torque amplitudes occurred for stripes with periods between 7.5° and 9°; peak roll torques required stripes with periods between 9.5° and 12°. The cutoff period was about 4° for roll torque, and 3° for yaw torque. These values agree with the limiting resolution for the locust's third cervical nerve electrophysiological responses to moving stripes. (The left and right, third cervical nerve innervate neck muscles that move the locust's head.)

Northrop (1975) found that below about a 4° stripe period, there was no significant firing on the third cervical nerve when the stripes were moved in front of one eye (the other was covered).

A freely-flying fly's body has six degrees of freedom: rotational — roll, pitch, and yaw — and linear translational — forward, sideways, and vertical. Dragonflies are adept at lateral movements; houseflies are not. In addition to the six degrees of freedom for body movement, the fly's head can also move with respect to the body. Such head movements are roll around the body axis, side-to-side movement (yaw), and up and down (pitch). Thus, to describe the motions of a freely flying fly, nine vectors or dimensions as functions of time are required. Clearly, tethered flight under conditions of head immobilization offers great simplification by restricting measured parameters to either yaw torque or thrust and lift forces.

In a series of papers beginning in 1956, Werner Reichardt and co-workers extensively investigated the optomotor responses of tethered, flying flies. A fly's back was glued to a vertical probe attached to a torque sensor that measures visually induced yaw torque. The fly is suspended in the center of a display cylinder having one or more contrasting stripes, and caused to beat its wings as if in flight. When the stripes or visual object are moved at constant velocity, or oscillated back and forth sinusoidally, the fly generates yaw torque trying to follow the stripe(s). Rei-chardt's experimental system was similar to Thorson's described above, except the fly flies in place, its head fixed rigidly to its body. An overview of Reichardt's work can be found in chapter 17 of Sensory Communication (1964). The underlying model for fly optomotor response in this and all of Reichardt's many papers on fly opto-motor response is his dyadic, directional correlator (DDC) model, first proposed in 1956 to describe optomotor behavior. The pooled outputs of many parallel DDCs are postulated to drive the motoneurons responsible for optomotor turning. A simplified example of the DDC is shown in Figure 7.1-8. Unlike the Zorkoczy models, the Reichardt directional correlator is a purely analog system. The two inputs are signals proportional to the luminous flux on two adjacent "receptors." The model receptors are aligned along the direction of the object motion that is to be sensed. They have a linear spacing of 8mm. The analog signal output of the A receptor is delayed by e seconds, and is multiplied by the direct analog output of receptor B. The output of the multiplier, y, is subtracted from the output of the right-hand multiplier, y', and the difference, Ay, is time-averaged by a low-pass filter to form Ay. The mathematics of this model system are examined below.

In deriving an output/input relation for the DDC, for simplicity, assume one-dimensional, linear geometry, rather than the natural angular coordinates. Also, assume that the two receptors determine a line in the direction of object motion; and assume the object is a one-dimensional, moving, sinusoidal distribution of intensity:

I(x, t) = Io{1 + sin[(2n/X)(x - v t)]} = Io{1 + sin[(2nx/X) - rat]} 7.1-10

where X is the sine period in mm, v is the translational velocity of the moving sinewave in mm/s, and the temporal frequency of the sinewave is ra = 2nv/X rad/s. The object intensity ranges from 0 to 2Io, depending on the spatiotemporal argument of the sin(*)

FIGURE 7.1-8 Block diagram of a DDC. It is assumed that many DDCs exist in a compound eye, and that their outputs are effectively added together. The two receptors of every DDC are aligned in the preferred direction, and separated by 8°. A detailed algebraic analysis of the operation of a DDC's is given in the text. The DDC was first proposed by Reichardt in the mid-1950s.

FIGURE 7.1-8 Block diagram of a DDC. It is assumed that many DDCs exist in a compound eye, and that their outputs are effectively added together. The two receptors of every DDC are aligned in the preferred direction, and separated by 8°. A detailed algebraic analysis of the operation of a DDC's is given in the text. The DDC was first proposed by Reichardt in the mid-1950s.

function. Note that the traveling wave can be decomposed by trigonometric identity to functions of time multiplied by functions of distance. Because sin(A - B) = sin A cos B - cos A sin B, one can write:

I(x, t) = Io{1 + sin(2nx/X) cos(rat) - cos(2nx/X) sin(rat)} 7.1-11

The output of receptor A is delayed by e seconds (pure transport lag); the output of receptor B is displaced in space by 8 mm, so one can write the left multiplier output, y, as y = k2l2 {1 + sin(2nx/ X) cos[co(t - e)j - cos(2nx/ X) sin[co(t - e)]}

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