## ABSTRACT

The house-fly exhibits the landing reaction when an expanding optical pattern is placed before it.

An expanding pattern which is easy to measure and which keeps the level of illumination constant is created by rotating spirals.

In analysing the dependence of landing reaction on the spiral pitch and multiplicity and on the frequency of local stimuli on each ommatidium, (

*a*) interference by the tangential characteristics of the spiral has been observed; (*b*) stimulus efficiency has been found to depend on stimulus wave-length; (*c*) the relevance of the above factors and of a temporal parameter influencing both optomotor and landing reactions has been evaluated.The fact that no Moiré effect has been observed is explained in terms of characteristics of stimuli used.

The dependence of pattern efficiency on the position of the stimulus in the visual field has been ascertained.

Spatial integration of local landing stimuli has been shown in various cases and its threshold excitability and saturation point have been obtained. The dependence of these two last parameters on the position of the stimulus in the visual field has been ascertained.

## INTRODUCTION

The functional structure of the mechanism underlying the perception of motion and its processing by the central nervous system has thus far been analysed in the beetle *Clorophanus viridis*(Hassenstein & Reichardt, 1956; Reichardt & Vaijú, 1959; Reichardt, 1962), in the fly *Musca domestica* (Hertz, 1934a; Fermi & Reichardt, 1963; McCann & McGinitie, 1965; Bishop & Keehn, 1967) and in other compound-eye animals (Hertz, 1934*b*; Burtt & Catton, 1954; Horridge, 1966; Thorson, 1966; Wallace, 1959).

It is of interest therefore to ascertain whether such a mechanism is applied to the perception of more complicated dynamical features.

One environmental situation in which flying insects need the perception of one of such dynamical features is during their preparation for landing.

When a fly approaches a surface on which to land, at a certain distance from the surface it lifts its first pair of legs to both sides of the head and stretches the last pair backward. This characteristic behaviour has been called the landing reaction; it occurs not only in the house-fly but commonly in all flying insects.

Some investigations (Goodman, 1960, 1964; Braitenberg & Taddei Ferretti, 1966) of the landing reaction have proved that there are at least two components of the stimulus for landing. One of them is the expansion of a pattern on the fly’s compound eye; and it is known that the greatest distance from the pattern at which the fly reacts is proportional to the expansion speed of the pattern. The other is the variation of the light flux coming from the pattern; and it is known that the fly reacts to the decrement of light flux.

Generally, patterns which can be expanded involve a change in light flux, thus preventing independent analysis of these two components of the stimulus. An example of such a stimulus is an object running up and away from the fly; change in object area, for a given contrast with the surrounding, involves a change in light flux. Another such stimulus is a cathode ray tube with expanding circles on its screen; the increase in the circumference of the circles increases the light flux.

In the present research the two components of the optical stimulus have been separated and the dependence of the landing reaction on each of them has been analysed.

## METHODS

The present investigation concerns the dependence of the landing reaction on the speed of the expansion.

An expansion pattern which does not involve any change in the light flux is obtained (Taddei Ferretti & Fernandez Perez de Talens, 1967) by rotating—on its axis—a disk on which 2 m arithmetical spirals are drawn; these spirals are defined by *r*(*θ*) *= pθ/*2*π*, where *r* and *θ* are angular co-ordinates and *p* is the constant pitch. The pitches used vary between 1, 6 and 8 cm. Furthermore, the spirals are separated in phase by *π*/m (where *m* is an integer called multiplicity: the multiplicities used varying between 1 and 8) and the zones between one spiral and the next are alternately painted black and white. Such a disk will be specified by the values of *p* and *m*, and will be designated *p*/*m*.

When the fly is placed at a certain distance from a disk, in front of the centre of the disk, the reaction will take place only if the rotational frequency of the disk is higher than a minimum value a *ω*.

If, alternatively, the rotational frequency is fixed, the fly reacts only if it stays at a certain distance from the disk which is smaller than a maximum value *d*.

It is first necessary to ascertain that the pattern is of sufficient size. For this purpose the exterior zones of the spiral are covered with cardboard masks having reflectivity intermediate between the reflectivities of the black and white zones of the spiral. Masks are used in succession as follows: two vertical masks on both sides of the disk (Fig. 1), one asymmetric mask (Fig. 2) and one annular mask having a fixed external diameter of 40 cm. and an internal diameter variable between 2 and 30 cm. (Fig. 3). All three procedures are used to determine a maximum efficiency radius whose value, although depending on the spiral pitch, is fully provided for in our disks; the criterion is that a further increase of the radius of the presented stimulus is not followed by a corresponding increase of the stimulus efficiency.

## RESULTS AND DISCUSSION

We first studied the dependence of the pattern efficiency on the spiral pitch and multiplicity. Each pattern has radial characteristics generating the expansion effect, but it has also tangential characteristics generating a tangential effect and a torsion effect which interfere with the expansion signal and decrease the stimulus efficiency. We have found that the stimulus efficiency depends on the frequency of the stimulus on each ommatidium. (First part.)

Secondly, we have analysed the Moiré effect. In our experiments on landing reaction no Moiré effect has been in any case noted; this fact (surprising because in the optomotor reaction of Diptera the Moiré effect occurs with a remarkable intensity (Götz, 1964)) is explained in terms of non-radial characteristics of the stimuli. (Second part.)

Finally, we have studied the dependence of the landing reaction on the stimulated area and on the position of the stimulus in the visual field of the fly, obtaining data on: (*a*) dependence on direction in which the pattern is presented; (*b*) isotropy of the fly’s visual field; (*c*) discrimination between an expansion stimulus and a lateral displacement stimulus; (*d*) spatial integration of local landing stimuli, showing a spatial threshold for excitability and a spatial saturation effect. (Third part.)

### First part

This first part refers to the dependence of *d* and *ω* on the pitch and on the multiplicity of the spiral.

*p*is the pitch (cm.) of the spirals drawn on the rotating disk, and

*ω*the rotational frequency of the disk (r.p.s.), the speed of the motion of the black and white zones along a radius (radial speed of expansion) will be

*V*

_{R}

*= ωp*cm./sec.; in fact where

*t*is time. Substituting then For each disk the diagrams showing the variation of

*d*versus

*V*

_{R}and of

*V*

_{R}versus

*d*are, for

*V*

_{R}up to 105 cm./sec., straight lines (see also Braitenberg & Taddei Ferretti, 1966). This shows a difference between landing reaction and optomotor reaction (Reichardt, 1962). In fact, as the angular speed of the stimulus is increased, the optomotor reaction increases, attains a maximum and then decreases, while the landing reaction continues to increase with the expansion speed. This could be explained in terms of different relations in the systems underlying the two types of reaction. If the optomotor stimulus speed is too high, it is unimportant for the fly to follow it; but the higher the speed of the stimulus to produce the landing reaction, the more important for the fly to be ready to land.

The fact that, with fixed *p* and *m, d* is a linear function of w and the perceived expansion speed, for each disk, is constant (*V*_{p}, = *ωp/d* = const.) may lead one to think that *V*_{p} is always constant independently of the stimulating disk (i.e. that the whole dependence of *ω* and *d* on the geometrical characteristics of the disk is included in the linear dependence of *V*_{R}). In order to verify this hypothesis, many series of experiments have been performed by varying *d, ω* and *p*. It is to be noted that the same value of *V*_{R} may be obtained from spirals of different pitches by letting them rotate at angular speeds inversely proportional to the pitches *V*_{R}*= p*_{i}*ω*_{i,} beings *:p*_{i} : *p*_{j} = *ω*_{j} : *ω*_{i}: for *i*,*j* ∈ 1,…, *n*. It is also to be noted that all the disks with the same value of *L = p/m* are radially equivalent, i.e. all of them show along a radius a sequence of alternate white and black zones of width *L/*2. Then it is not possible, by analysing only the modulation of the reflectivity along a radius, to discriminate between these disks or to distinguish them from disks with painted sets of alternate black and white circular rings of the same width *L/*2 and expanding at the same *V*_{R}. In fact, for any one of these disks and for a fixed value of *V*_{R}, we have, at each point and in the same time interval, the same number of changes black-white and the corresponding same perception of motion. In our experiments *m* also has been varied; for disks radially equivalent one may expect to find the same landing behaviour of flies.

But experimentally we find:

(i) For disks with the same *L = p/m*: (*a*) for a fixed *V*_{R} by increasing the pitch and multiplicity, the maximum distance *d*, over which the fly never reacts, decreases;(*b*) for a fixed *d*, by increasing the pitch and multiplicity, the minimum radial expansion speed *V*_{R}, under which the fly never reacts, increases (Fig. 4); then following from (*a*) and (*b*) above: (c) the minimum perceived expansion speed *V*_{p}, sufficient to generate a reaction, increases with the increase of the pitch and multiplicity, or, in other words, the stimulus loses its efficiency with the increase of the pitch and multiplicity.

To establish whether this finding was due to the increase of *p* or to the increase of *m*, two more types of experiments have been performed:

(ii) Having fixed the multiplicity, for each value of the radial speed of expansion *V*_{R} we have investigated the dependence of the reaction on the pitch. Obviously this condition requires that the disks are not radially equivalent and that—in accordance with the results of investigations on the dependence of the optomotor reaction on the stimulus wavelength, showing an increase of stimulus efficiency with the increase of stimulus wavelength (Graffon, 1934; Fig. 6 of Fermi & Reichardt, 1963; Fig. 7 of Bishop & Keehn, 1967)—it could be expected that spirals with higher *L* (that is with higher *p*) would be more efficient. On the contrary, the results of this experiment show that, in the same sense as above, the stimulus efficiency decreases with increasing pitch (Fig. 5).

(iii) Having fixed the pitch *p*, for each value of the radial speed of expansion *V*_{R}, we have investigated the dependence of the reaction on the multiplicity. For the same reason as in (ii), it could be now expected that spirals with higher *L* (that is, with lower *m*) would be more efficient. And in fact it has been found that, always in the same sense as above, the stimulus efficiency decreases with increasing multiplicity (Fig. 6). This would be in general obvious, because by increasing the multiplicity the pattern contrast decreases (Horridge, 1968) so diminishing the pattern efficiency; but in general our patterns are large enough.

All these results show that increases of *p* and *m* co-operate in generating the loss of efficiency in radially equivalent disks.

To explain the discrepancy between our results in (ii) and the results of authors mentioned above it is to be noted that a disk with a painted spiral causes, by rotating, the perception of an expansion in consequence of its radial characteristics, but that it causes also, in consequence of tangential characteristics, the perception of a tangential displacement of the wave front, in the same sense as that of the disk rotation, as well as the perception, along each radius, of a torsion of the wave front in a sense opposite to that of the disk rotation. In fact, the direction of the wave front translation, normal to the wave front itself, is not a radial direction, but it is shifted of an angle *μ =* tan^{-1}*p*/2*πr* (see Appendix 1); as can be seen, *μ* varies, increasing with the pitch and decreasing with the increase of the radius. The existence of such a *μ* produces the perception of a tangential displacement of the wave front (tangential effect). The variation of *μ* along the radius produces the perception of a torsion of the wave front (torsion effect). This variation is given by *∂μ/∂r =* − *2πp/*[(2*πr*)^{2} − *p*^{2}] (see Appendix 1); as can be seen, the intensity of this torsion effect decreases with the increase of the radius and increases with the pitch. For a set of expanding circles, obviously *μ* = 0 and *∂μ/∂r* = o, and the expansion would be purely radial. Thus the higher the pitch, the higher are *μ* and *∂μ/∂r* = 0 and the less comparable is the disk to the corresponding set of expanding circles. The tangential effect and the torsion effect have therefore to be considered as a form of interference on *V*_{R}, increasing with the pitch and decreasing with the increase of the radius. Further consideration of this interference and of other factors influencing the results of the above experiments is given in Appendix 2.

Considering what has been said hitherto, it appears logical that the whole dependence of *d* and *ω* on *p* is not included in the dependence on *V*_{R}, since the pitch influences not only the radial configuration of the spiral but also its tangential configuration.

A confirmation of this interpretation has been obtained through the following experiment. By sending the stimulus through a slit along a diameter, the perception of the angle *μ* and of *∂μ/∂r* may be eliminated. If the slit is widened, the intensity of perception of the parameters *μ* and *∂μ/∂r* increases, and, as a consequence, the scattering between the curves corresponding to different pitches should increase. As can be seen in Fig. 7, the results of this experiment agree, qualitatively, with this interpretation.

### Second part

A theoretical limit to the use of rotating spirals as expansion patterns could be the Moiré effect.

The Moiré effect consists in the fact that, when a moving periodic structure is observed by a periodic retina, the perceived motion does not ever coincide with the real one (Rayleigh, 1874).

*ϕ*be the interommatidic angle and 2

*λ*the angle under which is seen the radial period of the stimulus; then if the motion is not perceived at all; if 2

*λ*lies between (2

*λ*)

_{2n}and (2

*λ*)

_{2n−1}(

*n*= 1, …) the sense of the motion, as perceived, is reversed; and finally the sense of the motion, as perceived, is the same—but not its speed—if 2

*λ*lies between (2

*λ*)

_{2n−1}and (2

*λ*)

_{2n}(

*n*= 1,…).

The distance *d*(*γ*) from an ommatidium—the axis of which forms an angle *y* with the axis of the fly’s head—to the observed point of the disk varies with *γ* according to *d*(*γ*) = *d*_{0}/cos *γ*, where *d*_{0} is the distance between the fly’s head and the spiral (Fig. 8). On the other hand, the projection *L*(*γ*) of the radial period of the spiral on the plane of the ommatidium varies with *γ* according to *L*(*γ*) *= L*_{0} cos *γ*, where *L*_{0} is the radial period of the spiral.

*R*

_{n}−

*R*

_{n−1}, rotating alternately in the same and in the opposite sense with respect to the real rotation sense of the spiral, and therefore perceived as a pattern composed of alternate expanding and contracting zones. For this reason, for each set of spirals with the same radial period, i.e. radially equivalent, it is possible to plot a diagram giving, for each distance from the eye of the fly to the spiral, the reversal points due to Moiré effect (Fig. 9).

A set of experiments has been performed to measure the influence of the Moiré effect on the landing reaction. The results were: (*a*) inside the ranges of distance and speed used, no fly ever reacted to an effective contraction, whatever were the pitch and the multiplicity of the spiral and the shape and the position of the stimulus, even if this stimulus was a ring completely included in a zone which, according to the above-mentioned diagrams, would have been theoretically a zone of expansion by Moiré effect; (*b*) no specific effect due to the adding of a Moiré zone was ever found; (*c*) the flies reacted to an effective expansion of rings completely included in a Moiré zone in which a contraction should have been perceived.

These results do not agree with the theoretical considerations developed above in the Moiré radial effect.

One may take into account the spread of values of interommatidic angles. The extreme values of these angles are (Vowles, 1966) *ϕ =* 2*°* and *ϕ* = 4·6°. In consequence of that spread of *ϕ*, there is a spread of Moiré zones (for each set of spirals with the same radial period). Then one may consider the intersection of such zones in all such diagrams as reduced common Moiré zones (Fig. 10).

Not even in this case do the above-mentioned results agree with the theoretical considerations.

One has to note, as before, that the stimuli used have not only radial characteristics (for which the Moiré effect is valid, the effect depending on the intersection between radial periodicities of the eye and of the stimulus), and that by making use of the tangential characteristics the fly might avoid the confusion arising from the Moiré effect along the radius. This interpretation is in accordance with the existence of the above-mentioned tangential effects.

### Third part

Several methods have been used in the study of the dependence of the landing reaction on the stimulated area and on the position of the stimulus in the visual field.

(*a*) The dependence of the reaction on the direction in which the pattern is presented to the fly has been studied.

For this purpose the whole disk has been moved around the fly, leaving the fly in front of the centre of the disk in the horizontal plane of flight. The direction of the stimulus is indicated by the angle *β* formed by the axis of the disk and the flight direction. The results are plotted in Fig. 11. The reaction is maximum for *β =* 0°, that is for an expansion arising in front of the fly, and there is no reaction for *β*= 90°. But if, always leaving *β =* 90°, the fly is displaced from the axis of the disk toward the periphery, facing the axis of the disk, then there is a zone of reaction.

(*b*) An interesting characteristic of the structure on which the landing reaction is based is the existence of preferential directions in the visual field.

In order to determine these directions, the rotating spiral has been presented to the fly through a horizontal slit having a length of 30 cm. and a width variable between 30 cm. and 2 mm. As shown in Fig. 12, as the width is reduced the stimulus efficiency decreases. The smallest slits for which reaction generally occurred were 3–4 mm. wide (but some flies reacted also with one 2 mm. wide). Slits 3–4 mm. wide have been presented at various angles (every 10°) with reference to a horizontal plane.

This experiment has been performed in two different ways: by fixing the rotation speed and reducing the distance in order to obtain the reaction, and by maintaining the fly at a fixed distance and increasing the rotation speeds until the fly reacts. Following both procedures, it was found that the vertical direction is the only preferred one. In fact any given slit is most efficient if it is vertical, other parameters being fixed (Fig. 13).

This higher efficiency of a vertical fissure having been noted, a set of experiments was performed to verify it.

A slit 30 cm. long and 1 cm. wide was shown, in the horizontal and in vertical positions, to the same flies situated at 10 cm. from the spiral (therefore the angle under which the stimulus was seen was about 60), at three different speeds: 150, 180, 22orev./min. (Table 1). From this table a clearly higher efficiency of the vertical slit may be deduced.

In order to avoid that the eventual reaction to a type of slit could depend on the training received, we calculated the percentages of the responses obtained after different types of stimuli had been shown. The results for each number of pre-adaptation stimuli are given in Table 2. From this it can be seen, in addition to the already observed higher efficiency of the vertical slit, that there is a reduction in the efficiency of both the vertical and horizontal stimuli with repetition.

(*c*) During the experiments performed at the beginning of this work in which only one side of the spiral was covered (Fig. 2), it has been found that the reaction is eliminated only if the pattern is covered up to 2·5 cm. beyond its centre (with the fly in front of the spiral centre as in all other experiments), independently of the direction of the rim of the covering mask. This leads one to think that, in this case, the stimulating pattern was perceived as a lateral displacement.

It was also thought that the moving pattern would be perceived as a lateral displacement rather than as an expansion, if the fly would see the rotating spiral through the sector of a circle, of sufficiently small aperture. This has been experimentally obtained for sectors whose apertures are less than 155^{0}, the value being the same whatever the direction of the rim of the covering mask. The value of 155^{0} agrees well with the value of 161^{0} obtainable from the experiment described in Fig. 2.

(*d*) At the beginning of the work (Figs. 1,2, 3) it was found on the disk that—after a central zone of proportionality between stimulus radius and stimulus efficiency—there was a maximum radius beyond which an increase of the stimulus radius was not followed by a corresponding increase of the stimulus efficiency.

To discover whether this maximum radius corresponds to the limits of the visual field of the fly as far as the landing reaction is concerned, the fly has been stimulated with rings of the disk all having internal radius greater than the maximum efficient radius. The results have shown that a stimulus lying external to the maximum efficient radius, if presented alone, is able to generate the landing reaction.

These two results, together, show a spatial integration of local landing stimuli that presents a phenomenon of saturation: the increase in size of the stimulus beyond a certain value does not generate an increase of the reaction.

It is to be recalled that another spatial summation effect was found in the experiments under (*b*) above (Fig. 12), in which the point of saturation was not reached because there were no experimental points for stimulus width between 3 and 30 cm.

The result of a typical experiment, using an annulus as stimulus is shown in Fig. 14. As the external radius of the stimulus (*R*) is increased, there is no reaction until a threshold point (*R*2) is reached; then there follows a zone in which the stimulus efficiency increases with the stimulus width, to reach a maximum (saturation point –*R*3–); thereafter the saturation zone follows.

The dependence of both threshold and saturation radii on the position of the stimulus in the visual field of the fly (indicated by the value of the internal radius of the stimulus –*R*1–) is obtained by varying both the internal and external radii of the opening in the mask between 0 and 15 cm. This is shown in Fig. 15. The variation of the parameters involved is reported in Fig. 16. As the internal radius is increased, both the angles *α2* and a3 (under which the fly sees respectively the threshold and the saturation radii) increase; and both the angles *δ*2 and *δ*3 (under which the fly sees respectively the threshold and the saturation widths) decrease. It is also to be noted that these two last curves converge, showing that the proportionality zone, given by their difference, decreases.

One may note that the value of the angle, under which the fly sees the stimulus at the shortest distance in Fig. 3 (that is ), agrees well with the corresponding average value of 20° obtained for *R*1 = 0 in Fig. 16.

Successive analysis of this spatial integration of local stimuli, together with the results on the temporal organization of stimuli reported in the first part and in Appendix 2, and their comparison with anatomical data on the visual system of flies (Cajal & Sanchez, 1915; Braitenberg, 1967; Trujillo-Cenóz & Melamed, 1966; Kirschfeld, 1967) will lead to a model of the functional structure underlying the landing reaction and its relation with models of movement perception.

### APPENDIX I

*p*is The equation of its tangent in the point (

*x*

_{0},

*y*

_{0}) is The slope of the normal to the spiral will be is the slope of the tangent. So, being the angle that the normal to the spiral forms with the

*x*axis, where then The normal to a circumference is its radius, so its slope is where is the angle that such a normal forms with the

*x*axis.

### APPENDIX 2

The experiments of the first part show that stimulus efficiency decreases both by increasing *m* or *p*, as is schematized in Fig. 17. Recall that: (*a*) the same *V*_{R} is obtainable with spirals of different *p*; (*b*) the same number of stimuli per unit time on the same ommatidium may be obtained at the same *V*_{R} by spirals differing by *p*, but not by *L;* (c) in using spirals as expanding patterns the existence of two sources of interference (tangential and torsion effects) is implicit. The amount of such interference is constant if *p* is not changed.

Let *T*_{1} be the time necessary for a stimulus to pass from one ommatidium to the adjacent ommatidium. This time is the sole cause of any movement detection in Reichardt’s model describing stimulus-reaction relations of optomotor responses (Reichardt, 1962).

Let *T*_{2} be the time necessary for two successive stimuli to reach an ommatidium: *T*_{2} = [(*p/m*)/2]/*V*_{R} The variation of this time is the cause of the spread of curves showing the direct proportionality of optomotor reaction and stimulus wavelength in other experiments (Graffon, 1934; Fig. 6 of Fermi & Reichardt, 1963 and Fig. 7a of Bishop & Keehn, 1967).

Note that:

(I) The efficiency of stimulating patterns increases with the increase of

*T*_{2}and with the decrease of*T*_{1}and of interference.(II) If, on the curve of any single spiral (Fig. 17), one passes from one point to another, as

*V*_{R}increases both*T*_{1}and*T*_{2}decrease and interference remains constant.(III) If one passes from the curve corresponding to one spiral to another with the same

*p*, leaving*V*_{R}= const. (Fig. 17), as*m*decreases (i.e. from 8/5 to 8/1)*T*_{1}remains constant,*T*_{2}increases and interference remains constant.(IV) If one passes from the curve corresponding to one spiral to another with the same

*m*, leaving = const. (Fig. 17), asp decreases (i.e. from 8/1 to 1·6/1) 7] remains*T*_{1}constant,*T*_{2}decreases and the interference also decreases.(V) If one passes from the curve corresponding to one spiral to another with the same

*L*, leaving*V*_{R}*=*const. (Fig. 17), as both*p*and*m*decrease (i.e. from 8/5 to 1·6/1) both*T*_{1}and*T*_{2}remain constant and the interference decreases.(VI) If one passes from a point at speed

*V*_{Ri}on the curve corresponding to a spiral*p*_{i},*m*_{i,}to a point at speed*V*_{Rj}on the curve corresponding to a spiral*p*_{j},*m*_{j}(Fig. 17),*T*_{2}varies as

The motion-detecting cells in optic ganglia are systems in which several inputs are weighted differently in effectiveness (Bullock & Horridge, 1965).

The relevance the fly ascribes, for landing reaction, to the three above-mentioned parameters (*T*_{1}, *T*_{2}, interference) will be obtained from the fly’s responses at different *p, m* and *V*_{R}.

To compare the relevance for the fly of the variations of two of the three parameters, it is necessary to leave the third one constant.

*T*_{1}*and T*_{2}Interference remains constant if the spiral is not changed (see*c*). Take two points at different*V*_{R}on a single spiral curve (Fig. 17). As*V*_{R}increases*T*_{1}and*T*_{2}decrease (see II). These decrements are in percentage the same. As stated in I, a decrease of*T*_{1}raises pattern efficiency and a decrease of*T*_{2}depresses it. Experiments (Fig. 17) show that as*V*_{R}increases pattern efficiency increases. Then*T*_{1}variation is more relevant than*T*_{2}variation.*T*_{2}*and interference T*_{1}remains constant if*V*_{R}is constant. Take two curves of spirals with the same*m*, differing in*p*, and, on each of them, a point at the same*V*_{R}(Fig. 17). Asp decreases*T*_{2}and interference decreases (see IV). As stated in I, a decrease of*T*_{2}depresses pattern efficiency and a decrease of interference raises it. Experiments (Fig. 17) show that asp decreases pattern efficiency increases. Then interference variations are more relevant than*T*_{2}variation.*T*_{1}*and interference T*_{2}remains constant if*γ*= 1. Take two curves of spirals with the same*m*, and, on each of them, a point at different*V*_{R}, such that when*V*_{R}decreases*p*also decreases (Fig. 17). As*V*_{R}decreases*T*_{1}and*T*_{2}increase (see II). As*p*decreases*T*_{2}and interference decrease (see IV).*T*_{2}remains constant when

*T*

_{1}depresses pattern efficiency and a decrease of interference raises it. Experimental data in such conditions (

*γ*= 1) show that as

*V*

_{R}and

*p*decrease pattern efficiency decreases. Then

*T*

_{1}variation is more relevant than interference variations.

We have to add in finishing that in Fig. 6 of Fermi & Reichardt (1963), (1) along each curve varying *V*_{R}, *T*_{1} and *T*_{2} change simultaneously and by the same amount; (2) along a line of constant *V*_{R}, passing from one curve to another, *T*_{1} remains constant and *T*_{2} varies in the same sense as the stimulus wavelength; in both cases there is no interference; in the present experiments in (iii) for each *V*_{R}, *T*_{1} remains constant, *T*_{2} varies in the same sense as the stimulus wavelength and the existing interference does not change. So the only difference from (2) is the existence of a certain constant amount of interference; this constant interference changes the zero position of the efficiency axis, but not the relative position of curves in diagrams; in (ii) for each *V*_{R}, there is a change of interference and so the relative position of the curves is varied.

Comparing the results herein reported with the results obtained by other workers, the disagreement under (ii) and the agreement under (iii) are explained by the considerations outlined above.

### APPENDIX 3

*λ*(

*γ*), in place of the real spatial angle covered by the seen surface, i.e.

*λ*

_{1}+

*λ*

_{2}(in Fig. 8 taking

*AC*in place of

*BD*), is This error, in every case in our experiments, is only a few percent, reaching 10 % in some rare cases.

In fact:

*γ*= 1 (

*γ*= 45 °) In Fig. 18 are reported the Table and the diagram of the percentage error 100

*ε*for the various spiral radial periods at several distances.

Now, the maximum radial period used has been *L* = 8 cm. and the minimum distance attained with such a spiral has been of the order of 6–8 cm. For these data Fig. 18 gives a percentage error *ε* ≃ 11 % ÷ 6 %, q.e.d.

## ACKNOWLEDGEMENTS

We thank Professor Valentino Braitenberg (present address: Max Plank Institut für biologische Kybemetik, Tübingen) for having suggested the present research and supported it with fruitful criticisms and discussions, Dr Uja Pinardi Vota for her valuable collaboration in performing part of the experiments and the Laboratorio di Parassitologia of the Istituto Superiore di Sanità of Roma for supplies of flies. This research was sponsored in part by EOAR 68-0030.

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*Musca domestica*)

*Proc. Roy. Soc. Lond*. B

*J. exp. Biol*

## LIST OF SYMBOLS USED IN FIGURES

*d*, threshold value of fly’s distance from the disk; *D*, distance, from the centre of the disk, of a vertical chord limiting a circular segment; *V*_{B}, threshold value of radial speed of expansion; spiral *x/y*, spiral having pitch of *x* cm. and multiplicity *y*; *ϕ*, diameter of stimulating pattern; *p*, pitch of the spiral; *m*, multiplicity; *L*, radial period of the spiral (L = *p/m*); *w*, width of a unidimensional stimulus; *γ*, angle between the axis of an ommatidium and the fly-disk axis; *λ*_{1} + *λ*_{2} radial period of the spiral seen by an ommatidium whose axis is at angle *y* with the fly-disk axis; *d*_{0} = *d*(o), fly-disk distance; *d*(*γ*), fly-disk distance along the axis of an ommatidium, the axis being at angle *y* with the fly-disk axis; *L*o = *L*(o), radial period of the spiral = *L*; *L*(*γ*), projection of the radial period of the spiral on the plane of an ommatidium whose axis is at angle *y* with the fly-disk axis; *λ*_{0} half of the radial period of the spiral seen along the fly-disk axis; *λ* (*γ*), projection of half of the radial period of the spiral on the plane of an ommatidium, whose axis is at angle *γ* with the fly-disk axis, as seen by that ommatidium; *R*_{n,}*n*^{th}, Moiré reversal radius at distance *d*_{0}; *ψ*, interommatidic angle; *β*, angle formed by the axis of the disk and the flight direction; 100*ε*, percentage error; *R*, external radius of a stimulating annulus; *R*1, internal radius of a stimulating annulus; R2, threshold excitatory external radius of a stimulating annulus; R3, saturation external radius of a stimulating annulus; *β*1, angle at which the fly sees the internal radius of a stimulating annulus; *β*2, angle at which the fly sees the threshold excitatory external radius of a stimulating annulus; *β*3, angle at which the fly sees the saturation external radius of a stimulating annulus; *δ*2, angle at which the fly sees the threshold excitatory width of a stimulating annulus; *δ*3, angle at which the fly sees the saturation width of a stimulating annulus.