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Tweed, Douglas. Three-dimensional model of the human eye-head saccadic system. J. Neurophysiol. 77: 654–666, 1997. Current theories of eye-head gaze shifts deal only with one-dimensional motion, and do not touch on three-dimensional (3-D) issues such as curvature and Donders' laws. I show that recent 3-D data can be explained by a model based on ideas that are well established from one-dimensional studies, with just one new assumption: that the eye is driven toward a 3-D orientation in space that has been chosen so that Listing's law of the eye in head will hold when the eye-head movement is complete. As in previous, one-dimensional models, the eye and head are feedback-guided and the commands specifying desired eye position eye pass through a neural "saturation" so as to stay within the effective oculomotor range. The model correctly predicts the complex, 3-D trajectories of the head, eye in space, and eye in head in a variety of saccade tasks. And when it moves repeatedly to the same target, varying the contributions of eye and head, the model lands in different eye-in-space positions, but these positions differ only in their cyclotorsion about the line of sight, so they all point that line at the target—a behavior also seen in real eye-head saccades. Between movements the model obeys Listing's law of the eye in head and Donders' law of the head on torso, but during certain gaze shifts involving large torsional head movements, it shows marked, 8° deviations from Listing's law. These deviations are the most important untested predictions of the theory. Their experimental refutation would sink the model, whereas confirmation would strongly support its central claim that the eye moves toward a 3-D position in space chosen to obey Listing's law and, therefore, that a Listing operator exists upstream from the eye pulse generator.
In most large gaze shifts, eye and head work together to bring the line of sight swiftly onto its new target. Because of the complex, fast and accurate coordination involved, these movements—called eye-head saccades—have attracted the attention of theorists (Galiana and Guitton 1992; Galiana et al. 1992; Guitton et al. 1990; Laurutis and Robinson 1986; Tomlinson and Bahra 1990), but current models of eye-head control deal with motion in only one dimension and do not touch on the new issues that arise in three dimensions, such as curvature, noncommutativity and Donders' and Listing's laws.
From a three-dimensional (3-D) perspective, the central question is how the saccadic system converts its two-dimensional (2-D) input signal, retinal error, into 3-D rotations of the eye and head. Retinal error, defined as the location of the target image on the retina, is 2-D simply because the retinal surface is 2-D. Eye and head rotations are 3-D because each is free to move horizontally, vertically, and torsionally. Together, eye and head have 3 + 3 = 6 df, so the eye-head saccadic system must actually transform a 2-D input into a six-dimensional output. It is true that for some gaze shifts stereoscopic depth is important, and then one might speak of 3-D or even four-dimensional biretinal error driving a nine-dimensional system consisting of the eyes and the head, but this paper will not consider such vergence-saccades. It is also true that not all saccades are driven by retinal error—we can look toward auditory, tactile, remembered or olfactory targets—but there is evidence that auditory targets, at least, are converted to retinal coordinates within the brain and are stored, together with visual targets, in a single map in the superior colliculus, SC (Jay and Sparks 1984), a map of target direction relative to the eye (Robinson 1972; Schiller and Stryker 1972; Sparks and Mays 1980; Van Opstal et al. 1991). In this paper, therefore, I assume that the input to the saccadic system is always a2-D signal coding target direction in eye coordinates.
The saccadic system's job is to convert this 2-D input into 3-D rotations of the eye and head, which must fit certain constraints. Of course the most important constraint is that the gaze line land on the target, but the rotations must also obey Listing's law of the eye in head (Ferman et al. 1987; Radau et al. 1994; von Helmholtz 1867) and Donders' law of the head on torso (Radau et al. 1994). I shall argue that recent data on eye-head gaze shifts can be explained, and fundamental new predictions can be made, by a model that introduces just one new assumption, and that is otherwise based on ideas that are well established from one-dimensional studies. The new assumption is that the eye is driven toward a 3-D orientation in space that has been chosen so that Listing's law of the eye in head will hold when the eye-head movement is complete. Before presenting this3-D model and its predictions, I review the experimental data and previous models on which it is built.
Superior colliculus codes gaze shifts in space
Electrically stimulating the SC in a cat with its head free evokes an eye-head gaze shift (Guitton 1992; Guitton and Volle 1987; Guitton et al. 1990; Roucoux et al. 1980). Repeated stimulation of a single site causes repeated gaze shifts of approximately the same size and direction, but the contributions of eye and head may vary, so that for one gaze shift, the eye may move a lot and the head only a little, for another the reverse. And similar observations have recently been made in monkeys (Robinson and Cowie 1993). These findings suggest that there is a map within the SC that specifies the desired movement of the gaze line or, in other words, gaze error. Regarding terminology, note that if gaze error is specified in eye coordinates, as we shall assume, then it is the same thing as target direction in eye coordinates, which is equivalent, in turn, to retinal error (if the lights are on and the target in within visual range); e.g., if retinal error is 20° right, then the target direction relative to the eye is also 20° right, as is the gaze shift, in eye coordinates, required to foveate the target.
Any gaze shift can be parceled out to the eye and head in many different ways. If you want to shift your gaze point 20° rightward, you can turn your eyes 20° right in your head and hold your head fixed, or turn your head 20° right and leave your eyes fixed in your head, or turn your eyes and head 10° right, and so on. For sensible results, this division of labor must take into account the current positions of the eye and head. Thus if your retinal error is 20° rightward, your eyes are currently 20° left in your head and your head is 10° right in space, you may move your eyes 20° right in your head, i.e., center them, and leave your head where it is. But if you see the same retinal error when your eyes are already 30° right in your head and your head is centered, you are more likely to turn your head 20° right and leave your eyes fixed in your head. These scenarios show that the retinal error, or desired gaze shift, command cannot drive the eye and head by itself, but must interact, inside or downstream from the SC, with signals coding current eye and head position.
It is known that saccadic bursters in the SC code initial gaze error; e.g., the site representing 20° rightward error remains active throughout a 20° rightward saccade (Sparks and Mays 1980, 1983). Another group of SC neurons, the buildup cells, may code the evolving, or "dynamic," gaze error throughout the movement (Munoz and Wurtz 1995a,b), but the existence of dynamic gaze error signals and their role in saccade generation are controversial. Following Galiana et al. (1992), the model in this paper drives the eye and head with a dynamic gaze error signal computed within the SC, but it is important to note that the experimental evidence for this is inconclusive, and that the crux of the model—driving the eye to a 3-D orientation in space chosen so that the eye's final orientation in the head satisfies Listing's law—is independent of this question.
Eye is driven to a position in space
There is convincing evidence that both eye and head are driven to their desired positions by feedback in the form of efference copy and, likely, vestibular signals (Becker et al. 1981; Guitton and Volle 1987; Guitton et al. 1990; Laurutis and Robinson 1986; Pélisson et al. 1989). Further, it is known that the eye is driven toward its desired position in space, not its desired final position in the head. For example, if the visual target is 65° right in space-fixed coordinates, then at the end of the movement, the head will be ∼50° right and the eye 15° right in the head. But during the saccade, the eye does not simply move to its final position in the head, 15° right, and wait there for the head to finish its motion. Instead, it shoots well past 15° until the gaze line hits the target, and then, under the influence of the vestibuloocular reflex (VOR), it rotates back to 15° right as the head completes its motion (Guitton and Volle 1987; Laurutis and Robinson 1986; Tweed et al. 1995). This way, the gaze line reaches the target earlier, before the head finishes its motion.
VOR is shut off in one direction
During large saccades, the VOR is weakened or shut off, gradually turning back on as gaze error becomes small and becoming fully operational again during the final, VOR stage of the movement, when the gaze is on target but the head is still completing its motion (Guitton and Volle 1987; Laurutis and Robinson 1986; Roucoux et al. 1981; Tomlinson and Bahra 1986; Tweed et al. 1995). As Laurutis and Robinson pointed out, it is sensible to switch the VOR off during such saccades, because its function is to hold the gaze stable in space and that function is not desirable during a saccade. Optimally, the brain should shut off the VOR in the direction of the saccade but leave it on in other directions, because only head movements in the direction of the saccade should affect the motion of the eye in space. This design specification is at least roughly implemented in the actual VOR, which switches off in the direction of the saccade (Guitton and Volle 1987; Laurutis and Robinson 1986; Tomlinson and Bahra 1986), but remains on in the opposite (Pélisson and Prablanc 1986; Pélisson et al. 1988) and orthogonal directions (Tomlinson and Bahra 1986).
An important 3-D property of the VOR is that it does not preserve Listing's law of the eye (Crawford and Vilis 1991; Henn et al. 1992; Misslisch et al. 1994a). This means that, if the eye is to obey Listing's law when the eye-head gaze shift is over, it must not be in Listing's plane when the gaze line hits the target, because then the VOR would just drag it back out again. Instead, the eye saccadic system must place the eye in a position, outside Listing's plane, that has been chosen carefully so that the subsequent action of the VOR brings it into the plane. Further, this must happen despite large variations in the size and duration of the VOR stage (Fuller 1992; Tweed et al. 1995). As stated above, the model achieves this by driving the eye toward a desired final position in space that has been chosen such that eye position in the head will fit Listing's law when the head reaches its desired position.
Desired eye position is saturated
Guitton and Volle (1987) showed that the command driving the eye saccade passes through a neural "saturation box" before reaching the saccadic pulse generator, so that a desired eye position command of 90° right, for example, would emerge as ∼40° right. Saturation prevents the eye from running at full speed to the end of its leash, like the dog in the Bugs Bunny cartoon, a maneuver that might overstretch the muscles or damage the globe. Further, saturation prevents errors that might arise if the muscles were given commands they could not execute. But note that saturation, if it is to serve these purposes, must act on signals coding desired eye position relative to the head, not desired gaze or eye movement. For example, if the desired gaze movement is 60° to the right, is saturation required? Yes, if the eye is, say, 30° right, because then the desired eye-in-head position is 30° + 60 deg = 90° right, which is uncomfortably eccentric. No, if the eye is 30° left, because then the desired eye-in-head position is only 60° − 30° = 30° right. This shows that the saturation operator cannot act on retinal error or on commands coding only the desired movement of the eye but requires signals coding absolute eye position in the head. In the model, the input to the saturation function is a desired eye-in-head position signal, computed within or downstream from the SC.
Donders' and Listing's laws
Imagine a rotating object whose position is expressed using quaternion vectors. We shall say that the object follows Donders' law if its quaternion vector remains within a 2-D surface (Donders 1848). If this surface is flat,—i.e., if it is a plane—then the object also obeys Listing's law (von Helmholtz 1867; Westheimer 1957). It is known that eye-in-head positions obey Listing' law, with an inaccuracy of only a degree or so, during and between head-fixed saccades and between eye-head saccades (Ferman et al. 1987; Radau et al. 1994; Tweed and Vilis 1990; von Helmholtz 1867). Head positions obey Donders' law during spontaneous eye-head gaze shifts (Tweed and Vilis 1992), and recent studies have shown that the law applies much better to head-on-torso motion than to head-in-space (Misslisch et al. 1994b; Radau et al. 1994). Donders' law of the head fails during repeated gaze shifts between two targets (Tweed and Vilis 1992), and of course this law can be repealed voluntarily, i.e., you have the power to cock your head into almost any 3-D orientation you wish. This shows that Donders' law of the head is a default rule that can be altered or overridden.
The map in the burst cell layer of the SC specifies the initial target direction relative to the eye, T 0 e. In the buildup cell layer of the SC, a comparator driven by feedback signals coding head velocity (from the semicircular canals) and eye position and velocity (from efference copy), computes dynamic target direction, T e, as it changes throughout the saccade. This signal, which is 2-D, must be converted into 3-D head and eye rotations. The operations involved in this conversion are shown in flow diagram form in Fig. 1; for simplicity, the computation of dynamic target direction T e is not shown in the flow diagram but takes place off the left-hand side of the figure.
To drive the head, the dynamic target direction T e is converted from eye coordinates into space coordinates using feedback signals coding eye and head position. The resulting target direction in space signal, T s, which is still just 2-D, passes through a Donders operator to yield a 3-D desired head position in space command, q*h, which specifies a head position that is "comfortable" for viewing a target in that location and that obeys Donders' law of the head. The override command in the flow diagram reflects the fact, mentioned above, that Donders' law can be abrogated voluntarily, i.e., you can choose to make an eye-head saccade that leaves your head in almost any 3-D orientation imaginable. Most simulations in this paper use the Donders operator because Donders' law of the head is known to hold during normal eye-head gaze shifts (e.g., Glenn and Vilis 1992; Tweed and Vilis 1992; Tweed et al. 1995). For the simulations in Figs. 3-5 and 7, the operator is set so that head movement contributes ∼80% of horizontal gaze shifts and ∼50% of vertical gaze shifts, because these values are typical for normal subjects (Glenn and Vilis 1992). Figure 6 shows what happens when the Donders operator is altered, changing the contribution of the head (for mathematical details of this adjustment, see the appendix). Figure 8 gives an example where the Donders operator is overridden.
Whether it arises from the Donders operator or from the override, the desired head position command then passes to the head pulse generator P h where it is compared with an internal estimate of current head position, q h derived from vestibular, proprioceptive and/or efference copy signals, and yields a 3-D head velocity command, q˙ h, that drives the neck muscles until the head movement is complete. Note that this velocity command, q˙ h, is quaternion velocity, i.e., it is the rate of change of the head position quaternion; it is not the angular velocity vector of the head.
To drive the eye, the 2-D target direction in space signal T s interacts with the 3-D desired head position signal q*h to yield a 3-D desired eye position in the head, q*eh, which fits Listing's law and which will point the gaze line at the target when the head reaches its desired position. But as we saw earlier, the eye is not driven toward its desired final position in the head, so q*eh is combined with desired head position q*h to yield desired eye position in space, q*es. This signal then interacts with an internal estimate of head position to yield current desired eye position in the head, q + eh, i.e., the position that the eye would have to have in the head to be in its desired position in space, given the current position of the head. For example, if the desired eye position in space is 90° right and the current position of the head is 10° right, then the current desired eye position in the head is 80° right. As this example shows, current desired eye position may be highly undesirable because it lies beyond the oculomotor range, and so this signal passes through a 3-D version of Guitton and Volle's saturation box, labeled Sat, to yield saturated current desired eye position in the head, q s eh. Finally, this 3-D signal passes to the eye pulse generator P e, where it is compared with an internal estimate of eye position, q eh, as in Robinson's original feedback model for head-fixed saccades (Robinson 1975), to yield a saccadic eye velocity command which, together with the VOR, determines eye-in-head velocity q˙ eh. The equations for this flow diagram are given in the appendix.
In the model, saturation of desired eye position operates three-dimensionally: horizontally, vertically, and torsionally, a generalization of Guitton and Volle's (1987) one-dimensional scheme which raises some new issues. In one dimension, saturation is a simple matter of "clipping" desired eye position signals larger than some limit, say 40°. In 3-D, saturation involves projecting overly eccentric desired eye position vectors into some central, 3-D subvolume of the oculomotor range, called the effective oculomotor range (EOMR). We know very little about the 3-D shape of the EOMR. Ellipsoids or cylinders seem the likeliest possibilities, so in the simulations here the EOMR is a compromise between these two shapes (see Fig. 2): it is a clipped ellipsoid, or pill-shape, 80° in diameter in the horizontal-vertical plane and 16° thick in the torsional dimension in the center, a value chosen because it is roughly the torsional eye position range seen during roll VOR (Misslisch et al. 1994a).
Geometrically, there are many ways to project an eccentric desired eye-in-head position q + eh onto a saturated desired eye-in-head position q s eh within the EOMR, and different projection rules have different functional advantages and disadvantages. Tweed et al. (1995) suggested that the best strategy would be to project in such a way that q s eh is the point where q + eh (which is continuously changing because of the head's motion) is predicted to enter the EOMR. This strategy would get the eye-in-space to its final position at the earliest possible moment and with the least wasted eye-in-head motion. But it turns out that, in some cases when the head is prevented from completing its motion, this strategy could cause the gaze line to fall short of the target.
For this reason, the present paper uses a slightly different strategy, driving the eye to the point on the boundary of the EOMR where, it is predicted, the target will first become foveable. In other words, it drives the eye to that position within the EOMR where it will first be possible to get the target image onto the fovea. Thus whereas Tweed et al. (1995) suggested driving the eye to the position within the EOMR where the target will first be foveable with the correct eye-in-space torsion, the present model drives the eye to the position in the EOMR where the target will first be foveable, period. I should add, however, that for normal, unperturbed gaze shifts, there is little difference between the saccades produced by the two saturation schemes and, in particular, the patterns in Figs. 3-8 below are insensitive to the details of the saturation.
The new saturation strategy is illustrated in Fig. 2. If current desired eye position, q + eh, lies outside the EOMR in the horizontal-vertical plane, then it is projected into range along the straight path joining it to the final desired eye position q*eh; i.e., it is projected along the path that it is predicted to follow as a result of the head movement. This projection brings it into the EOMR in the horizontal-vertical plane, but the projected point still may lie outside the EOMR in the torsional dimension. If so, as shown in the top half of the figure, it is projected torsionally until it meets the boundary of the EOMR. Strictly, it should be projected along an isogaze line (drawn in Fig. 6) rather than a straight torsional path, but the straight line is a reasonable approximation.
Gaze point and facing direction
We start the model on a relatively simple task: predicting gaze paths. This is simpler than predicting eye and head rotations, because gaze paths are 2-D whereas rotations are 3-D. Figure 3 is a computer simulation showing the predicted paths traced by the gaze point as it moves across a large spherical viewing screen centered on the subject. The simulated subject makes 10 saccades: the first moves from center to 70 deg up and left; the next 8 saccades take the eye first counterclockwise (CCW) and then clockwise (CW) around a square—where CW and CCW are defined from the subject's viewpoint—and the tenth saccade returns to center. Thicker lines depict the paths of the head's facing direction, which can be thought of as a spot cast by a laser attached to the subject's nose, pointing straight ahead when the head is in its reference position. These simulated paths match at least five key characteristics of the 3-D data collected by Glenn and Vilis (1992) and Tweed et al. (1995). 1) The box traced by the facing direction is wider than itis tall, meaning that the head contributes more to horizontal than to vertical gaze shifts. This feature was explicitly built into the model (for details see Eq. A5) to mimic Glenn and Vilis's (1992) data, so its confirmation doesn't provide any new support for the model, but the remaining four points are genuine predictions, confirmed by the data in Tweed et al. (1995). 2) Horizontal paths of both the gaze point and facing direction are roughly straight, but vertical paths bow out like the sides of a barrel. 3) The subtle figure-eights in the horizontal gaze point paths in Fig. 3 also are seen in all human subjects. 4) Oblique paths of the facing direction are straight. 5) Oblique paths of the gaze point are not straight, and their curvature changes with the direction of the gaze shift, so that centrifugal and centripetal saccades into this quadrant trace out a CCW loop. The mechanisms behind these curving paths, and behind the similar patterns we shall see in plots of 3-D eye-in-space position, are dealt with in the discussion.
Eye-in-space trajectories
Simulated oblique, eye-head saccades in all four quadrants are shown in Fig. 4, which plots eye-in-space orientations, depicted as quaternion vectors and viewed from behind so that the horizontal and vertical components of the motion are seen. A complex pattern of loops appears: centrifugal and centripetal saccades form CCW loops in the first and third quadrants, and CW loops in the other two quadrants. This pattern is echoed in the data shown at the bottom of the figure and indeed was present in all subjects tested by Tweed et al. (1995). But although the qualitative looping pattern is the same, data and model do not agree perfectly on the detailed paths taken by the eye. In particular, centripetal saccades usually are curved more in the data than in the model. Reasons for this discrepancy, as well as for the correct prediction of systematic looping, are covered in the discussion.
Loops also are predicted during purely horizontal and vertical saccades, but in these cases the looping is mainly in the torsional dimension. Figure 5 shows eye-in-space quaternion vectors in a right side view, so that the torsional and horizontal components are seen. The gaze shifts shown are horizontal, back and forth between targets 70° up and left (UL) and up and right (UR), and between 70° down and left (DL) and down and right (DR). The simulation shows torsional loops that reverse direction depending on the elevation of the gaze shifts. Again, this pattern is also present in the data at the bottom of the figure and was seen in all subjects tested by Tweed et al. (1995).
Isogaze curves
A central feature of the model is the way internal variability is manifested in its overt behavior and, in particular, in its final gaze direction and eye-in-space position. Recall that any one gaze direction can be achieved using infinitely many distinct eye-in-space positions, differing only in their orientations about the line of sight. If we plot all the eye-in-space quaternion vectors corresponding to one gaze direction, they form a curve in 3-D quaternion vector space, an isogaze curve. When the model makes a number of saccades between center and a fixed, eccentric target (e.g., the down-and-right target DR), varying the contributions of eye and head each time, then it lands in different eye-in-space positions, but all these positions lie on the same isogaze curve, so the gaze line nevertheless reaches the target every time.
This behavior is simulated in Fig. 6, which shows final eye-in-space positions for nine saccades to each of four targets, 90° eccentric along the oblique meridians: up and left, up and right, down and left, and down and right. The nine saccades to each target differ in the contributions made by the head and eye. That is, the Donders operator (Fig. 1) was altered from saccade to saccade so that it generated different final head positions even though the target location stayed the same (for mathematical details, see Eq. A5). These alterations can be viewed as simulating either noise in the Donders operator or voluntary decisions to change or override that operator. Figure 6 shows the result: changing the head's contribution changes the final position of the eye in space, but the eye-in-space vector continues to lie on the same isogaze curve. In other words, internal variability in the contribution of eye and head does not affect the final direction of the gaze line in this model but it does affect the final3-D orientation of the eye in space. Experimental evidence that the human eye-head-torso saccadic system has this same property is provided by Radau et al. (1994).
Eye-in-head trajectories
Simulated eye-in-head trajectories for centrifugal and centripetal saccades are plotted in Fig. 7. Eight saccades are shown in all, moving between center and the targets from Fig. 3, oblique targets at 70° eccentricity. Centrifugal movements are plotted in thin lines and centripetal in thick. These simulated saccades show the features seen in the human data of Glenn and Vilis (1992) and Tweed et al. (1995), e.g., the eye overshoots its desired final position in the head, q*eh, and then glides back. How far and in what direction the eye overshoots depends on how much the head contributes to the saccade; in this simulation, as in most human subjects, the head contributes more to horizontal than to vertical gaze motion, and so the overshoot is largely horizontal, as seen by Tweed et al. (1995).
Figure 7 shows another confirmed prediction of the model: that the eye's initial path is aimed at neither the visual target nor the desired final position of the eye in the head. Arrows at the center of the graph show the initial directions of current desired eye position, q + eh (white arrow) and final eye position q*eh (black arrow) for one centrifugal saccade. Note that the eye's initial path lies between these two directions, as in the data of Tweed et al. (1995).
The model also predicts eye-in-head overshoot in the torsional dimension. These overshoots are small for normal eye-head saccades of <90°, but they can be quite marked for larger saccades, or for saccades involving unusual head postures, as in the simulation in Fig. 8. Here the simulated subject starts out with the head tilted 30 deg CCW (i.e., left ear down) and the eyes 10° up relative to the head. After the gaze shift, the head is tilted 30° CW and the eyes are again 10° up relative to head. To get these unusual head postures, the initial and final head positions were generated by overriding the Donders operator. As shown in Fig. 8, the head trajectory is simply a straight line in the torsional dimension, going from 30° CCW to 30° CW. The eye-in-space trajectory is also mainly torsional but with a small rightward component. What is interesting in the figure is the eye-in-head trace: it begins and ends at the same position in Listing's plane, but during the gaze shift, it loops a full 8° out of the plane in the CW direction. To maximize the loop, I allowed the eye to begin moving 50 ms before the head, a typical lead time for human subjects (Tweed et al. 1995). We return to this simulation, which is the central untested prediction of the model, in the discussion.
The model is good at predicting the complex, curving paths of head-in-space, eye-in-head, and eye-in-space, even though it was not devised with these 3-D trajectories in mind but was built from a handful of "axioms," all with clear functional justification: 1) the VOR is switched off in the direction of eye-in-head error because its gaze-stabilizing function is counterproductive during saccades; 2) eye and head are driven by internal and vestibular feedback for speed and accuracy; 3) desired eye-in-head position is saturated to avoid driving the eye to its mechanical limits; 4) the eye moves toward its desired position in space, rather than its final position in the head, to bring the gaze line to the target as fast as possible; and 5) this desired eye position in space is chosen so that Listing's law of the eye in head will hold at the end of the head motion. As discussed above, the first four axioms are inherited from previous models and are well confirmed by 1- and 2-D eye-head data (Galiana and Guitton 1992; Guitton and Volle 1987; Guitton et al. 1990; Laurutis and Robinson 1986; Pélisson and Prablanc 1986; Pélisson et al. 1988; Tomlinson and Bahra 1986). The fifth will stand or fall based on its key prediction that the eye-in-head should leave Listing's plane during eye-head saccades (see Site of the Listing operator, below). In what follows, I discuss some issues arising from the simulations in Figs. 3-8 and describe some further predictions of the model.
Trajectories, saturation and Donders' law
The looping paths in Figs. 3 and 4 occur, at least in the model, because the eye is bearing for a desired position in space, q*es, but is deflected systematically by the saturation function. As described earlier, saturation prevents the eye leaving the oculomotor range; i.e., if q*es is so far eccentric, relative to the head, that the current desired eye position in the head, q + eh, lies outside the effective oculomotor range, EOMR, then the brain computes a saturated desired eye position, q s eh, which is the position within the EOMR where it will first be possible to foveate the target. This saturated position depends on how the head is going to move, because it is head movement that causes q + eh to move toward the EOMR. Because the head moves more horizontally than vertically in most subjects (Glenn and Vilis 1992; Tweed et al. 1994), q s eh lies on a more vertical meridian than the unsaturated target, q + eh (see Fig. 2). In other words, the eye initially moves more vertically than the direction of the target because it predicts that the head will do much of the horizontal work. It may seem like a lot of work to predict where the target will enter foveation range, but Eq. EA10 shows that a rather simple algorithm does the job.
The model does not perfectly predict the detailed paths of the eye in Fig. 4; in particular, centripetal saccades are usually more curved in the data than in the model. Part of the problem is that paths vary idiosyncratically among subjects (Tweed et al. 1995), so any model would need personalized parameter settings to mimic individuals' patterns. Another problem is that in the model, curvature is mostly due to the saturation function, whereas in reality, additional factors are at work. For instance real subjects show systematic curves even during head-fixed saccades (Bains et al. 1992), where the current desired eye position is always inside the EOMR, and so saturation should play no role, or at best a small role if we imagine a "soft" saturation. But even if saturation is not the whole explanation for curved saccade paths, Figs. 3 and 4 suggest that it does play an important role, because the 3-D saturation function in Fig. 2, designed on purely functional grounds, predicts the qualitative pattern that is shared by all subjects: CCW loops in the first and third quadrants, CW loops in the second and fourth. This same saturation mechanism explains why the eye-in-head paths in Fig. 6 start out on a path intermediate between the direction of the visual target and the final position of the eye in head:the saturated target, q s eh, lies on a path between q + ehand q*eh.
Figure 5 shows that eye-in-space positions violate Listing's law both statically and dynamically, i.e., between and during eye-head saccades, in both simulations and data. Thus eye-in-space torsion is not fixed at 0: in the simulation traces, the saccade from down and left (DL) to down and right (DR) involves a change in torsional position of ∼25° CW, whereas the saccade from UL to UR moves ∼25° CCW. As a result, the static, intersaccadic positions DL, DR, UL, and UR cannot all fit in one plane, i.e., Listing's law fails for the eye in space. The main reason is that the head does most of the horizontal work in eye-head saccades, whereas the eye does the vertical, and so the eye in space moves as if mounted on Fick gimbals (see Glenn and Vilis 1992), an arrangement which produces the kind of twisted, nonplanar distribution of static positions seen in Fig. 5. This arrangement is also responsible for the barrel-shaped gaze paths in Fig. 3: with Fick gimbals, horizontal paths are flat, following lines of latitude, whereas vertical paths bulge out, tracing lines of longitude.
The looping paths in Fig. 5, like those in Fig. 4, are due to the saturation function, but here it is the saturation in the torsional dimension that matters. For example, the movement from DL to DR plotted in Fig. 5 involves a 25° CW change in eye-in-space position. Torsional saturation prevents the eye-in-head from moving too far CW, with the result that the eye-in-space path initially has too little torsion and later arcs toward its target. One consequence of this torsional looping is that the eye in space violates Donders' law during these movements: it is not possible to fit all these looping paths into any single 2-D surface.
Further, the model predicts that the eye in space also will break Donders' law statically between gaze shifts. For example, if a subject stands facing a tangent screen, turns the head 20° left, and makes head-fixed saccades between targets on the screen and then does the same with the head turned 20° right, the eye-in-space quaternion vectors are predicted not to lie in a single surface, but to fill out two distinct planes: a plane tilted ∼10° left when the head is 20° left, and a plane tilted ∼10° right when the head is 20° right. This prediction has recently been confirmed experimentally by Hubert Misslisch (1995; see also Tweed et al. 1993).
Site of the Listing operator
The central new claim of the model is that the eye is driven to a position that has been chosen to fit Listing's law, and therefore this position must be set by a Listing operator upstream from the comparator that computes eye-in-head motor error. Theoretically, this is not the only possible way to implement Listing's law. For example, if the oculomotor integrator is leaky in torsion, as suggested by Seidman et al. (1995), eye positions gradually will leak into Listing's plane, thereby restoring Listing's law whenever it is broken. However, it is clear that humans and monkeys do not rely on this mechanism for Listing's law because when supine subjects are rotated in roll, they make torsional quick phases to return to Listing's plane (Crawford and Vilis 1991; Seidman et al. 1995); i.e., they do not wait for a leaky integrator or any other slow mechanism to restore Listing's law, rather they make saccades aimed at Listing's plane. Clearly, then, there is a saccadic mechanism for Listing's law. This mechanism appears in the model as the Listing operator in Fig. 1, but that operator goes one step further: during eye-head saccades, it aims the eye, not at a position fitting Listing's law, but at a position that will fit Listing's law when the entire eye-head movement is complete. This predictive feature can explain Crawford and Vilis's (1991) observation that monkeys, when they are rotated so that their VOR slow phases carry them out of Listing's plane, make quick phases that carry them well past the plane.
Another important consequence of this predictive Listing operator is that the eye should leave Listing's plane transiently during eye-head saccades. The reason, again, is that the desired eye position in space is chosen so that Listing's law will hold when the head movement is finished. In the middle of the gaze shift, when the head is still far from its final position, the desired eye-in-space position will not generally be consistent with Listing's law; i.e., to put the eye in its desired position in space, q*es, given the current position of the head, the eye in head must violate Listing's law. Therefore current desired eye-in-head position, q + eh, generally does not obey Listing's law during an eye-head saccade, and so the eye is predicted to leave Listing's plane.
Tweed et al. (1995) observed the predicted effects in human eye-head saccades, but the observed and predicted departures from Listing's plane were small for the saccade tasks they recorded. A more dramatic test will be to have a subject perform the gaze shift that is simulated in Fig. 8. During this movement, the head rotates purely CW, so in the middle of the saccade, the desired eye position in space is rotated far CW relative to the head. As a result, the eye is driven to the CW torsional boundary of the EOMR, a full 8 deg out of Listing's plane. A complicating factor, neglected for simplicity in the simulation, is that Listing's plane moves in the head depending on head orientation (Haslwanter et al. 1992), so a 60° CW head rotation, as in this simulation, shifts Listing's plane ∼5° CCW. Therefore the actual prediction is a large CW motion, still reaching the forward boundary of the EOMR, but then looping back to ∼5° behind the starting position. If experimental data bear out this prediction, they will provide direct support for the fifth axiom of the model, which implies a Listing operator upstream from the eye pulse generator.
Further predictions and extensions
1) According to the model, head-only saccades, in which the eyes stay fixed in the head, should be impossible; i.e., when subjects attempt head-only saccades, their eyes should still shoot toward the target and then roll back during the VOR stage of the movement. The reason is that the eye is driven toward a target in space and therefore cannot be ordered to stay still in the head.
2) Many human subjects routinely make large gaze shifts with several small eye saccades superimposed on one smooth head saccade. This can be explained by the model if a series of gaze shift commands are sent from the saccade related bursters in the SC to the eye saccadic system, while the head is driven to a single desired position chosen by the override mechanism (see Fig. 1). Alternatively, perhaps a set of gaze targets are coded simultaneously in the SC and all of them are sent, in series, to the eye saccadic system whereas only the most eccentric gaze target is used to drive the head. Either way, the model predicts that the eye in head should deviate systematically from Listing's law between and during the consecutive eye saccades, landing in Listing's plane only when the head finishes its motion.
3) Real gaze shifts involve movements of eyes, head, torso, and limbs, and some of these motions are translational as well as rotational. A starting point for modeling translational saccades is provided by the "target locator" equations for 3-D translational and rotational VOR in Viirre et al. (1986).
4) As far as it can, the model avoids guessing about the anatomic loci for various computations, because it aims to describe quantitatively what the eye-head saccadic system as a whole is doing rather than how or where. The what question comes logically before the others and enjoys a much stronger database; i.e., current behavioral data give major clues to the overall input-output behavior of the system, whereas the electrophysiological and anatomic data are controversial regarding, e.g., the locations of the comparators relative to the SC, the signals emerging from the SC, the site(s) of the Listing operator(s), and the 3-D kinematic variables coded by the short lead burst neurons and the extraocular motoneurons. As the anatomic database grows, it will fill in the details in whole-system models such as this one or its successors.
I thank S. Au, M. Fetter, T. Haslwanter, H. Misslisch, and T. Vilis for valuable comments.
This study was supported by the Medical Research Council of Canada Grant MT-12847.
Address for reprint requests: Dept. of Physiology, University of Western Ontario, London, Ontario, N6A 5C1, Canada.
- Copyright © 1997 the American Physiological Society
Mathematical representations
For computational convenience, I express the model equations using quaternions, vectors, and matrices. There is no evidence that the brain uses these representations, but then there is no evidence that it uses any other standard notation either. Probably the brain uses some exotic representations, as yet unknown to mathematicians. In this paper, however, I am concerned primarily not with how the computations underlying gaze shifts are implemented neurally, but with expressing clearly what I think the computations are, so I shall stick to standard, quaternion-vector-matrix notation. [For readers unfamiliar with the relevant 3-D geometry, the necessary background is given in Brand (1948), or any kinematics text with a section on quaternions or Clifford algebra.] Formulas in this paper use arithmetic operations, such as multiplication and square root finding, that are somewhat unusual in oculomotor models but this is no cause for concern. It is clear that individual neurons are versatile information-processing devices (see e.g., Poggio and Torre 1978), and even if they were not, work on artificial neural networks has shown that assemblies of very simple elements can perform arbitrarily complex mathematical operations. And finally, I have no doubt that the model described here can be simplified considerably, without significant loss of performance, by using approximations to its equations. But approximations can blur the essential structure of the model, and therefore it is better to describe the theory in exact form first. If the framework is sound, simplification can come later.
Gaze comparator
A comparator in the SC computes dynamic target direction relative to the eye, T e (= retinal error = gaze error, or better: desired gaze shift in space expressed in eye coordinates). This, I am assuming, is the command coded by the buildup cells. In Eq. EA1 below, T e is represented as a unit vector pointing in the direction of the saccade target. By a theorem of kinematics (see any text, e.g., Goldstein 1980), the time rate of change of T e, called T˙ e, equals the vector cross product of T e and ωesc, the angular velocity of the eye relative to space, expressed in an eye-fixed coordinate system. This cross product formula is shown in the first line of Eq. EA1, but because ωese is not one of the variables shown in the flow diagram in Fig. 1, the subsequent lines in Eq. EA1 derive an expression for T˙ e based on variables that are available Equation A1Here the quaternion q eh is a neural representation of eye position in the head, q˙ eh is its time rate of change, and q −1 eh is its inverse. ωhsh is the angular velocity vector of the head relative to space, expressed in head-fixed coordinates; in other words, ωhsh is the head velocity estimate delivered by the vestibular system. (Strictly, q eh should be written q ehh, meaning the quaternion of eye position relative to the head in head-fixed coordinates, but because q eh never is expressed in any other coordinates in this paper, I have omitted the final h for simplicity.) The symbol × indicates the vector cross product, and the juxtaposition ωhsh q eh is the quaternion product of the vector and the quaternion, in which the vector is treated as a quaternion with a scalar component of 0. Once T˙ e has been obtained, it is integrated within the SC to obtain the updated estimate of T e.
Desired head position and the Donders operator
Desired head position is computed from Te by the formulas Equation A2 Equation A3 Equation A4 Equation EA2 multiplies head position in space qh with eye position in the head qeh to yield eye position in space, qes , which then conjugates Te in Eq. EA3 , converting it from eye to space coordinates to yield Ts , target direction in space. The latter then feeds into the operator called Donders to yield q*h, the quaternion of desired head position (relative to space, expressed in space-fixed coordinates).
As described earlier, Donders' job is to choose a head position that is comfortable for viewing the target. Because the head usually moves more horizontally than vertically (Glenn and Vilis 1992; Tweed et al. 1995), Donders scales the horizontal and vertical components of desired head position differently, and then it scales the torsional component to fit Donders' law of the head. For clarity, Donders is best defined in three steps Equation A5a Equation A5b Equation A5cHere i, j, and k are the unit vectors pointing along the three axes of the coordinate system, i pointing forward, j left, and k up; x 2 and x 3 are the coordinates of the vector x along the j and k axes, and ⋅ indicates the dot product of two vectors. Equation A5a quaternion-multiplies −T s with i and then takes the quaternion square root. (If p is the quaternion square root of q, then the quaternion product pp = q; in terms of rotations, p has the same axis and direction as q but only half the amplitude.) This operation yields a quaternion x representing the shortest rotation taking the vector i to the vector T s. In other words, x is the head orientation that would point the nose at the visual target while obeying a head version of Listing's law, with Listing's plane of the head orthogonal to the vector i. But because the head doesn't normally rotate far enough to point the nose at the target, Eq. A5b scales the rotation by the factors δV and δH. In all simulation plots except Fig. 6, δH is near 0.9 and δV is near 0.3, reflecting the fact that the head contributes more to horizontal than to vertical gaze shifts; (in Fig. 6, δH is varied between 1 and 1.4 and δV between 0 and 0.4 to simulate noise or alteration in the Donders operator). And because the head doesn't obey Listing's law, its position quaternion needs a torsional component, δT x 2 x 3 i, where δT is set at −0.15 (approximately equal to −δVδH/2) to yield a quadratic Donders surface for the head that matches those seen in the data (Glenn and Vilis 1992) (δT = 0 would yield a planar surface, i.e., Listing's law). Finally, Eq. A5c converts the vector y into a unit quaternion representing desired head position in space.
Again, the values given for δT, δV, and δV, and indeed the whole computation in Eqs. A5a–c, are merely defaults. It is clear that we can control voluntarily how much motion the head contributes in all three dimensions; i.e., the Donders operator can be voluntarily altered or overridden. Further, a more realistic model, particularly one involving torso movements, would require a more sophisticated way of choosing the final head position than just scaling by three factors δT, δV, and δH as in Eq. A5b, but this simple scheme seems satisfactory for torso-fixed saccades.
Head pulse generator
The head is driven by a velocity command q˙ h representing the rate of change of head position Equation A6where Vq is the operator that takes the vector part of a quaternion. If we write ν for Vq (q*h q −1 h), then the head pulse generator P h can be defined by Equation A7Pulse generator nonlinearities usually are modeled using exponentials (Robinson 1975; van Gisbergen et al. 1981), but the division by 1 + 20‖ν‖ used here is nearly equivalent and computationally simpler. Also, a more realistic model would have the head driven by a signal coding some mixture of jerk, acceleration, velocity, and position, rather than simply velocity, and would pass this signal through a higher-order plant, but such complications would not alter the basic kinematics under study here. Note that the output of the pulse generator is the quaternion velocity of the head, q˙ h—i.e., the time derivative of the quaternion of head position, q h—and not the angular velocity of the head [which would be given by ωhss = 2q˙ h q −1 h = 100 ν/(1 + 20‖ν‖)].
Desired eye position and the Listing operator
Desired final eye position in the head, q*eh, is computed by the formula Equation A8The Listing operator takes as input the unit vector q*−1 h T s q*h, representing the direction that the saccade target will have relative to the head when the head is in its desired position q*h, and yields as output a quaternion q*eh representing desired eye-in-head position, which fits Listing's law. To create this output, Listing quaternion multiplies −q*−1 h T s q*h by g p, which is the primary gaze direction of the eye expressed in head-fixed coordinates, and takes the quaternion square root, to yield a desired eye-in-head orientation, q*eh, that obeys Listing's law and that points the gaze line in the desired direction. In the simulations, g p is set equal to i, the forward pointing coordinate axis although in reality it moves slightly depending on head position (Haslwanter et al. 1992).
Eye saturation and pulse generation
For clarity, the computation yielding saturated desired eye position in the head can be broken down into three steps Equation A9a Equation A9b Equation A9c Equation A9a combines desired head-in-space position with desired eye-in-head position to yield desired eye-in-space position, q*es. This is then divided by q h, the system's internal estimate of actual head position, to yield the current desired eye-in-head position, q + eh, in Eq. A9b.
Equation A9c introduces the important function Sat, a3-D version of Guitton and Volle's (1987) one-dimensional saturation box. As we saw in Fig. 2 above, 3-D saturation involves projecting an overly eccentric current desired eye position q + eh into the effective oculomotor range, EOMR, so that the saturated desired eye position q s eh is that position on the boundary of the EOMR where the target will first be foveable. The model's algorithm for this projection rule is given by Equation A10What is happening here is that, if the squared horizontal-vertical eccentricity α of current desired eye position q + eh is larger than the squared radius of the EOMR [where radius = sin(40°/2) ≈ 0.12], then the code within the curly braces projects q + eh into the EOMR in the horizontal-vertical plane by solving a quadratic equation whose coefficients are a, b, and c. As this projected point may still lie outside the EOMR in the torsional dimension, the next two lines after the curly braces compute the maximum allowable torsion, given the horizontal and vertical coordinates of the projected point, and, if necessary, clip the torsional component of q s eh at that value. The 0.15 in the third last line is a parameter affecting the shape of the EOMR; it is set slightly larger than radius to give the EOMR a pill-like, rather than a purely ellipsoidal, shape (see Fig. 2). Similarly, the 0.25 in the same line is equal to 0.2 × 0.15/radius, and reflects the fact that the torsional "thickness" of the EOMR at its thickest point is only 20% of its horizontal or vertical diameter, i.e., the allowable torsion lies between ±0.2(40°) = ±8 deg. The last line in Eq. EA10 transforms the vector q s eh into a unit quaternion, which is the output of the saturation operator.
Finally, this saturated desired eye position signal is used to compute the saccadic eye velocity command q˙ eh Equation A11Here the eye pulse generator P e is defined similarly to P h in Eq. EA7; i.e., writing ν for V q(q s eh q −1 eh), we have Equation A12And q˙ VOR, the last term in Eq. EA11, is the eye velocity command from the VOR, which is described in the next section. Equation EA12 was used to model the pulse generator because it yields eye-only saccades with fixed axes and a realistic amplitude-duration relation.
VOR shutoff
As noted earlier, we know from 1- and 2-D studies that the VOR is weakened or shut off in the direction of the saccade. In 3-D, new questions arise, e.g., should the VOR be shut off in the direction of gaze error or head-in-space motor error or eye-in-head motor error? There are no experimental data on the precise direction of VOR shutoff, and, in any case, the model in this paper operates very well even if this direction is very imprecisely controlled. But it is an interesting theoretical point that if the VOR were switched off in the direction of current eye-in-head motor error, then it could do a surprising amount to steer the gaze line to the target, even in the absence of other signals guiding the eye. In other words, this sort of precisely directed VOR could provide a valuable backup if other parts of the saccadic system were malfunctioning. The following equations achieve precise VOR shutoff of this type. Equation A13 M is a 3-by-3 matrix, I is the 3-by-3 identity matrix, and uu T is the 3-by-3 matrix obtained by matrix-multiplying the column vector u by its transpose u T. The quaternion x represents current, unsaturated eye-in-head motor error, and C is a constant equal to cos(A/2), where A is the VOR shutoff amplitude. That is, when eye-in-head error exceeds A, the VOR is switched off completely. Then as eye-in-head error falls from A toward 0 (i.e., as its scalar component x 0 rises from C toward 1), the VOR gradually turns on again. In the simulations, A = 20° and therefore C = 0.985. It is a simple matter to verify that the above equations switch off the VOR in the direction of eye-in-head motor error x, and turn it on again gradually as motor error falls.
The following 2 m-files, esdot.m and eyespace.m, implement the model in Matlab. In these files, quaternions are expressed as 4-by-4 matrices of real numbers, with the result that all the quaternion algebra is done using Matlab's built-in matrix operators, so no extra m- or mex-files are needed. It is also possible to represent quaternions in Matlab as 2-by-2 matrices of complex numbers, but then the code runs slightly slower and certain operations involving quaternion vectors are more complicated. function xdot = esdot(t,x) % Convert input vector x into 4-by-4 matrices (quaternions) global i j k o % Basis quaternions, defined in eyespace.m (below) te = x(1)*i + x(2)*j + x(3)*k; % Target direction in eye coordinates = retinal error = retinotopic gaze error qh = x(4)*i + x(5)*j + x(6)*k + x(7)*o; % Head position qeh = x(8)*i + x(9)*j + x(10)*k + x(11)*o; % Eye-in-head position % Desired head position in space and head pulse generator qes = qh * qeh; ts = qes * te * qes′; % Eye-in-space position; Target direction in space q = real((−ts * i)∧ .5); % "real" removes imaginary parts brought in by "∧" xqh = −.15*q(2,4)*q(3,4)*i + .3*q(2,4)*j + .9*q(3,4)*k; m = xqh(:,4)′ * xqh(:,4); % Donders if m < = 1, xqh = xqh + sqrt(1 − m)*o; else, xqh = xqh/sqrt(m); end % Desired head position q = xqh * qh′; q = q − q(4,4)*o; dqh = q * qh * 40/(1 + 20 * norm(q(:,4))); % Head motor error, Head velocity % Desired eye position in space fxqeh = real((−xqh′ * ts * xqh *i)∧ .5); % Desired final eye-in-head position (obeys Listing's law) xqes = xqh * fxqeh; % Desired eye-in-space position % VOR cxqeh = qh′ * xqes; q = cxqeh * qeh′; % Current desired eye-in-head position; Eye-in-head error if q(4,4) < .985, m = 1; else, m = (1 − q(4,4))/.015; end; v = q(1:3,4); vor = m * v * v′/(v′ * v) − eye(3); whsh = 2 * (qh′ * dqh); % VOR matrix; Head angular velocity v = vor * whsh(1:3,4); weh = v(1)*i + v(2)*j + v(3)*k; dqVor = .5 * weh * qeh; % VOR eye velocity command % Saturation s = cxqeh(1:3,4); f = fxqeh(1:3,4); alpha = s(2) * s(2) + s(3) * s(3); % Squared eccentricity of current desired eye position in hor-vert plane radius = sin(40 * pi/360); % Radius of EOMR in hor-vert plane if alpha > radius * radius, beta = s(2) * f(2) + s(3) * f(3); gamma = f(2) * f(2) + f(3) * f(3); a = alpha − 2 * beta + gamma; b = 2 * (alpha − beta); c = alpha − radius * radius; % Polynomial coefficients root = (−b + sqrt(b * b − 4 * a * c))/(2 * a); s = s + root * (s − f); % Project s into EOMR in hor-vert plane alpha = radius * radius; end maxTorsion = sqrt(1.25 − alpha/(radius * radius)) * sin(4 * pi/180); if abs(s(1)) > maxTorsion, s(1) = sign(s(1)) * maxTorsion; end % Torsional component of s s(4) = sqrt(1 − s(1) * s(1) − s(2) * s(2) − s(3) * s(3)); % Scalar component of s sxqeh = s(1)*i + s(2)*j + s(3)*k + s(4)*o; % Convert saturated desired eye in head position, s, to a quaternion % Eye pulse generator q = sxqeh * qeh′; q = q − q(4,4)*o; dqeh = q * qeh * 80/(1 + 20 * norm(q(:,4))) + dqVor; % Gaze comparator dte = te * (qeh′ * (whsh * qeh + 2 * dqeh)); % Convert to derivative vector xdot for output xdot(1:3) = dte(1:3,4); xdot(4:7) = dqh(:,4); xdot(8:11) = dqeh(:,4); function eyespace global i j k o % Basis quaternions expressed as 4-b-4 matrices i = [0 0 0 1; 0 0 −1 0; 0 1 0 0; −1 0 0 0]; j = [0 0 1 0; 0 0 0 1; −1 0 0 0; 0 −1 0 0]; k = [0 −1 0 0; 1 0 0 0; 0 0 0 1; 0 0 −1 0]; o = eye(4); a = sin(70 * pi/180) * sqrt(.5); b = sqrt(1 − 2 * a * a); x0 = [b a a 0 0 0 1 0 0 0 1]; % Initial retinal error, head position and eye-in-head position [t, x] = ode23('esdot', 0, .5, x0); plot(x(:,9), x(:,10), 'c'), m = sin(60 * pi/360); axis([−m m −m m]), axis('square'), hold on, plot(x(:,5), x(:,6), 'r')


text: Daniel Daniel is 39 years old male from Trier, Germany. He is interested in meet new people. 39, male, Leipzig Likes: 30x Facebook profile Who s/he liked? Send message Favorite May she/he ask you for a date? Never Maybe Sure Listening to Music French Fries Hardstyle I Am Legend Příroda a procházky Kisses Relaxing Shoes Jokes Paris Report photo Sign up to see all photos Facebook Twitter Google+ Pinterest Similar users Matthew 41, Ravena 503 Edgardo 42, Chino 90 Kai 42, New York 262 mike 42, Potwin 188 john 39, Knoxville 334 Joe 40, Lafayette 317 Steve Duncan 42, Lake Forest 614 Joshua 40, Tonasket 262 Andre 41, Spartanburg 295

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text:
IRS Says No More Free Lunch for Silicon Valley Tech Companies
Short version: Numerous companies offer various 'perks' to employees to attract and retain them. In addition to the 'free lunch' mentioned, some others that come to mind are onsite free exercise centers, concierge services and more.
Should all fringe benefits be considered as taxable income?
Here are a few articles showcasing additional perks that may be targeted (*note some are already taxable):
14 Companies with Incredible Employee Perks
18 Of The Best Perks At Top Employers


text:
Those two are made for each other.
People disappear when they die. Their voice, their laughter, the warmth of their breath. Their flesh. Eventually their bones. All living memory of them ceases. This is both dreadful and natural. Yet for some there is an exception to this annihilation.
– Diane Setterfield, The Thirteenth Tale
No matter how deeply you go, my voice will go with you.
Natsumezaka Shiki (棗坂シキ) | Dance with Devils E08
How you think you look like VS How you really are
Collecting names of the lovers that went wrong. The lovers that went wrong.
*insert Russia's laugh here*
My creepy edit 👻🎃🕸
Happy 3rd Formation Announcement Anniversary, Sexy Zone
Hope we will be always able to see your smile.
Gillian Jacobs at the Variety Power of Women Event
artist kook
Sloppy edit sorry :/
BEN_Drowned edit inspired by the-real-ben-drowned
submitted by anonymous
this is something i found back in my files omg why
2014 Australian Open 1R: Novak Djokovic def. Lukas Lacko 6-3, 7-6(2), 6-1


text:
Fortinet Inc. (FTNT) PT Lowered to $31.00
Fortinet Inc. (NASDAQ:FTNT) had its target price reduced by equities research analysts at Citigroup Inc. from $35.00 to $31.00 in a research note issued on Wednesday. The firm currently has a "neutral" rating on the stock. Citigroup Inc.'s target price would suggest a potential downside of 9.06% from the company's current price.
Several other brokerages also recently weighed in on FTNT. Goldman Sachs Group Inc. downgraded shares of Fortinet from a "conviction-buy" rating to a "buy" rating and lowered their target price for the company from $43.00 to $36.00 in a report on Wednesday. Wells Fargo & Co. reissued a "buy" rating on shares of Fortinet in a report on Sunday, September 25th. Stephens downgraded shares of Fortinet from an "overweight" rating to an "equal weight" rating in a report on Wednesday. Royal Bank Of Canada decreased their price objective on shares of Fortinet from $37.00 to $30.00 and set a "sector perform" rating on the stock in a report on Wednesday. Finally, Wunderlich started coverage on shares of Fortinet in a report on Tuesday, September 13th. They set a "buy" rating and a $45.00 price objective on the stock. One equities research analyst has rated the stock with a sell rating, nine have assigned a hold rating and eighteen have given a buy rating to the company's stock. The company currently has an average rating of "Buy" and a consensus price target of $38.48.
Shares of Fortinet (NASDAQ:FTNT) opened at 34.09 on Wednesday. The firm has a 50-day moving average of $35.98 and a 200 day moving average of $33.64. Fortinet has a 12-month low of $23.16 and a 12-month high of $44.63. The stock has a market capitalization of $5.89 billion, a price-to-earnings ratio of 921.35 and a beta of 1.62.
This story is the sole property of American Banking News and it was originally published by American Banking News. If you are reading this story on another website, that means this article was illegally copied and re-published to this website in violation of U.S. and International copyright law. The original version of this article is available at https://www.americanbankingnews.com/2016/10/12/fortinet-inc-ftnt-pt-lowered-to-31-00.html
Fortinet (NASDAQ:FTNT) last issued its earnings results on Thursday, July 28th. The company reported $0.14 earnings per share for the quarter, meeting the Thomson Reuters' consensus estimate of $0.14. The business had revenue of $311.40 million for the quarter, compared to the consensus estimate of $304.71 million. Fortinet had a net margin of 0.07% and a return on equity of 0.68%. The business's quarterly revenue was up 29.9% compared to the same quarter last year. During the same quarter in the previous year, the firm earned $0.11 earnings per share. Equities analysts forecast that Fortinet will post $0.70 earnings per share for the current fiscal year.
In other news, VP John Whittle sold 13,663 shares of the company's stock in a transaction on Monday, August 1st. The shares were sold at an average price of $35.00, for a total value of $478,205.00. Following the completion of the transaction, the vice president now directly owns 18,969 shares in the company, valued at $663,915. The sale was disclosed in a legal filing with the Securities & Exchange Commission, which is available through the SEC website. Also, Director Ming Hsieh sold 23,900 shares of the company's stock in a transaction on Monday, August 22nd. The shares were sold at an average price of $34.84, for a total transaction of $832,676.00. The disclosure for this sale can be found here. Company insiders own 16.40% of the company's stock.
Several hedge funds and other institutional investors have recently made changes to their positions in the stock. US Bancorp DE increased its position in shares of Fortinet by 3.8% in the second quarter. US Bancorp DE now owns 3,342 shares of the company's stock valued at $106,000 after buying an additional 123 shares during the period. Nordea Investment Management AB increased its position in shares of Fortinet by 10.4% in the second quarter. Nordea Investment Management AB now owns 3,865 shares of the company's stock valued at $122,000 after buying an additional 365 shares during the period. Advisor Group Inc. purchased a new position in shares of Fortinet during the second quarter valued at $152,000. Tower Research Capital LLC TRC increased its position in shares of Fortinet by 179.7% in the second quarter. Tower Research Capital LLC TRC now owns 6,464 shares of the company's stock valued at $204,000 after buying an additional 4,153 shares during the period. Finally, Compagnie Lombard Odier SCmA increased its position in shares of Fortinet by 62.0% in the second quarter. Compagnie Lombard Odier SCmA now owns 6,500 shares of the company's stock valued at $206,000 after buying an additional 2,487 shares during the period. Institutional investors and hedge funds own 70.65% of the company's stock.
Fortinet Company Profile
Fortinet, Inc provides cyber security solutions to enterprises, service providers and government organizations across the world. The Company's flagship network security solution consists of FortiGate physical, virtual machine and cloud platforms, which provide integrated security and networking functions to protect data, applications and users from network-and content-level security threats.
Receive News & Ratings for Fortinet Inc. Daily - Enter your email address below to receive a concise daily summary of the latest news and analysts' ratings for Fortinet Inc. and related companies with MarketBeat.com's FREE daily email newsletter.


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July 28, 2013
Reaching Your Goals Realistically
Pisces Daily Horoscope
Your approach to matters pertaining to your life situation could take on a more down to earth slant today and this could mean that you may be much more likely to make decisions based on a realistic outlook. Maybe your desire to see things reasonably might be due to your need to plan more carefully for your future, and this feeling may provide you with the motivation you need to really assess what your overall long-term goals are and how you can most effectively reach them. Try making a list of the things that you know you hope to attain in your life within the next 3 years. As you think of these things today, you can ask yourself what skills and resources you currently have that you can reasonably put into your goals at this moment. The objectives that seem the most practical, you could find, may be the ones that are the most worthwhile exploring.
Giving ourselves a specific target to focus on allows us to see how we can wisely and cogently use our present situation to manifest our future. Since we have so many mixed messages about what we should do with what we have, it can be difficult to put our time, energy and savings into something that will really mean the most to us. Consciously thinking about your future in a logical and rational way, however, will help you to become more aware of how you will best be able to allocate what you have to do with what is really meaningful for you.
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A new species of Heteroclinus is described from Victoria, South Australia and Western Australia. The species is distinctive from other Australian clinids in having two well-developed segmented pelvic rays, first dorsal fin on head originating just behind or at posterior end of eye, a reduced lateral line and in fin-ray counts.
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To read the latest tweets from @museumvictoria
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Hi Judith. I arrived in Southhampton on the Fairsea on 8 September 1966 from Melbourne. You boarded the ship that I had just left a few days earlier. I hope ...


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After an initial phone call that still makes them cringe, the television odd couple of Andy Cohen and Anderson Cooper became good friends.
Both men are putting their easy camaraderie on display in a live lecture series this spring. The idea came from Cohen's mother and Cooper's agent, who watched in November when Cooper interviewed Cohen about his book onstage in New York. The series, subtitled "Deep Talk and Shallow Tales," starts Mar...
Also read: 'Better Call Saul' filming third season in Albuquerque
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17:04 Rock-smashing monkeys make flakes like early stone tools
22:56 Perrie Edwards Admits She Was 'Like A Woman Possessed' During Nasty Phone Fight With Zayn Malik!
18:50 Police in Texas search for two teens who mysteriously vanished


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Arts, Visual Arts, Games & Hobbies, Hobbies, Technology, Tech NewsFollow
|Visual Arts
Join Jim Harmer and the hosts of all the Improve Photography Network podcasts for the weekly roundtable each week. Photography nerds around the world listen to the Improve Photography Podcast and praise the show for including enough unique tips to appeal to the professional while not leaving the beginning photographer behind.
December 17th, 2013
Episode 19 of 182 episodes
Click the title of the article to read this post on Improve Photography, which includes all media files mentioned. World famous photographer Joel Grimes shares how to develop your own unique style in photography, how he got out of a rut in his career, and his tips and techniques for shooting 32 bit photos, strobing HDRs, and more in episode 44 of the Improve Photography Podcast. What's in this episode How Joel Grimes got CONTINUE The post Joel Grimes on Developing Your Style [IP44] appeared first on Improve Photography.
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Have you trembled 'round the campfire as the ghost stories are told? Do tales of horrors wrought by nature and beyond strike fear in your soul? Is it best when you are afraid to turn out the light? Thrill to stories that strike at your deepest fears, as host Stephen Kilpatrick brings the best of horror fiction to your ears and your mind, read to you by the most chilling narrators that podcasting has to offer.

 Podcasting the finest in genre fiction, Tales to Terrify is where the depths of horror reveal the truths of good, evil, and the human spirit in the District of Wonders podcast network. Like all shows in the District of Wonders, Tales to Terrify is supported by a welcoming community of dedicated fans and contributors. Subscribe today, and begin your journey through the spine-tingling depths of storytelling.

 Everyone has a story in the District of Wonders. Come and find yours.


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I am trying to get a particular item value from the dropdown list in the WebList object. But unable to get the value dynamically.
I tried with the below code:
Val = Browser("Browser").Page("Page").WebList("Weblist Name").GetItem(3).
But this returns the 3rd item of the list captured during recording only.
If the values in the weblist changes during runtime, i am not getting the new value at the third position.
Kindly help me with this as i am new to QTP.
Thanks in advance.
Try to print all location and see exactly it which location do you get the new item?
Tarun Buy the NEW QTP Descriptive Programming Unplugged Book ** Download QTP Unplugged 2nd Edition eBook for FREE **
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Guess How Much I Love You in the Summer
Written by Sam McBratney
Part of the Guess How Much I Love You Series
Lovereading4kids Price £4.79
£5.99 saving £1.20 (20%)
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The Lovereading4Kids comment
From the award-winning creators of Guess How Much I Love You comes this delightful summertime tale about colours. Seasonally themed and beautifully illustrated throughout, little ones will also have fun learning about colours while discovering what Big Nutbrown Hare's favourite one is! This title is one of a four-part series featuring the seasons - why not collect them all!
"The best thing about books like Guess, books which are very popular, is this: I know that somewhere in the world a mum or dad will reach for a copy of something I've written, and read it to the most precious thing they have in the world. That thought really pleases me." Sam McBratney
*** Celebrate twenty years of sharing love to the moon and back with this special anniversary paperback edition of one of the world's bestselling picture books and there is a fantastic Storytime Event Kit full of ideas for your own Little Nutbrown Hare party with games, suggested activities and a lovely party invite - download it here!
Guess How Much I Love You in the Summer by Sam McBratney
Which blue do you like best? asked Little Nutbrown Hare. It's summer and Little Nutbrown Hare learns about all of the different colours that are around him - the red of the berries, the green of the leaves, the blue of the sky...
Celebrating 20 years of Guess How Much I Love You - for games, activities and more about the books visit www.guesshowmuchiloveyou.com
Praise for Guess How Much I Love You
'This is a book which both parents and children adore and will be a book any adult will happily read over and over again at bedtime!' The Guardian
'The pictures, which are uncluttered, light, expressive and full of movement, match the poignancy of the text. They are delightful to children and the simple, poetic language lends itself to re-reading.' The Times
'Gorgeous watercolours illustrate this affectionate story. A perfect picture book' The Good Book Guide
'A super book!' Home and Country
'Charming and captivating' The Observer
'Delightfully illustrated' The Spectator
About the Author
Other Formats
Book Info
24 pages
Interest Age: From 3
Sam McBratney
More books by Sam McBratney
Walker Books Ltd
Publication date
7th May 2015
Lovereading is just a convenient way to find new books and hear others opinions on them.Sarah Murray – age 15
Love "Lovereading4kids" as my son gets to hear about & read new books before his mates which keeps him interested in reading=a very happy MumLiz Evans
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It gives me a chance to read types of books that I would not normally try, and it motivates me to read every night to finish it!Alice Horncastle, age 14


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The Golden State Warriors and Atlanta Hawks have regained control of their respective series, doing their best to restore order in the NBA playoffs.
CHARLOTTE, N.C. (AP) - Robert Streb shot a 7-under 65 on Thursday to take a one-shot lead over Patrick Reed and Kevin Chappell after the first round of the Wells Fargo Championship.
SANTA CLARITA (AP) - Britain's Mark Cavendish raced his third Tour of California stage victory and Latvia's Tom Skujins retained the overall lead Thursday in the rainy fifth stage.
CINCINNATI (AP) - One of the most famous All-Star endings won't be repeated when the game returns to Cincinnati this summer.
OAKLAND (AP) - Stephen Curry turned in an MVP-worthy performance, Klay Thompson snapped out of his shooting funk and the Golden State Warriors rolled past the Memphis Grizzlies 98-78 on Wednesday night to take a 3-2 lead in their Western Conference semifinal series.
HOUSTON (AP) - The San Francisco Giants have been superb in one-run games to start the year.
OAKLAND, Calif. (AP) - Bob Melvin knows Sonny Gray's winning streak should still be going. More missed chances, more mistakes defined the day once again.
HOUSTON (AP) - Chris Heston was glad he was given the opportunity to go back out to the mound in the ninth inning.
OAKLAND (AP) - For one night at least, the Oakland Athletics put everything together.
SAN JOSE (AP) - Latvian Tom Skujins pedaled to a surprising mountain stage win by more than a minute and took the lead Tuesday in the Tour of California.
FOXBOROUGH, Mass. (AP) - Tom Brady and the New England Patriots are preparing for a fight.
LAS VEGAS (AP) - Floyd Mayweather Jr. and Manny Pacquiao proved a box office smash, even if their welterweight title fight didn't live up to the hype.
HOUSTON (AP) - James Harden had 26 points, 11 rebounds and 10 assists, Dwight Howard added 20 points and 15 rebounds, and the Houston Rockets bounced back from two lopsided losses to avoid elimination with a 124-103 victory over the Los Angeles Clippers on Tuesday night in the Western Conference semifinals.
TAMPA, Fla. (AP) - Nikita Kucherov had two goals and an assist and the Tampa Bay Lightning advanced to the Eastern Conference finals with a 4-1 victory over the Montreal Canadiens on Tuesday night.
NEW YORK (AP) - The NFL came down hard on its biggest star and its championship team, telling Tom Brady and the Patriots that no one is allowed to mess with the rules of the game.
Page 235 of 842
SANTA CLARA (AP) - Colin Kaepernick has brought some of his San Francisco teammates closer with his national anthem protests against social injustice.
OAKLAND (AP) - Hisashi Iwakuma pitched into the seventh inning to match his career high with 15 wins, and the Seattle Mariners beat the Oakland Athletics ...
ATLANTA (AP) - Mets general manager Sandy Alderson is defending the team's deal with Tim Tebow, saying the decision is about baseball and not any ...
NEW YORK (AP) - For quite a while, Novak Djokovic's opponent in the U.S. Open semifinals, Gael Monfils, looked as if he didn't ...
CARMEL, Ind. (AP) - U.S. Open champion Dustin Johnson doesn't care about the physics behind his new putter. Seeing more putts drop is enough ...
CHARLOTTE, N.C. (AP) - Panthers coach Ron Rivera wants Cam Newton to receive the same protection on helmet-to-helmet hits as other quarterbacks get in the ...
CLEVELAND (AP) - One show, one match at a time, CM Punk climbed to the top of pro wrestling, acquiring unimagined fame and fortune along the ...
EDEN PRAIRIE, Minn. (AP) - The rehabilitation of Minnesota Vikings quarterback Teddy Bridgewater's left knee will require recovery from multiple ligament reconstruction.
Chris Buescher's task, should he accomplish it Saturday night at Richmond International Raceway, is no small feat.
DENVER (AP) - Cam Newton will get a few extra days to recover from his latest beating by Von Miller and crew. At least he won ...
SPRINGFIELD, Mass. (AP) - The Naismith Memorial Basketball Hall of Fame will soon welcome two of the sport's game-changing stars - one big and one small.
NEW YORK (AP) - Having flopped with the New York Jets, Tim Tebow will try to revive his career as an outfielder with the Mets.
CARSON (AP) - Landon Donovan is returning to the LA Galaxy, ending a 21-month retirement to suit up for his Major League Soccer team for at ...
CARMEL, Ind. (AP) - Roberto Castro wants nothing more than to go home to Atlanta, and he took a soggy step in the right direction Thursday ...
Ryan Lochte will forfeit $100,000 and a chance to swim at next year's world championships as part of the penalty for his drunken ...
Page 23 of 31
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Jodie Foster
Jodie Foster. #Davids05 #LAD #LADavids https://www.facebook.com/LDSTO-1709014606047668/ https://www.facebook.com/Sensualidad-1402482520062913/?ref=hl https://relaxliveblog.wordpress.com/ https://www.facebook.com/Disfruta-el-Momento-Enjoy-the-Moment-750346691726285/?ref=hl
Jodie Foster: brainy, gorgeous badass at every age.
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Jodie Foster
Alicia Christian "Jodie" Foster (born November 19, 1962)[1] is an American actress, director and producer who has worked in films and on television. She has often been cited as one of the best actresses of her generation.[2][3] Foster began her career at the age of three as a child model in 1965, and two years later moved to acting in television series, with the sitcom Mayberry R.F.D. being her debut. In the late 1960s and early 1970s, she worked in several primetime television series and…
Jodie Foster Famous People multicityworldtravel.com We cover the world over 220 countries, 26 languages and 120 currencies Hotel and Flight deals.guarantee the best price
Jodie Foster by Annie Leibovitz, art, portrait, strong, woman, female, actress, versatile, personality, celeb, famous /// A few less fawning adjectives come to mind for me: boring and pretentious among them, mostly spawned by her ridonkulous Oscar speech a few years ago, but also by her gravitas in silly roles. It's not a death camp; it's entertainment, seriously...
Jodie Foster (b 1962)multi award winning American actress, film director and producer.
Taxi Driver (1976). Dirigida por Martin Scorsese y protagonizada por Robert De Niro, Cybill Shepherd, Jodie Foster y Harvey Keitel. Taxi Driver es una excepcional obra maestra por su extremo riesgo y coherencia estética, porque no busca soluciones fáciles
Taxi Driver, 1976 - original vintage poster listed on AntikBar.co.uk


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27.10.16 – 28.10.16 (1 night)
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text: Help...I'm totally confused..... we are moving to Adelaide in 2009 & looking to rent a property for 6 months until we get settled into the right area. I will be entering as a student until I obtain an accountancy degree. Are we able to buy a property before obtaining a residential visa? Also, my Partner has investments in London, how do we transfer the income into Australia without paying double tax - does anyone recdommend a good IFA?? Any input would be much appreciated. Sue


text:
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by POWDERMAKER, HortenseBoston : Little, Brown , (1950)
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Vettel lays down marker ahead of US Grand Prix with fastest lap in first practice
Sebastian Vettel issued an early warning to Fernando Alonso that he is not prepared for their title fight to go down to the wire.
In the opening practice session at the £250million Circuit of The Americas that plays host to the first United States Grand Prix for five years, Vettel left his rivals in the shade come the end.
It is, of course, still early days on a track still very green and needing to be rubbered in, but as a potential marker, it will have had an effect.
Fastest: Vettel has laid down a marker for Sunday's Grand Prix
Vettel needs to finish 15 points clear of Alonso on Sunday to become only the third driver in Formula One history - Juan Manuel Fangio and Michael Schumacher the others - to claim three titles in a row.
Organisers predicted a lap time of one minute 37 seconds for the 5.516km track that has 'borrowed' a number of features from other circuits.
Vettel, however, finished the initial 90 minutes with a time of one minute 38.125secs, and that on the hard, slower tyre compared to the option of the faster medium this weekend that was never used by any driver during FP1.
Waiting in the wings: Alonso is looking to catch the German champion
The opening laps were naturally cautious, with the track noticeably slippy, and with the first timed lap set by Sauber's Kamui Kobayashi of one minute 55 seconds.
Eventually McLaren's Lewis Hamilton took over proceedings and led the way for a lengthy period until the final 10 minutes when Vettel broke the back of the track.
The 25-year-old Red Bull star, who is currently 10 points clear of Alonso, finished 1.418secs clear of Hamilton, the biggest margin at the end of any practice session this season.
Debut: Austin is ready to host the American Grand Prix
Naturally, it is not known what fuel level Vettel was running, but even so it was still an impressive margin, with Alonso third on the timesheet, 2.212secs adrift in his Ferrari.
The top three teams occupied the top six places, with Jenson Button in his McLaren, Red Bull's Mark Webber and Ferrari's Felipe Massa fourth, fifth and sixth respectively, all around 2.5secs down.
Force India's Nico Hulkenberg was the best of the rest ahead of Kobayashi and team-mate Sergio Perez in eighth and 10th, with Nico Rosberg in his Mercedes splitting the Sauber duo.
Three and easy: Vettel is looking to secure his third consecutive crown
Paul di Resta, apparently struggling badly with the balance of his Force India car, was 11th quickest, 3.658secs behind Vettel, and 0.8secs down on team-mate Hulkenberg.
The Lotus duo of Kimi Raikkonen, who gave the team their first win this season last time out in Abu Dhabi, and Romain Grosjean were a lowly 14th and 15th but with the prospect of more to come.
At the bottom, and clearly out of his depth on this occasion, was HRT reserve Ma Qing Hua, the Chinese finishing 10.434secs off the pace, and 1.7secs behind team-mate Pedro de la Rosa.
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3 Bedroom Flat For Sale
Drayton Park, London, N5
Luxury penthouse featuring the best of contemporary materials, with generous space and versatile layout. Located just a step away from the trendy Angel and Islington. This 3 bedroom penthouse is available to view straightaway and viewing is highly advised. Perfectly...
3 Bedroom Flat For Sale
Petherton Road, London, N5
This charming three bedroom maisonette is set over the lower and upper ground floor of this attractive period building on a tree lined road in Canonbury with a rear garden. EPC:D
2 Bedroom Flat For Sale
Petherton Road, London, N5
An impressive two bedroom apartment over the raised ground and first floor of a period house in Canonbury. The apartment has undergone a wealth of re-furbishment to offer a perfect blend of contemporary and modern elegance whilst maintaining and restoring much of its...
2 Bedroom Ground Floor Flat For Sale
Highbury Stadium Square, London, N5
A superb two double bedroom two bathroom ground floor apartment located at Highbury Stadium Square, the former site of Arsenal Football Club. Benefiting from two spacious double bedrooms, a large open plan kitchen and sitting room, as well as two generous sized...
2 Bedroom Flat For Sale
Morea Mews, London, N5
A two bedroom and two bathroom lateral apartment within a newly built block.
3 Bedroom Flat For Sale
Loxford Gardens, London, N5
A wonderfully spacious three bedroom apartment split across two floors in the heart of Higbury Barn. Benefits include a rear patio garden, extensive communal garden and 24 hour concierge. EPC:B
2 Bedroom Flat For Sale
Highbury Park, London, N5
A two bedroom period conversion located on the third floor of this early 1900s build. EPC:D
2 Bedroom Flat For Sale
Sotheby Road, London, N5
This two bedroom period conversion is located in one of Highbury's most sought after streets moments from Highbury Barn. EPC:D
2 Bedroom Flat For Sale
Canning Road, London, N5
A fully refurbished and immaculately presented two bedroom garden apartment with its own entrance. Superbly located for lots of transport links and shopping. Boasting it's own entrance and share of freehold.
1 Bedroom Flat For Sale
Aberdeen Road, London, N5
Positioned on the first floor of this imposing period building is this grand and spacious one bedroom flat complete with an abundance of period features.
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