The relationship between curvature and velocity in two-dimensional smooth pursuit eye movements

Abstract

Curvature and tangential velocity of voluntary hand movements are constrained by an empirical relation known as the Two-Thirds Power Law. It has been argued that the law reflects the working of central control mechanisms, but it is not known whether these mechanisms are specific to the hand or shared also by other types of movement. Three experiments tested whether the power law applies to the smooth pursuit movements of the eye, which are controlled by distinct neural motor structures and a peculiar set of muscles. The first experiment showed that smooth pursuit of elliptic targets with various curvature-velocity relationships was most accurate when targets were compatible with the Two-Thirds Power Law. Tracking errors in all other cases reflected the fact that, irrespective of target kinematics, eye movements tended to comply with the law. Using only compatible targets, the second experiment demonstrated that kinematics per se cannot account for the pattern of pursuit errors. The third experiment showed that two-dimensional performance cannot be fully predicted on the basis of the performance observed when the horizontal and vertical components of the targets used in the first condition were tracked separately. We conclude that the Two-Thirds Power Law, in its various manifestations, reflects neural mechanisms common to otherwise distinct control modules.

Fig. 1.

Shape and kinematics of the targets, first experimental condition. A, Trajectories; the major axis of the trajectory was the same in all cases. B, Horizontal (HOR) and vertical (VERT) velocity components of the target as a function of time for Σ = 0.936 and each value of β. For β = 2/3, the components are sine and cosine functions (Lissajous movement). Only one cycle is shown. C, Tangential velocities corresponding to the components in B (the vertical spacing between traces is arbitrary). Time and velocity calibration bars are common to B and C.D, Polar plot of the tangential velocity distribution along the trajectory for Σ = 0.936 and the extreme values of β. The velocity at each trajectory point A is proportional to the length of the segment A–B.

Fig. 8.

Tracking compatible targets, second experimental condition. Left column, Tangential velocity and derivatives of the cartesian components. Averages and 95% confidence bands calculated on three subjects; same format as in Figure 5. Target tangential velocities are identical to those in the first condition for Σ = 0.968 and for the associated β (A, β = 4/3;B, β = 1; C, β = 2/3;D, β = 1/3). The velocity of the components depended on the trajectory of the stimuli (right column) and were not the same as those in the first condition. By design, all stimuli in this condition complied with the Two-Thirds Power Law.

Fig. 2.

Tracking eye movements, first experimental condition. Left, The four bottom traces inA (β = 4/3) and B (β = 1/3) show representative examples of the horizontal and vertical position of the target (TARG) and the tracking eye movements (TRACK) for Σ = 0.936 (for clarity, only 6 cycles are plotted). HOR, Horizontal component;VERT, vertical component. Movement directions for both components are indicated by arrows (RW, Rightward; LW, leftward; UW, upward;DW, downward). In the four top traces, tracking components were dissociated into saccadic (SACC) and smooth pursuit (SP) contributions. Right, Representative examples of tracking trajectories (POS, including both saccades and smooth pursuit) and the polar plot of the smooth pursuit velocity (VEL), superimposed to the corresponding curves of trajectory and velocity for the target (thick continuous line). Target shapes were reproduced fairly accurately. Target velocity was generally underestimated.

Fig. 3.

Top panel, Mean and 95% confidence interval of the number of saccades per cycle. Middle andbottom panels, Mismatch index (mean and 95% confidence interval) for smooth pursuit and saccades, respectively. Results for all targets and all subjects in the first experimental condition. Gaze-target distance never became zero during smooth pursuit (positive values of the smooth pursuit index). Most saccades were compensatory (positive values of the index). Between β = 5/6 and β = 2/3, smooth pursuit was most effective, and the contribution of compensatory saccades was smallest.

Fig. 4.

Polar plots of the average retinal position error (RPE) along the trajectory for the indicated values of β at Σ = 0.968. Instantaneous values of the RPE for all subjects were averaged within each of 16 angular sectors. The average RPE within each sector is proportional to the radial distance from the inner ellipse taken as zero reference. Radial bars indicate 95% confidence intervals of sector means. Note the nonuniform distribution of the RPE along the movement cycle.

Fig. 5.

Tangential velocity and derivatives of the cartesian components of the smooth pursuit. Averages for all cycles and subjects for the indicated values of β at Σ = 0.968 in the first experimental condition. The 95% confidence bands (shaded areas) were computed from individual means over nine cycles.Thick lines indicate tangential and component velocities of the target. Thin horizontal lines indicate zero reference.

Fig. 6.

Relationship between radius of curvature and tangential velocity during smooth pursuit in the first experimental condition. Data points indicate instantaneous values ofR and V in logarithmic scales. Results for the indicated selected β at Σ = 0.968 and all subjects.Thick lines indicate covariation of R andV of the targets. A regression analysis estimated β from the slope of the major axis of the confidence ellipse (thin lines) and the so-called normal coefficient of correlation (E_max −E_min)/(E_max +E_min) (E_max andE_min, larger and smaller eigenvalues of the variance–covariance matrix, respectively). β of the smooth pursuit was only weakly dependent on target’s β.

Fig. 7.

Smooth pursuit β values (ordinate) as a function of target β values (abscissa). Averages and SD values were calculated from individual regressions. If smooth pursuit reproduced the stimuli faithfully, all data points would fall on the dashed line. This happens only for Lissajous targets (arrow).

Fig. 9.

Tracking compatible targets, second experimental condition. Average number of saccades per cycles (top panel) and mismatch indexes (middle andbottom panels) as a function of target type. Results for three subjects are presented in the same format as in Figure 3. Target types are identified by letters referring to the β values for the corresponding targets in the first condition (A, β = 4/3; B, β = 7/6;C, β = 1; D, β = 5/6;E, β = 2/3; F, β = 1/2;G, β = 1/3). For comparison, the dotted lines report again the data in Figure 3 for Σ = 0.968.

Fig. 10.

Tracking 1-D targets, third experimental condition. Smooth pursuit velocity while tracking independently the horizontal and vertical components of the targets used in the first experimental condition (Σ = 0.968 for the indicated β). The reconstructed tangential velocity was computed by vector sum of the velocity components. Averages and 95% confidence bands were calculated on three subjects; same format as in Figure 5.

Fig. 11.

Analysis of the smooth pursuit system in the frequency domain. A, B, Gain and phase characteristics (Bode plots) of the smooth pursuit velocity components in the first experimental condition. Results for one target (Σ = 0.968, β = 4/3). Averages and 95% confidence intervals computed from individual Bode plots. For both horizontal and vertical components, the transfer function of a four-pole low-pass filter (continuous lines) fits well the gain data points up to ∼5 Hz. At all frequencies, the same transfer function predicts a much larger phase lag than the data points, suggesting a predictive component in the pursuit system. Beyond 5 Hz, stimuli did not have appreciable power. Harmonic components of the oculomotor response in this range were generated internally. C, D, Comparison in three subjects of the horizontal and vertical gain characteristics between the first and second experimental conditions. First condition, results for one target (Σ = 0.968, β = 4/3, topmost tracesin Fig. 5). Second condition, compatible target with the same tangential velocity distribution (topmost traces in Fig.8). The dynamic range of the system was much wider when pursuing compatible targets than when pursuing targets that violate the Two-Thirds Power Law. E, F, Comparison in three other subjects of the gain characteristics between the first and third experimental conditions. First condition, target as inC and D; second condition, components of the same target but tracked separately. The dynamic response of the pursuit system in the 1-D case was significantly better than in the 2-D case. In all three conditions, responses along the vertical direction were more sluggish than responses along the horizontal direction.