Article Figures & Data
Figures
- Fig. 1.
Similarities of MSD and MV model trajectories (see Materials and Methods). Shown are velocity profiles of minimum acceleration, jerk and snap (MA, MJ, MS), minimum variance second and third order (MV2, MV3 with third time constant = 10 msec); the descriptive Yarbus model (Y) is also shown for its similarity to MA. The time-origin is centered at peak velocity, which has been normalized to unity. Trajectories have been scaled in time so that velocity is 0.25 at ±0.5 time units, except in MV3 where a slight asymmetry precluded the alignment at +0.5 time units. For clarity, the order with which the profiles reach zero velocity has been mirrored in the legend. The MV profiles are based on an amplitude of 5° and duration 38 msec.
- Fig. 2.
Fourier energy spectra of velocity profiles plotted on log10-linear axis. Energy plots are shown for the models in Figure 1 and for the rectangular pulse bang-bang model (BB). For clarity the sharpness of the minima has been reduced, and the ordinates have been offset. The minima frequencies and their ratios are summarized in Table 3.
- Fig. 3.
Illustration of bivariate confidence regions. Ellipses show 95% (inner) and 99% (outer) confidence regions for the estimate of the population slope and intercept of the individual M1 versus 1/T regressions (○). The center, ( x̅, y̅), is shown by thecross-hair.
- Fig. 4.
Illustration of self-similarity using a parabola as an example. A, The basic temporal shape (curvea) remains the same with arbitrary scaling in time (curveb) or in velocity (curvec) or both (curved). B, The Fourier transform of the curves in A. Scaling in velocity amplitude scales the overall energy without affecting the frequency at which the minima occur, whereas scaling in time has an inverse relationship on the minima frequencies. The ratios between the minima are unaffected by the time or amplitude scaling for a given shape, as shown in theinset for the first two minima (see Materials and Methods).
- Fig. 5.
Illustration of bang-bang control with a second-order ocular plant. A, Simplest control is a rectangular pulse in which maximum agonist signal is maintained. Different amplitudes are achieved by changing the duration of the rectangle (dotted line); hence pulses are self-similar (see Materials and Methods). B, Velocity trajectories resulting from A are not self-similar.
- Fig. 6.
Time-domain analysis of saccades showing typical temporal main sequence relationships. A, Peak velocity versus amplitude. B, Duration versus amplitude.C, Peak velocity × duration versus amplitude. Regression line constrained through origin gives the ratio of peak velocity to mean velocity, Q.
- Fig. 7.
Typical velocity trajectories of saccades with superimposed MV trajectory. Shown is near symmetry for low amplitudes, becoming more skewed for large amplitudes. Plots were aligned approximately with peak velocity, and amplitudes were 5.4° (▪), 10.3° (▴), and 20.0° (●). The lines show the MV (with third time constant of 4 msec) profiles for matched amplitudes and durations.
- Fig. 8.
Typical energy spectrum of the velocity profile of a saccade (inset) with amplitude 9.5°, durationT = 47 msec (1/T ∼ 21 Hz). Log energy is plotted against linear frequency. Only the first three minima are shown.
- Fig. 9.
Typical SMS for an individual subject.A, Plots of minima frequencies M1, M2, and M3 versus reciprocal duration of saccade (1/T) for the same subject as in Figure 6. B, Plots of M2 and M3 versus M1, and M3 versus M2. Note near-linear relationships in all plots.Dotted lines show bivariate linear regressions.Solid lines indicate predicted harmonic relationships for the rectangular bang-bang control pulse.
- Fig. 10.
Comparison of SMS confidence regions with models. Ellipses show 95% (inner) and 99% (outer) confidence regions for the group SMS (see Materials and Methods). Predicted rectangular BB model (square), MSD models (circles, size in increasing order of derivative: acceleration, jerk, snap), Yarbus model (cross-hair), and MV models (triangles, size in increasing order of third time constant:t3 = 0, 4, 10 msec) are plotted. The equations for the 95% (d = 1.004) and 99% (d = 1.946) ellipses are also shown. All model slopes and intercepts are summarized in Table 3.
- Fig. 11.
Neural control signals for MV models. Predicted combined agonist and antagonist motoneuronal firing rate for the second-order MV (solidlines) and third-order MV (dottedlines,t3 = 4 msec) models.
Tables
Model Velocity Fourier energy spectrum Yarbus (1967) 
Minimum acceleration Minimum jerk Minimum snap Rectangular pulse Equations of velocity trajectories and their Fourier energy spectra for Yarbus, MSD, and rectangular pulse models. Trajectories are normalized for unit amplitude and unit duration with time origin at peak velocity.
ID n T-A Q M1–1/T M2–1/T M3–1/T M2–M1 M3–M1 M3–M2 Incpt Slope r Incpt Slope r Incpt Slope r Incpt Slope r Incpt Slope r Incpt Slope r Incpt Slope r 1 70 28.3 3.37 0.91 1.80 −7.29 2.02 0.93 −7.76 3.24 0.90 −15.5 4.82 0.86 3.90 1.60 0.91 4.16 2.30 0.86 −0.99 1.42 0.94 2 77 20.5 3.15 0.95 1.71 −9.40 2.12 0.98 5.52 2.46 0.95 0.62 3.69 0.92 16.4 1.16 0.96 17.2 1.73 0.95 −7.73 1.50 0.95 3 70 23.5 2.22 0.96 1.61 −9.58 2.04 0.97 −6.97 3.05 0.88 −13.4 4.51 0.83 7.60 1.52 0.88 8.40 2.24 0.85 −2.74 1.47 0.90 4 73 21.6 3.00 0.95 1.60 −0.19 1.56 0.91 0.89 2.72 0.89 1.34 3.77 0.82 1.03 1.75 0.89 4.76 2.33 0.86 3.18 1.33 0.87 5 71 20.3 2.83 0.98 1.64 −2.53 1.70 0.99 1.79 2.64 0.97 13.7 3.14 0.90 5.68 1.55 0.96 18.4 1.83 0.90 11.8 1.18 0.93 6 73 29.5 2.54 0.91 1.64 −7.25 1.99 0.97 0.33 2.67 0.93 −5.55 1.09 0.88 10.2 1.34 0.94 10.8 2.00 0.90 −4.13 1.49 0.93 7 79 20.7 2.42 0.96 1.60 −1.12 1.60 0.97 1.08 2.71 0.97 17.7 3.10 0.92 2.92 1.70 0.96 19.8 1.93 0.91 16.6 1.14 0.94 8 57 28.0 2.71 0.90 1.54 −2.36 1.55 0.96 −16.8 3.54 0.88 −16.2 4.72 0.86 −11.3 2.28 0.88 −8.64 3.02 0.87 6.22 1.33 0.93 9 80 30.4 2.66 0.96 1.60 −9.47 2.18 0.98 −3.02 2.93 0.87 2.41 3.55 0.87 10.0 1.32 0.87 18.6 1.60 0.86 6.48 1.21 0.90 10 78 24.2 3.14 0.91 1.72 −7.05 1.95 0.98 −3.48 2.83 0.96 −6.11 4.05 0.91 6.74 1.45 0.96 8.76 2.06 0.91 −0.77 1.42 0.92 Mean 72.8 24.7 2.80 1.65 −5.62 1.87 −2.84 2.88 −2.11 3.94 5.32 1.57 10.2 2.11 2.78 1.35 Regressions of duration–amplitude main sequence (T-A) intercept (incpt) and slope, the ratio of peak to mean velocity Q, and the spectral main sequence bivariate regressions (see Materials and Methods) are shown along with the respective correlation coefficients and number of saccades scored for each subject (n).
Rectangle Minimum acceleration Yarbus model Minimum jerk Minimum snap Minimum variance T3 = 0 msec T3 = 4 msec T3 = 10 msec Incpt Slope Incpt Slope Incpt Slope M1 versus 1/T 1.00 1.44 1.50 1.84 2.24 −1.97 1.44 −4.66 1.81 −3.84 1.89 M2 versus 1/T 2.00 2.46 2.50 2.90 3.32 −1.52 2.48 −4.10 2.92 −2.83 2.95 M3 versus 1/T 3.00 3.48 3.50 3.94 4.38 −1.21 3.49 −3.43 3.95 −2.19 3.97 M2 versus M1 2.00 1.71 1.67 1.58 1.48 1.87 1.72 3.42 1.61 3.15 1.56 M3 versus M1 3.00 2.42 2.33 2.14 1.96 3.57 2.43 6.77 2.18 5.87 2.10 M3 versus M2 1.50 1.41 1.40 1.36 1.32 0.93 1.41 2.12 1.36 1.63 1.35 The predicted regression slopes and intercepts (incpt) are summarized for all eight models. Only the MV models have non-zero intercepts. The zero intercepts for the other models are not shown.
