The main sequence of saccades optimizes speed-accuracy trade-off

Abstract

In primates, it is well known that there is a consistent relationship between the duration, peak velocity and amplitude of saccadic eye movements, known as the 'main sequence'. The reason why such a stereotyped relationship evolved is unknown. We propose that a fundamental constraint on the deployment of foveal vision lies in the motor system that is perturbed by signal-dependent noise (proportional noise) on the motor command. This noise imposes a compromise between the speed and accuracy of an eye movement. We propose that saccade trajectories have evolved to optimize a trade-off between the accuracy and duration of the movement. Taking a semi-analytical approach we use Pontryagin's minimum principle to show that there is an optimal trajectory for a given amplitude and duration; and that there is an optimal duration for a given amplitude. It follows that the peak velocity is also fixed for a given amplitude. These predictions are in good agreement with observed saccade trajectories and the main sequence. Moreover, this model predicts a small saccadic dead-zone in which it is better to stay eccentric of target than make a saccade onto target. We conclude that the main sequence has evolved as a strategy to optimize the trade-off between accuracy and speed.

Fig. 1

a The fixation cost (see Eq. (1)) of making a unit amplitude saccade as a function of movement duration, T, plotted on a log-log scale (F = 300 ms). Note that cost decreases for longer duration for a given amplitude. b A linear-linear plot of the total cost of a 10 degree saccade as a function of duration, T. The total cost (solid line) is made up of two components (see Eq. (1)): the movement cost which increases linearly with duration (dotted line) and the fixation cost which decreases with duration, as in (a) (dashed line). The optimum cost is the minimum total cost, shown by the circle. c The total cost for movements of different amplitudes (5°, 10°, 20° and 30°) shows an increase in optimal duration (circles) with amplitude, as seen empirically in the duration main sequence

Fig. 2

Comparison of theoretical optimal main sequence (solid lines) to empirical data (dots). a Duration against movement amplitude. b Peak velocity against movement amplitude. c The product of duration and peak velocity against movement amplitude. This typical main sequence was recorded from a healthy adult using an infra-red limbus eye-tracker at 1 kHz. The data were recorded in a previous study (Harwood et al. 1999). The ratio α/(β’k²) was set to 25,500 (see text). d duration against amplitude plotted together with data from (Baloh et al. 1975) with α/(β’k²) = 12, 200 (see text). e Optimal saccadic speed profiles for 5°, 10° and 20° saccades

Fig. 3

The optimal duration main sequence for mean fixation periods, F = 100, 200 and 1000 ms

Fig. 4

A comparison of total costs for two strategies of movement. The dotted line shows the cost for saccadic movements and the solid lines show the cost for drifting movements with different tolerated drift rates d. The intersection of each solid line with the dotted line represents the saccadic deadzone for the given drift rate d