Optimal control of gaze shifts

Abstract

To explore the visible world, human beings and other primates often rely on gaze shifts. These are coordinated movements of the eyes and head characterized by stereotypical metrics and kinematics. It is possible to determine the rules that the effectors must obey to execute them rapidly and accurately and the neural commands needed to implement these rules with the help of optimal control theory. In this study, we demonstrate that head-fixed saccades and head-free gaze shifts obey a simple physical principle, "the minimum effort rule." By direct comparison with existing models of the neural control of gaze shifts, we conclude that the neural circuitry that implements the minimum effort rule is one that uses inhibitory cross talk between independent eye and head controllers.

Figure 1.

Effort and movement duration. a, Time course of effort for a medium-sized (50°) gaze shift. Solid line, Combined effort for both eye and head. Dashed line, Effort associated with eye component. Dashed-dot line, Effort associated with head component. Note that the total effort of the movement is the cost value of the combined effort at the end of the movement. b, Iso-effort contours for head-fixed saccades. Abscissa, Saccade amplitude; ordinate, saccade duration. Lines indicate the amplitude–duration relationship obtained from Fuchs (1967), King et al. (1986), van Gisbergen et al (1981), and the one used in our optimal control model (dashed line, dotted line, dashed-dot line and solid line, respectively). c, Iso-effort contours for head-free gaze shifts. Abscissa, Amplitude; ordinate, duration. Lines indicate the amplitude–duration relationship obtained from Tomlinson and Bahra (1986), Phillips et al. (1995), Freedman et al. (1997), and the one used in our simulations of eye-head gaze shifts (dashed line, dotted line, dashed-dot line and solid line, respectively). Numbers indicate the total effort required to accomplish the movement (on the log-scale).

Figure 2.

Eye velocity profiles. a, Time course of head-fixed eye velocity for saccades ranging from 5–40°. Inset, Data extracted from Figure 2B of the study by Freedman (2008), showing a 10°, 20°, and 30° saccade. b, Eye-velocity profiles of head unrestrained gaze shifts of 20°, 40°, and 70° starting with the eyes centered in their orbit. Inset, Data extracted from Figure 8B of the study by Freedman and Sparks (1997) showing a 24° and 35° gaze shift, and from Freedman (2008) showing a 70° gaze shift.

Figure 3.

Metrics of gaze shifts with the eyes and the head facing straight ahead. a, b, Eye amplitude (a) and head contribution (b) as a function of gaze amplitude for gaze shifts ranging from 5 to 75°. Open diamonds, Simulation results; dots, experimental data extracted from Figure 6 of the study by Freedman and Sparks (1997). c, Amplitude of head-unrestrained gaze shifts as a function of retinal error.

Figure 4.

Metrics of gaze shifts with the eyes starting from different initial positions. a–c, Size of eye (a) and head (b) contributions to rightward head-free gaze shifts (c) of constant amplitudes equal to 30° (open circles), 50° (open squares), and 70° (open diamonds), as a function of initial eye position (abscissa). Negative values indicate leftward initial eye positions. Data were fit with least-squares regression lines the slopes of which were the following: (a) 30°: −0.3; 50°: −0.5; 70°: −0.7 and (b) 30°: 0.3; 50°: 0.5; 70 °: 0.7. Insets, Data extracted from Figure 15B (in b) and Figure 15C (in a) of the study by Freedman and Sparks (1997).

Figure 5.

Motor commands controlling gaze shifts (ΔG) of 30, 50, and 70°. a, Signals derived from the minimum effort rule. b, Left, Schematic illustrating the major building blocks of a neural model that assumes independent eye and head control and inhibitory cross talk between the head related and eye related neural circuitry (Phillips et al., 1995; Freedman, 2001; Moschovakis et al., 2008). Right, The eye and head commands it generates for the gaze shifts illustrated in a. c, Schematic (left) illustrating the major building blocks of a neural model that assumes gaze feedback driving both the eye and head controllers (Guitton et al., 1990) and the control signals it generates (right). Negative signs next to the arrowheads indicate inhibitory connections. All other connections are excitatory. The VOR has been ignored because of its negligible role during the quick phases of gaze shifts. Both neural models (b) and (c) have been implemented in Simulink of the MatLab environment. Time bar (200 ms) applies to all waveforms. The amplitude of the motoneuronal eye and head units (measured in spikes per second) vary because of differences among the gains of their corresponding eye and head plants.