Optimal control of saccades by spatial-temporal activity patterns in the monkey superior colliculus

Abstract

A major challenge in computational neurobiology is to understand how populations of noisy, broadly-tuned neurons produce accurate goal-directed actions such as saccades. Saccades are high-velocity eye movements that have stereotyped, nonlinear kinematics; their duration increases with amplitude, while peak eye-velocity saturates for large saccades. Recent theories suggest that these characteristics reflect a deliberate strategy that optimizes a speed-accuracy tradeoff in the presence of signal-dependent noise in the neural control signals. Here we argue that the midbrain superior colliculus (SC), a key sensorimotor interface that contains a topographically-organized map of saccade vectors, is in an ideal position to implement such an optimization principle. Most models attribute the nonlinear saccade kinematics to saturation in the brainstem pulse generator downstream from the SC. However, there is little data to support this assumption. We now present new neurophysiological evidence for an alternative scheme, which proposes that these properties reside in the spatial-temporal dynamics of SC activity. As predicted by this scheme, we found a remarkably systematic organization in the burst properties of saccade-related neurons along the rostral-to-caudal (i.e., amplitude-coding) dimension of the SC motor map: peak firing-rates systematically decrease for cells encoding larger saccades, while burst durations and skewness increase, suggesting that this spatial gradient underlies the increase in duration and skewness of the eye velocity profiles with amplitude. We also show that all neurons in the recruited population synchronize their burst profiles, indicating that the burst-timing of each cell is determined by the planned saccade vector in which it participates, rather than by its anatomical location. Together with the observation that saccade-related SC cells indeed show signal-dependent noise, this precisely tuned organization of SC burst activity strongly supports the notion of an optimal motor-control principle embedded in the SC motor map as it fully accounts for the straight trajectories and kinematic nonlinearity of saccades.

Figure 1. Properties of saccades and models of saccade generation.

A) Left: main sequence relationship between saccade amplitude, duration and peak velocity. Right: component stretching. Here, an oblique saccade (blue) has very different component amplitudes (top), but horizontal and vertical velocity profiles ( and ) have equal durations and similar shapes (bottom). The horizontal saccade (red) has a much shorter duration and higher velocity than the equally large horizontal component of the oblique saccade. B) Classic one-dimensional model which assumes that the superior colliculus (SC) specifies a desired displacement vector (after [12]). Main sequence properties are attributed to a saturating nonlinearity of the burst generator which is controlled through local feedback. Cross-coupling between horizontal and vertical components (not shown) is needed to produce straight saccades. C) Linear two-dimensional model of the SC – brainstem saccade generator (after [33]). In this scheme, spatial-temporal activity patterns in the SC specify an intended movement trajectory, which is decoded downstream by ‘spike-vector’ summation: each spike from each neuron adds a site-specific vectorial contribution to movement command. The actual movement is generated by pulse-step activation of the extra-ocular motor neurons, as in B, but the burst generator, which produces the pulse, is linear. ΔE, desired eye displacement; Δe(t), current eye displacement; me(t), dynamic motor error; , current eye velocity; w(R), exponential weighting function; ∫dt temporal integration; Burst, brainstem burst generator; NI, neural eye position integrator; NDI, resettable neural eye displacement integrator; MN, motor neurons; innerv., eye plant pulse-step innervation signal.

Figure 2. Movement field scan of a saccade-related SC neuron illustrating the presence of signal-dependent noise.

A) Spike density (color code) as a function of time for saccades in and near the center of the movement field, with trials sorted by amplitude of the movement (amplitude scan; fixed direction Φ≅237°). Tick marks indicate spike-count windows (20 ms before onset to 20 ms before offset of the saccade). Superimposed is the average eye position of saccades towards the movement field center. Spike counts (open symbols) are displayed as running averages across 6° wide bins. Error bars indicate the trial-to-trial variability in spike counts (±1 SD). B) Same for a direction scan through the center of the movement field (fixed amplitude R≅13°). Spike counts were averaged across 13° wide bins. Average velocity profile of saccades towards the center is superimposed. C) Spatial extent of the movement field together with the endpoints of saccades (re. to initial fixation, +) included in the amplitude and direction scan. Color code: movement field description (Methods, Eq. 3) of the number of spikes in the burst as function of saccade amplitude and direction. Contour lines are drawn at [0.5, 1.0, 1.5 and 2.0]⋅σ_mf. D) Spike-count standard deviations as a function of mean number of spikes in the burst. Linear regression line (solid) was calculated from the pooled data of the amplitude (squares) and direction (circles) scans.

Figure 3. Signal-dependent noise across the population of recorded SC neurons.

A) Movement field data from an individual neuron was parsed into clusters (here, n = 15) containing at least 5 saccades to nearby locations. Preferred vector of the cell shown here was [R,Φ] = [12,183]°. Endpoints of saccades belonging to the same cluster are gray-coded and connected by line segments. B) For each cluster, mean and standard deviation of the number of spikes in the burst were computed (Methods). The relation between these variables is characterized by the coefficient of variance, C_v, which we determined from the slope of the linear regression line (solid, C_v = 0.43±0.08). C) Stacked histograms show the distribution of C_v for the population of cells in which the regression was statistically significant (bright; n = 96) and for the population of cells for which it was not significant (dark; n = 12). Legend: p-value of the regression (two-tailed t-test); n.s., not significant. D) Same for the distribution of linear correlation coefficients.

Figure 4. Burst magnitude across the SC motor map.

Three different activity measures were used to quantify the magnitude of the saccade-related bursts of SC neurons for movements towards the center of their movement field as a function of their rostral-to-caudal location in the SC motor map: A) number of spikes in the burst, B) mean firing rate and C) peak spike density. Cells were selected for having at least five saccades into the center of their movement field (within 0.5⋅σ_mf). Highlighted cells (asterisks) in four clusters of cells (dashed circles) were selected for producing ∼18 spikes for their preferred saccade (c.f., Fig. 5). Linear regression lines (solid) were fitted to the data from all n = 103 cells.

Figure 5. Systematic rostral-to-caudal changes in temporal burst profiles.

A–D) Illustrated are the temporal firing patterns of saccade-related SC neurons at four different rostral-to-caudal locations in the motor map (insets). Movements (top) corresponded with their preferred vector. Eccentricities of the cells (as indexed by their preferred vectors) were about 5°, 13°, 21° and 32°, respectively. All cells (n = 5, n = 11, n = 8, and n = 7, respectively) were selected for producing ∼18 spikes for their preferred saccade (i.e., highlighted cells in Fig. 4). Thin and thick lines are cell and cluster averages, respectively. Importantly, spike density functions were not normalized in any way.

Figure 6. Burst profiles show amplitude-dependent skewness.

A) Average spike-density profiles of the four clusters of cells normalized with respect to their peaks. Cells in each cluster were selected for producing ∼18 spikes for their preferred saccade (c.f., Figs. 4 and 5). Insets: average eye position traces and eye velocity profiles for the each of the four cell clusters. B) Skewness of saccade-related bursts for preferred saccades as a function motor map coordinates. The relationship was quantified with linear regression (solid).

Figure 7. Burst shapes depend on planned movement.

Temporal firing patterns of a neuron during small and large saccades for which the cell fired about same number of spikes. A–C) Spike density (color code) as a function of time for saccades of different amplitude in the preferred direction of the cell. Data are sorted by saccade amplitude. Tick marks indicate spike-count windows. Gray traces: average eye position, eye velocity and spike density functions for small, 6° saccades (n = 5). Black traces: averaged data for large, 17° saccades (n = 16) for which the cell fired about the same number of spikes (N_s≅25). Inset in A: running average of the number of spikes as a function of saccade amplitude. Vertical line segments: amplitude range of saccades included in the two datasets. Horizontal line segments: mean±SD of the corresponding spike counts. Insets B and C: schematic drawing of population activity relative to the recording site. D) Location and extent of the movement field (color code) plus saccade endpoints (symbols). E) Normalized spike density functions for the two saccade vectors. Note that the shapes of the bursts for small saccades (light-gray traces) and large saccades (dark-gray traces) are clearly different. Peak firing rates occurred at about the same instant relative to movement onset regardless of the movement amplitude, but burst durations were shorter and peak firing rates were higher when the cell took part in the small saccades than when it participated in the large saccades.

Figure 8. Same analysis of burst activity as in Fig. 7 , but now applied to data from a different neuron in a different animal.

Note different burst profiles for the small (n = 10) versus large (n = 17) saccades for which the cell fired the about same number of spikes (N_s≅20), indicating that the burst parameters depend on the actual saccade in which the neuron is participating, rather than on its topographic location in the motor map.

Figure 9. Population dynamics within the motor map.

Temporal discharge patterns of saccade-related neurons along the rostral-to-caudal extent of the SC for five different saccade amplitudes. Saccade direction matched the preferred direction of each cell. A) Instantaneous firing rates (color code) of individual cells, sorted according to their rostral-to-caudal location in the SC motor map. Average eye position traces are superimposed. B) Mean discharge profiles of cells at different locations in the map. Each spike density function is the average activity of cells recorded in restricted region of the SC. Bin centers at 2.5° intervals. Burst profiles are shifted upward and color-coded according to the motor map coordinates. C) Site dependent discharge profiles computed at 0.2 mm intervals. As in B, hues of the individual traces refer to the rostral-to-caudal location, with green corresponding to rostral sites and cyan to caudal sites while color saturation is proportional to the instantaneous spike density. Discharge profiles of individual cells are not normalized. Note that for a given saccade amplitude the burst profiles along the rostral-to-caudal extent of the SC appear to have very similar shapes, which change systematically as function of saccade amplitude (and duration).

Figure 10. Neurons within the recruited population have similar burst dynamics.

Temporal discharge patterns of saccade-related neurons for 14° saccades in five different directions ψ relative to their preferred vector. Positive values of ψ correspond with counter-clockwise rotations. Same layout as Fig. 9 but note that each row now represents the burst profiles of cells along a rostral-to-caudal iso-direction line though the active population. ψ = 0° runs through the center of the population. The other four iso-direction lines show firing patterns at different medial-lateral locations. Note that the peak firing rates decrease systematically with increasing rostral-to-caudal and increasing medial-to-lateral distance from the center of the population activity while the shape of the temporal burst profiles remains remarkably similar. All discharge profiles are approximately scaled versions of each other indicating that the temporal dynamics of burst activity is very similar throughout the population of recruited cells.

Figure 11. Recruited SC neurons synchronize their burst profiles.

Temporal cross-correlation analysis of saccade-related burst activity for saccades of different amplitudes. A) Site-dependent burst profiles (from Fig. 9C) normalized with respect to their peak. Burst profiles in each panel are drawn at 0.2 mm intervals, starting 1 mm rostral (dark) to the center of the recruited population, and ending 1 mm caudal of it (bright). B–C) The central burst profile was taken from 20 ms before saccade onset until saccade offset (dashed lines in A) and cross-correlated with the population activity (solid lines) at different rostral-to-caudal distances from the center (negative values are more rostral, positive values are more caudal), and with the spike density functions of the individual cells (open symbols; from top to bottom: n = 35, n = 71, n = 99, n = 104, and n = 85). Positive delays in C indicate that the burst at a given location occurs later than at the center of the active population. Gray lines and symbols in B are the cross-correlation values obtained at time lag zero. Black lines and symbols are the correlations obtained at the optimal delay (i.e., peak of the cross-correlation function). U_c: motor map coordinates of the center of the population taken along the horizontal meridian.

Figure 12. Burst shapes depend on planned movement.

Shown are the saccade-related discharges of 16 neurons located in a small, central region of the SC for five different saccade amplitudes. Preferred amplitudes of the cells ranged from 13 to 15°. A) Instantaneous firing rates of the individual cells (color code) averaged across trials with the population average superimposed. Insets: schematic drawing of population activity relative to the recording sites. B) Eye position traces and eye velocity profiles of the corresponding eye movements show the main sequence behavior. C) Averaged spike density functions for the different saccade amplitudes normalized to their peak. Note the systematic increases in burst duration and skewness as saccade amplitude increases from 8 to 32 degrees, indicating that the shape parameters of the burst depend on the actual saccade in which the neurons participate, rather than on their topographic location in the motor map.