Signal-dependent noise

One of the central premises in the optimal control theories proposed by Harris and Wolpert [5] and Tanaka et al. [7] is that the amount of noise in the control signal is proportional to signal amplitude, but experimental evidence for this assumption is still limited [39]–[41]. Here we test whether such signal-dependent noise exists at the level of the SC motor map. Towards that end, we quantified the trial-to-trial variability of saccade-related burst activity of SC neurons for visually guided saccades into their movement field.

Figure 2 illustrates the results for a typical cell. The left-hand panels show the instantaneous firing rate of the cell (color code) as a function of time for selected saccades of different amplitudes (Fig. 2A, ‘amplitude scan’) and directions (Fig. 2B, ‘direction scan’). Individual trials are sorted according to saccade amplitude and direction, respectively. For each saccade, the burst magnitude was quantified from the raw data by counting all spikes in the time window between the two tick marks (identifying saccade onset an offset with a 20 ms lead time), and the resulting spike counts are displayed in the adjacent panels as running averages. Saccade endpoints of the responses in Figs. 2A and B (squares and circles, respectively) are plotted in Fig. 2C, together with a two-dimensional representation of the cell's movement field (Methods). This movement-field plot shows that the number of spikes in the burst (color code) varies systematically with saccade amplitude and direction. However, as can be seen in Figs. 2A and B, it is not only the mean number of spikes in the burst (open symbols) that depends systematically on the amplitude and direction of the saccade vector; also the trial-to-trial variability in the spike counts changes systematically (error bars indicate ±1 SD). More specifically, when the spike-count variability is plotted as a function of the average number of spikes in the burst (Fig. 2D), it appears for this cell that the standard deviation increases almost linearly with the mean. To quantify this relation, we fitted a regression line to the data. Note that the intercept is practically zero. The slope of the regression line thus provides a good measure of the so-called coefficient of variation, which is the ratio between the standard deviation and the mean (Methods). For the neuron in Fig. 2, the coefficient of variation was C_v = 0.278±0.016 (mean±SEM).

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g002

To quantify the signal-dependent variability of saccade-related bursts for the entire population of cells, we selected for each SC neuron a series of non-overlapping clusters of closely matched saccade responses. This analysis, which is illustrated in Fig. 3A and B, typically included the entire movement field. Figures 3C and D summarize the results for all 108 cells for which we obtained sufficient data (see Methods, for details). Note that for the vast majority of cells (89%, n = 96) spike-count variability increased significantly with the mean number of spikes (two-tailed t-test, p<0.05). The mean (±SEM) value of the coefficient of variation across the cell population was 0.306±0.014. These findings thus indicate that motor commands generated by individual SC neurons are endowed with a considerable amount of signal-dependent noise.

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g003

Because signal-dependent noise alone cannot fully account for the variability in observed saccade trajectories, it has been suggested that cell activity may also be endowed with signal-independent noise [42]. We therefore quantified the level of signal-independent noise in the SC by analyzing the intercepts of the regression lines (c.f., Fig. 2D and Fig. 3B). Averaged across our sample of cells, the mean intercept was indeed significantly different from zero (two-tailed t-test, p<0.001), but with an average value of 0.54±0.03 spikes (mean±SEM) it reached only 3.0±0.2% of the cells' average peak response.

Spatial variation of burst properties

As mentioned in the introduction, we recently proposed on the basis of model simulations that a topographic organization of burst properties within the SC motor map could underlie the nonlinear kinematics of saccades [37]. More specifically, our simulations showed that the SC population could specify the main sequence if the saccade-related bursts vary from brief and intense in the rostral zone (small-saccade area), to less intense and of longer duration in the caudal zone (encoding large saccades) while keeping the number of spikes constant.

To test these theoretical predictions, Fig. 4 quantifies several burst properties of saccade-related SC cells for saccades towards the center of their movement field as a function their anatomical rostral-to-caudal location in the motor map: number of spikes in the burst (Fig. 4A), mean firing rate (Fig. 4B), and peak firing rate (Fig. 4C). Spike counts and mean firing rates were calculated from the raw data, while peak firing rates were estimated from the spike-density functions. We indexed the rostral-to-caudal location of each cell in the motor map by the amplitude of its preferred vector. Cells (n = 103) were selected for having at least 5 saccades into the center of their movement field (Methods).

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g004

Note that the spike counts are remarkably constant across the rostral-to-caudal extent of the SC as cells at each location fire on average about 18 spikes for their preferred saccade. Mean and peak firing rates, on the other hand, decrease systematically with the rostral-to-caudal location of the cells. To quantify these topographic relations we fitted linear regression lines (solid) to each of the three datasets. The numerical results are listed in Table 1 (top entries). This regression analysis showed there is no significant correlation between spike counts and preferred amplitudes, and thus anatomical location, of the neurons (Pearson correlation, p>0.3). Correlations between firing rates and preferred amplitudes, however, were significantly different from zero (t-test, p<0.005). Slopes of the regression lines (mean±SEM) in Fig. 4B and C were −7.6±1.9 and −7.3±2.3 spk/s per deg, respectively. The data thus indicate an almost two-fold decrease in firing rates from small to large saccades within the oculomotor range.

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Table 1. Summary of topographic variations in burst parameters.

https://doi.org/10.1371/journal.pcbi.1002508.t001

Because mean firing-rates were computed from the number of spikes divided by saccade duration (Methods), a reduction in mean firing-rate with amplitude could, in principle, result from the confounding factor that saccade duration increases linearly with amplitude (c.f., Fig. 1A). However, this potential confound does not play a role for the peak firing-rate. In our analysis, the absolute peak firing-rate values are partly determined by the width of the Gaussian smoothing kernel (see Table 1 for results with different values of ), but for a fixed kernel width (here,  = 5 ms), the changes in peak firing-rate across the population of cells are entirely due to changes in the temporal distribution of spikes within their bursts. The data in Fig. 4C thus confirm the presence of a real rostral-to-caudal gradient in the cells' activity patterns, which – to our knowledge – has not been documented before.

This spatial gradient is further illustrated in Figs. 5A–D which show the full temporal burst profiles of four clusters of cells that fired ∼18 spikes for their preferred saccade (dashed circles in Fig. 4A). Insets illustrate, schematically, the location of the recording sites in the motor map. It is important to note that this analysis does not involve any normalization of activity. Thin lines represent the mean spike-density functions of the individual cells, where data are aligned with saccade onset and averaged across at least 5 saccades into the center of the movement field (within 0.5⋅σ_mf). Amplitudes of the preferred vectors were about 5°, 13°, 21°, and 32°, respectively, as may be inferred from the corresponding eye-position traces. Thick lines are the grand averages of the responses. Note that the systematic rostral-to-caudal changes in mean and peak firing rates are quite obvious from these discharge profiles too. Cells recorded in the rostral region of the SC (Fig. 5A) clearly showed much higher peak firing rates and shorter bursts for their preferred vector than the ones in the caudal SC (Fig. 5D), while cells found at intermediate locations (Fig. 5B and 5C) had intermediate firing rates and burst durations.

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g005

In Fig. 6A we normalized the average burst profiles from the four clusters of cells (Figs. 5A–D) with respect to their peak. The top-right insets show the main sequence behavior and velocity profile skewness [38] of the corresponding average eye movements (data replotted from Fig. 5). This analysis shows that the peak firing rate occurs at about the same instant relative to saccade onset (see Table 1, for further quantification) while the burst duration increases systematically with the preferred amplitude of the cells (indexed by the gray-scaling). Hence, these results indicate that, just like the saccade velocity profiles, the skewness of the bursts increases systematically with saccade amplitude (and duration).

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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).

https://doi.org/10.1371/journal.pcbi.1002508.g006

Note, however, that the shapes of the spike-density waveforms in Fig. 6A also depend on the width of the Gaussian smoothing kernel that is used to compute the instantaneous firing rates (here,  = 5 ms). To circumvent this problem, in Fig. 6B, we therefore calculated burst skewness directly from the distribution of spike moments in the burst (Methods), rather than from the spike density functions. As in Fig. 4, cells (n = 103) were selected for having at least five saccades into the center of their movement field. Note that bursts associated with small saccades (produced by cells in the rostral region of the SC) are nearly symmetric (skewness about zero) while bursts associated with large saccades (produced by cells in the caudal SC) have longer tails towards the end of the saccade (positive skewness). Linear regression analysis showed a significant, positive correlation (t-test, p≪0.001) between the skewness of the bursts and the preferred amplitudes of the cells. This shows that burst skewness indeed increases systematically with the cells' rostral-to-caudal location in the motor map. Table 1 summarizes the numerical results of this regression analysis (fifth entry).

The systematic increase in burst skewness with saccade amplitude was not related to any change in spike-count variability; the coefficient of variation computed for these bursts (Eq. 4) was more or less constant across the map (Table 1), and did not significantly correlate with burst skewness (t-test, p>0.28).

The topographic variations in burst properties that are revealed by the analyses in Figs. 4–6 were consistent across animals. In all three animals for which we obtained a sufficient number of cells at different locations in the map (n>10; smallest eccentricity <4°; largest eccentricity >40°), we observed a nearly constant number of spikes across the map, a systematic rostral-to-caudal decrease in mean and peak firing rates, and a systematic rostral-to-caudal increase in burst skewness. In the fourth animal, recording sites did not span a large enough range of eccentricities (11°<R<15°) to obtain reliable fit results.

Burst dynamics depend on the actual saccade vector

So far the analyses quantified the burst properties of SC neurons for saccades towards the center of their movement field, i.e., when the cells are part of the central region of the recruited population. However, SC cells are recruited for a wide range of saccades (the movement field) that all have different kinematics depending on their amplitude. The question therefore arises what happens to the burst of a given cell when it is recruited for saccades that deviate from its preferred vector.

To address this question, we examined the shape of the burst profiles for saccades towards more eccentric locations within the movement field. More specifically, we compared the burst dynamics when the same cells are part of the rostral region of the recruited population (i.e. they are recruited for saccades that are larger than their preferred movement), with the burst dynamics when they belong to the caudal region of the recruited population (for saccades that are smaller than the preferred one; see motor-map insets Figs. 7 and 8, for schematic illustration).

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g007

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g008

If saccade-related SC cells indeed participate in specifying the optimal saccade trajectory [33], one would expect that the shape of their bursts is different depending on whether the cells are recruited for saccades that are smaller than their preferred vector, versus whether they are recruited for saccades that are larger than their preferred vector, because the shapes of the velocity profiles for smaller and larger saccades differ (Fig. 6, inset). Alternatively, the shape of the bursts could remain the same regardless of the actual saccade vector. Such behavior is expected if the burst activity of each cell represents only the weight of its preferred vector in a downstream center of gravity computation of the saccade goal [43]–[45]. In this case, the number of spikes produced by each cell is determined by its input, but the temporal firing properties only depend on its location in the motor map.

To dissociate between these two possibilities, Figs. 7 and 8 show the measured burst profiles of two cells in for a series of large and small saccades for which the cell fired the same number of spikes. We chose two cells from the central region of the SC (preferred amplitudes 9.5° and 13.5°, respectively) because these cells fired vigorous bursts for saccades with distinctly different kinematics. Note that the shapes of the bursts for small saccades (light-gray traces) and large saccades (dark-gray traces) are clearly different. For both cells, 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. These examples thus suggest that the shape parameters of the burst depend systematically on the actual saccade to which the neuron contributes, rather than on its topographic location in the motor map.

Population dynamics

If the SC acts as an optimal controller, one would expect that cells in the recruited population synchronize their burst profiles because this would ensure that the net SC movement command specifies a straight trajectory with optimal horizontal and vertical velocity components. The example cells in Figs. 7 and 8 suggest that the burst shape of individual SC cells is indeed tuned such that all cells in the recruited population become synchronized for the saccade in which they participate. To determine whether burst synchronization in the recruited population is actually found across the SC, we studied the spatial-temporal burst dynamics of all recorded cells in the motor map for a range of different saccade amplitudes (Fig. 9) and directions (Fig. 10), cross-correlated the burst shapes of all recruited cells to determine their similarity (Fig. 11), and analyzed the burst shapes of a fixed set of neighbouring neurons for different saccade vectors (Fig. 12).

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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).

https://doi.org/10.1371/journal.pcbi.1002508.g009

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g010

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g011

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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.

https://doi.org/10.1371/journal.pcbi.1002508.g012

We first examined the temporal discharge profiles of saccade-related neurons along the rostral-to-caudal extent of the SC for saccades of five particular amplitudes. Because we applied the rotation algorithm (Methods), cells having different preferred directions could all be pooled.

Figure 9 shows this analysis for saccade amplitudes ranging from 5 degrees (top panels) to 31 degrees (bottom panels). Saccade directions always corresponded with the direction of the cells' preferred movement (Methods). The different plots in Fig. 9 thus provide estimates of the rostral-to-caudal cross-sections through the center of the population. As in previous figures, we indexed the rostral-to-caudal location of each cell by the amplitude of its preferred vector. The color-codes in Fig. 9A reflect the averaged spike density of individual cells as a function of time relative to saccade onset and their location in the SC motor map. The associated mean eye movements are superimposed. We calculated the activity profiles in Fig. 9B from the raw data in Fig. 9A by averaging the spike-density functions of nearby cells according to a Gaussian weighting function (width σ = 0.25 mm) and sorting them according to their location in the motor map (Methods). Bin centers were chosen at 2.5 deg intervals in visual space. In Fig. 9C, the site-dependent spike density functions are collapse onto a single pair of axes, and bin centers were chosen at 0.2 mm intervals on the SC motor map. The latter produced spike density functions of the population activity at equally spaced distances from the center of the active population. To facilitate visual inspection, the hue of the individual spike density functions indicates the rostral (green) to caudal (cyan) location of the cells while the color saturation is proportional to the firing rate (i.e., traces become darker when activity increases).

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). More specifically, peak firing rates in the recruited population decrease systematically with increasing saccade amplitude while burst duration and skewness increase with increasing saccade amplitude. The data thus demonstrate that not only the location of the recruited population changes systematically with saccade amplitude; also the dynamics of the burst activity within the active population changes systematically.

The observation that cells at different locations within the active population all have very similar burst dynamics is further illustrated in Fig. 10. As in Fig. 9, we plot the responses of cells as a function of their rostral-to-caudal location in the SC, but now for 14 degree amplitude saccades in five different directions, ψ, relative to their preferred vector. The plots in Fig. 10 thus provide rostral-to-caudal cross sections through the population for a 14 degree saccade at five different iso-direction lines. Iso-direction line ψ = 0 deg runs through the center of the active population, so the plots in the center row of Fig. 10 are equivalent to the ones in the center row of Fig. 9. The other iso-direction lines, however, characterize the temporal discharge profiles 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. In fact, it is not difficult to see that practically all discharge profiles in Fig. 10 are approximately scaled versions of each other. Our findings thus demonstrate that the temporal dynamics of burst activity is very similar throughout the population of recruited cells.

To quantify the temporal synchrony and shape similarity of the burst profiles at different locations within the active population, we performed a series of temporal cross-correlation analyses (see Methods, for details). As shown in Fig. 11A, we first normalized the site-dependent spike density functions from Fig. 9C with respect to their peak. For each saccade amplitude, we then cross-correlated the population activity at the center of the activated region of cells with the population activity at different rostral-to-caudal distances from the center (solid lines), and with the activity of the individual cells (open symbols).

Note that for each of the five movement amplitudes the normalized population responses fall on top of each other (Fig. 11A). Accordingly, the cross-correlation analyses performed on the population data (solid lines) produced correlation values at lag zero that were close to one (Fig. 11B), and optimal delays that were close to zero (Fig. 11C). These results thus indicate that the response profiles are indeed synchronized, scaled versions of each other. Even at the level of the individual cells it is observed that the cross-correlation values at lag zero are very high (gray squares; typically r(τ = 0)>0.8). For about 50–70% of the cells, the cross-correlation values at the optimal delay (black circles) were significantly higher than at lag zero (Fig. 11B), but the burst delays of the individual cells were not systematically related to their rostral-to-caudal location within the recruited population (Fig. 11C). The same pattern of results was obtained along the medial-lateral dimension of the population (data not shown). The cross-correlation values themselves were of course influenced to some extent by the width of the Gaussian smoothing kernel (here, = 5 ms; Methods), but the resulting optimal delays were not. When we repeated the analysis with different kernel widths ( in the range of 2 to 10 ms), they were virtually identical. The robust analysis of relative burst delays in Fig. 11 thus demonstrates that there is no systematic spread of activity, neither in the rostral-to-caudal direction [31], [46] (which would produce a systematic increase of the lag), nor from the center towards the periphery of the population [27], [47] (which would produce a V-shape pattern).

Note, however, that the response profiles for saccades of different amplitudes are clearly different. I.e., burst profiles within the active population become more and more skewed as saccade amplitude and duration increases. The interesting question then arises whether these systematic changes in burst shape are also reflected in the population activity at a given location in the SC when the cells at that location participate in the generation of saccades of different amplitudes (and durations). The results of Figs. 7 and 8, in which we analyzed the responses of two example cells for small versus large saccades, would indeed suggest such changes.

To address this question, we selected in Fig. 12 a cluster of 16 neurons located in the central region of the SC. Preferred amplitudes of these cells were closely matched, ranging between 13–15 degrees. From the responses of the individual cells (color code in Fig. 12A) we first calculated the mean population response at that location for five different saccade amplitudes (solid lines in Fig. 12A), and we then normalized the resulting response profiles with respect to their peaks (Fig. 12C). Note that there are systematic increases in burst duration and skewness as the saccade amplitude increases from 8 to 32 degrees. Also note the main-sequence behavior in the corresponding eye movement traces of Fig. 12B.

Dynamic population code

We previously showed that a simple linear ensemble-coding model of the SC motor map could fully account for the nonlinear properties of saccades [33]. By driving a linear model of the brainstem with actual recorded spike trains from a large population of SC neurons widely distributed across the motor map we obtained realistic, straight saccades with the correct kinematic properties (Video S1). The assumptions of this linear ensemble-coding model (Fig. 1C) are the following:

1. In line with neuroanatomical data [48]–[51], each spike of each recruited cell contributes a fixed movement (‘spike vector’) to the saccade, , which is only determined by the cell's location, , in the motor map according to the efferent mapping function:with a scaling constant that is the same for all cells. The fixed parameters [A, B_u, B_v] = [3.0 deg, 1.4 mm, 1.8 mm/rad] describe the anisotropic and inhomogeneous geometry of the motor map [26], [28]; ρ is the cell density, N₀ the maximum number of spikes in the burst of SC cells, and σ₀ the width of the Gaussian population of recruited cells. Our recordings indicate that both N₀ and σ₀ are constant across the map (N₀≈18–20 spikes, and σ₀≈0.5 mm). Thus, if cell density does not vary along the motor map either, our current results imply that the total number of collicular spikes associated with saccades would be constant too.
2. Linear summation of all spike vectors determines the desired two-dimensional saccade trajectory:with P the number of active cells in the population, and N_i the number of spikes generated by cell i. This crucial assumption states that the SC output is a dynamic desired eye-displacement signal that grows from zero to the final saccade amplitude in synchrony with the spike trains (i.e., the spatial-temporal integration stage in Fig. 1C). This assumption differs radically from alternative models, which assume that the SC population encodes the goal, not the motor command, for the upcoming saccade by weighted averaging of individual cell contributions [43]–[45], [52]. In weighted-averaging schemes, the saccade amplitude is specified virtually instantaneously and does not develop during the course of the saccade-related burst, i.e. (Fig. 1B). These alternative models therefore provide no explanation as to why caudal cells would have to fire at much lower peak rates than rostral cells (the difference is, on average, about 50%), or why the shape of the motor burst of a given cell would have to depend systematically on the saccade for which it is recruited.
3. The brainstem saccade generator decomposes the dynamic vectorial SC signal into its horizontal and vertical eye-velocity components, but is described by a linear input-output characteristic (Fig. 1C). In virtually all previous models (see [53], for review) the brainstem saccade generator embodies the nonlinear kinematics of saccades through a nonlinear, saturating characteristic. The saturation was therefore not explained, but added to the model to fit eye velocity profiles. By removing this nonlinearity from our model we could demonstrate that the SC output contains the information needed to generate the main-sequence properties and component stretching without additional assumptions.
4. The motor neurons of the extra-ocular eye muscles receive eye-velocity commands from the brainstem pulse generator as well as eye-position step signals from parallel neural-integrator pathways (Fig. 1C). As in previous models [11]–[13] this so-called pulse-step innervation acts an inverse model of the oculomotor plant (which has a linear low-pass filter characteristic) to fully compensate for its sluggish dynamics.

Note that our scheme does not make any prior assumptions about the activity patterns of individual cells in the SC motor map. For example, the finding that the number of spikes in the burst is invariant for fast and extremely slow eye movements [54], and that it is invariant across the motor map (Fig. 4A) are not properties of the model, but appear to be properties of the motor map. The same holds for our current findings that the peak firing rate of cells, their burst duration and their burst skewness all vary in a systematic way with their location in the motor map. Finally, that all cells within the population are synchronized during the saccade, and that the burst properties are determined by the saccade in which the cell participates, rather than by its location in the map, is not a model assumption either. Yet, all these features taken together fully explain how the SC population could encode the nonlinear kinematics, and at the same time generate straight saccades in all directions.

The nonlinear behavior of saccade kinematics is due to two opposing factors: (i) peak firing rates in the caudal SC are lower than in the rostral SC, but (ii) each spike in the caudal SC has a much larger impact on the brainstem than a rostral cell, due to the exponentially growing efferent mapping function. Thus, although spike firing-rates decrease in the caudal SC, the increase in eye displacement provided by each spike is much larger at these locations. Straight saccades result from synchronization of burst profiles, especially along the medial-lateral dimension of the SC (Fig. 10). This synchronization ensures that the horizontal and vertical velocity commands are scaled versions of each other as is required for producing straight oblique saccades. Our findings thus strongly support the idea that the SC motor map acts as a nonlinear vectorial pulse generator.

Because the saccade results from the linear contribution of a large ensemble of recruited cells, our linear ensemble-coding model does not necessarily predict that each individual cell should encode the details of the saccade kinematics. Nevertheless, the saccade kinematics are to a large extent reflected at the level of single cells, as the shape of the saccade-related burst follows a similar skewness relationship with burst duration as the saccade velocity profile to saccade duration (Fig. 6). In addition, the burst shape of any individual cell is to a large extent determined by the saccade metrics, and can thus vary substantially between small and large saccades into its movement field (Figs. 7, 8 and 12). Interestingly, these features are predicted by the notion that saccade-related SC neurons have dynamic movement fields which determine the dynamic relationship between the activity of individual cells and the ensuing eye displacement as a function of the saccade metrics [33]. This concept also predicts that for movements to a single visual target the recruited cells act together as a ‘common source’ by synchronizing their burst profiles. This behavior is indeed observed (Figs. 9–11).

Taken together, our findings provide strong support for the argument that the nonlinear saccade kinematics are not due to a passive saturation of brainstem burst neurons, e.g., as a result of neural fatigue, but reflect a deliberate design property within the saccadic system to produce the main-sequence characteristics.

Optimal control

We believe that such a strategy aims to optimally cope (in a statistical sense) with conflicting constraints that impede the fovea from getting as fast and as accurately as possible on a peripheral target of interest. Several constraints may be identified within the system: neural noise, considerable processing delays, and the highly inhomogenous organization of the retina that introduces considerable uncertainty about stimulus locations within the visual periphery. Indeed, theoretical studies on optimization have provided an elegant explanation for different features of saccadic behavior, such as the tendency to systematically undershoot visual targets [55], but also the main-sequence nonlinearity [4]–[7]. In these studies, the noise is assumed to be multiplicative, i.e. signal-dependent. To optimize such a system for speed and accuracy, the control signal should obey the nonlinear main sequence, and at the same time employ an undershooting strategy. Our data show for the first time that SC cells indeed possess multiplicative, signal-dependent noise (Figs. 2 and 3), and that this property is invariant across the motor map (Table 1).

Since the coefficient of variation is, on average, constant across the map, it follows from our spike-vector summation theory that the variance in the resulting displacement vector, , as function of desired amplitude and direction is given by:Where N_i(R,Ф) is the mean number of spikes fired by cell i, its spike-vector contribution, and C_v the coefficient of variation (population average). Clearly, the covariances in this equation cannot be determined from our single-unit recordings, so we cannot be 100% sure that the net collicular output has multiplicative, signal-dependent noise. Simulations with a uniform motor map (i.e., fixed cell density, , and location-invariant widths σ_mf and heights N_pref of the Gaussian movement fields) showed, however, that both correlated and uncorrelated cell activities produce elliptical endpoint distributions (e.g., [56]) with standard deviations that increase linearly with eccentricity (not shown).

The optimal control models described above all dealt with the generation of horizontal saccades. However, in two dimensions the fastest response should also follow a straight line; thus an extended optimal control theory would predict straight oblique saccades [8]. In alternative models, in which the brainstem is driven by separate horizontal and vertical nonlinear pulse generators, the programming of straight oblique saccades is highly nontrivial as it requires a tedious scheme of cross-coupling between the horizontal and vertical systems [21]–[23]. The problem is even more complex when considering head-free gaze shifts [57]–[59], because a fixed cross-coupling scheme will no longer work when the eye-head coupling varies considerably from trial to trial. However, if the nonlinear burst generator resides in the SC motor map, straight saccades (and head-free gaze shifts) become an emerging property of the system, requiring no further cross-coupling than a simple linear horizontal/vertical decomposition (i.e. sine and cosine projection) of the vectorial gaze-shift command.

Bursting mechanisms

Our simple linear ensemble-coding model is still incomplete. It accounts for the generation of saccadic eye movements once the spatial-temporal distribution of SC activity is known, but it does not explain how the tuned burst patterns come about in the first place (an omission in most models of the saccadic system). In principle, the saccade-related burst of SC neurons could be derived from two alternative mechanisms. First, the type of input signal, such as that from the frontal eye fields (FEF), might impose the spatial-temporal pattern of excitation [60]. However, this explanation is neither attractive nor plausible; it merely shifts the problem to a different area of the brain, and there exists (at least to our knowledge) no evidence that the FEF is involved in the dynamic, online control of saccade trajectories under normal conditions. Second, the burst might arise from intrinsic membrane properties of saccade-related SC neurons [61], or from properties of the local circuits within the SC motor map. Recent in vitro experiments indeed suggest that the synchronous bursting command observed in our data could result from a local excitatory network, in combination with NMDA receptor activation in the deeper layers of the SC [62]–[64]. These in vitro experiments also demonstrated a strong nonlinear signal amplification process in the SC, which is interesting because it might account for the nonlinearity (i.e., vector averaging) of responses obtained with certain types of electrical double stimulation [65]–[67].

In theory, feedback from the brainstem saccade generator could also contribute to the shaping of the burst dynamics. For example, Van Opstal and Kappen [68] suggested that a linear model of the brainstem together with weighted feedback projections to the SC motor map reproduces straight saccades with the correct kinematics. However, in their scheme caudal cells receive the weakest feedback, and the model therefore predicts a strongly asymmetric distribution of burst durations and skewness within the recruited population. This is clearly not observed in our data (Figs. 9–11). Moreover, previous perturbation studies have shown that activity in the SC is also not consistent with other types of feedback models, as the SC activity does not encode dynamic motor error [32], [54], [69]–[74].

Conclusion

Further research is needed to elucidate the mechanisms that shape the spatial-temporal firing patterns during saccades, and the behavior of the SC population in more complex motor behaviors, like during head-free gaze shifts, curved double-step saccades, or electrical microstimulation. Nevertheless, the burst properties reported in this study strongly support the idea that the deeper layers of the SC act as an optimal controller: the systematic organization of peak firing rates and burst durations as function of saccade amplitude along the motor map, the synchronous change in firing rate of recruited cells in the population, and the shaping of the temporal burst profile of a given cell with the currently planned saccade, all contribute to the generation of straight eye-movement trajectories with optimal kinematics.

Experimental procedures

We collected data from four rhesus monkeys (macaca mulatta) that were trained to follow a small visual target with saccadic eye movements. The setup, surgical procedures and behavioural paradigms have been described elsewhere [33], [54], [71]. In short, the animals were seated in a primate chair facing a spherical array of light-emitting diodes (LEDs) in an otherwise completely dark room. The head was restrained, and movements of the eye were measured with the double-magnetic induction technique [75], [76]. Single-cell recordings were made through a recording cylinder using tungsten microelectrodes that were advanced into the SC with the use of a hydraulic stepping motor.

We recorded activity of 146 saccade-related neurons that were found in the intermediate and deep layers of the SC (about 0.5–3 mm below the dorsal surface). Cells were considered saccade-related if they showed an increase in firing rate around the onset of saccades towards a particular region of the visual field. All of these cells were studied with a standard saccade task in which the animal made saccades from an initial fixation LED to a peripheral target LED which was presented for 500 ms. The movement field of each neuron was determined by eliciting saccades to targets inside and neighboring the response field (‘movement field scan’). In addition, saccades were evoked to a fixed series of targets across the visual field (R between 2–35 deg, Φ∈[0,30,…,360] deg; ‘rose scan’). Eye movement data, spike data, and movement field parameters of all 146 recording sites were stored in a database for further processing. The file contained data from a total of 32,147 trials.

Data analysis

Saccades were detected off-line on the basis of the calibrated eye-position signals using custom software (see [71] for details). Subsequent analysis was done in Matlab 7.9 (version R2009b).

Single-cell activity was displayed in spike rasters and peri-stimulus time histograms (PSTHs) that were aligned on specific events such as target onset and the onset of a saccade. Spike trains from individual trials were represented as a sequence of δ pulses at the time of spike occurrence, (1 ms resolution): . Spike density functions, , were calculated from these raw spike trains by convolving with a Gaussian smoothing kernel that had a default width of  = 5 ms and a height of spk/s, but analyses were also repeated for different kernel widths (see text).