### Patient behavior

To test the sensorimotor eye movement function of a patient, the first paradigm traditionally used is the visually guided saccade test. During this protocol, targets were presented in sequence at different places and the patient was asked to look at the target as soon as she saw it. Several variations of this protocol exist, depending on the timing of target display: the next target is shown while the last one is still visible (overlap condition), has disappeared for a certain duration (gap) or disappeared at the same time (synchronous condition). This test is informative because saccade kinematics are very stereotyped and simple analyses can be done to characterize saccadic eye movements. In the next sections, we will first show the main sequence to characterize leftward and rightward average saccadic behavior. After, we will present a typical rightward and a typical leftward saccade done by the patient with particularly large overshoots. Then we will present saccadic movements with horizontal oscillations made by the patient. Finally, from the behavioral observations we made, we then show how small adjustments to the model’s parameters cause it to change from a healthy configuration to a configuration that reproduces the majority of the patient’s conditions.

#### Main sequence

In this section, we characterize the general saccadic behavior of the patient. The upper graph in Figure 4 shows the relationship between saccade amplitude (e.g. difference between eye position at point c and eye position at point b in Figure 5 for the first saccade) and saccade peak velocity known as the main sequence [23]. Leftward (rightward) saccades are represented with a negative (positive) amplitude. Lower graph shows the peak velocity as a function of the maximum displacement during the saccade. The maximum displacement is defined as the difference between eye position at the extrema of the position hook (reversal of the saccadic trajectory, e.g. point b in Figure 5 for the first saccade) and the eye position at the onset of the saccade (e.g. point a in Figure 5 for the first saccade). One can see that the dispersion is smaller for the peak-velocity vs. maximum-displacement relationship than for the peak-velocity vs. saccade-amplitude relationship. To test this, we fitted an exponential model to characterize the different relationships in Figure 4:

\begin{array}{rl}{V}_{R,\mathit{\text{Max}}}& =(579\pm 35)\ast \left(1-{e}^{-(0.128\pm 0.018){A}_{R}}\right)\\ \mathit{\text{MSE}}& =5643,\end{array}

(4)

\begin{array}{rl}{V}_{L,\mathit{\text{Max}}}& =(938\pm 237)\ast \left(1-{e}^{(0.043\pm 0.017){A}_{L}}\right)\\ \mathit{\text{MSE}}& =11562,\end{array}

(5)

\begin{array}{rl}{V}_{R,\mathit{\text{Max}}}& =(617\pm 35)\ast \left(1-{e}^{-(0.077\pm 0.009){M}_{R}}\right)\\ \mathit{\text{MSE}}& =3517,\end{array}

(6)

\begin{array}{rl}{V}_{L,\mathit{\text{Max}}}& =(1133\pm 303)\ast \left(1-{e}^{(0.029\pm 0.011){M}_{L}}\right)\\ \mathit{\text{MSE}}& =7400.\end{array}

(7)

The parameters of each fit is given with their 95% confidence interval. *A*_{
R
} (*A*_{
L
}) corresponds to the amplitude of rightward (leftward) saccades. *M*_{
R
} (*M*_{
L
}) corresponds to the maximum displacement of rightward (leftward) saccades. *V*_{R,M a x} (*V*_{L,M a x}) represents the peak velocity during rightward (leftward) saccades. Finally, MSE represents the mean squared error of each fit. Fits (4)-(7) are shown in Figure 4 using blue thick lines for leftward fits and red thick lines for rightward fits. The 95% confidence interval for each fit is represented in Figure 4 by the corresponding thin lines.

The main sequence expresses that saccadic eye velocity saturates for large-amplitude saccades. For a healthy subject, there is no statistically significant difference between the saturation velocity for leftward and rightward saccades. For our patient, there is an asymmetry between the peak velocity for leftward and rightward saccades: rightward saccades have a velocity saturation approximately half the size of that of leftward saccades. Normal subject have a peak eye velocity saturating between 500 and 700 deg/s [24]. Therefore, this asymmetry does not arise from slow rightward saccadic movements but from extremely fast leftward saccadic movements. This is the first characteristic that the model should reproduce.

Comparing MSEs between equations (4) and (6) and between equations (5) and (7), fits using the maximum displacement as the independent parameter explain more variability than those using saccadic amplitude as the independent parameter. Two-tailed f-tests between residual distributions indicate that this difference between MSEs is statistically significant (leftward saccades: F(163,163)=1.606, p <0.05. Rightward saccades: F(103,103)=1.526, p <0.05). The better fit using the maximum amplitude suggests to us that the command sent to the burst neurons has a normal saccadic shape but that the discharge that stops the saccade is too large.

#### Dynamic overshoot

A saccadic dynamic overshoot corresponds to a fast reversal of the saccadic trajectory before the end of the movement. It is different from a pulse-step mismatch because of the time course of the reversal movement (see [35] for a study of pulse-step mismatch). This can be observed in Figure 5. To characterize the dynamic saccadic overshoot of the patient, we computed a linear regression between rightward and leftward maximum displacements and saccadic amplitudes:

\begin{array}{rl}{A}_{R}& =(0.790\pm 0.013){M}_{R}-(0.698\pm 0.2129)\\ {R}^{2}& =0.96,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}p<0.001,\end{array}

(8)

\begin{array}{ll}\hfill {A}_{L}& =(0.928\pm 0.018){M}_{L}-(0.528\pm 0.086)\hfill \\ \hfill {R}^{2}& =0.98,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}p<0.001.\hfill \end{array}

(9)

Regressions (8) and (9) show that the dynamic overshoot made by the patient corresponded to ≈21% of the maximum displacement for rightward saccades and to ≈7% of the maximum displacement for leftward saccades. The coefficient of variation of the two regressions shows that a linear relationship accurately captures the relationship between the maximum displacement and the amplitude of the saccade. The dynamic overshoot is the second major characteristic that the model should simulate.

#### Post-saccadic drift

Figure 5 shows 25 deg rightward (upper row) and leftward (bottom row) saccades made by the patient during the second session. Target position is represented in black, horizontal eye position in orange and vertical eye position in green. Saccades as determined by our algorithm are colored in blue. As shown in Figure 5, a position hook was visible at the end of each saccadic trajectory. The overshoot of the first rightward saccade in Figure 5 has an amplitude of 9.05 deg. The overshoot of the first leftward saccade in Figure 5 has an amplitude of 8.86 deg. Additionally, when saccades were directed rightward, the patient drifted towards the center but this drift was strongly reduced (even absent sometimes) when the patient executed leftward saccades. To quantify the time constant and the amplitude of the drift, we fitted an exponential function to the movements between two saccades. We excluded fits with a time constant larger than 20 seconds and consider them as non-decaying (no rightward movements, 5 leftward movements). The average time constant for the drifts following rightward saccades was equal to 90 ±75 ms and for the drifts following leftward saccades was equal to 142 ±171 ms. The average amplitude of the drifts following rightward saccades was equal to 3.4 ±1.9 deg (statistically different from zero, two-tailed t-test, t(63)=14.22, p <0.001). The average amplitude of the drifts following leftward saccades was equal to -0.5 ±0.9 deg (statistically different from zero, two-tailed t-test, t(60)=-4.60, p <0.001). Finally, we computed the correlation between the amplitude of the drifts and the orbital position at the onset of the drift. We found a significant positive correlation between the amplitude of the rightward drifts and the orbital position (*ρ*=0.418, p <0.001). In contrast, no significant correlation was observed between the amplitude of leftward drifts and the orbital position (*ρ*=-0.098, p=0.450). These statistical analyses confirmed that there was a strong drift following rightward saccades and a marginal leftward drift following leftward saccades. The asymmetrical drift is the third major characteristic of the patient’s saccadic behavior that the model should simulate.

#### Saccadic oscillation

Figure 6, upper panel, shows a horizontal saccade towards a target located 15 deg on the right. As for the rightward saccade in Figure 5, this saccade has a hook in the position trace at the end of the movement. However, unlike the case of Figure 5, the eyes started to oscillate horizontally at the end of the saccade. No oscillations were observed on the vertical channel. To test if the oscillation was linked to the saccadic command, we tested the behavior during saccades. The bottom row of Figure 6 shows a vertical saccade towards a downward 13 deg target. No oscillations were observed on the vertical channel at the end of the saccade. However, horizontal oscillations were observed during the largest vertical saccade. These two behavioral observations indicate an oscillation mechanism based on the cross-inhibition of the short lead inhibitory burst neurons similar to the one previously reported by [6]. We quantified the frequency of the horizontal oscillations following horizontal saccades and during vertical saccades. We computed the oscillation frequency based on the time between successive peak positions during the oscillations. The average frequency in our patient is equal to 14.5 ±3.4 Hz after horizontal saccades and to 13.1 ±3.1 Hz during vertical saccades. We found no significant difference between the two frequency ranges (two-tailed t-test, t(68)=-1.459, p=0.142), pointing towards an identical mechanism in both cases. As in [6], those oscillations are only possible if the omnipause neurons are held off. Therefore, the oscillatory saccadic behavior of the patient indicates that OPN are not reactivated correctly at saccade offset. This behavior is the fourth major characteristic that model should reproduce.

### Model simulations

In this section, we will explain how we reproduced the four major characteristics of the patient saccadic behavior: the dynamic overshoot, the pronounced rightward drift and the attenuated leftward one, the saccadic oscillations and the asymmetry in the peak velocity. First we present how the model can reproduce general characteristics of healthy saccades.

#### Healthy saccade

To simulate healthy human subjects, we tuned the parameters of the model (*c* *S* *C*_{
M
a
x
}, *c* *F* *N*_{
M
a
x
}, *i* *F* *N*_{
o
} and *i* *F* *N*_{
M
a
x
}, see methods) to reproduce the main sequence represented by equation:

\begin{array}{l}{V}_{\mathit{\text{Max}}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathit{\text{sign}}\left(A\right)\xb7601.4\xb7\left(1-{e}^{-0.103\parallel A\parallel}\right)\end{array}

(10)

This main sequence is extracted from a fit we performed on the data presented in Figure 1 of [24]. Compared to the patient situation, there is only one expression of the main sequence because there is neither a dynamic overshoot nor a left-right asymmetry. Thus *A* in eq. (10) corresponds to the amplitude of the saccade, whether it is rightward or leftward. To account for the natural undershoot behavior of saccades, we set a gain of 0.9 for the saccadic displacement.

The upper panel of Figure 7 shows the time course of a simulated saccadic movement toward a rightward 25 deg target. Because of the undershoot, the model generated two saccades. The first saccade ended at 22.5 deg (peak velocity: 545 deg/s) and a corrective saccade of 2.5 deg (peak velocity: 138.5 deg/s) was triggered by the model of the cortex to cancel the remaining visual error.

The solid gray line in the lower panel of Figure 7 represents the main sequence of eq. (10) while the blue dots represent simulated saccades with a range of amplitudes between two and 40 degrees in steps of one degree. This panel shows that, once tuned, the model reproduced correctly the desired behavior.

#### Patient simulation of average behavior: asymmetric peak velocity and main sequence

To reproduce the main sequence of the patient, we increased the activity of the contralateral caudal fastigial nucleus and the contralateral caudal superior colliculus (*c* *F* *N*_{M a x, p a t i e n t}>*c* *F* *N*_{M a x, h e a l t h y} and *c* *S* *C*_{M a x, p a t i e n t}>*c* *S* *C*_{M a x, h e a l t h y}). Those parameters were tuned to reproduce the peak velocity-maximum displacement relationship presented in Figure 6. To simulate the drift, first we increased bilaterally the time constant of the NPH (from 20 seconds to 22.5 seconds). Second, we increased the gain of the projection from the right EBN and left IBN to the right abducens nucleus (step gain from 0.15 to 0.185). This second modification disturbs the compensation of the longest time constant of the eye plant on one side, and thus generates an asymmetrical drifting behavior as observed in the patient. It must be stressed that the effect of the drift could not be observed in the main sequence but was present in the patient behavior. Thus, we already included the drift modifications in those simulations but the results of the changes will be discussed in the next section. To reproduce the dynamic overshoot, we increased the maximum discharge of the ipsilateral caudal fastigial nucleus (*i* *F* *N*_{M a x, p a t i e n t}>*i* *F* *N*_{M a x, h e a l t h y}) and we triggered the ipsilateral caudal fastigial nucleus activity sooner (*i* *F* *N*_{o, p a t i e n t}<*i* *F* *N*_{o, h e a l t h y}). Through those changes, the ipsilateral EBNs start to discharge too soon, and thus reverse the movement. The higher and/or sooner the ipsilateral caudal fastigial activity, the bigger the saccadic overshoot made by the model. Each of the piecewise functions was tuned independently for leftward and rightward movements to match the dynamic overshoot amplitude for leftward and rightward saccades expressed by eq. (8) and (9).

Once the parameters were tuned to match this relationship, we added a 25% random gaussian noise on *i* *F* *N*_{M a x, p a t i e n t} to account for a part of the variability observed in the patient data (noise amplitude arbitrarily chosen). Then, we simulated 117 leftward and 117 rightward saccades with varying amplitudes between 2 and 45 deg. Figure 8 shows the main sequences generated by the model. Upper panel of Figure 9 represents the saccade-amplitude vs. peak-velocity relationship while the lower panel shows the peak-velocity vs. maximum-displacement relationship. Red dots represent the rightward saccade simulations and blue dots represent the leftward saccade simulations. As expected by the tuning of the parameters, the model reproduces correctly the maximum displacement-peak velocity relationship.

Finally, we computed the regression between the maximum displacement and the saccade amplitude for leftward and rightward simulated saccades:

\begin{array}{rl}{A}_{R,S}& =(0.773\pm 0.014){M}_{R,S}-(0.611\pm 0.436)\\ {R}^{2}& =0.98,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}p<0.001,\end{array}

(11)

\begin{array}{rl}{A}_{L,S}& =(0.936\pm 0.005){M}_{L,S}-(1.098\pm 0.138)\\ {R}^{2}& =0.99,\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}p<0.001.\end{array}

(12)

Comparing regressions (8)-(9) with regressions (11)-(12), one can see that the model correctly approximates the patient behavior. A t-test showed that the slope of eq. (8) is not statistically different from the slope of eq. (11) for rightward saccades (two tailed t-test. t(115)=0.4513, p=0.3264). Similarly for leftward saccades, a t-test showed that the slope of eq. (9) is not statistically different from the slope of eq. (12) for rightward saccades (two tailed t-test, t(115)=0.8768, p=0.1909).

#### Patient simulation: asymmetric drift and dynamic overshoot

Simulating the average behavior of the patient is important for the model, but it is also important to show that it can reproduce extreme conditions. The examples of Figure 5 present a fairly large dynamic overshoot for the first saccade compared to the average behavior. Therefore, we tuned the model with a new set of the four parameters (*c* *S* *C*_{
M
a
x
}, *c* *F* *N*_{
M
a
x
}, *i* *F* *N*_{
o
} and *i* *F* *N*_{
M
a
x
}) to reproduce the larger dynamic overshoot of the first saccades of the trials in Figure 5. To reproduce the patient behavior, we used the behavioral observation that the main sequence is better defined if one used the maximum displacement instead of the saccadic displacement. Therefore, the parameters were tuned as a function of the maximum displacement instead of the saccadic amplitude. We used the inverse of relationships (8) and (9) to compute the amplitude of the saccade sent to the cerebellum and the colliculus by the cortex. All the other parameters were kept constant. Figure 8 shows a rightward simulated saccade (upper panel) and a leftward simulated saccade (bottom panel) that reproduce the patient behavior presented in Figure 5: the dynamic overshoot in both directions and the asymmetric drift at the end of the movement. The upper panel shows the simulation of a rightward saccade toward a 25 deg visual target. The saccadic gain in the cortex was set to 1 to reproduce the behavior shown in Figure 5. Compared to the upper panel of Figure 5, one can see that the general behavior is reproduced: the first saccade overshoots the target and subsequent saccades are triggered even though the eye is close to the target. In addition, a drift can be observed between rightward saccades. The lower panel shows the simulation of a saccade towards a visual target located 25 deg on the left of the center. For this leftward movement, the saccadic gain in the cortex was set to 0.8. Comparing this simulation with the patient behavior presented in the bottom panel of Figure 5, one can see that the model reproduces correctly the desired behavior. The drift between the saccades is greatly reduced compared to rightward movements but the dynamic overshoot is still present. The amplitude of the dynamic overshoot of the first saccade is identical in the simulations (rightward simulation: 9.1 deg, leftward simulation: 8.3 deg) compared to the ones observed in Figure 5.

#### Patient simulation: saccadic oscillations

Figure 10 shows the model behavior when the OPN activity is not reactivated at the end of a 15 deg rightward saccade (to reproduce the patient behavior in the upper panel of Figure 6). The model starts to oscillate if the OPN are not reactivated at saccade offset. The simulated oscillation mechanism is similar to the one reported in [6] and can be reproduced by the model because of the cross-inhibition of the short-lead inhibitory burst neurons and the post-inhibitory rebound of the neurons. To generate the oscillation pattern of Figure 10, we decreased only the input gain of the OPN and we kept all the other average parameters as in Figure 9. Therefore, at the end of the saccade when the OPN should have fired to prevent the sIBN_{L}- sIBN_{R} circuit from oscillating, the OPN inhibition by the long-lead inhibitory burst neurons could not be stopped and an oscillatory movement started. The main differences between our simulation and the patient observation is the variable amplitude of the oscillations. To generate a variable amplitude of the oscillations, we could included some variability in the amplitude of the input gain of the OPN, but that is beyond the scope of this paper. The patient also exhibited shorter oscillatory periods. To simulate those situations, the input OPN gain must be amplified sooner. This will excite the OPN and stop the oscillations.