## Abstract

The oculomotor integrator is usually defined by the characteristics of decay in gaze after saccades to flashed targets or after spontaneous gaze shifts in the dark. This property is then presumed fixed and accessed by other ocular reflexes, such as the vestibuloocular reflex (VOR) or pursuit, to shape motoneural signals. An alternate view of this integrator proposes that it relies on a distributed network, which should change its properties with sensory-motor context. Here we demonstrate in 10 normal subjects that the function of integration can vary in an individual with the imposed test. The value of the time constant for the decay of gaze holding in the dark can be significantly different from the effective integration time constant estimated from VOR responses. Hence analytical tools for the study of dynamics in ocular reflexes must allow for nonideal and labile integrator function. The mechanisms underlying such labile integration remain to be explored and may be different in various ocular reflexes (e.g., visual versus vestibular).

## INTRODUCTION

The oculomotor system has long been represented as relying on sensory processes, which then converge on a shared “oculomotor integrator.” Many publications have addressed the issue of estimating the time constant of this gaze-holding integrator (*T*_{g}), as most recently illustrated in Goldman et al. (2002). A common presumption is that this gaze holding capability is a localized (anatomically) process the function of which is *invariant* and necessary for the appropriate formation of motoneural drives to the ocular muscles. That is, the oculomotor integrator is presumed to rely on a filtering process with very large time constants (>20 s) (Cannon and Robinson 1987; Mettens et al. 1994). This integrator is then presumed to interact with other systems to produce proper reflexes such as the vestibular ocular reflex (VOR). Although it is accepted that the VOR can change its dynamics (gain, time constant) with vergence context (Paige et al. 1998), the gaze integrator is always presumed to be fixed and with large time constant unless lesioned (Cannon and Robinson 1987; Cheron and Godaux 1987; Cheron et al. 1986; Goldman et al. 2002; Mettens et al. 1994). In an alternate representation of the oculomotor system and the VOR, Galiana et al. have proposed that the process of integration could rely instead on a distributed process (Galiana 1991; Khojasteh and Galiana 2003); this research was recently supported by Aksay et al. (2003). Because of central nonlinearities, the connectivity of this distributed process should change with sensory-motor context, and hence this hypothesis predicts that the oculomotor integrator should have a time constant that is variable with test conditions (dark/light, fixed/free head…). Figure 1 summarizes the different views of central integration in the oculomotor system (see methods).

Here we report on the first direct test of oculomotor integration performance in normal human subjects. The ocular responses to three conditions are examined: the decay of fixation in the dark to flashed targets at various eccentricities, the responses during slow phases of the VOR to passive harmonic rotation with one sinusoid, or a sum of two uncorrelated sinusoids. The VOR produces compensatory eye movements during the slow phases of induced nystagmus with head perturbations. The reflex pathways are known to converge on an oculomotor integration process, located in the VN-prepositus hypoglossi (PH) complex used by all horizontal ocular reflexes, including the maintenance of gaze direction in the dark after fixation of a flashed target (Cannon et al. 1987; Mettens et al. 1994; Robinson 1968). In a previous paper (Green and Galiana 1998), we demonstrated that sharing of the integrator would be facilitated by sensory projections on different points on the network. Hence we selected a comparison of integrator function with and without head motion to test the validity of our hypothesis: integration function should vary with recruited sensory pathways. We have already demonstrated in a model study that integrator function could be labile with binocular context (orbital positions) given nonlinearities in vestibular neural responses during the VOR (Khojasteh and Galiana 2003): this suggests a mechanism for the prior observations of Crawford and Vilis (1993), showing that gaze direction decays with different time constants depending on eye eccentricity. Preliminary results supporting labile integration have appeared elsewhere in abstract form (Chan et al. 2004).

## METHODS

### Data acquisition

Ten human volunteers with no known vestibular dysfunction participated in the experiment. All subjects signed a consent form describing the protocol, which was approved by the Institutional Review Board of McGill’s Faculty of Medicine. The experimental procedure has been described in detail elsewhere (e.g., see Galiana et al. 1995). In brief, to avoid invasive procedures, horizontal conjugate eye movements were measured by electrooculography (EOG) with an accuracy of ±1.5° estimated in the calibration data. Passive head rotations around a vertical axis were performed en block using a servo-controlled chair with either a single sinusoid (1/6 Hz, peak: 150°/s) or a sum of two sine waves (0.03 and 0.1 Hz, peak: 120°/s). The intent here is to also test for possible changes in reflex dynamics with changes in input bandwidth. EOG and turntable signals were filtered to 40 Hz with an 8-pole Butterworth filter and stored on a PC at 500 Hz for later analysis.

After a 20-min rest period in the dark to stabilize the EOG electrodes, the recording procedure in the dark included: calibration with a laser target over ±50° range to quantify any bias and recording sensitivity (°/V), recording of eye movement with rotational stimuli (VOR, no target), recording of self-paced fixations intermingled between flashed targets (gaze, no rotation), and a final re-calibration. Pauses between tests were used to ensure the disappearance of any residual effects from the previous test. The dark-fixation trials were designed with imbedded catch-trials (flashed targets) to control for the well-known drift properties of EOG (Schlag et al 1983). First a horizontal target was presented randomly on a circular screen within 60° from the mid-point for 5 s and back to center for 5 s; and then a new target was *flashed* at another random location, and the subject was allowed to perform spontaneous saccades over 60 s. This sequence was repeated four times and allowed calibration of the EOG throughout the gaze-holding trials, particularly for any changes in zero offset. In our experience, the *amplitude* calibration (volts to saccade *size* in degrees) does not change over our trials. Trends in the zero offset during catch trials were used to detrend the EOG gaze-holding trials and did not exceed 5–10° over the whole procedure. (i.e.∼0.04°/s over the gaze trials)

### Data analysis

The raw data sampled at 500 Hz was first digitally filtered to 20 Hz before down-sampling to 100 Hz for analysis. Slow phase segments of the eye movements were then selected from records in each test with a minimum duration of 0.2 s to allow for the estimation of large time constants in the presence of noise. The selection was based on the classification routines developed by Galiana and colleagues (Radinsky and Galiana 2004; Rey and Galiana 1993) that propose a simple reduced model for the relationship between stimulus and eye response and accept as slow phase only those segments that are behaving within a range of the model. This often appears as selection based on an eye-velocity criteria but in general allows classification for any stimulus trajectory and robustly rejects low-velocity saccades.

To quantify the dynamics of eye responses, we rely on a global schematic generally accepted in oculomotor control for the conjugate slow phases of the VOR (Fig. 1): angular head velocity (*Ḣ′*) is the stimulus and angular eye position (*E*) is the response, assuming the eye plant is well enough described as a simple low-pass filter with time constant *T* and gain *K.* In Fig. 1*A*, the classical approach uses a first stage typically called velocity storage (VS) (Raphan et al. 1985; Robinson 1968) to describe the centrally derived vestibular signal, followed by direct and neural integrator (NI) parallel pathways to compensate for eye plant dynamics—it implies a near-perfect integrator in all eye reflexes. Figure 1*B* describes an alternative (Galiana 1991; Guitton et al. 1990) that achieves the same integration and eye plant compensation with a single feedback loop through a model of the eye plant—here integrator function depends on loop gains through nonlinear vestibular cells (VN) (Khojasteh 2003; Khojasteh and Galiana 2003). Both approaches can be lumped into equivalent dynamic processes (Fig. 1*C*), where now we also include potential initial conditions on eye position *E*° at the beginning of slow phase segments. Algorithms were developed in Matlab to estimate the value of the integrator time constant (*T*_{g} in Fig. 1*C*) in each protocol.

##### For gaze holding in the dark.

On a stationary chair, the input head velocity in Fig. 1 is set to zero, and two approaches were compared to evaluate integrator function. The first used a previously published equation (Becker and Klein 1973; Goldman et al. 2002), where the integrator time constant can be extracted from the slope of an eye velocity versus eye position plot (Fig. 1) with *T*_{g} = −*E*_{p}/*E*_{v}, in seconds. The second method relied on a subset of our NARMAX estimation method (Kukreja et al. 2005; Rey and Galiana 1993; Smith et al. 2002) (see Fig. 2 and 4*B*). In fact, the Rey and Galiana (1993) algorithm is a more general superset of that proposed by Becker and Klein (1973), later re-validated by Goldman et al. (2002). The relationship in Fig. 1*C* is mapped into the discrete domain for all pooled slow-phase segments *n* {*n* = *1*…*N,* the number of slow-phase data points} refers to the sample number in a given slow-phase segment, and *b* allows for a nonzero null point in some subjects. For sampling interval *T*_{samp}, *T*_{g} = −*T*_{samp}/ln(*a*), in seconds. Because the eye-position measurements (*E*^{m}) are noisy, estimates of real eye position: *E*^{m} = *E* + *r*, with assumed white noise *r.* As a result, the regression problem in the equations above, in terms of *E*^{m} (the data) becomes

This is a linear ARMA problem: after substitution for the noisy eye-position observations in the equations, there is a moving-average over the innovations (noise) but no exogenous input (such as head velocity). The estimates are found by extended least squares over all the slow-phase segments and do not require differentiation. Note that both methods are less accurate (larger SDs) when the time constants are large and/or when noise increases. As a result, it becomes more and more difficult to numerically resolve changes in the slopes as time constants get large in either method.

##### For VOR in the dark.

Both dynamic processes in Fig. 1*C* must be taken into account. Analysis of VOR data were performed with our ARMAX method with an iterative regression approach to simultaneously estimate the optimal gain and time constant for the VS stage (*T*_{v}), and the time constant of the NI stage (*T*_{g}). As in the preceding text, the dynamics of the VOR in Fig. 1*C* can be mapped to the discrete domain with where *Ḣ _{n}* denotes the internal estimate of head velocity from the VS stage into the NI stage, and {

*a, b,*and

*c*} denotes the coefficients for the one step delay of

*E*, the bias, and the head velocity input into the regressor, respectively. Again, substituting for the measured data, the regression problem in the VOR becomes

This is an ARMAX problem, where the exogenous input *Ḣ _{n}* must also be determined from the known head-velocity profile. We relied on an iterative search where potential

*T*

_{v}over a range from 1 to 50 s were used to generate candidate

*Ḣ*profiles by simulation of the VS stage in Fig. 1 prior to applying each regression. The regression optimally finds the parameters in each case using all samples and then also estimates the optimal initial condition

_{n}*E*

^{0}for individual slow-phase segments. This last step caused a change in the optimal time constants of no more than 5% and was only necessary to provide accurate simulations in the selection of VS time constants, especially when

*T*

_{g}is less than a few seconds. The optimal {

*T*

_{v},

*T*

_{g}} pair was selected as the one that generated the best quality of fit (QF) in the validation (see following text). This algorithm has been extensively tested on simulated data and shown to converge correctly (Kukreja et al. 2005; Smith et al. 2002). It has the advantage of being applicable to any head velocity profile.

##### Validation of models and statistical tests.

A Simulink (Matlab) model of Fig. 1*C* generated predicted eye-position profiles in each VOR protocol, using the parameters estimated from the data, the measured head velocity profile and the optimal initial condition at the beginning of each slow phase (e.g., Fig. 3). The QF for estimated models was calculated from where *z*_{fit} is the predicted trajectory (model simulation) and *z*_{meas} is the measured response. QF served to select the cases that minimized the modeling error with the best {*T*v,*T*g} combination for VOR tests.

To estimate the confidence interval of time constant estimates in both gaze holding and VOR protocols, a *t*-test was first performed on the regression coefficients (θ). In general, the regression problem can be written as: *Y* = *R*θ, where *Y* is a vector of observations (*E _{n}*

_{+1}in the preceding text),

*R*is the matrix of input/output observations (regressor

*without*initial conditions) that multiplies the vector of desired coefficients (θ of dimension

*k;*a–c in the preceding text). The regression algorithm provides a solution θ̂, which is used to generate the prediction

*Ŷ*=

*R*θ̂. The residual vector, or errors in prediction, is defined as ε =

*Y*−

*Ŷ*, with SD σ̂. From this, we expect the coefficient estimates to belong to a Normal distribution with mean θ̂ and with confidence intervals defined by the

*t*-statistic for the

*i*th estimate as where

*c*is the

_{ii}*i*th diagonal element of (

*R′R*)

^{−1},

*N*is the number of data points, and

*k*is the number of estimated coefficients. The confidence interval so computed at the 99% level for each parameter was then converted into the range of associated time constants (

*T*), according to the function postulated in each method [e.g., 1/

*a*or –

*T*

_{samp}/ln(

*a*)]. This approach relies on the fact that the cumulative probability of a coefficient interval must equal the probability of the associated range after a nonlinear mapping.

## RESULTS

### Gaze holding in the dark

Figure 2 provides a sample of the spontaneous saccades and postsaccadic drift for *subject JN48.* Like two other subjects, he had a rather small gaze holding time constant in the dark (9.1 s). All other subjects had larger time constants reaching even 100 s in one case as would be expected from the classical view of an ideal oculomotor integrator (Fig. 4*A*). Integrator time constants evaluated with either the Goldman or the ARMA method were equivalent (no statistically significant differences, Fig. 4*B*). Despite the presence of noise on the EOG records, the SDs for integrator estimates are reasonable and allow statistical tests against the VOR results in the following text.

### Integration function in the VOR

Figure 3 illustrates part of the VOR response for *subject DC06* during rotation with a sum of sines. Using the time constant estimates from the ARMAX method, the fits for the VOR responses in both eye position and velocity are excellent. The integrator time constants (*T*_{g}) in the single sine and sum of sinusoids protocol were found to be, respectively, 2.6 and 17.4 s. Yet, the estimated integrator time constant during gaze holding was much larger for this subject, at *T*_{g} = 31 s for the Goldman method and 33 s for the ARMA method (no significant difference, *P* < 0.005). Integrator function during VOR tests is summarized for all subjects in Fig. 4*A* and in Tables 1–4. This was typical for almost all subjects: weaker integration during rotation than during gaze shifts in the dark. A large gaze holding time constant is not necessarily associated with a strong integrator function during rotation tests, when examining estimates for a *given* subject (Fig. 4*C*). Although *T*_{g} estimated in the VOR can vary between subjects from 1 to ∼20 s, the mean across all subjects in the single sinusoid protocol is 4.5 s, whereas the mean for the sum of two sinusoids is 8.2 s. This can be compared with the mean for the stationary protocol (gaze holding in the dark) at 31 s for the Goldman method and 33 s for the ARMA method. Therefore the averages across subjects for the VOR protocols are lower than the generally accepted range for the oculomotor integrator during gaze holding (20–30 s). This trend is also observed within the estimates for a *given* subject. The integration time constant during rotation is systematically reduced from that during gaze holding in the dark for almost all subjects (* in Fig. 4A, *P* < 0.005). Thus it appears that the functional level of oculomotor integration (*T _{g}*) varies with sensory context.

To further support these changes in different protocols, rotational data in the low eye-velocity range (|*Ė*|<15°/s) were selected to generate *T*_{g} estimates at eye speeds comparable to the gaze holding protocol; similarly, integrator estimates from VOR data at these low speeds were compared with those extracted only from high-speed segments (|*Ė*|>25°/s). Figure 4*D* illustrates the results in the three scenarios: in the eight subjects that posses a significant change in *T*_{g} from stationary protocol to rotational protocol, all but one subject retain the same trend in *T*_{g} deficits (decrease) in the VOR, whether eye velocities are small or large. There is a trend for stronger decreases in integrator time constant with larger eye speeds, which will be covered in the discussion.

## DISCUSSION

Classically, the oculomotor integrator is assumed to be a very effective filter with a large time constant in all ocular reflexes. It was first hypothesized by Robinson as a global concept to transform velocity signals from sensors into position signals for the eye plant. As a result, many analysis procedures to study ocular reflexes rely on this presumption of near-ideal integration in premotor ocular circuits: for example, eye velocity is assumed to allow unmasking of sensory stimuli because differentiation of eye position cancels the effect of an ideal central integrator. This study illustrates in 10 normal subjects that the general presumption of ideal integration can be totally unfounded. The results here support a distributed integrator process with very labile properties, which will have significant impact on the analysis of ocular reflex dynamics, for both neuroscientists and clinicians.

### Accuracy of estimates

The estimated “integrator” time constant in our subjects varies in a context-dependent manner. The time constant is significantly larger in the stationary protocols when compared with rotational protocols (Fig. 4*B*). The changes in *T*_{g} during rotation tests might be ascribed to inaccurate algorithms that converge on biased values. However, we have tested the ARMAX algorithm extensively with simulations over a broad range of {*T*_{v}, *T*_{g}} combinations and found it robust and unbiased even in the presence of the noise levels associated with EOG. Furthermore, estimates of dark gaze-holding from the Goldman et al. approach are not significantly different from those obtained with the ARMA method (Fig. 4*B*). Although one might be tempted to argue that the ARMAX algorithm for the VOR mistakenly assigned large time constants to the vestibular system and smaller ones to integration, this is not possible: in addition to our prior tests with simulated data, we use the validation of model predictions compared with experimental data to verify the high quality (QF) of fits—if one reverses the time constant estimates, the result is a very poor fit for the VOR data. Hence, the measured changes in the oculomotor integrator with test conditions are not likely due to our analysis algorithms. In fact, there is also an indication of differences in integrator time constants in the article by Goldman et al. (2002; their Fig. 3) with search coil data, when comparing fixations to VOR, but it was not discussed at that time. The main difference between the Goldman method and the ARMA method described here is that the Goldman method does not include an estimate for the filtered noise term (MA) that is generated by the differentiation process of eye position. This can introduce biased estimates unless the noise level on eye records is extremely low.

Finally, one might argue that the changes in estimated integration could be due to failure of the model used in regression. For example, Goldman et al., in commenting on their estimation technique, add the cautionary note that it should only be applied at low head velocities to avoid corruption by head-velocity signals in a pathway parallel to the integrator. This is because they rely on the characteristics of eye velocity. In our method, we simply assume that the premotor pathways serve to cancel eye-plant dynamics in whatever form they may take (Fig. 1*C*), and so the only restriction for valid estimates is that the model be valid for any head velocity profile, i.e., that the eye plant is well compensated in all conditions and that the assumption of a linear model (Fig. 1*C*) is valid. First, according to Sylvestre and Cullen (1999), there is no reason to believe major changes are needed in eye plant compensation in these protocols: the *r*:*k* ratios in their Fig. 14 remain between 100 and 200 ms (dominant time constant for the eye plant) for eye velocities up to and beyond 200°/s. Second, our analysis supports changes in the integrator time constant that not only depend on the protocol (e.g., gaze holding vs. VOR) but also appear to be sensitive to other variables such as eye speed. This is clearly a nonlinear property so that the model in Fig. 1*C* is not sufficient, and some estimates will be biased. However, changes in the integrator time constant must be real given the huge and statistically significant differences observed here even after restricting eye velocity ranges where a linear model should hold. The results are not likely due to simple model failure.

### Mechanisms for labile integration

Changes in *T*_{g} with protocols, and even with different subsets in the same protocol, are perfectly compatible with the concept of a distributed oculomotor integrator the filtering properties of which will vary with recruitment of feedback or recurrent pathways. Activating vestibular processes in the dark can affect the recruitment level of brain stem loops around the vestibular nuclei, and the operating point of nonlinear vestibular cells. In Fig. 1*B* for example, with the assumed shape of the nonlinearity, one would expect changes in the gain and time constant of the behavioral VOR with set point due to either sensory level *Ḣ′* or eye eccentricity *E** (Khojasteh and Galiana 2003). This hypothesis has recently been tested successfully by Wu Zhou (personal communication) using acoustic clicks during head rotation. Similarly, a protocol in the light will add a visual loop around the brain stem filter(s) and again change the overall global performance of the integrator process. Because visual signals such as slip also converge on premotor loops, one would expect the effective integration to vary with all sensory contexts and with motor context (vergence/version set points). More experiments will be required to explore all the factors affecting integration in both normal subjects and patients.

### Implications for the estimation of sensory dynamics

Traditional estimates of vestibular time constants from eye-velocity trajectories in the VOR can be seriously biased (Galiana 1991). Unless the integrator is near ideal, the estimated *T*_{v} (Fig. 1) will actually be an average of the concurrent vestibular and integration function. In the 10 subjects studied here for example, the average vestibular time constant *T*_{v} was found to be ∼35 s. This is larger than the traditionally accepted VOR time constants of ∼20 s, but it is compatible with the expected underestimation of vestibular time constants in the presence of degraded integration (*T*_{g} ∼ 5.7 s). The argument of biased sensory estimates would hold true for any ocular reflex because the integrator is also shared by the pursuit, optokinetic, saccadic, etc. systems. One might be tempted to conclude that the VOR is deficient in subjects with a small integrator time constant during rotation (near 1 s in 1 subject), but this would be false. The global VOR performance at the behavioral level remained perfectly equivalent in all these normal subjects in terms of the slip levels during slow phases in the dark. It is possible to achieve appropriate reflex dynamics at the behavioral level with different combinations of sensory and integrator dynamics, especially with the help of nystagmus.

In summary, the results here point to a need to fully investigate this phenomenon in a much larger group of both normal subjects and patients. More importantly, we cannot continue to use analytical methods in the study of ocular reflexes that rely on the *assumption* of a near-ideal integrator. More general statistical approaches must be applied to at least allow for the *possibility* of dynamic changes in the integrator with context. The first step will require a model framework that incorporates potential *nonlinear* equations for the integration stage (Fig. 1*B*) in the regression problems, using NARMAX approaches (Kukreja et al. 2005).

## GRANTS

This work was supported by the Canadian Institutes for Health Research.

## Footnotes

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- Copyright © 2005 by the American Physiological Society