Short-Term Adaptive Control Processes in Vergence Eye Movement

Cahiers de Psychologic Cognitive/ Current Psychology of Cognition 2002, 21 (4-5), 343-375

Short-term adaptive control processes in vergence eye movement

John L. Semmlow,1'2 Weihong Yuan,2 and Tara 3 Alvarez

1. UMDNJ, New Brunswick, NJ, USA 2. Rutgers University, Piscataway, NJ, USA 3. New Jersey Institute of Technology, Newark, NJ, USA

Abstract The "Dual Mode" theory for the control of disparity vergence eye movements states that two control components, a preprogrammed "transient" component and a feedback control "sustained component, mediate the motor response. Although prior experimental work has isolated and studied the transient component, little is known of the contribution of the sustained component to vergence response. The timing between the two components and their relative magnitudes are of interest as they have implications on the control strategies used to coordinate the two components. Modeling studies can provide an estimate of component magnitudes, but cannot uniquely identify the component timing. The work presented in this paper applies Principal Component Analysis (PCA) to ensemble data both to confirm the presence of two major components and for data reduction. A novel, ensemble version of Independent Component Analysis (1CA), is then exploited to estimate the contribution of the two control components to the eye movement. Other recent experiments have shown that the vergence system is capable of rapidly modifying its dynamic characteristics (short-term adaptation) when exposed to specially designed "adapting" stimuli. Adapted

Correspondence should be sent to Department of Surgery (Bioengineering), Robert Wood Johnson Medical School - UMDNJ, New Brunswick, NJ 08903, USA (email: [email protected]) 344 J. L.Semmlow et al. responses were characterized by faster dynamics often featuring large overshoots. ICA analysis of adapted and normal responses show that the enhanced dynamics of adapted responses are due to an increase in transient component amplitude. In addition, the sustained component of adapted responses often showed double-step behavior in the later portion of the response. Finally, the extent of adaptation produced appeared to be related to the unadapted transient component amplitude.

Key words: eye movements, disparity vergence response, independent component analysis, vergence components.

INTRODUCTION

The complex strategies used by the brain to implement eye movement behavior have evolved in the nervous system to solve a difficult control problem: the need to produce fast and highly accurate responses in the face of substantial computational delays. One of the most dramatic of these strategies is the lavish use of independent neural control centers to mediate various oculomotor tasks. The major motor centers that are thought to support the control of eye movements are represented schematically in Figure 1. Note that the centers shown in Figure 1 represent only the motor side of the visuomotor system: complex tasks such as target identification and spatial localization would be mediated by different centers that are simply lumped together in Figure 1 under the heading "Stimulus Influences". This organization is a synthesis of extensive eye movement research in many laboratories involving both behavioral and neurophysiological approaches. The neural structures and their organization were originally inferred from behavioral data, but many have since been identified or confirmed in neurophysiological studies (shown as shaded). As shown, the centers can be divided into two broad categories: those that mediate version (tandem) movements and those that mediate vergence (opposition) movements. Although the version and vergence subsystems mediate different behaviors, and have different levels of performance, these is some structural similarity between them. The two subsystems are composed of a high speed and low speed component, with the low speed component under the influence of visual feedback. In addition, each subsystem receives a contribution from processes that are driven by stimuli other Adaptive control in vergence eye movement 345 than the retinal image position: the version subsystem receives a signal from the vestibular apparatus that is sensitive to head movement while the vergence sub-system receives an input from accommodative processes that is related to blur.

Accommodative Prediction ( Vergence

Adaptation ( )•• '" * Prediction (

Adaptation

Figure 1. Schematic representation of the major motor centers active in the control of eye movements. For simplicity, feedback pathways are not explicitly shown, but the "Stimulus Influences" of both smooth pursuit and slow vergence are modified by visual feedback. Shaded neural centers and their associated pathways have been confirmed in neurological studies. Dashed lines indicate pathways not yet experimentally verified or controversial.

The control of version eye movements has received much attention, particularly the saccadic and smooth pursuit components. The appeal of these two systems is likely due to the extraordinary performance of the saccadic system. Saccadic movements appear to be time-optimal (Bahill & Stark, 1979; Robinson, 1975) and can reach velocities of nearly a 1,000 deg/sec (Bahill, Clark, & Stark, 1975). Prediction is known to occur in both the saccadic and smooth pursuit systems (Young, 1971; 346 /. L.Semmlow et al.

Stark 1971). In the former, prediction is observed as a marked decrease in the response latency of a saccadic response (Young, 1971), while in the latter, prediction improves the tracking performance (Stark, 1971; Robinson, Gordon, & Gordon, 1986). Prediction can be activated simply by using target motions that are predictable. If the timing of the stimulus is regular, such as a square wave stimulus pattern, prediction in the saccadic system will reduce the saccadic response latency, normally around 200 msec, to near zero (Stark, 1971). Similarly, if the stimulus trajectory follows a simple periodic function, such as a sine wave, the smooth tracking error decreases substantially (Young, 1971; Stark, 1971). The saccadic system is also influenced by both long-term and short-term adaptive processes (McLaughlin, 1967; Semmlow, Gauthier, & Vercher, 1989; Optican & Miles, 1985; Optican, Zee, & Chu, 1985). Smooth pursuit adaptive processes are also known to exist (Optican et al., 1985), but have not been extensively studied. Short-term saccadic adaptation can be evoked experimentally using training stimuli that force post-saccadic errors (McLaughlin, 1967; Semmlow et al., 1989), but this protocol requires a large number of adaptive training saccades. Since both saccadic and smooth pursuit systems mediate only version eye movements, they cannot be used to change the depth of the eye fixation point. This task is relegated to the vergence motor control system which drives the two eyes in opposition. The vergence system receives information on both the "disparity" of the retinal image of interest (the difference in the position of the two retinal target images with respect to a set of corresponding retinal reference positions), and the blur of the target image. Image blur comes from an error in the focusing of the lens and is related to target distance. Blur-induced vergence (termed "accommodative vergence") may play a role in maintaining vergence position, but has been shown to be too slow to contribute significantly to the dynamic vergence response (Hung, Semmlow, & Ciuffreda, 1983). The vergence system shares some of the same features as found in the saccadic system. For example, closely-spaced step-like responses have been observed in response to single step stimuli in both saccades and vergence as discussed below. Vergence control processes also include a prediction operator, and the behavior of prediction in vergence is quite similar to that seen in saccades: prediction reduces latency and improves smooth tracking (Yuan, Semmlow, & Munoz, 1998, 2000). Another paper in this series (Alvarez, Semmlow, Yuan, & Munoz, 2002) ad- dresses the behavior of prediction in vergence eye movements. Vergence Adaptive control in vergence eye movement 347 moments also respond adaptively to certain stimulus conditions. In fact, adaptive changes in vergence produced by special adaptive stimuli are both faster and more dramatic than those produced in saccades.

Vergence control concepts

Abnormalities in the control of disparity vergence eye movements underlie many binocular clinical conditions such as strabismus (Schor & Ciuffreda, 1983), yet the basic neural control strategy which guides these movements is only just beginning to be revealed. Though scientific interest in disjunctive eye movements can be 'traced to the early 19th century, it was not until 1961 that Rashbass and Westheimer (1961) presented a quantitative theory for the neural control of these movements. The movements produced by the vergence control system are not as dynamically exciting as those produced by the saccadic system, reaching peak velocities only one quarter of that of saccades (Hung, Ciuffreda, Semmlow, & Horng, 1994). That behavior, coupled with the very low position errors achieved by this system (sustained binocular position errors are only several minutes of arc), has led control oriented re- searchers to conclude that vergence responses are under the guidance of a single feedback control system (Rashbass & Westheimer, 1961; Zuber & Stark, 1968; Krishnan & Stark, 1977). Subsequent support was provided by experimental (Zuber & Stark, 1968) and analytical (Krishnan & Stark, 1977) studies. Elegantly simple, the feedback control theory provides a complete description of control strategy: it can predict behavior and suggest underlying control processes. Recent experimental evidence suggests that a more complicated control structure drives vergence eye movements. This evidence is summarized in the next section, but the basic problem with feedback control can be demonstrated in a single vergence step response such as shown in Figure 2. After an initial delay, or latency, of approximately 200 msec, most of the movement is completed in a time frame only slightly longer than this latency. If this response were visually guided, control instability, characterized by over-shooting behavior, would be unavoidable. Indeed, we have found that models based solely on feedback control are not able to generate a rapid, stable response such as shown in Figure 2. As in many physiological motor responses, vergence control processes are faced with two major challenges: the need for a quick 348 /. L.Semmlow et al. response despite substantial processing delays, and the need to attain accurate positioning despite errors inherent in the neural and muscular apparatus. Unfortunately these two challenges are best met with different control strategies: feedback can produce extremely accurate responses, but for stability, response velocity must be reduced by de- creasing gain when delays are present in the feedback loop. Conversely, preprogrammed (i.e., open-loop) control can generate rapid responses, even in systems with long delays, but these responses suffer in accuracy. Recent experimental evidence indicates that the neural processes which control disparity vergence eye movements employ both these strategies: a rapid, preprogrammed "transient" response (which may be sufficiently accurate to provide limited binocular vision) is followed by a much slower, "sustained" component thought to be guided by feedback (Semmlow, Hung, & Ciuffreda, 1986; Semmlow, Hung, Horng, & Ciuffreda, 1993, 1994). This evidence led to the development of the "Dual-Mode" Theory where disparity vergence, like version, is mediated by two separate sub-systems: a transient, open-loop system and a sustained, probably closed-loop system, Figure 1. Initial Component Amplitud

e 0.0 0.5 1.0 1.5 2.0 Time (msec)

Figure 2. Time plot of a typical vergence movement to a 4 deg. step change in target vergence. The difference between left and right eye movement is plotted with convergence as upward. Other potential vergence stimuli such as blur have been eliminated. A scaled velocity trace (lower curve) is also shown to emphasize dynamics.

Adaptive control in vergence eye movement 349

Experimental evidence for the Dual-Mode Theory

Considerable evidence supports the Dual-Mode theory (Semmlow et al., 1986, 1993, 1994; Hung, Semmlow, & Ciuffreda, 1986; Horng, Semmlow, Hung, & Ciuffreda, 1998). The earliest indication of the presence of two control components came from experiments of West-heimer and Mitchell (1969) who reported substantial vergence movements in response to short presentations of stimuli, 50 to 100 msec; specifically, the transient response to a briefly presented step stimulus presented was quite similar to a standard step response. Even nonfusable targets (a vertical line paired with a horizontal line), produced responses similar to the initial transient of a normal vergence step response (Westheimer & Mitchell, 1969; Jones, 1980). Jones first suggested that the vergence system was composed of two different control processes: a "fusion-initiating phase" and a "fusion-sustaining phase" (Jones, 1980; Jones & Stephen, 1989). Somewhat later, our group found dichotomous behavior in response to ramp stimuli: slow ramps produced smooth tracking movements, while faster ramps introduced step-like behavior (Semmlow et al, 1986). Finally, there is neurophysiological evidence for two signal components from recording of cells in the midbrain by Mays, Porter, Gamlin, and Tello (1986). They found two cell types: burst neurons that code for velocity and tonic cells that code for position. The most straightforward demonstration of the action of two components can often be found in a single step response. In the response of Figure 2, the final position is not achieved by the initial rapid movement, but rather by a slow movement which takes nearly 2 sec to bring the eyes to their final position. Note that in these experiments, all other potential stimuli to vergence such as blur and proximity were eliminated so that only a pure vergence stimulus (termed disparity vergence) was operating. Evidence also exists regarding the control structure of the two components. To demonstrate that the transient component was preprogrammed, a constant error vergence response was generated using a stimulus manipulation technique. (This is also referred to as an "open-loop" stimulus condition.) The constant error response can definitively differentiate between preprogrammed and continuous feedback control: a continuous feedback control system will exhibit a smooth, integrative response, while a preprogrammed, or intermittent feedback, control process will respond with a sequence of predetermined movements that

350 /. L.Semmlow et al. would give the appearance of multiple steps. To generate the constant error stimulus condition, the eye movement signal was continuously monitored, and this signal was used to manipulate the stimulus target in such a way as to keep vergence error constant, an approach first used by Young and Stark (1963) for saccades. Essentially, an external, positive feedback loop is used to cancel the internal negative feedback. A "repeated calibration" and "random stimulus" protocol was developed to guard against measurement errors (Semmlow et al., 1994). Figure 3 shows a typical individual vergence response to a 2 deg open-loop stimulus. The corresponding velocity trace is also presented to aid in identifying the dynamics. The multiple step-like responses clearly evident in Figure 3 indicate the operation of preprogrammed control, probably with intermittent feedback.

O I— 3 M

0.6 TIME (sec)

Figure 3. A typical individual vergence response to a 2 deg constant error stimulus is shown with the corresponding velocity trace presented to aid in identifying the dynamics. The multiple step-like responses clearly evident in the velocity trace indicate the operation of preprogrammed control. Adaptive control in vergence eye movement 351

Another behavior that might be found in a preprogrammed control component is multiple steps in response to a single stimulus. Closely-spaced step behavior is sometimes seen in saccades (Bahill & Stark, 1975). An analysis of the timing of saccadic doubles indicates that the two steps are programmed together based on some internal monitoring process. While the vergence eye movement system normally responds to step stimuli with a single high-velocity response, occasionally double high-velocity movements are seen (Alvarez, Semmlow, & Yuan, 1998). Such double responses were found to occur when the first high-velocity component was less than approximately 80% of the required amplitude (Alvarez et al., 1998). Double high-velocity movements are not common: they occur less than 3% of the time in some subjects, while the most prolific subjects exhibit double responses only 10% of the time. Despite their rarity, double high-velocity responses have important implications on neural processing as they clearly demonstrate switching or behavior characteristic of open-loop control as seen in saccades. Factors leading to the generation of vergence doubles are unknown; however, more double responses were generated in response to larger stimuli and when the subjects stated they were less attentive (Alvarez et al., 1998). Comparing vergence doubles evoked by an experimental protocol that induces post-movement visual error with those that occur normally, Alvarez, Semmlow, Yuan, & Munoz (1999) provided evidence that the double steps were not mediated by visual feedback, but rather, as in the saccadic system, by an internal monitoring process that triggers these secondary movements. Much less is known about the sustained component. The small posi- tional errors achieved by this sub-system indicate that feedback control is involved. This conjuncture is supported in the analysis below that shows the isolated sustained component to have slow dynamics as would be expected from a feedback control system.

Component isolation

In order to study a multiple controller system in any detail, it is essential to have a technique for experimentally isolating the output of each controller. Ideally, a stimulus can be found which evokes only one component. This is the approach used to study version subsystems: sac-cades are produced by step stimuli while smooth pursuit is evoked using 352 /. L.Semmlow et al. continuous stimulus waveforms of slow or moderate velocity. In the experiments described here, carefully controlled stimulus conditions were used to isolate disparity vergence from blur and proximal vergence components. While experimental stimuli have been developed that isolate the transient component of vergence (Semmlow et al., 1993), the isolation or identification of the sustained component is problematic. Due to the presence of feedback, simply subtracting the transient component found in isolation experiments from the combined response will not give an accurate representation of the sustained component's role in a combined response. The transient component does have a threshold related, in part, to target velocity which has been exploited to isolate the sustained component (Semmlow et al., 1986). Unfortunately, the slow ramp responses produced by this stimulus tell little about the control characteristics of this response component and nothing of its interaction with the transient component. Ideally we would like to isolate the sustained component contribution within a normal binocular step response such as occurs when target vergence is quickly changed, or when fixation is voluntarily switched from a far to near target. A new technique developed in our laboratory, applies Independent Component Analysis (Cardoso & Souloumiac, 1993; Cardoso, 1994; Comon, 1994; De Lathauwer, De Moor, & Vandealle, 2000; Hyvarinen & Oja, 1997; Hyvarinen, Karhunen, & Oja, 2001) to multiple, or ensemble, response data to identify underlying components. While the technique has been applied here only to vergence motor responses, it can be used to identify the underlying components of any response provided multiple observations of the response can be obtained. Independent Component Analysis (ICA) has already seen numerous applications in biomedical studies including fetal ECG extraction (De Lathauwer et al., 2000), ECG analysis (Vetter, Virag, Vesin, Celka, & Scherrer, 2000), analysis of EEG and MEG (Vigario, Sarelaa, Jousmaki, Hamalainen, & Oja, 2000), optical imaging (SchieBl, Stetter, Mayhew, McLoughlin, Lundand, & Obermayer, 2000), and fMRI analysis (Biswal & Ulmer, 1999; McKeown, 2000). The conventional application of ICA requires several different measurements of the combined signals usually taken from different physical locations. In this application, each of a number of vergence responses produced by the same stimulus is treated as a separate signal source. The approach takes advantage of the fact that underlying control components have some inherent response-to-response Adaptive control in vergence eye movement 353 variability and this variability can be used to identify component con- tributions to the total response. ICA analysis requires only that component signals arise from independent sources and be actual signals (i.e., nongaussian). Assuming that separate neural processes generate the transient and sustained components, the resultant signals can be taken as independent. However, they also share a common triggering event, the stimulus. Since both components are initiated by a common event, there will be a temporary loss of independence from stimulus-induced synchronization. This could produce an error in the ICA-based decomposition of the early portion of the response. To circumvent this error, the ICA analysis was restricted to the latter portion of the response as described below. Note that the decomposition of the vergence response will identify the motor contri- butions of the underlying neural signals to the overall response, not the neural signals themselves. These motor components will be related to the neural signals through the properties of the extraocular muscles and related orbital mechanics (see Discussion).

Vergence adaptation

Adaptation is an important control strategy found in many motor control systems, and has been particularly well studied in saccadic eye movements (McLaughlin, 1967; Semmlow et al., 1989; Deubel, 1995) Recently our group has demonstrated short-term adaptive modification in disparity vergence eye movements (Munoz, Semmlow, Yuan, & Alvarez, 1999; Semmlow & Yuan, 2002). Figure 4 shows an ensemble of vergence step responses before and after adaptation. The two sets of responses were produced by identical stimuli, but the adapted responses were recorded following an adaptive training protocol developed by Munoz et al. (1999) and briefly described in the Methods section. Adapted vergence responses, as shown in Figure 4B, have much larger peak velocities and greater overshoots than found in normal vergence responses, Figure 4A. It was also found that the main sequence ratio (a measure related to the first-order dynamics of a response) increased dramatically in adapted responses compared with normal responses indicating a fundamental change in the basic response dynamics. Recent studies have confirmed that disparity vergence adaptation involves complex modification of the dynamics,of the response, not just a rescaling of 354 /. L.Semmlow et al. the desired amplitude (Takagi, Oyamada, Abe, Zee, Hasebe, Miki, Usui, & Bando, 2001).

i- A. Before Adaptation - B. After Adaptation

0.5 1.0 1.5 0.5 1.0 1.5 2.0 Time (sec) Time (sec)

Figure 4. A: An ensemble ofvergence responses to a 4 deg change in vergence stimulation before adaptation (n = 19). B: An ensemble of responses from the same subject to the same stimulus after exposure to approximately 50 adapting stimuli (n = 19; see text for adaptive protocol).

Component verification

Various methods exist to estimate the number of components contained in a multiple data set. Several popular methods use the eigenvalues of the correlation matrix. The "Scree" test plots eigenvalue against component number and the cutoff point is chosen where the curve flattens (Johnson, 1998). A more informative method based on temporal Principal Component Analysis (PCA) has been developed by our laboratory for time data such as vergence eye movements. This approach compares the data ensemble standard deviation with the deviation represented by the sum of a given number of principal components.1 In

1. Since appropriately scaled principal components represent standard deviation, they add as the square root of the sum of squares. The comparison could have also been based on variance in which case they would simply add. Adaptive control in vergence eye movement 355 this application of PC A, each response is treated as a separate observation and each time sample becomes a variable. Since the variables are points along the vergence response trajectory they are highly correlated; however, PCA does not require variable independence (Reyment & Joreskog, 1996). As verified through simulations, this analysis technique shows the contribution of each of the components to the overall ensemble standard deviation. Both this analysis and the more traditional Scree plots will show that two components are sufficient to represent most of the variation contained in an ensemble of vergence movements.

METHODS Subjects, stimulus presentation, and data recording

Four subjects, three males and one female, with normal binocular vision were involved in the experiments. One subject (JS, age 56) was highly experienced and knew the goals of experiment. The other three subjects (YC, LA, and WO, ages 25, 25, and 27) were naive to the goals of the experiment. All had normal uncorrected binocular vision and could perform the experiments without difficulty. The research followed the tenants of the declaration of Helsinki: informed consent of the subjects was obtained after the nature and consequences of the experiment were explained, and the experiment was approved by the IRB's of both Robert Wood Johnson Medical School and Rutgers University. The stimulus was generated by a pair of oscilloscopes (P31 phosphor and a bandwidth of 20 MHz) arranged to present two separate vertical lines to each eye. The stimulus device was calibrated by two physical targets viewed directly by the subjects. Symmetrical inward target movements were generated by a PC-type microcomputer to elicit convergence eye movements. The stimulus patterns used are described below. The responses of each eye were detected by a Skalar infrared eye movement monitor (Model 6500), which has a bandwidth of 200 Hz, well above the Nyquist frequency for vergence movements. Based on measurement noise, the resolution of this monitor was estimated to be approximately 3 min of arc. Linearity was evaluated empirically using a repeated series of three-point calibrations. Based on the deviation of the center-point calibration from the theoretical linear value, the average experimental deviation from linearity was found to average 3 % of the 356 /. L.Semmlow et al. total movement with a maximum of 5 %. Since this linearity is acceptable, the experimental trials used only a simple two-point calibration, where the calibration points covered the full response range. The positions of each eye in response to fixed target positions were recorded immediately before and after each experimental trial for use in the 2-point calibration. The calibration data were used to recreate the two eye movements during off-line analysis. The vergence response was taken as the computed difference between the two eye movements. Eye movements and calibrations were recorded and stored in the laboratory computer using a standard 12 bit analog-to-digital converter sampling at 200 Hz well above the Nyquist frequency for vergence eye movements. All associated parameters such as stimulus and calibration information were saved with the individual eye movement records. The velocity data were calculated using a two-point central difference algorithm during off-line analysis. Data analysis and display were done using the MATLAB and Axum software packages. The Independent Component Analysis technique used here requires a number of repetitive responses for the behavior being analyzed. Sim- ulations indicated that 10-20 individual responses were sufficient to determine accurate estimates of the two components (under certain con- straints detailed below). Our analysis uses from 15-30 individual responses to vergence step stimuli. All stimuli were presented randomized in time to discourage prediction. Common artifacts that necessitated rejection of a response included large or badly timed saccades and occasional blinks. Completely saccade-free vergence responses are rare; however, if the stimulus is carefully adjusted to be symmetrical, sac-cades during the transient vergence response can be avoided. Typical 4 deg vergence step responses in both normal and adapted conditions are shown in Figure 4A and B.

Adaptive experimental protocol

The adaptive protocol used two different visual stimulus patterns: a training stimulus and a test stimulus. The training stimulus was a step-ramp signal; i.e., an initial 4-deg step followed immediately by a 16 deg/sec ramp. This special stimulus pattern was found to be very effective in generating adaptive changes in previous studies (Munoz et al., 1999) although double steps, as used for saccadic adaptation, are also Adaptive control in vergence eye movement 357 fairly effective. The test stimulus was a standard 4-deg step change in disparity vergence. An adaptive experimental run consisted of four modes: pre-adapt mode, adapt mode, sustain mode, and recovery mode. In pre-adapt mode, 20-30 test stimuli were presented to the subject and the responses were recorded to obtain baseline dynamics. In the adapt mode, about 50 training stimuli and 10 test stimuli were randomly presented to the subjects. Sustain mode was similar to the adapt mode except that the ratio between the two types of stimuli was 3 training to 1 test. The responses to the test stimuli in the latter portion of the adapt mode and in the sustain mode were used to analyze the influence of adaptation on vergence dynamics. In the recovery mode, 10 test stimuli provided responses that were used to study how an adapted system recovered to the normal state. Recovery responses were not used here.

Analysis - independent components

Independent Component Analysis (ICA) is a form of "blind source separation" that can isolate individual components from a mixture provided the components are nongaussian and sufficiently independent (Cardoso & Souloumiac, 1993; Cardoso, 1994; Comon, 1994; Hyvarinen & Oja, 1997; Hyvarinen et al., 2001). ICA has also been described as a nongaussian version of factor analysis and somewhat similar to PCA (Hyvarinen & Oja, 1997; Hyvarinen, 1999; Hyvarinen et al., 2001). In fact, an extensive study of the application of factor analysis to ensemble vergence response data was undertaken, but none of the standard approaches such as "varimax" were able to identify the underlying components in simulated data. The basic principles behind ICA are well described in number of references (Cardoso & Souloumiac, 1993; Cardoso, 1994; Comon, 1994; Hyvarinen & Oja, 1997; Hyvarinen et al., 2001) and will be only briefly mentioned here. The ICA model is a generative model: it attempts to explain how the components are mixed to generate the observed signals based on a linear mixing model (Hyvarinen et al., 2001): x = As + noise where x and s are random vectors of size m (the number of mixtures, or individual responses in our application), and n (the number of underlying sources, or components in our application). The noise vector rep- 358 /. L. Semmlow et al. resents the disturbances in the form of additive noise independent of the source vector s. The goal of 1C A is to identify the linear mixing matrix A. Inverting the mixing matrix produces an "unmixing" matrix, U = /4~1, that can be used to estimate the unobservable source vector s (s = Ux). This is accomplished by linear transformations of the data set (i.e., rotations and scalings) with the goal of optimizing some objective function related to statistical independence, such as a measure of nongaussianity. There are a quite a number of different approaches for estimating A, differing primarily in the objective function that is optimized and the optimization method (Hyvarinen et al., 2001). Since both the mixing process A and the sources s are unknown, these techniques are part of a larger family known as blind source separation (BSS) (Cardoso, 1994; Hyvarinen etal., 2001). In this application, the signals produced by the neural control components of vergence eye movements constitute the latent variables s, and the mixing matrix A accounts for their movement-to-movement variability. The critical assumptions in 1C A are that the variables are statistically independent and have nongaussian distributions. This latter is essential since it is the nongaussianity of the data set that is often optimized. While vergence responses are certainly nongaussian, the initial portions of these responses may not be completely independent due to stimulus induced synchronization of the driving neural sources. In other words, even if the underlying neural sources are independent, their activation by a common stimulus could induce a temporary correlation between their responses. As these responses continue this "stimulus effect" diminishes so that the components become independent during the latter portion of the response. To avoid this stimulus induced synchronization, the evaluation of the mixing matrix, A, was performed only on the latter portion of the responses. The time period following maximum ensemble variance (close to the time of peak velocity) was found to provide sufficient component independence to permit accurate determination of the mixing matrix. The mixing matrix, A, obtained from the truncated responses was inverted to give the unmixing matrix, U, which was then applied to the entire response (including the initial portion) to estimate the underlying motor components, s. Another important assumption in the ICA model is that the independent sources not only exist, but undergo instantaneous linear mixing to produce the sensor signal. While no biological process is likely to be truly linear, extensive eye movement data indicate that separate neural Adaptive control in vergence eye movement 359 signals, such as those from version and vergence neural centers, do combine more-or-less linearly. Moreover, most models of the oculomotor plant are linear (Robinson, 1964; Zee, Fitzgibbon, & Optican, 1992; Van Opstal, Van Gisbergen, & Eggermont, 1985). The 1C A model generally requires that the number of mixtures, or responses, be the same as the number of sources (i.e., that the mixing matrix, /A, must be square). When there are more responses than sources, as is the case here, the data are pre-processed using PC A to reduce the number of responses to equal the number of sources or independent components. Several popular ICA algorithms can be downloaded from the Web as MATLAB script files. In this study, we investigated two such algorithms: the "FastICA" algorithm developed by the ICA Group at the Helsinki University (available at: http://www.cis.hut.fi/projects/ica/ fastica/fp.html) and the "Jade" algorithm for real-valued signals developed by J.-F. Cardoso (available at: http://sig.enst.fr/~cardoso/stuff. html). Both these algorithms can, if requested, provide preprocessing that uses PCA to reduce the dimensionality of the data set. In our analysis, data dimensionality was based on results from the number-of-components analysis described below. Although the two algorithms performed identically on real data, we selected the Jade algorithm as it gave slightly more accurate decompositions on simulated data. To apply ICA to ensemble vergence response data, each 2 sec response is treated as an observed signal as shown in Figure 4 for both normal and adapted conditions. Simulations showed that the algorithms produced more accurate results if the data sets were symmetrical, so each response was modified by adding the inverted response to the end of the actual response to make the ensemble data symmetric along the time axis (that is, the responses have the format of periodic functions). While this operation does not add any new information to the data set, it does change its statistical properties. Specifically, a modified, symmetrical data set showed more than one order of magnitude greater difference in the ratios between the first three eigenvalues as compared to the original data set. This improvement influenced the selection of principal components during the pre-processing data reduction operation and gave much more accurate component decompositions on simulated data. After analysis, the inverted responses were discarded. 360 /. L. Semmlow et al.

Due to inherent ambiguities, 1C A cannot determine the scale of the components. The initial pre-response period was used to establish a zero reference for the components. To determine the amplitude, we note that the sum of the two components should equal the average response. Hence, the amplitude of the individual components was adjusted until their sum equaled the average response. Since there were only two components with quite different time characteristics, the amplitude scaling was uniquely determined by matching the average response. Amplitude scaling was implemented using the MATLAB basic optimization routine "fminsearch" which uses the Nelder-Mead simplex, or direct search, method. After scaling, the sum of the two components matched the ensemble average response almost exactly.

ICA evaluations and model simulations

Modeling and simulation was used to confirm that this application of ICA was able to identify the underlying control components and that ensemble PC A can be used to establish the number of components. The

Pulse Signal

Step Signal

Figure 5. Block diagram of the pulse-step model. The model represents only the motor components, and feedback pathways are not explicitly shown. The seven model parameters that were adjusted to fit experimental data, or to produce response variability, are shown in bold italics within their associated element. Adaptive control in vergence eye movement 361 model used to represent the vergence system was the well established component model developed in our laboratory and shown schematically in Figure 5. The model was simulated using the MATLAB Simulink software package. With appropriate adjustment of model parameters, the model is capable of accurately simulating the individual vergence response of all subjects studied thus far (Yuan, Semmlow, Alvarez, & Munoz, 1999). Details of the model's behavior are well described elsewhere (Yuan & Semmlow, 2000, 2001; Yuan et al., 1999). The advantage of this model-based evaluation is that the underlying control components are directly available as model outputs. Ensemble averages were determined for each component after the component was filtered by the oculomotor plant. Usually 40 model, responses were simulated, although this number was varied to evaluate the minimum number of responses required to give an accurate simulation. Component variability was produced by randomly varying seven model parameters associated with the model processes shown in Figure 5 and these parameters are also identified in Figure 5 (bold italics). Specifically, variability in the transient component was simulated by randomly varying pulse width, pulse slope, and gain. Variation in the sustained component was produced by randomly varying the step slope and the gain. The onset time of both components and the time constant of the plant was also randomly varied. The extent of this random variation was empirically determined to provide an approximate match to the variability seen in the experimental data. Figure 6A shows a typical ensemble of simulated disparity vergence responses. Figure 6B (dashed line) shows the ensemble standard deviation computed from the simulated responses along with the ensemble standard deviation computed from experimental response ensembles of two subjects (solid lines). The ensemble standard deviation of the simulated data falls between the standard deviations from the two subjects indicating that the variability in model responses is in the same range as the experimental data. Figure 7 (solid lines) shows the Independent Components evaluated from an ensemble of model responses such as seen in Figure 6A. Unlike experimental data it is possible to obtain the normally hidden components from the model. These are shown as averages Figure 7, dashed lines. A very close match is observed between the actual component and those found by 1C A. 362 /. L. Semmlow et al.

Adaptive control in Mergence eye movement 363

Principal component evaluation of the number of independent components

As noted above, a common approach for selecting the number of significant components in combined data uses the data set eigenvalues, usually searching for a knee or breakpoint in the Scree plot, a plot of eigenvalue against number of components (Kline, 1994). Scree plots for simulated and real data are presented in the Results section. A specially developed PCA-based method provides a more informative method of assessing the number of components present. If appropriately scaled, the variance of all the principal components must equal the total data variance. Moreover, the first principal component represents the maximum variance possible, the second component, the maximum amount of the remaining variance, and so on. To evaluate the number of components present in the data set, our graphically-based method plots the total ensemble variance (or standard deviation, as used here) represented by a given number of components against the total variance in the data set. Essentially, we assume a given number of components and examine how much of the total variance is accounted by these components over the duration of the response. This approach not only shows how well the selected number of components accounts for the experimental variability, but also how this accountability varies over the time response. Principal components were obtained using the MATLAB "princomp" routine which is based on singular value decomposition. After evaluation, the principal components are scaled by multiplication by the square root of their respective eigenvalues (Reyment & Joreskog, 1996). This scales the corresponding components to their roots (Johnson, 1998) and makes the variance of the principal components equal to the data variance. The variance accounted for by the selected number of components is then found by summing over the number of components at each point in time. Here the square roots of both PC A and data variance were taken, so the comparison is of standard deviations. As mentioned above, the advantage of the approach is that it shows the standard deviation match over the entire response allowing a more informed assessment. For example, if the match is poor over a specific time period, this indicates that the selected number of components are not sufficient to explain the data variability in that region; however, if the poor match occurs during a time period when noise or artifact might be expected 364 J. L.Semmlow et al.

(for example, before the onset of the response during the latent period), then the selected number of components may still be deemed adequate. This approach was confirmed through model simulations. Plots of simulated standard deviation and the sum of the first two principal components overlaid perfectly as expected since the simulated data contains only two components by design.

RESULTS

Number of components

Figure 8 (upper left) shows the Scree plot obtained from simulated data. The curve descends steeply then flattens for eigenvalue numbers greater than three. This would indicate that the data set contains only two uncorrelated sources, as is known to be the case with this simulated data. The remaining plots in Figure 8 show the Scree plots for the four subjects for both baseline step responses (closed circles) and adapted responses (open triangles). Since the absolute values of the eigenvalues are not of interest, the eigenvalues have been normalized so that they encompass more-or-less the same range. Note that all subjects show curves similar to the one seen in the simulated data for both adapted and baseline data: the curves tend to flatten above the second eigenvalue. This provides traditional support indicating that both baseline and adapted responses consist primarily of two components. This also indicates that the adaptation process does not evoke additional components, but modifies the existing components. Results of the PCA-based method for evaluating the number of components are shown in Figure 9. These figures plot the ensemble standard deviations (solid line) obtained from the a set of vergence responses, along with the standard deviation accounted for by the first two principal components (dotted line). For all subjects, two components account for most of the ensemble standard deviation with small differences occurring just before the onset of movement and at the end of the response. Since the ensemble standard deviation should be very small both before the movement begins and when the movement ends, we attribute the deviation during this portion of the response to measurement artifacts or other noise. It is possible that a small, third component exists in the late portion of disparity vergence; however, for this study, we assume that

368 J. L.Semmlow et al. rapid rise in the first 600 to 800 milliseconds followed by a gradual rise to the final value during the subsequent 1.5 seconds. In three of the four subjects, the sustained component shows an additional onset delay approximately 50 msec, longer than the transient onset delay. The adapted responses show larger transient component amplitudes for all subjects. Adapted transient component amplitudes very between 1.2 to 2.7 times the unadapted transient component amplitude (Table 1), although a paired T-test showed significance only at p < 0.055 due to the small number of subjects. Two subjects, JS and YC, showed a strong double-step response in the sustained component. Note that the multi-step behavior is latent and not observable in the combined responses (dashed lines). One subject, LA, shows only a slight increase in the dynamics of the transient component; however, this subject's responses are only slightly modified by the adaptation process.

Table 1 Transient component peak amplitudes in unadapted and adapted responses

Subject Unadapted (deg) Adapted Ratio* Unadapted time (deg) const, (sec)

JS 1.67 4.46 2.7 0.13 WO 1.52 3.34 2.2 0.16 YC 1.36 3.00 2.1 0.17 LA 1.29 1.52 1.2 0.19 Average SD 1.46 0.15 3.08 1.05 2.1 0.54 0.16 0.02

* Ratio of adapted to unadapted transient component amplitudes.

DISCUSSION AND CONCLUSION

While ICA provides the isolation of individual components from a combined response, it can only be applied to an ensemble of responses, and it provides estimates only of the components averaged over a number of responses. This precludes its application to those problems where

Adaptive control in vergence eye movement 369 only a single, or a small number of responses can be obtained. Another problem is due to the averaging inherent in the ICA approach. If the variability in onset (i.e., the response latency) is large in comparison to the response dynamics, then the averages will not reflect the true dynamics of the component since these averages will be constructed from responses displaced in time. In the last stage of the analysis, the amplitudes of the two components are adjusted so their sum fits the ensemble average. A wide variation in onset time will tend to reduce the dynamics of the ensemble average, resulting in an inaccurate estimate, particularly of the fastest component, in our case, the transient component. Simulations in which the variability in the latency was large in comparison to the response dynamics (25 % of the approximate time constant) showed that the transient component was underestimated by 20% with a corresponding overestimation of the sustained component. In vergence responses, the variability in latency is small compared to the dynamics of the response and alignment was not required. However, for responses that have large latency variations compared to their dynamics, such as saccadic eye movements, the variability will have to be reduced either by shifting the response into better alignment or carefully editing the data to eliminate responses with widely varying onset latencies. ICA-based component separation of the unadapted (i.e., baseline) vergence step responses show that the transient component peaks between 300 and 500 msec following the stimulus, then decays to zero after approximately 800 milliseconds. The sustained component begins at approximately the same time as the transient component, indicating that it undergoes similar processing delays. The sustained component is relatively smooth and approaches the final value slowly, reaching the sustained level (4 deg) usually near the end of the 2 sec record. Note that these traces represent the motor components, not the underlying neural signals; that is, they represent the neural signals after modification by the vergence plant. Since the vergence plant has been shown to be approximately first-order with a time constant of around 220 msec, the components shown are essentially low-pass filtered versions of the neural signals. Table 1 shows that the amplitude of the transient component varies somewhat across subjects as does the modification in amplitude brought about by the adaptive stimulus. A simple measure of the dynamics of the average baseline response is given in Table 1 (right-most column) as the time constant, the time to reach 63 % of the final value not including the 370 J.L.Semmlow et al. latent period. This dynamic measure shows an inverse relationship between speed of the response and the peak amplitude of the transient component. This is expected since the speed of the response is primarily determined by the fast transient component (in conjunction with the dynamics of the oculomotor plant). The ability of the adaptive stimulus to modify the transient component amplitude is also seen in Table 1 to be related to the value of the unadapted transient component amplitude. Thus, the subject with the largest unadapted transient component amplitude (Sub. JS) also showed the greatest change in amplitude with adaptation and vice versa. This suggests that the larger the neural signal that produces the transient component, the more strongly it can be modified. In two subjects (WO and LA), the adapted and unadapted sustained components were similar; however, in the two other subjects the sustained component showed distinctive double-step behavior. A previous model-based analysis of adapted responses predicted an increase in transient component amplitudes (Yuan & Semmlow, 2000), but also predicted a delay between the transient and sustained components not found in the adapted subjects, except for the short delay seen in subject JS. The model-based analysis did not uncover the double step behavior of the sustained component as the model was not designed to generate multiple sustained component steps. From the perspective of the new ICA-based analysis, it now appears that the sustained component delays predicted by the model were an artifact produced by a constrained model attempting to represent the double step behavior. Double step behavior can also be occasionally found in normal ver-gence responses. In an analysis of naturally occurring double responses, Alvarez et al. (1998) found response doubles occurred whenever the initial velocity response was less than 80 percent of a normal (i.e., single step) response. Analogous behavior has also been found in the saccadic system in the form of overlapping or double saccades (Bahill & Stark, 1975). In a subsequent study, Alvarez et al. (1999) compared the timing of naturally occurring double responses to those produced by forced errors induced by external feedback. They found that naturally occurring doubles were significantly faster and concluded that they must be generated by an internal feedback mechanism. We speculate that a similar process is occurring here for the adapted responses of JS and YC. Specifically, the enhancement of the transient component leads, in some subjects, to a reduction in the accompanying sustained component. This could occur if transient component enhancement is achieved by the Adaptive control in vergence eye movement 371 recruitment of neurons from a pool usually used by the sustained component. The resulting reduction in sustained component amplitude would then lead to the generation of a compensatory secondary sustained component, probably through the action of an internal feedback monitoring process.

RESUME

La theorie du double mode pour le controle des mouvements oculai-res de vergence de disparite etablit que deux composantes de controle, une composante transitoire pre-programmee et une composante soutenue en retroaction, conditionnent la reponse motrice. Bien que des resultats experimentaux anterieurs aient isole et etudie la composante transitoire, on sail peu de chose quant a la contribution de la composante soutenue a la reponse de vergence. Le timing entre les deux composantes et leurs amplitudes relatives sont interessants car ils ont des implications sur les strategies de controle utilisees afin de coordonner les deux elements. Les etudes de modelisation peuvent permettre d'estimer 1'amplitude des composantes, mais ne permettent pas d'identifier le timing de facon unique. Le travail presente ici applique 1'analyse en composante principale (ACP) afin a la fois de confirmer la presence des deux composantes majeures et de compresser les donnees. Une nouvelle version de 1'analyse en composante independante (ACT) est alors utilise afin d'estimer la contribution des deux composantes de controle sur le mouvement ocu-laire. Des experiences plus recentes ont montre que le systeme de vergence peut changer rapidement de caracteristiques dynamiques (adaptation a cours terme) s'il est expose a un stimulus adaptatif specialement concu. Les reponses adaptees etaient caracterisee par des dynamiques plus rapides, presentant souvent de grands depassements. L'ACI des reponses normales et adaptees montre que les dynamiques etendues des reponses adaptees sont dues a une augmentation de 1'amplitude de la composante transitoire. De plus, la composante soutenue des reponses adaptees montre souvent un comportement en double-saut, dans la der-niere partie de la reponse. Enfm, 1'amplitude de 1'adaptation semble etre liee a 1'amplitude de la composante transitoire non-adaptee. 372 J. L.Semmlow et al.

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Received 8 November, 2001 Accepted 18 July, 2002