Dynamics Of The Disparity Vergence Step Response - Information ...

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 46, NO. 10, OCTOBER 1999

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Dynamics of the Disparity Vergence Step Response: A Model-Based Analysis Weihong Yuan, Student Member, IEEE, John L. Semmlow,* Fellow, IEEE, Tara L. Alvarez, and Paula Munoz

Abstract—A new method to analyze the dynamics of vergence eye movements was developed based on a reconstruction of the presumed motor command signal. A model was used to construct equivalent motor command signals and transform an associated vergence transient response into an equivalent set of motor commands. This model represented only the motor components of the vergence system and consisted of signal generators representing the neural burst and tonic cells and a plant representing the ocular musculature and dynamics of the orbit. Through highly accurate simulations, dynamic vergence responses could be reduced to a set of five model parameters, each relating to a specific feature of the internal motor command. This dynamic analysis tool was applied to the analysis of inter-movement variability in vergence step responses. Model parameters obtained from a large number of response simulations showed that the width of the command pulse was tightly controlled while its amplitude, rising slope, and falling slope were less tightly regulated. Variation in the latter three parameters accounted for most of the movement-to-movement variability seen in vergence step responses. Unlike version movements, pulse width did not increase with increased stimulus amplitude, although the other command signal parameters were substantially influenced by stimulus amplitude. Index Terms— Dynamic analysis, eye movements, eye movement models, oculomotor vergence, vergence motor control.

I. INTRODUCTION

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N disjunctive, or vergence eye movements, the two eyes rotate in opposite directions, either inward or outward in response to a target moving toward or away from the subject. The vergence response can be triggered by several types of stimuli; however, disparity between retinal images is acknowledged as the primary stimulus [1], [2]. Vergence is only one type of eye movement, but when stimulated in isolation, it is believed to be controlled primarily by processes unique to that type of eye movement. The dynamic behavior of vergence and other eye movements can be used to gain insight into the neural control processes that generate those movements. In the past, the quantification of response dynamics has relied on relatively simple firstorder measurements such as the equivalent time constant of the Manuscript received August 14, 1998; revised April 8, 1999. This work was supported in part by the National Science Foundation (NSF) under Grant IBN-9511655. Asterisk indicates corresponding author. W. Yuan, T. L. Alvarez, and P. Munoz are with the Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08855-0909 USA. *J. L. Semmlow is with the Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08855-0909 USA. He is also with the Department of Surgery (Bioengineering), Robert Wood Johnson Medical School-UMDNJ, New Brunswick, NJ 08901 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9294(99)07410-8.

response [3], [4] or the ratio of peak velocity to amplitude, the so-called main sequence [5]–[9]. While enjoying widespread use, the main sequence, as a two-parameter measurement, does not provide a definitive representation of the complex dynamics formed in vergence responses. As we show here, it is easy to find two quite different responses that have the same main sequence ratio [9]. The need for a more comprehensive measure of response dynamics motivated the development of the model-based tool described here. The purpose of any dynamic analysis tool is to accurately represent the transient behavior by a limited, preferably minimal, set of parameters. Examples based on linear system analysis are common in engineering practice, but the inherent nonlinearity of oculomotor behaviors make these approaches less appealing for eye movement responses. Here we develop a tool that accurately encodes the vergence step response into a few physiological parameters, parameters that also relate, at least approximately, to the underlying neural control signals. A simple model that mimics the motor command signals will be used to convert from response space to command-signal space; that is, convert the dynamic time response to an equivalent motor command signal. Iterative adjustments of the model parameters will be made until the simulated responses of the model match the experimental response of interest. When this occurs, the model parameters provide a concise quantitative representation of the dynamics and they have a relationship with the underlying motor command signal. Model-based representations of neural control structure have a long and rich history dating back to a negative feedback system proposed by Rashbass and Westheimer [9] in 1961. Since then, many researchers have tried to improve upon the representation of the vergence system using feedback models [11]–[15]. Krishnan et al. [11] added a velocity element in parallel with the integrator to improve simulation performance. Schor and colleagues [12]–[14] developed a feedback control model and have applied this model primarily to studies of the interaction between accommodation and vergence. Pobuda et al. [15] proposed a model with multiple channels, each of which corresponded to a certain disparity range. A model based on interaction between version and vergence incorporating weighted sums of individual neuronal signals was developed by McChandless et al. [16] to explain adaptive processes in vertical vergence. Patel et al. [17] simulated the vergence control system using a neural network model instead of the servo-analytic approach. Finally, a dualcomponent model involving a fast open-loop and a slow

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feedback control component was developed by Hung et al. [18] and recently extended by Horng et al. [19]. For the open-loop control component (open-loop in the sense that visual feedback was not involved), Horng et al. [17] adopted a slightly modified version of Zee et al.’s model [20], and for the closed-loop component he used the model developed by Krishnan et al. [11]. The purpose of most previous models was to represent the overall structure of the vergence control system in order to study basic control issues such as dynamics and stability [4], [11], [16], [19], [20], or to evaluate neural control theories through simulation [12]–[16], [18], [20]. The model developed here has a different and more limited purpose: to transform dynamic vergence behavior into equivalent motor command signals, and to represent those signals quantitatively by a small number of parameters. To achieve the former, the model’s controller will be constructed based on known features of the vergence premotor and motoneurons. The later dictates that the model contains a small number of adjustable parameters to provide concise representation of the motor command signal. For example, to model the control process of the fast component, Horng used a local feedback mechanism [21] to generate a pulse signal, a feature borrowed directly from the model by Zee et al. [20]. This representation resulted in a model with a large number of adjustable parameters (eight). While it produced highly accurate simulations of experimental responses, its complexity made it difficult to relate the model’s parameters to specific features of the motor command signal. Once developed, the model-based dynamic analysis tool will be used to study the influence of stimulus features on the motor command signal and to analyze movementto-movement variability. Response variability is an integral feature of all motor behaviors and the new dynamic analysis tool will also be used to identify aspects of the command signal most responsible for this variability.

standard step stimuli or disappearing step stimuli. In the latter, the stimulus was presented for a brief period (50–300 ms) following the initial step, then the target disappeared and no stimulus was visible for the remainder of the 3-s data acquisition period. The eye movements were recorded using a Skalar infrared eye movement monitor (Model 6500). The monitor had a published resolution of 1.5 min of arc and a bandwidth of 200 Hz, which was well beyond the requirement for recording vergence dynamics. The linear range ( 3%) was 25 . Due to the limitations of the stimulus device, the effective target range was only 8 for each eye for a maximum change in vergence stimulation of 16 . To minimize motion artifact and avoid influence from the vestibular system, a dental impression bite bar was used to stabilize the head and minimize the head motion relative to the target. The subject was requested to fuse the two lines during each 3-s trial. Subjects would push a button when they felt ready for each trial. In order to avoid prediction, both the duration between the button press and the stimulus onset and the stimulus pattern itself were randomized. Calibration was carried out by recording the eye movement monitor output at two known eye positions, one immediately before, and the other at the end of each trial. During the off-line analysis, the vergence response was constructed by the computer as the difference between left and right eye movements after calibration of each eye based on these two points. Eye positions were sampled for each eye at 5-ms intervals 200 Hz) and stored in the computer for later ( analysis. During the experiment, the individual responses of the two eyes were displayed on the computer screen to insure accuracy of the data acquisition process. Responses with blinks were discarded. Since we were inspecting binocular behavior, responses with small saccades were not necessarily discarded, as long as the saccades in two eyes occurred after the initial transient and were equal in amplitude.

II. METHODS III. MODEL DEVELOPMENT AND SIMULATION A. Data Acquisition Experiments were conducted on three human subjects: one male and two females. All of them had normal binocular vision. Subject JS has been participating in oculomotor control experiments for many years and has substantial knowledge of oculomotor behavior and the goal of the research. The other subjects, BS and CC had periods of training and were aware of the general goals of the experiments, but had no knowledge of the specific objective of any given experiment. Two oscilloscopes (P31 phosphor and a bandwidth of 20 MHz) were arranged to produce a stereo pair of vertical lines (0.15 in width and 5 in height) against an otherwise dark background. The subject and stimulus apparatus were in a separate darkened enclosure although absolute darkness was not essential provided there are no distracting visible images. A PC-486 computer was used to generate different patterns of stimuli and the stimulus signal was sent to the two oscilloscopes to control the position of the vertical targets. All of the responses analyzed in this study were produced by either

As mentioned previously, the system components represented by the model include only the vergence plant and the neural signal generator that produces the motor command signal. Moreover, the pulse-like transient signal and the step signal are generated by elements selected for their simplicity and do not represent the way in which neuronal processes create the signal. The model attempts to represent the neural command signals, but not the structures that produce those signals. (For a model that more closely represents the neural processes involved in the generation of pulse and step signals see Zee et al. [20] or Horng et al. [19]). These constraints greatly simplify the model and direct our focus to the relationship between the dynamics and the neural processes that generate motor command signals. Previous neurophysiological research, particularly that from Mays’ laboratory, [22]–[25], led to a model of the disparity vergence system based on the role of convergence tonic and burst cells (CBC), Fig. 1(a). Mays and his colleagues found that the neural activity of convergence tonic cells was highly

YUAN et al.: DYNAMICS OF THE DISPARITY VERGENCE STEP RESPONSE

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(c) Fig. 1. (a) A model for vergence eye movement control proposed by Mays et al. [22] based on neurophysiological evidence. CBS represents convergent burst cells. TC represents tonic cells. (b) Schematic diagram of the new model. It has only five adjustable parameters. The upper pathway represents the burst cells and produces a shaped pulse signal via the pulse generator, rate limiter and gain terms. The lower pathway represents the tonic cells and produces a step signal responsible for maintaining steady state position. The nominal signal amplitude of both the pulse and step generators is equal to the stimulus amplitude. The plant is a second-order element with unity gain. The delay element is not shown in this model since it has no impact on the dynamics of the system. The pulse and step gains include all the gain elements in these pathways. (c) Schematic representation of an approximate time course of the activity of a burst cell and tonic cell during convergence. (Approximation taken from data [22].)

correlated with the coding of the position of convergence eye movements. They also found that the burst cells fired before and during a convergence movement. The output of these burst cells, the burst firing rate, was closely correlated with the profiles of vergence velocity indicating that burst cells were responsible for coding velocity signals. Some of these cells also had a tonic firing rate that was positively related with vergence angle [24]. Mays et al. [24] proposed that the burst cells provided an input to a vergence integrator whose output established a new steady-state position. In addition, Mays et al. [24], [25] suggested that these velocity signals went directly to the plant to improve frequency characteristics during the initial portion of response. The velocity signal itself would have little steady-state influence according to Mays’ model. This is also compatible with an earlier model by Krishnan et al. [11] that included a separate velocity-dependent pathway (i.e., derivative element). The model signal generator represents the output of burst neurons (CBC) as described by Mays et al. [22]. A schematic representation of a typical burst cell signal is shown in Fig. 1(c). In the model, this signal is fed directly to the plant without passing through an integrator, Fig. 1(b). The tonic signal is likely produced by integration of the pulse signal, but for simplicity and flexibility is represented here by a separate step generator in the model, Fig. 1(b). In addition to the pulse and step generators, the controller contains pulse and rate limiters, step and step gain elements,

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and a second-order plant, Fig. 1(b). Since the model does not include representation of the sensory processes (nor the external feedback pathway), the input to the step and pulse generators is assumed to be an output signal from sensory processes that activates the two generators. This signal triggers the two signal generator elements, which then produce an output that is equal in amplitude to their respective gain terms. While it is conceivable that there is a delay between the activation of the pulse and step generators, model simulations showed that such a delay could not be uniquely identified from behavioral data. Moreover, if the step is generated by integration of the pulse signal, as is the case in the saccadic system, then the step and pulse signals will activate at the same time. Hence, in our model the two generators were assumed to activate simultaneously. The rate limiters are the only nonlinear elements in the model and shape the rectangular pulse and step signals by restricting their rate of change (i.e., maximum velocity) in rising and falling (for the pulse generator) directions. Any signal with a slope beyond the limits set by the parameters will be constrained to those limits. While the model shows separate pulse and step rate limiters, in fact the rising limit value of each was set to be the same. Hence, the controller contained five parameters that quantify its behavior: pulse width, pulse gain, step gain, and rising and falling slew rate. The pulse gain element was placed after the nonlinear rate limiter to preserve the shape of the pulse (or step); hence, the gain provides a simple scaling of the rate limited pulse (or step) signal. The controller parameters have a direct relationship to the dynamics of the response. Pulse gain strongly influences the value of peak velocity while pulse width determines the time when peak velocity is attained. The rising slew rate influences the up-slope of the velocity curve (i.e., acceleration) and has a secondary influence on peak velocities, while falling slew rate shapes the falling portion of the velocity curve (i.e., deceleration). Since the plant has a gain of one (all feed forward gains were lumped with the generator gains), the step gain determines the final steady-state position and, hence, is the same as the stimulus amplitude. To represent the plant, a second-order element was adopted from the model developed by Robinson et al. [19]. The minor time constant of the plant was decided empirically during preliminary simulations and was set to 0.03 s for all three subjects. The major time constant for the plant was also empirically selected; however, while this constant was fixed for each individual subject, it varied among the three subjects. Specifically, the major time constant was 0.2 s for JS and 0.4 s. for BS and CC. In Robinson et al.’s [19] model, the average of the major time constant was 285 ms. However, the individual recordings showed that this constant could range from 200–400 ms. Hence, the two constants chosen here are reasonable. Plant time constants will also influence dynamic features such as peak velocity and velocity profiles. By holding plant time constants fixed for each subject, they exerted a consistent influence on dynamics. Simulations were carried out using Matlab and plotted using Axum. Each simulation was done by manually adjusting the parameters of the model until the root mean squared

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(c) Fig. 2. Representative individual vergence responses to 4 step stimuli (solid line) along with model simulations of these movements (dashed line). Both position (left side) and velocity (right side) time plots are presented. Note that the dynamics vary between subjects, but are still accurately represented by model simulations. (a) Subject JS, (b) subject BS, and (c) subject CC.

(RMS) value of the difference between the simulation and response was less than 3% of the steady-state amplitude. This error was approximately the same as the measurement error. For a 4 step response, this corresponded to a RMS error of 0.12 averaged over the period of the simulation. To avoid subjectivity in the parameter adjustment process, an optimization procedure was applied after each manual adjustment, with the manual simulation result being used as initial values for optimization. The optimization routine, a local minimizer, was taken from the Matlab software package (FMINS) and was based on the Simplex search method. Since there is a response delay or latency in the response data, but no latency in the model, the simulated data was shifted to best coincide with the actual response. The simulation was optimized to provide the best fit to the eye movement data by adjusting the simulated data shift and the five adjustable parameters mentioned above. IV. RESULTS Fig. 2 shows typical 4 individual vergence step responses from three subjects along with model simulations. The solid line represents the disparity response and the dashed line represents the simulation result. The left and right subplots correspond to position and velocity traces, respectively. Note that after parameter adjustment, the model simulation results

fit the experimental responses very well for both position and velocity traces. Hence, the model parameters constitute an accurate, quantitative representation of the transient dynamics. To study dynamic response variability, 15–30 responses to the same stimuli were fitted by model simulations for each subject. The average value and standard deviation of the five model parameters obtained from these simulations are shown in Fig. 3. Values for four stimulus amplitudes, 2 , 4 , 8 , and 12 , were obtained for subject JS. The other two subjects were unable to respond to as wide a range of stimulus amplitudes, so only two stimulus amplitudes (4 and 8 ) were used. From Fig. 3, an interesting trend can be observed in the parameters. When the stimulus amplitude increases from 2 to 12 , subject JS’s pulse width parameter value shows only a slight increase while both the pulse and step gain values increase in proportion to stimulus amplitude. Both RSR and FSR decreased with increased stimulus amplitude. A decrease in the slew rate parameters indicates slower rising and falling slopes of the pulse signal that, in turn, imply slower acceleration and deceleration response dynamics. The simulation results of subjects CC and BS showed similar trends. In all three subjects, the standard deviation of PW for each stimulus amplitude was relatively small, ranging from 5% to 15% of the average value. The other three parameters, especially FSR, showed large standard deviations. The standard deviation for


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(c) Fig. 3. Bar chart with error bars showing the average and standard deviation of model parameters and their variation with stimulus amplitude. pulse width (PW) (s/100), pulse gain (PG), step gain (SG), rising slew rate (RSR) ( /s), and falling slew rate (FSR) ( /s). (a) Subject JS, (b) subject BS, and (c) subject CC.

FSR could be as high as 45% of the average value, showing large movement-to-movement variation in this parameter. V. DISCUSSION The model described here was developed as a research tool for studying vergence dynamics. Accordingly, the model had very specific and limited functions: to map the dynamic response into an equivalent motor command signal and to represent that signal quantitatively using a small number of parameters. The first goal could have been achieved using an inverse model of the plant, but the second would still have required a parameter estimation technique such as the one used here. The approach used here was selected since it is more direct and also provides some insight into the motor command signal. When the model’s simulated output matches a given experimental response, the model parameters provide a concise representation of the dynamics of a vergence step response. An analysis that is based on transforming a dynamic response into an equivalent command signal assumes that plant dynamics are known and relatively stable. The relationship between individual premotoneuron activity and response dynamics has been well studied by Mays and colleagues [22]–[25]. They found that while the functional relationship between premotoneuron activity and response dynamics varied from neuron-to-neuron, the relationship was relatively constant for any given premotoneuron. Specifically, a regression analysis of the relationship between vergence burst cell activity (i.e.,

Fig. 4. Scatter plots showing the relationship between FSR and PG for different values of stimulus amplitude (2 , 4 , 8 , and 12 ). The impact of stimulus amplitude on the slope of the best-fit straight line of the scatter plot is substantial. Subject: JS.

firing rate) and vergence response velocity identified a number of neurons with correlation coefficients around 0.90 or above [22]. Such high correlation rates imply not only that the neuron’s activity is tightly coupled with the response, but that variability produced by the vergence oculomotor plant was minimal. An important criterion of the model was that it contain a small number of adjustable parameters, yet still meet the objective of accurate simulation. Simulation results showed that all stimulus dependency as well as movement-to-movement variability could be well represented by variations in only five parameters. Most of the parameters showed no significant correlation for any given stimulus; however, two of the parameters pulse gain and FSR, had a clear relationship. As shown in Fig. 4, the relationship between pulse gain and FSR varied for different stimulus amplitudes. For small responses (2 stimulus amplitude), there is a substantial change in FSR as a function of pulse gain. However, for large responses (8 and 12 stimulus amplitudes) the value of FSR is relatively constant irrespective of the pulse gain component of the movement. A highly speculative, but intriguing explanation for this is that the increases in pulse gain produced by larger stimuli depend to a greater extent on neural recruitment. Larger neuronal pools might possibly produce pulse signals that decay more slowly due to de-synchronization of individual neural decay rates. Slower decay of the pulse signal would be reflected as a reduction in the FSR parameter. Assuming that the vergence motor command signal consists primarily of step (tonic) and pulse (phasic) components (substantial evidence to support this assumption has been presented earlier), then the model signal generator outputs should reflect, at least approximately, the analogous motor command signals. Under this assumption, the behavior of the model parameters provides some information on the behavior of the associated neural processes. For example, the variation of the parameters with stimulus amplitude showed only a small change in the pulse width parameter, but a large change in the gain and slew rate parameters. This indicates that

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neural processes for generating the pulse timing are tightly controlled, possibly by a local feedback network [21] while the other signal generator characteristics, such as maximum pulse firing rate or number of burst cells recruited, are less tightly regulated. In saccadic eye movement, both pulse amplitude and pulse width can be modified in order to generate motor signals that produce movements of different amplitude [27]. Abel et al. [28] showed that the signal was modified primarily through the change of pulse width rather than the pulse height. However, as shown in Fig. 3, not only was pulse width little affected by stimulus amplitude, its standard deviation was low at any given amplitude. This tighter control of the pulse timing suggests that the underlying mechanism for vergence and saccade eye movements are substantially different from one another. Evidence relating to these results can be found in neurophysiological studies of Mays et al. [24]. Mays and his colleagues recorded, in monkey, the activity of a single convergence burst cell in response to step stimuli of six amplitudes [24, Fig. 3]. In this particular cell, the increase in stimulus amplitude was accompanied by a trend toward increased pulse amplitude just as we found. This cell also showed a trend toward increased pulse duration with larger stimulus amplitudes, in conflict with our results. However, considerable variation from this trend was found in both pulse amplitude and duration. Pulse amplitude actually decreased in two instances, and overall pulse duration was more-or-less constant over three different stimulus levels. Most striking, however, was the change in pulse shape. Of the six recordings, four different pulse shapes could be observed. In summary, the single cell recordings of Mays et al. [24] showed a great deal of variability in amplitude, duration and, particularly, pulse shape. We conjecture that the variability seen in RSR, FSR, and gain, which represent the aggregate behavior of these neuron cells, is a reflection of the variability in shape found for individual cells. Conversely, the tightly controlled aggregate pulse width may be the result of a neuronal structure such as local feedback mechanism [19] that imposes greater order on the output of the overall population. Note that the action of such a process would not be observable in the behavior of any given single cell. The use of identical rising slew rates for both burst and tonic pathways was made primarily to minimize the number of parameters. Nonetheless, the values obtained are in agreement with neurophysiological evidence. Mays et al. [23] showed the activity of a typical tonic cell during a 4 vergence movement [their Fig. 7(b)]. This cell increased its firing rate to the final value over a period of about 300 ms, giving a rising slew rate of approximately 13 –14 /s. Our analysis, as shown in Fig. 3, 1.6 , produced rising slew rates for 4 responses of 12.6 4.3 , and 16.3 4.7 /s for subject JS, BS, and CC, 16.7 respectively, in close agreement with the tonic cell from Mays and colleagues’ work [23]. The model of Fig. 2 contains two major signal components, the step and pulse signals, and the relative contribution of these two signal components to any given simulation can be determined from the ratio of PG to SG in the results. Again, assuming the model signals reflect the analogous neural signals, then the PG/PS ratio reflects the relative contribution of phasic and tonic components in the actual response.

TABLE I PULSE/STEP RATIO VERSUS PLANT TIME CONSTANT

JS BS CC

(a)

200 ms

300 ms

400 ms

2.00 1.96 1.18

2.73 3.00 1.75

3.80 4.24 2.70

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Fig. 5. A typical response (solid line) to 4 disappearing step stimulus with a 50-ms target presentation time along with associated simulation curves (dashed line). The position curve and its simulation are in the left-hand plot. Velocity curve and its simulation are in the right-hand plot. Notice the close fit between the response curve and simulation curve throughout the response.

As shown in Fig. 3, for all subjects, the pulse gain was always larger than the step gain at every stimulus amplitude level indicating a strong contribution from the burst cells. However, while the step gain is uniquely determined by the steady-state position, the pulse gain and, hence, the pulse/step ratio, will depend somewhat on the value chosen for the plant time constants. The influence of plant constants on pulse-step ratio is presented in Table I. Note that as the plant’s major time constant decreases, the pulse/step ratio needed to achieve accurate simulations also decreases. Nonetheless, even at the lowest physiologically plausible major plant time constant, the pulse/step ratio is greater than one in all subjects, demonstrating a significant contribution from the burst cells. The high degree of variability seen in the initial transient response and the related pulse generator parameters indicates that the very accurate final position attained by vergence response must be due to another component probably acting later in the movement. The “slow component” hypothesized by the Dual-Mode theory [18] and represented by the tonic and step-like signal would be a likely candidate for this additional component. While it is difficult to isolate the two components of the Dual-Mode theory, it is possible to emphasize the role of the “initial” component by using step stimuli that are presented for only brief periods. Fig. 5 shows a typical response to a step stimulus that disappeared after only 50 ms. The initial portion of this response is similar to the initial portion of a step response, except that the response is not sustained and the eyes return to approximately their prestimulus position.


Disappearing step responses should be produced primarily by the pulse component since the step component has little time to become active. This is confirmed in the simulation shown in Fig. 5 (dashed line) which was done activating only the model’s pulse generator pathway.

VI. CONCLUSION A model-based dynamic analysis technique has been developed and applied to the vergence step response. Because of its specific role as an analysis tool, the model was limited to the motor command components and plant of the vergence eye movement system. Although the model has only five adjustable parameters, it provided highly accurate simulations of the transient portion of the vergence step response. The pulse and step-like signal generated by the model controller were consistent with neurophysiological evidence regarding actual brainstem signals. From an analysis of movement-to-movement dynamics, it was shown that the width of the pulse signal was tightly controlled and was not correlated with other elements of the control signal. Stimulus amplitude had a strong influence on pulse gain and slope parameters, but it had no observed influence on pulse width. The decay of the pulse signal as described by the falling slew rate had the largest variability and was related to the gain of the pulse signal through a relationship that depended on response amplitude. This model provides a concise representation of the vergence response behaviors to step inputs and will be used in future research to quantify this behavior under various stimulus conditions. ACKNOWLEDGMENT The authors would like to thank one of the reviewers for an unusually careful review. REFERENCES [1] B. G. Cumming and S. J. Judge, “Disparity-induced and blur-induced convergence eye movement and accommodation in monkey,” J. Neurophysiol., vol. 55, pp. 896–914, 1986. [2] L. Stark, R. Kenyon, V. V. Krishnan, and K. J. Ciuffreda, “Disparity Vergence: A proposed name for a dominate component of binocular vergence eye movement,” Amer. J. Optometry Physiolog. Opt., vol. 57, pp. 606–609, 1980. [3] B. T. Zuber and L. Stark, “Dynamical characteristics of the fusional vergence eye-movement system,” IEEE Trans. Syst., Man, Cybern., vol. SSC-4, no. 1, pp. 72–79, Jan. 1968. [4] C. Rashbass and G. Westheimer, “Disjunctive eye movement,” J. Physiol., vol. 159, pp. 339–360, 1961. [5] A. T. Bahill, M. R. Clark, and L. Stark, “The main sequence, a tool for studying human eye movements,” Math. Biosci., vol. 24, pp. 191–203, 1975. [6] A. T. Bahill and L. Stark, “The trajectories of saccadic eye movements,” Scientific Amer., vol. 240, pp. 108–117, 1979. [7] H. C. Collewijn, C. J. Erkelens, and R. M. Steinman, “Voluntary binocular gaze-shifts in the plane of regard: Dynamics of version and vergence,” Vision Res., vol. 35, pp. 335–338, 1995. [8] G. K. Hung, K. J. Ciuffreda, J. L. Semmlow, and J.-L. Horng, “Vergence eye movements under natural viewing conditions,” Investigat. Ophthalmol. Visual Sci., vol. 35, pp. 3486–3492, 1994. [9] T. L. Alvarez, J. L. Semmlow, W. Yuan, and P. Munoz, “Dynamic analysis of disparity vergence eye movements: Beyond the main sequence,” in Proc. 19th Annu. Int. Conf. IEEE Engineering in Medicine and Biology Society, 1997, pp. 2755–2759.

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[10] G. K. Hung, H. Zhu, and K. J. Ciuffreda, “Convergence and divergence exhibit different response characteristics to symmetric stimuli,” Vision Res., vol. 37, pp. 1197–1205, 1997. [11] V. V. Krishnan and L. Stark, “A heuristic model for the human vergence eye movement system,” IEEE Trans. Biomed. Eng., vol. BME-24, pp. 44–49, 1977. [12] C. M. Schor, “The relationship between fusional vergence eye movements and fixation disparity,” Vision Res., vol. 19, no. 12, pp. 1359–1367, 1979. [13] C. M. Schor, “A dynamic model of cross-coupling between accommodation and convergence: Simulations of step and frequency responses,” Optometry Vision Sci., vol. 69, no. 4, pp. 258–269, 1992. [14] C. M. Schor, J. Alexander, L. Cormack, and S. Stevenson, “Negative feedback control model of proximal convergence and accommodation,” Ophthalmolog. Physiolog. Opt., vol. 12, pp. 307–317, 1992. [15] M. Pobuda and J. Erkelens, “The relationship between absolute disparity and ocular vergence,” Biolog. Cybern., vol. 68, pp. 221–228, 1993. [16] J. W. McChandless, C. Schor, and J. Maxwell, “A cross-coupling model fo vertical vergence adaptation,” IEEE Trans. Biomed. Eng., vol. 43, pp. 24–34, Jan. 1996. [17] S. S. Patel, H. Ogmen, J. M. White, and B. C. Jiang, “Neural network model of short-term horizontal disparity vergence dynamics,” Vision Res., vol. 37, no. 10, pp. 1383–1399, 1997. [18] G. K. Hung, J. L. Semmlow, and K. J. Ciuffreda, “A dual-model dynamic model of the vergence eye movement system,” IEEE Trans. Biomed. Eng., vol. BME-33, pp. 1021–1028, 1986. [19] J.-L. Horng, J. L. Semmlow, G. K. Hung, and K. J. Ciuffreda, “Initial component control in disparity: A model-based study,” IEEE Trans. Biomed. Eng., vol. 45, pp. 249–257, Feb. 1998. [20] D. S. Zee, E. J. Fitzgibbon, and L. M. Optican, “Saccade-vergence interactions in humans,” J. Neurophysiol., vol. 68, pp. 1624–1641, 1992. [21] D. A. Robinson, J. L. Gordon, and S. E. Gordon, “A model of the smooth pursuit eye movement system,” Biolog. Cybern., vol. 55, no. 1, pp. 43–57, 1986. [22] L. E. Mays, “Neural control of vergence eye movements: Convergence and divergence neurons in midbrain,” J. Neurophysiol., vol. 51, no. 5, pp. 1091–1108, 1984. [23] L. E. Mays and J. D. Porter, “Neural control of vergence eye movements: Activity of abducens and oculomotor neurons,” J. Neurophysiol., vol. 52, no. 4, pp. 742–761, 1984. [24] L. E. Mays, J. D. Porter, P. D. R. Gamlin, and C. A. Tello, “Neural control of vergence eye movements: Neurons encoding vergence velocity,” J. Neurophysiol., vol. 56, pp. 1007–1021, 1986. [25] P. D. R. Gamlin and L. E. Mays, “Dynamic properties of medial rectus motoneuron during vergence eye movements,” J. Neurophysiol., vol. 67, pp. 64–74, 1992. [26] J. L. Semmlow, G. K. Hung, J.-L. Horng, and K. J. Ciuffreda, “Initial control component in disparity vergence eye movements,” Ophthalmic, Physiolog. Opt., vol. 13, pp. 48–55, 1993. [27] F. K. Hsu, A. T. Bahill, and L. Stark, “Parametric sensitivity analysis of a homeomorphic model for saccadic and vergence eye movements,” Computational Programs, Biomed., vol. 6, no. 2, pp. 108–116, July 1976. [28] L. A. Abel, D. Schmidt, L. F. Dell’Osso, and R. B. Daroff, “Saccadic system plasticity in humans,” Ann. Neurol., vol. 4, no. 4, pp. 313–318, Oct. 1978.

Weihong Yuan (S’98) was born in Dalian, China, in 1968, and received the B.S. degree of biomedical engineering from the Zhejiang University, Hangzhou, China, in 1991. In 1995 he entered Graduate Program in Biomedical Engineering at Rutgers University and Robert Wood Johnson Medical School (UMDNJ), New Brunswick, NJ, receiving the M.S. degree in 1997. Currently he is pursuing the Ph.D. degree in neural mechanisms of oculomotor control. After receiving the undergraduate degree, he worked as an Assistant Engineer at Dalian Medical University, Dalian, China. He is a student member of the Association for Research in Vision and Ophthalmology.

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John L. Semmlow (M’79–SM’89–F’94) was born in Chicago, IL, in 1942 and received the B.S.E.E. degree from the University of Illinois in Champaign in 1964. Following several years as a design engineer for Motorola, Inc., he entered the Bioengineering Program at the University of Illinois Medical Center in Chicago, receiving the Ph.D. degree in 1970. He has held faculty positions at the University of California, Berkeley, and the University of Illinois, Chicago, and currently holds a joint position as Professor of Surgery, UMDNJ-Robert Wood Johnson Medical School and Professor of Biomedical Engineering at Rutgers University, New Brunswick, NJ. In 1985, he was a NSF/CNRS Fellow in the Sensorimotor Control Laboratory of the University of Provence, Marseille, France. His active research areas include physiological motor control and medical instrumentation with an emphasis on noninvasive instrumentation.

Tara L. Alvarez received the B.S. degree in electrical engineering from Rutgers University, New Brunswick, NJ, in 1994 and the M.S. and Ph.D. degrees in biomedical engineering from Rutgers University in 1996 and 1998, respectively. She is currently a Member of the Technical Staff at Bell Laboratories-Lucent Technologies, Whippany, NJ. Her interests are in signal processing and neural control. Dr. Alvarez is a member of Tau Beta Pi and Eta Kappa Nu.

Paula Munoz is currently a Ph.D. degree graduate student of biomedical engineering, Rutgers University, New Brunswick, NJ. She received the B.S. and professional degrees in electrical engineering from the University of Chile, Santiago, Chile, in 1990 and 1993, respectively. Also she received the M.S. degree in biomedical engineering from the same university in 1994. Her interests are in signal and image processing as well as functional neuroimaging.