A Minimal Model to Simulate Dynamics of Initial Vergence Component. Weihong YUAN1, John L. Semmlow1'2 ,Tara L. Alvarez1 and Paula Munoz1.
A Minimal Model to Simulate Dynamics of Initial Vergence Component Weihong YUAN1, John L. Semmlow1'2 ,Tara L. Alvarez1 and Paula Munoz1 Department of Biomedical Engineering, Rutgers University1,Department of Surgery, Bioengineering, Robert Wood Johnson, Medical School-UMDNJ2 New Brunswick, NJ ABSTRACT According to the Dual-Mode theory, the two different dynamic components seen in the step response of disparity vergence eye movements are under separate control mechanisms. The initial, fast rising component is controlled by an open loop, preprogrammed system while the later, slow component is feedback controlled. The ability to separate the two components, or to isolate the initial component, is critical to the study of both control systems. To approach this goal, a simple open-loop control model, containing four independently adjustable parameters, was designed to simulate the disparity vergence movement. This model provided a substantial simplification of the control system, yet demonstrated remarkable accuracy in its ability to simulate the dynamic details of the initial component vergence response.
INTRODUCTION Over the past three decades, there has been much work done on the dynamics of disparity vergence eye movements. To explain the operation of the system that controls and regulates vergence eye movement, several quantitative theories have been proposed. Rashbass & Westheimer (1957) proposed a simple feedback model, which included a first-order lag component in the feed-forward loop. The eye position was fed back to be compared with the referential target position. The position error was translated into disparity and processed by an integral element (represented as a first order lag) with variable gain followed by a time delay and a second order plant. Krishnan & Stark (1977) , building on the simple feedback model mentioned above, suggested a parallel operator that contributes only to the fast initial response. A derivative unit in parallel to the first order lag unit was added to improve the stability of the response during the initial period. The major concern regarding the dynamics of vergence eye movement is how to explain two seemingly contradictory phenomena: the system can produce responses that have high accuracy and high speed (implying high feedforward gain), yet remain stable despite the presence of long processing delays in the feed-forward loop. The vergence response can approach steady-state within 100-200 msec 0-7803-3848-0/97/$10.00 ©1997 IEEE
with a steady-state error of only several minutes of arc despite feed- forward latencies totaling 120-180 msec. Semmlow et al. (1986) observed that the vergence response to fast ramps showed staircase-like behavior and they suggested a switching process within the initial response period. Based on this observation, Hung & Semmlow (1986) presented a new theory: a dual-mode structure that included an initial and a late component in the feed-forward loop. The initial component operated under open-loop conditions and was composed of a predictor and a sampler. The late component operated under feedback control and included an error magnitude and velocity limitter that restricted the range of this slow tracking component. Based on the dual-mode structure mentioned above, Horng (1994) proposed a more detailed model for the two components. It was believed that the open-loop component dominates in the first several hundred milliseconds, while the slow component is responsible for reducing the error to a minimum in the period that follows the initial transient. Given enough parameters, we can simulate very complicated dynamic processes. But this approach might obscure the dominant control elements in the system. The objective of the work presented here is to present a simple model, which has only four adjustable parameters, yet which accurately represents the initial response components in a number of subjects.
METHODS Two oscilloscopes were arranged as a haplascope with two vertical lines generated on these oscilloscopes used to create a stereo pair. The disparity stimulus generated by these lines was constructed and controlled by a laboratory computer. The eye movements were recorded by Skalar infrared eye movement monitor (Model 6500) which had a resolution of 1.5 min of arc. The linear range of the eye movement monitor (3dB) was + 25 degree. Three taamaa subjects (IS,CC,BS) with binocular vision participated the project. During the experiments, a standard 4 degree step stimulus was presented and the subject was requested to fuse the two lines to perceive them as one image. A two-point calibration was carried out online for each trial. The eye positions were sampled at 5 msec intervals, digitized, and stored along with calibration information on disk. Off-line data analysis and simulations 52
were carried out using the Matlab software system and plotted using Axum.
MODEL
responses. We used the RMS value of the difference between the position of eye and the simulated model trace to aid in optimizing the model fit to the data.
The model is presented in Figure 1. It includes a DISCUSSION pulse generator, a rate limitor , a leaky integrator, and a first The goal of this research is to use minimum number of order plant. The pulse generator can be adjusted in terms of parameters to simulate the control processes of the initial pulse-width and gain. The rate limitor is the only nonlinear vergence component, and to estimate their relative importance. element in the model and restricts its output to a maximum While the model may not necessarily be unique, previous (or minimum) slew rate. The time constant in the leaky simulation experience indicates that it is minimal. The integrator was decided empirically and was set to 0.03 sec. nonlinear element appears to be critical both in improving the fit for all three subjects. The time constant for the plant was and reducing the number of components. also empirically selected; however, its value is comparable with neurophysiological evidence (Mays, 1984). This constant The four parameters behaved differently in terms of was fixed for each individual subject, but varied between the their range and variance. The mean values of pulse-width were three subjects. For JS, the constant was 0.2 sec. and for CC and similar in the three subjects and this parameter seems to have the BS it was 0.4 sec. smallest range. The standard deviation of pulse-width was only 10.5%,6.7% and 7.7% of mean value for JS, CC, and BS, 1 Pulse Rate Limitor 1 respectively. The other three parameters all showed relatively Generator 0.25 + 1 0.03s + 1 large standard deviations and the differences among subjects were larger than for pulse-width. The largest variation was Figure 1: The simplified model for the initial component. found for falling slew rate, its standard deviation to mean value ratio was of the order of 40%. This indicates a much less RESULTS tightly controlled process. While the step disparity stimulus triggers the neural system to send out a pulse signal tightly controlled in width, the mechanisms that shape its rising and falling dynamics show considerable variability. Hence, the variation in initial component dynamics comes from the processes associated with the gain and slew rate, which can change dramatically from movement to movement. This variation of initial component makes it less likely to attain a small steady state error, and this task is left to the slow component. Figure 2: Time course of vergence step response and model simulation. Figure 2 shows the time traces of a typical response to a 4 degree step stimulus by subject JS. The solid line represents the disparity response and the dashed line represents the simulated result. The two plots correspond to position and velocity traces, respectively. A good fit is obtained for the early portion of the response where the initial component dominates. Table 1: Model Parameters Pulse Rising Slew Gain Width(sec) rate 91.9 (21.64) JS n=30 0.1310 9.92 (0.0136) (1.67) CC n=14 0.1510 10.26 85.36 (0.0100) (2.01) (18.24) BS n=12 0.1300 14.20 180.90 (0.0100) (1.45) (20.80)
Falling Slew Rate -74.2 (28.69) -41.43 (20.42) -103.0 (41.19)
Table 1 shows the mean value and standard deviation of each model parameter obtained after fitting a number of
Although the model has only four adjustable parameters, the precision of simulation and the robustness of the parameters presented were unexpectedly high. This model provides a concise representation of the initial component behavior and will be used hi future research to identify and isolate this behavior from the combined response.
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