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1Medical School, University of Patras, Patras, Greece; 2Department of Anatomy and Neurobiology, Virginia Commonwealth University, Richmond, Virginia; and 3Department of Psychology, University of Sheffield, Sheffield, United Kingdom
Submitted 7 June 2005; accepted in final form 14 October 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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0.01, 0.1, 1, and 10 s. To demonstrate directly the presence of long (1,10 s) TC elements and to assess their contribution quantitatively, horizontal eye displacement was induced in Cynomolgus monkeys under deep barbiturate anesthesia that prevented interference from spontaneous eye movements. The displacement was maintained for either a prolonged (30 s) or brief (0.2 s) period before release. Return to resting position took 20–30 s after prolonged displacement but only 1–2 s after brief displacement, consistent with the presence of long TC elements that would only be substantially stretched in the former condition. Quantitative fitting of the release curves after prolonged displacement indicated that the two long TC elements contribute a substantial proportion (30%) of the total plant compliance. A model based on the estimated compliance values is shown to account quantitatively both for our release data and for Goldstein and Robinson's data on hysteresis of ocular motoneuron firing rates measured after centripetal saccades following prolonged eccentric fixation. Long time-constant elements in the plant thus make a substantial contribution to some types of eye movement, and their inclusion in plant models can help interpret the firing patterns of single units in the oculomotor system. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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) where a Voigt element has stiffness k, viscosity c, and time constant (TC) of c/k. Most models of the human or primate plant have two Voigt elements, one with a TC of 10–60 ms and the other of 250 –370 ms (Goldstein and Reinecke 1994; Goldstein and Robinson 1984; Optican and Miles 1985; Robinson 1964; Stahl and Simpson 1995).
However, a recent analysis of data on the globe's return from horizontal displacement in lightly anesthetized monkeys has suggested that these models are incomplete (Sklavos et al. 2005). The use of light anesthesia allowed eye-position traces to last on average 1–2 s without interruption from saccades, and their analysis indicated that the viscoelasticity of the plant could not be approximated by two Voigt elements. Although individual traces could be reasonably well fitted with two TCs, the values of these TCs varied by up to sixfold as trace duration increased. Robust fitting required a plant of four Voigt elements, with TCs ranging from 0.01 up to 10 s.
Adding long TC elements to the plant model substantially changes its behavior. A Voigt element with a TC of 10 s takes 23 s after release to return to within 10% of resting position, whereas an element with a 0.5 s TC, the longest previously used in a model of the human plant (Robinson 1965), takes only 1.15 s. Moreover, only certain types of eye movements are likely to involve elements with TCs as long as 1 and 10 s, because these elements will be not be substantially stretched by the brief (up to 0.1 s) forces typical of saccades, but only by the longer-duration forces typical of "slow" eye movements such as 0.1-Hz sinusoidal smooth pursuit and gaze-stabilization reflexes. As pointed out by Stahl and Simpson (1995), the addition of long TC elements to plant models might therefore help to overcome a long-standing problem in the interpretation of OMN firing, which is that one- or two-element plant models require different parameters to explain both fast and slow eye movements (Fuchs et al. 1988; Sklavos et al. 2005; Sylvestre and Cullen 1999).
Given the potential significance of the new plant model, it is important to obtain further experimental evidence to test and refine it. Although there have been previous reports suggesting the presence of long TC elements in the plant (Collins 1971; Pfann 1993; Robinson et al. 1969), these have been brief or anecdotal, and have concerned the behavior of the globe with the medial and lateral rectus muscles detached. The present study exploited the properties of deep barbiturate anesthesia to investigate long TC behavior in the intact oculomotor plant, as follows.
First, deep anesthesia permits both prolonged displacements of the globe and prolonged return traces uninterrupted by saccades. As mentioned in the preceding text, the average duration of traces under light ketamine anesthesia was 1–2 s before interruption by eye movements (Sklavos et al. 2005). Deep anesthesia allows direct testing of an important predication of the new model, which is that the eye should take up to 20 s to return to resting position if the long TC elements are stretched. To ensure such stretching, the globe was subjected to prolonged displacement (30 s). As a control condition, return trajectories were also measured after brief (0.2 s) displacement. Because prolonged displacement stretches long TC elements much more than a brief displacement, comparison of the two release curves should provide unequivocal evidence for the existence of such elements.
Second, a 30-s displacement gives elements with TCs 
10 s time to reach equilibrium. In equilibrium, the degree of stretch depends only on an element's compliance (that is, 1/stiffness), so that estimating each element's length at the moment of release is equivalent to estimating its relative compliance (further details in METHODS). Prolonged displacements therefore allow more accurate estimates of the relative compliances of long TC elements than the previous study, where only brief (1–2 s) displacements were possible resulting in an underestimate of the contribution of long TC elements (Sklavos et al. 2005).
Third, the new compliance estimates can be incorporated in a more accurate plant model, which can be then be applied to both the present data on brief displacements and to previously obtained data on hysteresis in OMN firing rates (Goldstein and Robinson 1986). According to the model, sustained eccentric fixation stretches the long TC elements, which affects the neural commands required to keep the eye stable after its subsequent return to resting position. The dependence of OMN commands on previous fixation ("hysteresis") has been shown to be qualitatively consistent with the four-element model (Sklavos et al. 2005). The new compliance estimates allow quantitative modeling of hysteresis, an important advance because such modeling addresses the issue of whether the long TC behavior is primarily a property of orbital tissue rather than extraocular muscle. Only in the former case is it likely that a model derived from the anesthetized preparation would account quantitatively for behavior in the alert animal.
A brief account of the results has previously been presented as an abstract (Sklavos et al. 2003).
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METHODS |
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Two cynomolgus monkeys (M1 and M2, aged between 2 and 3 yr, weights: 2.4 and 3.4 kg) were premedicated with 15 mg/kg ketamine and 0.01 mg/kg glycopyrrolate administered intramuscularly and then anesthetized with pentobarbital sodium at 25 mg/kg intravenously into the saphenous vein. Additional doses of pentobarbital sodium were provided intravenously during the experiment to maintain deep anesthesia as assessed by the absence of the blink reflex and withdrawal to digit pinch. After topical anesthesia of the larynx with 4% lidocaine, the animals were intubated with a 3.5- to 5.0-mm endotracheal tube. End-tidal CO2, respiratory rate, and heart rate were continuously monitored and maintained within a normal range. Body temperature was kept at 37°C with a heating pad. Lactated Ringer solution was administered via an intravenous drip (10 mg · kg–1 · h–1).
The animal was placed in a Kopf stereotaxic frame, and a midline incision was made in the skin overlying the scalp. An anterior right parietal craniotomy was performed for placement of a stimulating electrode in the abducens nerve, located at coordinates A 1.5 L 2.0 and a depth that varied between animals (Contreras et al. 1981). On completion of the experimental protocol, which included a number of procedures in addition to those described here (Dimitrova et al. 2003), the animals were killed with an overdose of pentobarbital administered intravenously. All procedures and protocols for animal care and use were approved by Animal Care and Use Committee of Virginia Commonwealth University.
Displacement, stimulation, and recording procedures
Prolonged displacements of the right eye were produced by a suture in the cornea, pulled either laterally (5 displacements) or medially (5 displacements) by 10–15° and held in place for 
30 s. The suture was either cut (animal M1) (Childress and Jones 1967; Collins 1971) or released manually (M2). Brief displacements (n = 24) were produced electrically by stimulation of the abducens nerve with a stainless steel bipolar electrode (0.2-mm tip diameter, separated by 1.5 mm). Individual pulses lasted for 0.2-ms and ranged in intensity from 400 to 800 µA. They were delivered as constant frequency trains lasting for 200 ms with frequencies ranging from 50–250 Hz. The pulse trains were produced by a programmable pulse generator (AMPI Master-8) and were delivered at 5-s intervals. Movements of the right eye were recorded with an eye search coil (3-dimensional eye-movement monitor model EM7, Remmel Labs) glued to the eyeball, at a sampling rate of 10 kHz. The coil was calibrated by simultaneously recording a set of stimulation evoked movements with the coil and with a digital camcorder (Canon Elura2) at 30 frames/s. A small wooden rod (1–1.5 cm in length and weighing 6 mg) was also glued to the eyeball and served as a marker to track the movement with the video camera. Twitch responses of the lateral rectus muscle were recorded as described in Shall, Dimitrova and Goldberg (2003).
Different techniques for brief versus prolonged displacements were applied because a method for precise brief mechanical stimulation was not available and very prolonged electrical stimulation was thought likely to affect the state of the muscle. The actual use of electrical stimulation for the "brief" condition was conservative with respect to the hypothesis under test: the artifact produced by the time course of muscle activation in response to brief electrical stimulation would act to prolong the effects of that stimulation.
Data analysis and parameter estimation
The calibration procedure was as described previously (Dimitrova et al. 2003). The 10-kHz coil output was subsampled at 1 kHz by averaging over successive blocks of 10 data points.
The relaxation curves obtained after prolonged displacements (n = 10 per animal) contained mixed frequency noise, probably as a result of the external mounting of the eye-recording coil and attached wiring, This noise produces severely biased estimates of TCs with the methods described in Sklavos et al. (2005) but has little effect on the estimates of relative compliance for fixed TCs. The curves were therefore fitted using the model shown in Fig. 1. This model ignores the globe's inertia and treats the plant as a series of four Voigt elements with time constants of 0.01, 0.1, 1, and 10 s (INTRODUCTION). When this system is released after a displacement each element returns to its resting position with an exponential time course [instantaneous extended length of ith element 
i(t)], giving the equation
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i at t = 0) and Ti is its time constant. If the system is held at its initial position long enough for the elements to have stopped moving, then for each element
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i are the fitting errors for each data point. This can be written as a matrix equation by introducing column vectors x = (xi), A = (Aj), and = (ei), and the m x 4 matrix E with elements Eij = Aje-ti/Tj
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2 is then given by
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). The four estimated values of Aj were converted to relative compliances ai by dividing each by their sum 
Aj (Eq. 3). Goodness of fit was measured as proportion of variance explained (PVE) for by the fitting trace.
Modeling
The linearized visco-elastic behavior of an element can be represented by its transfer function in the Laplace domain
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However, the 0.2-s displacement was produced by electrical stimulation of the nerve to the lateral rectus muscle, instead of mechanically. Such stimulation introduces dynamical properties of its own, most simply represented by the twitch response of isometric muscle force to a single pulse. The shape of this response can be approximated by a function of the form
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). For the present model, we found that Ta = Tb = 0.01 s gave a twitch response similar to that observed experimentally in M1 and M2 (data not shown). The 0.2-s stimulation pulse was passed first through this function then the plant model.
To simulate hysteresis in OMN firing rates, the model must be used to calculate the control signal required to produce a particular eye movement, that is, in inverse rather than forward mode. The transfer function in Eq. 7 was therefore inverted (further details in Sklavos et al. 2005). Its input was a set of simulated saccades following the sequence described in Goldstein and Robinson (1986), p.1045: 10 iterations of four saccades in the order 1) saccade from primary position 0° to 20° right, 2) right 20° back to 0°, 3) 0° to left 20°, and 4) left 20° back to 0°. There were 3 s of steady fixation between saccades. For technical reasons (the inverted plant has more zeros than poles, and so is disallowed by the MATLAB Control System Toolbox), the desired inputs were converted to velocities. The saccadic velocity profile was represented by a Gaussian function chosen to produce a saccadic duration of 30 ms. The output of the model was compared with the firing rates for n = 39 OMNs measured 2.5–2.8 s after return to the primary position, and shown in Table 1 of Goldstein and Robinson (1986).
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RESULTS |
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Displacement duration affects globe's return
A representative return trajectory after prolonged displacement is shown in Fig. 2A. The globe had been rotated laterally for 14°, held for 30 s (see METHODS), then suddenly released. Although the initial movement to within 6° of the resting position was very rapid (<30 ms), the remainder of the return to resting position was much more gradual, occurring over 20–30 s. In contrast, the return to baseline after brief displacement (200 ms) was much more rapid (Fig. 2B). The individual eye-position record shown in B was chosen because the amplitude of the displacement (14°) is similar to that displayed in A. In the case of brief stimulation, however, the initial very rapid movement took the globe to within 2° of baseline, and the subsequent return to resting position was completed within 1–2 s, which is 10 times faster than after prolonged displacement. The difference between the two conditions can be seen particularly clearly when they are plotted on the same time scale (Fig. 2C). This graph shows the mean ± SD of the combined normalized traces for the two animals at 100-ms intervals for the first 3 s after release. The two traces diverge 50 ms after release. The SDs for the poststimulation traces are less than those for the traces after prolonged displacement, probably reflecting the greater standardization of the displacing stimulus in the former case. The difference between the two mean traces is large compared with their SDs, and in fact, there is no overlap between the two sets of traces for the period 0.1–3 s after release [data for a given time point, Mann-Whitney nonparametric U test, n1 = 20, n2 = 48, U = 0, P < 0.002 (Siegel 1956)].
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Estimates of compliance of long TC elements
The release traces after prolonged displacement showed some low-frequency interference (5–35 Hz, 0.1–1.5 s after release), possibly caused by vibration of the wires from the eye-coil (see DISCUSSION). The presence of this interference rendered the traces unsuitable for precise estimation of TC values. However, the traces could be fitted with TC values derived elsewhere (Sklavos et al. 2005), in this case 0.01,0.1,1, and 10 s (see METHODS). For animal M1, the average goodness-of-fit (PVE) produced by this method was 94%, for animal M2, 93%. An example of a fit is illustrated in Fig. 3A for the individual trace previously shown in Fig. 2A. The fitted curve is the sum of four exponential curves, with time courses corresponding to the four TCs, and initial values (at t = 0) of 11, 2.7, 1.8, and 1.7°. These initial values are the coefficients of the curves (see METHODS), and they can be normalized so that their sum (=1) is the same for each trace. The mean ± SD of the normalized coefficients ai for each animal are shown in Fig. 3B. The mean values are similar for the two animals with a tendency for the coefficients of the shorter time constants (especially the shortest at 0.01 s) to be more variable, probably reflecting the difficulty of reproducible mechanical displacement at this time scale.
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32% of the overall compliance of the oculomotor plant in the deeply anesthetized cynomolgus monkey (estimated relative compliances for elements in order of ascending TC: 0.52, 0.16, 0.19, 0.13). Because the stiffness of an element = total stiffness/element's relative compliance, these values can be translated into the stiffnesses of the model elements provided the stiffness of the plant as a whole is known. To our knowledge plant stiffness has not been measured for the cynomolgus monkey. However, substituting the value of 0.45 gf/° for the stiffness of orbital tissue in rhesus monkey [obtained from measurements with force transducers attached to the tendons of the horizontal rectus muscles (Miller et al. 2002)] gives estimates of 0.66 and 1.41 gf/° for the two short and two long TC elements in the present study (estimated stiffnesses of individual elements in order of ascending TC: 0.87, 2.9, 2.4 and 3.4 gf/°). Applications of model
Release Data.
The behavior of the four-element model (Fig. 1) with the values for relative compliance taken from Fig. 3 can be compared with the observed release traces. As expected, the correspondence with the mean normalized release trace for prolonged displacement is good (not shown: goodness-of-fit PVE = 99%). Figure 3C shows model behavior and mean normalized data for release after a 200-ms displacement as shown in Fig. 2C. The fit of model output to data are reasonably close (goodness-of-fit PVE = 79%), although the data are slightly underestimated at times less than 0.15 s and slightly overestimated thereafter. A possible reason for the early underfit is that the model does not allow for the dynamics of electrical stimulation acting on the muscle. A single pulse in fact produces a twitch of isometric force with a time course similar to that shown in Fig. 4A (see METHODS). If this time course is incorporated in the model (details in METHODS), the fit to the stimulation data for t < 0.15 s is markedly improved (Fig. 4B) and for t >0.15 s is little altered (not shown). Overall goodness-of-fit (PVE) increased from 79 to 97%.
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Single-unit recording has shown that after prolonged (2–3 s) eccentric fixation followed by a saccade to the primary position, OMN firing rates change over a long period of time before they reach a stable value. The direction of the change depends on the direction of the preceding fixation: if it was in the on direction of the OMN, the rate gradually increases to stability, whereas if the fixation were in the off direction, the rate gradually decreases. This phenomenon, termed hysteresis, is consistent with the presence of long TC elements in the plant (Fig. 5). Prolonged eccentric fixation stretches these elements (illustrated for the model in Fig. 5A). Rapid displacement of the whole plant back to the primary position mainly affects the short TC elements, leaving the long TC elements still stretched. The long TC elements then slowly relax, which requires a slowly varying control signal to keep overall plant position steady (Fig. 5, B and C). It has been shown previously that the four-TC model is qualitatively consistent with the pattern of OMN firing shown in Fig. 5A (Sklavos et al. 2005). Here we investigate the quantitative performance of the four-TC model (with TCs of 0.01, 0.1, 1, and 10 s, and the compliances indicated in Fig. 3) during hysteresis.
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; Table 1). The same sequence of eye movements was used to generate the control signals required by the model (see METHODS). Hysteresis was measured in the same way for the control signals as for OMN firing rates, and the two measures compared. Results of the comparison are shown in Fig. 5D. The performance of the model is close to the mean performance for the n = 39 OMNs. The agreement might be considered surprising given that the compliance estimates came from data collected in deeply anesthetized animals, whereas the firing rate data were collected in alert animals. However, the firing rates were measured >2.5 s after the saccade, so would be influenced only by the need to cancel creep in the long TC elements. It is possible that these elements primarily reflect the viscoelastic contribution of orbital tissue rather than eye muscle (DISCUSSION), in which case their quantitative behavior will be little affected by anesthetic.
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DISCUSSION |
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). This was an unexpected finding, for although there had been previous reports suggestive of such elements (Collins 1971; Pfann 1993; Robinson et al. 1969), they were brief and unsystematic and described the behavior of the globe when the horizontal recti muscles were detached. And as indicated in the INTRODUCTION, no viscoelastic model of the intact primate or human oculomotor plant included a Voigt element with a TC >0.5 s (Fuchs et al. 1988; Goldstein and Reinecke 1994; Goldstein and Robinson 1984; Optican and Miles 1985; Robinson 1964, 1965; Stahl and Simpson 1995). There is a model of the rabbit plant that includes a Voigt element with a TC of 3.4 s, but the authors considered this to be homologous to the longer (0.25 s) TC typical of monkey two-element models, invoking more sluggish dynamics rather than a larger number of elements to explain the difference between the rabbit and monkey data (Stahl and Simpson 1995).
The present study therefore sought to extend the results of Sklavos et al. (2005) by demonstrating directly the presence of long TC elements, estimating their relative compliance, and incorporating the compliance estimates into a plant model and applying it. Each of these aims is discussed in the following text. A final section considers the implications of the present findings for future interpretation and modeling of data on oculomotor control.
Demonstration of long TC behavior
The displacements of the globe that were analyzed by Sklavos et al. (2005) were produced manually, with no systematic variation of duration. Here, in contrast, two very different durations of displacement were used. The expectation was that long TC elements in the plant would be substantially stretched by the prolonged (30 s) but not brief (0.2 s) displacements, so that the subsequent release trajectories would be characteristically different. The expected differences were indeed observed (Fig. 2) and are consistent with a substantial contribution of long TC elements. To our knowledge, this is the first explicit demonstration of such long TC behavior in the intact oculomotor plant of primates.
The deep level of anesthesia needed for the prolonged displacements eliminated active muscle tone. Because the stiffness of the passive lateral rectus muscle in this study around the primary position was very low (data not shown), the behavior illustrated in Fig. 2 primarily reflects the properties of the nonmuscular orbital tissues (Sklavos et al. 2005). This conclusion is supported by the observations referred to above that suggested the presence of long TC elements when the horizontal recti are detached.
Estimation of long TC contribution
Prolonged displacements of the globe are required to stretch the long TC elements fully and so allow their contribution to plant dynamics to be estimated. The release traces analyzed by Sklavos et al. (2005) were not produced by such prolonged displacements, and moreover the average duration of the traces was only 2 s before interruption. They were not therefore suitable for deriving an accurate estimate for the summed relative compliance of the two longest TC elements.
In the present study, the use of deep anesthesia allowed both the required prolonged displacement and long-duration release traces. However, the external mounting of the eye-recording coil and attached wiring resulted in noisy traces that made precise characterization of release TCs difficult, because direct estimation of TC's from relaxation data are highly sensitive to very small amounts of measurement noise. Fortunately, as argued in Sklavos et al. (2005), discrete TC values in themselves may be of limited usefulness because biological materials typically comprise elements with a continuous range of time constants (Hall 1968). In contrast, the compliance associated with a broad TC interval is both robustly estimable and meaningful in the continuous case. This is most simply achieved by the method we have described, employing discrete TC's located at decade intervals (0.01, 0.1, 1, and 10 s) to estimate the total compliance of elements in that TC range.
Using this approximation with the present data gave an estimate for the relative compliance of the long TC elements of 32% [compared with a value of 19% obtained from fitting TCs of 0.01, 0.1, 1, and 10 s to the previous data in Sklavos et al. (2005)]. It is possible that this estimate errs on the low side: there is evidence to suggest that the globe with horizontal recti detached may have components with TCs >10 s (e.g., Pfann 1993), and use of the estimate to predict hysteresis gives a value that is slightly too low (next section). More accurate estimates of plant characteristics in deeply anesthetized animals could be obtained with implanted eye coils to avoid measurement noise and with a more precise mechanical means of rotating and releasing the eyeball for both brief and prolonged displacements.
Application of plant model
The plant model based on estimates of compliances derived from the prolonged-displacement data provided a reasonable fit to the release curves obtained after 0.2-s electrical stimulation. Perhaps more surprisingly, it also gave a fairly good quantitative estimate of hysteresis in OMN firing rates as measured in alert rhesus monkeys by Goldstein and Robinson (1986). EOM viscosity and elasticity might be expected to be very different in alert animals than in the deeply anesthetized animals that provided the data for the model. However, explicit modeling indicates that the long TC behavior of the plant as a whole is little affected by plausible variations in muscle elasticity and viscosity (Sklavos et al. 2005). The success of the model with regard to hysteresis may therefore depend on that phenomenon primarily arising from the properties of orbital tissue rather than extraocular muscle.
In more general terms, the fact that the same model could fit the trajectories produced by either a 30- or 0.2-s displacement, and also account for hysteresis, is a good argument for the linearity of the system in the range of conditions investigated.
Interpretation and modeling of experimental data
It is important to know when the presence of long TC elements needs to be taken into account for interpreting or modeling experimental data. The proportion of the overall model response that is contributed by long TC elements, as estimated from the data in the present study, is shown as a function of frequency in Fig. 6A. Given the limitations of the present estimates as discussed in the previous section, this figure can only be an approximate guide. However, it suggests that long TC elements contribute >5% of the total response at frequencies <1 Hz and so might be taken into account when interpreting eye-movement dynamics in that frequency range. The eye movements in question could include smooth pursuit, the slow phases of the vestibuloocular and optokinetic reflexes, and (as already discussed) prolonged eccentric fixation—the 2- to 3-s fixation periods used in the measurement of hysteresis (Goldstein and Robinson 1986) correspond to fundamental frequencies of 0.17–0.25 Hz. In contrast, the contribution of the long-TC elements to saccadic eye movements (without prior eccentric fixation) is likely to be small: a 100-ms pulse has a fundamental frequency of 5 Hz, a frequency at which the long TC elements make up 2% of the overall response. The implication that long-TC elements are relevant to the dynamics of slow but not fast eye movements is consistent with the findings that, for one- or two-element models relating the firing rates of OMNs to eye movements, different parameters have to be found for movements of different frequency (Fuchs et al. 1988; Sklavos et al. 2005; Stahl and Simpson 1995; Sylvestre and Cullen 1999).
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). The gain of the model was then set to correspond to the observed value of K (= 1/gain) at very low frequencies (static K). The performance of the model at higher frequencies depends on the value assumed for muscle stiffness. It can be seen there is a reasonable fit between data and model when muscle stiffness is 0.6 times the orbital stiffness (thick line). If orbital stiffness is in fact 0.45 gf/° (see preceding text), this value corresponds to a muscle stiffness of 0.27 gf/°, close to indirect estimates of 0.3 gf/° (Sklavos et al. 2005). However, these estimates are uncertain, and Fig. 6B illustrates the performance of the model for both lower (0.3 times orbital stiffness, upper thin line) and higher (0.9 times, lower thin line) values. Even with substantially altered values for muscle stiffness, the overall model performs robustly: its K values change appropriately as frequency increases, illustrating how the presence of long TC elements affects system stiffness and thus the required control signals.
The final point concerns how such control signals might be generated by neural mechanisms. If the plant was effectively a single Voigt element, then the necessary control signal could be generated from a desired eye-velocity command by integration (Skavenski and Robinson 1973). The addition of a second Voigt element requires a new exponential "slide" term in the control signal, as shown by Goldstein and Robinson (1984) and Optican and Miles (1985). The long TC elements described here require further, long-duration, slide terms (which correspond to the additional zeros in the plant, Fig. 5). It seems likely that the cerebellar flocculus calibrates both integration and slide terms (Optican et al. 1986; Zee et al. 1981), so the question arises of whether it is realistic to expect the cerebellum to cope with the complexity of a four-element plant that has TCs ranging from 0.01 to 10 s. Recent theoretical models suggest that a plausible learning mechanism, based on decorrelating retinal slip from neural commands, could in fact generate the required commands (Dean et al. 2002, 2004; Porrill et al. 2004).
Conclusions
In the 40 yr since Robinson first described the basic viscoelastic nature of the oculomotor plant, the longest time constant used in a formal model seems to have been 0.5 s. Our recent study suggesting that the oculomotor plant in fact exhibits dynamic behavior up to at least a 10-s time scale (Sklavos et al. 2005) thus represented a major departure from the existing consensus. However, that study deduced the existence of long TC elements from traces that were on average 1–2 s long, rather than directly demonstrating their properties. Here we provide such a direct demonstration, by showing that the intact oculomotor plant in primate can take 20–30 s to return to baseline after release. We also show this long duration of return depends on prior treatment: it is seen after prolonged (30 s) but not brief (0.2 s) displacements, a very clear and statistically significant effect.
Previous Voigt element models with two or three TCs between 0.01 and 0.5 s cannot account for the preceding findings. At least four elements are required, with TCs of the order of 0.01, 0.1, 1, and 10 s. An accurate estimate of the relative compliances of the long-TC elements can be made from the release trajectories that follow prolonged displacement, possible in the present study using deep anesthesia but not possible with the light anesthesia used previously (Sklavos et al. 2005). The resultant quantitative model fits not only both sets of new data obtained here but also results from other laboratories concerning the firing rates of ocular motoneurons, measured in two unrelated experimental situations (Fuchs et al. 1988; Goldstein and Robinson 1986). This model, and its future refinements, may therefore be useful in helping to interpret the dynamic behavior of neurons in the oculomotor system.
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FOOTNOTES |
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Address for reprint requests and other correspondence: P. Dean, Dept. of Psychology, University of Sheffield, Western Bank, Sheffield S10 2TP, UK (E-mail: p.dean{at}sheffield.ac.uk)
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