Analysis of the Frequency Response of the Saccadic Circuit: System Behavior

James W. Gnadt,1 Mark E. Jackson,1,2 and Oleg Litvak1

 1Department of Neurobiology and Behavior, State University of New York at Stony Brook, Stony Brook, New York 11794; and  2Department of Psychiatry, Yale University Medical School, Veterans Affairs Medical Center, West Haven, Connecticut 06516


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Gnadt, James W., Mark E. Jackson, and Oleg Litvak. Analysis of the Frequency Response of the Saccadic Circuit: System Behavior. J. Neurophysiol. 86: 724-740, 2001. To more thoroughly describe the system dynamics for the saccadic circuit in monkeys, we have determined the frequency response by applying a frequency modulated train of microstimulation pulses in the superior colliculus. The resulting eye movements reflect the transfer function of the saccadic circuit. Below input modulations of 5 cycles/s, the saccadic circuit increasingly oscillates with multiple high-frequency, low-amplitude movements reminiscent of the "staircase saccades" evoked during the sustained step response. Between 5 and 20 cycles/s, the circuit entrains well to the input, exhibiting one saccadic response to each sinusoidal input. Within this range there are systematic frequency-dependent changes in movement amplitudes, including super-normal saccades at some input frequencies. Above 20 cycles/s, the saccadic circuit increasingly exhibits periodic failures at rates of 1:2 or higher. In addition, the circuit exhibits predictable amplitude-modulated interference patterns in response to a combined step and frequency-modulated input. These experimental results provide insight into several biological mechanisms and serve as benchmark tests of viable models of the saccadic system. The data are consistent with negative feedback models of the saccadic system that operate as a displacement controller and inconsistent with theories that put the superior colliculus within the lowest-order, local feedback loop. The data support theories that the circuit feedback operates with dynamics that simulate a "leaky integrator." In addition, the results demonstrate how the temporal output of the superior colliculus interacts with recurrent inhibition to influence the eye movement dynamics.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Saccades are the high-velocity, conjugate eye movements used by primates and many other vertebrates to realign the angle of regard for high-acuity central vision. Saccades are so fast that the relatively long latency of visual responses in the brain excludes the possibility that vision can supply dynamic feedback during the movements. Thus either saccades are feedforward ballistic movements or there must be a short-latency "local" feedback mechanism (Robinson 1975). It is now recognized that saccades are dynamically compensated during mid-flight experimental perturbations (Brown and Glimcher 2000; Keller 1977; Sparks and Mays 1983; Sparks et al. 1987), thus proving that the movements are not ballistic. Most current models of the saccadic control system assume that the movements are driven dynamically by a signal that compares the actual eye position with a desired eye position (e.g., Arai et al. 1994; Breznen and Gnadt 1997; Gancarz and Grossberg 1998; Lefèvre and Galiana 1992; Massone 1994; Moschovakis 1994; Optican 1995; van Opstal and Kappen 1993). According to this local feedback concept, the saccadic circuit generates an output proportional to the size of this difference, which is called motor error. This output is then progressively reduced by dynamic negative feedback of a copy of the neural command to the eyes, a nonsensory re-afferent signal known as efference copy.

It is also well accepted that the saccadic circuit operates as a displacement controller (Jürgens et al. 1981); that is, one that generates a change-in-position largely independent of the initial starting position. According to this theory, the feedback loop must accumulate, or "integrate," the movement trajectory as a re-afferent signal. To accept a new input command for the next movement, this feedback signal must return, or reset, back to the initial null state. Several labs have presented evidence that the reset occurs relatively slowly (Breznen et al. 1996; Kustov and Robinson 1995; Nichols and Sparks 1995). Nichols and Sparks (1995) assumed the discharge process was exponential and estimated a decay constant of ~45 ms. We present new data here that supports this "leaky integrator" concept but that suggest more complex decay dynamics.

Numerous physiological experiments have studied the constituent neurons of the saccadic circuit during natural behavior (for review, see Moschovakis et al. 1996; Sparks and Mays 1990; Wurtz and Optican 1994). This has provided an effective circuit diagram for the saccadic circuit and has served to investigate signal transformations by sequential description of neuron activity within the circuit. For part of our recent studies, we have adopted a separate and complementary strategy to understanding this circuit---a strategy based on engineering systems control theory (Breznen and Gnadt 1997; Breznen et al. 1996; Gnadt et al. 1997; Jackson and Gnadt 1998; Jackson et al. 2001). By injecting defined inputs at critical points within the circuit, one can test explicit predictions of control system organization by a thorough study of the circuit's dynamics under experimentally controlled conditions (Breznen and Gnadt 1997; Jackson et al. 2001). From systems control theory there are three well-defined, characteristic responses to consider (Phillips and Harbor 1991): the impulse response, the reaction to a large and brief input pulse, which reveals the system's initial conditions; the step response, the reaction to an instantaneous step from one input value to a new steady-state value, which describes the system's response to a sudden change in input; and the frequency response, which yields the system dynamics to a prolonged sinusoidal input. Quantitative study of these characteristic responses in vivo provides benchmark tests for hypothetical models and mechanisms of the biological circuit.

Characteristic responses of generalized recurrent feedback systems

We have presented analytical and numerical considerations of the characteristic responses as they apply to the saccadic circuit elsewhere (Breznen and Gnadt 1997; Jackson et al. 2001). Briefly, we can summarize that in response to a sudden perturbation (the impulse response), a generic recurrent feedback system will briefly oscillate at the system's damped natural frequency as it settles back to the null state. When the system input suddenly changes to a new steady-state value (the step response), a negative feedback system will shift to a new output with dynamics determined by its damping ratio, zeta . Critically damped systems (zeta  = 1.0) approach their new steady-state value with an exponential time course. Underdamped systems (zeta  < 1.0) tend to be unstable, exhibiting a periodic series of oscillatory dynamic overshoots at the damped natural frequency as it settles to the new output level. Overdamped systems (zeta  > 1.0) are well behaved but can be sluggish when zeta   1.0. In response to sustained sinusoidally modulated input (the frequency response), a linear feedback system responds as a delayed sinusoid, a phenomenon called entrainment. However, when the input modulation matches the system's natural frequency, the input sums in phase with the natural tendency to oscillate and produces a kind of constructive interference. Thus near the system's natural frequency the output amplitude becomes very large, a phenomenon known as resonance. Another form of constructive/destructive interference occurs by simultaneously invoking a step response added to a frequency response. Each input function produces an oscillatory response that collides with the other to produce a frequency- and phase-dependent amplitude beating in the output.

Therefore from purely analytical considerations, there are several general properties that can be investigated empirically in the saccadic circuit of monkeys. Importantly, unlike traditional behavioral neurophysiology, this experimental strategy is not limited to the range of behaviors available by volitional activation and can reveal system properties not apparent during normal behavior. This systems dynamics methodology is particularly useful for motor circuits, which generate machine-like outputs. However, it is important to understand that although we borrow techniques from linear systems tools, this strategy is not a linear systems analysis. Analytical solutions are convenient because they generalize, but nonlinear (i.e., biological) systems can only be studied by experimental and numerical solution throughout the entire range of operation, a strategy sometimes known as "reverse engineering." In terms of experimental biology, the technique allows us to reveal biological mechanisms of a mammalian brain circuit in situ with a level of experimental control usually available only in reduced preparations.

Characteristic responses of the saccadic system in monkeys

Experimentally, the saccadic circuit in monkeys fails to exhibit an impulse response. Unlike an ideal linear system, single pulses of stimulation applied at the descending inputs to the saccadic circuit (i.e., superior colliculus, frontal eye fields) do not produce any movement response. This deviation from ideal behavior reveals an important biological mechanism for the initial state of the saccadic circuit. We now know that the saccadic circuit is "latched" into quiescence between movements by a tonic inhibitory input from so-called omnipause neurons (OPN) (Curthoys et al. 1984; Keller 1974). Thus saccadic movements are thought to be triggered by the sudden pause of this inhibitory latch. The absence of the impulse response reveals that single pulses of stimulation are inadequate to inhibit the pause cells and thus release the saccade generator into action.

Experimental assessment of the step response in monkeys is available from Breznen et al. (1996). Prolonged constant stimulation at the superior colliculus produces a periodic series of saccade-like movements. Thus the eyes move along in a ratchet-like fashion, a behavior that has been described as "staircase saccades" (Robinson 1972; Stryker and Schiller 1975). These periodic movements to a sustained input in fact are predicted from systems control considerations of recurrent feedback systems. However, we also showed that the biological circuit behaved in ways not captured by contemporary feedback models of the saccadic mechanisms (Breznen and Gnadt 1997). At progressively higher stimulation frequencies, the periodic movements squeezed closer together in time and became smaller as the interval between movements decreased. Analytical considerations showed that the frequency of movements would be dependent on the input frequency only if the gains and time constants of the system change dynamically during the movements. In addition, the smaller movements with shorter inter-movement intervals would be expected if the recurrent feedback loop acted like a leaky integrator, where reset of the accumulated feedback activity would be accomplished gradually by the biologically plausible mechanism of a slowly adapting relaxation.

This paper continues this reverse engineering approach by reporting the frequency response of the saccadic circuit in monkeys. Since monkeys' neurons operate with a frequency of action potentials, we applied input at the superior colliculus using a frequency-modulated (FM) stream of stimulation pulses. The analytical considerations and a quantitative analysis of our systems dynamics model are presented elsewhere (Jackson et al. 2001), but these experimental results in the biological system serve as a comparison to essentially all published models that are explicit enough to allow quantitative comparison. This report also provides some new experimental data for the step response that is used in comparison with the frequency response. These characteristic responses must be accounted for in any viable model of the saccadic system and serve to offer new predictions at the cellular and systems integration levels for experimental verification.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Two Rhesus monkeys (Macaca mulatta) were trained to fixate and follow small visual targets on a tangent screen at a distance of 64 cm. Targets were small spots (0.1° visual angle) back-projected from an oscilloscope driven by a custom-made, vector-plotting driver on a PC lab computer. Subjects were implanted with scleral eye coils used for measuring eye position by means of the magnetic field coil technique. According to standard methods, the lab computer recorded the positions of both eyes at 500 Hz by monitoring the magnitude of the current induced in the eye coils with amplifiers phase-locked to the oscillating magnetic fields surrounding the monkeys. To compare movements of different directions at different stimulation sites, the positions of the eyes at each 2-ms time sample were calculated as a direction-independent trajectory using the geometric mean of the horizontal and vertical position in Cartesian coordinates. The eye speed was calculated by time differentiating the trajectory. The monkeys' eye movement behavior was maintained by rewarding the subjects with small sips of fruit juice when the computer detected that they had properly acquired and tracked the visual targets.

During experimental sessions, the monkey's head was restrained by clamping a small stainless steel head post to an attachment on the primate chair. The head posts were mounted to the calvarium using a cap of bone cement and surgical bone screws. The head cap also included a stainless steel recording cylinder mounted over a craniotomy positioned over the right superior colliculus. During daily experiments, insulated tungsten microelectrodes (Microprobe) were introduced painlessly into the brain by replacing a sterile delrin plug from the recording cylinder with a custom-designed X/Y micropositioner fitted with a hydraulic microdrive. All experimental protocols were performed according to the National Institutes of Health guide for the care and use of laboratory animals and were consistent with the principles approved by the Council of the American Physiological Society.

Neurophysiological methods

From transdural penetrations of the microelectrodes, the superior colliculus was identified according to its stereotaxic location and characteristic visual and motor activity. For microstimulation, we used low-impedance electrodes (15-150 kOmega ) and 0.15-ms biphasic, constant-current electrical pulses. Stimulation sites were selected at the lowest-threshold depth within the tectum (usually ~2.5 mm below the top of the superior colliculus) within the systematic map of eye movement metrics within the primate superior colliculus. Because our goal was to elicit several sequential movements from our stimulations, we had to restrict our studies to relatively small movements elicited from the anterior 1/3 of the collicular motor map. Movements larger than 15° would have exceeded the 40° range of the eye monitoring system by the third sequential movement. This also served to minimize the recruitment of head movements that we might expect if the subject's head was unrestrained (Freedman et al. 1996). Stimulation currents were presented at 10-50 µA.

Input patterns of collicular stimulation

Trains of stimulus pulses were delivered to the superior colliculus as one of three different stimulus functions: a step input (STEP; instantaneous change in pulse rate from zero to a new steady-state rate, usually 200, 400, or 600 pulses/s); a sinusoidal, frequency-modulated (FM) input (peak pulse rate of 600 pulses/s at FM values from 0.25 to 25 cycles/s); or a linear combination (STEP + FM) of a STEP input (400 pulses/s) and an FM input (400 pulse/s peak-to-peak modulation at 5-25 cycles/s). During recording sessions, different stimulus patterns were presented in randomly interleaved short blocks of trials. Figure 1 shows a schematic example of each stimulus condition. The STEP example shows a train of pulses with a constant rate of 400 pulses/s. The FM example illustrates a train of pulses with a FM of 1 cycle/s. Because the stimulus input is pulsatile, it can only approximate the continuously variable, idealized input function. Furthermore, the instantaneous frequency function is half-wave rectified at zero pulses/s. The STEP + FM example demonstrates a train of stimulus pulses that is the sum of a STEP input at 400 pulses/s and a 400-pulse/s peak-to-peak FM input at 1 cycle/s. The resulting instantaneous pulse frequency produces a function that oscillates between 200 and 600 pulses/s with an RMS (root-mean-square) average of 400 pulses/s.



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Fig. 1. Schematic diagram of the 3 input functions. The Step input is an instantaneous change of input at time 0 to a new constant value. The microstimulation pulses are indicated by event markers as a train of pulses as a function of time. The histogram below the pulse raster indicates the instantaneous pulse frequency (1/inter-pulse interval) as a function of time. The frequency-modulated (FM) input is a sinusoidal modulation of instantaneous pulse frequency with a peak frequency of 600 pulses/s. Below the pulse raster is a plot of the instantaneous frequency for this train of pulses () superimposed on an idealized, sinusoidal FM signal (). For obvious reasons, the instantaneous frequency of the stimulation pulses is truncated to only positive values at 0 pulses/s. The Step + FM input shows the stimulus pulse pattern for a 400-pulse/s step function added to a 400-pulse/s (peak-to-peak) FM function. The plot below the pulse raster compares the actual instantaneous FM of the stimulus pulses () against the idealized STEP + FM input function (). So that the stimulation events (pulses/s) can be seen on the same time scale as the FM (cycles/s), the pulse rasters shown are 1/10 the actual pulse rate.

For the FM input and the STEP + FM input, the critical values for experimental control were stimulus current per pulse (µA), the peak instantaneous frequency of the sinusoid (pulses/s), and the modulation rate, FM (cycles/s). The stimulus current was held constant for a given site at values that varied from 1.2 to 3.5 times the threshold for saccades at that site. For the FM input functions, we always set the value of the peak instantaneous frequency to 600 pulses/s. This applied a similar range of pulse frequencies to that we had used previously to study the step response from the superior colliculus (Breznen et al. 1996). Furthermore, we can compare the frequency response for this sinusoidal input to STEP values of 200 pulses/s (which is approximately the RMS time-averaged value of the half-wave rectified sinusoidal function) and to 600 pulses/s (which is a peak value of stimulus frequency within the upper limit of what collicular burst neurons should be able to follow) (Sparks et al. 1976). For similar reasons, we used an FM between 200 and 600 pulses/s for all the STEP + FM stimulations. This 200- to 600-pulse/s range not only avoids the half-wave rectification at 0 pulses/s but also operates above threshold for producing saccades from collicular stimulation. This presumably avoids the nonlinearity of having the brain stem omnipause neurons cycle ON-OFF during each movement. This function also has an RMS value comparable to a 400-pulse/s step.

Behavioral conditions

Three different behavioral conditions were employed during the collicular stimulation: stimulation during active fixation; postfixation stimulation; and stimulation during a no-task condition. Most of the data were obtained in the active fixation condition while the subjects engaged a visual target in a dimly lit room. Stimulations were randomly applied to roughly 2/3 of the trials. As long as the subjects performed adequately, they received juice reward at the end of each trial regardless of whether the eyes were on the visual target at the end of a the trial. This task-driven condition provided for favorable behavioral motivation and allowed maximal control over the starting position, which was varied among positions in the far ipsilateral field. For data analysis, when we encountered obvious examples of voluntary eye movements during the train of stimulus-induced movements (e.g., inconsistent movements of grossly different direction and amplitude, especially at a latency of 200-250 ms from the start of stimulation), we excluded that trial from analysis. However, we attempted to avoid investigator-influenced selection of data trials and instead relied on analysis of multiple examples of each stimulus condition to minimize the effects of occasional contamination of the stimulus-induced movements with voluntary movements.

In the postfixation condition, we timed the collicular stimulation to occur within 200 ms of the extinction of a visual target that positioned the eyes in the far ipsilateral field. During the stimulation, the room was completely dark, although it is likely that the subjects experienced a persistent after-image at the fovea. In the no-task condition, we stimulated eye movements while the subjects sat in a completely dark room with no active task demands.

Data analysis

Off-line computer analysis of the eye movements was accomplished interactively by displaying the trajectory and eye speed of each series of stimulus-induced movements along with the pattern of stimulation pulses. Each movement was selected using an investigator-controlled cursor and a computer algorithm determined the beginning and end of each movement using velocity thresholds adjusted for the relative sizes of the movements (35-60°/s for larger movements; 15-25°/s for small movements). For each movement, the computer calculated the position (0.1° resolution), time (2-ms resolution), and eye speed (cutoff frequency at 35 Hz, -20 dB) for the beginning and ending of the movement and for the time of the peak eye speed. From electrical transients of the stimulating electrode, stimulus pulses were recorded by a window discriminator at a resolution of 0.1 ms.

Statistical comparisons between experimental conditions were evaluated using the ANOVA. Post hoc comparisons between groups within a condition employed the analysis of Scheffe (Sokal and Rohlf 1981). Nonlinear regression was accomplished by iterative minimization of least-squares error and tested for significance by the ANOVA. For sinusoidal curve fitting, the cycle phase was a circular parameter space that was constrained to one cycle of 2pi radians.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

We recorded data from 63 stimulation sites in the right superior colliculi of two monkeys. Because the usual task demands were that the subject must accurately look to the target for reward, there were clear signs that the monkeys would attempt to counteract the stimulus-induced movements for the initial presentations. However, with a few days of training with stimulation, each subject appeared to accept the stimulus-induced movements without actively fighting them. If the target was still visible at the end of the stimulated movements, the subjects would usually attempt to reacquire the target with voluntary movements after a latency of >= 200 ms. In the no-task behavioral paradigm, the subjects tended to become drowsy, and one subject had a large, confounding dark nystagmus. This was reduced by the postfixation stimulation paradigm. However, we found no evidence that the various task demands changed the patterns of movements elicited by the collicular stimulation. Thus most of the data were collected using the active fixation task, and the analyses are combined here without regard to the task state at the time of data collection.

The eye-movement metrics characteristic for each site within the collicular motor map were determined from the average of the first movement for many stimulations at 400 pulses/s. Some sites were tested for values between 200 and 800 pulses/s at 100-pulse/s increments or some subset of that range. Figure 2A shows a typical example of the "staircase" step response. The first characteristic movement was larger than subsequent movements. As reported previously (Breznen et al. 1996), the size reduction for the subsequent movements and the frequency of the continuing movements were determined by the current and frequency of stimulation pulses. For most experiments, we kept the stimulation parameters between 1.25 and 1.5 times threshold. This produced reliable movement patterns with movement frequencies of 15-23 movements/s, which represents the circuit's damped natural frequency at those stimulation parameters. We specifically avoided high-stimulation parameters that could cause rapid small movements that ran together as a continuous movement of the eyes (Breznen et al. 1996).



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Fig. 2. The STEP response. A: a typical example of the "staircase" pattern of eye position (Eye Pos) and the corresponding eye speed profiles in response to the STEP stimulation function (Stim). B: movement amplitude as a function of inter-movement interval (period between peak eye speeds) from 7 representative stimulation sites with STEP input values from 200 to 700 pulse/s at 100-pulse/s increments.

Because the inter-movement intervals decrease with increasing stimulation frequency (Breznen et al. 1996), we plotted the movement amplitudes as a function of time interval between saccades for seven stimulation sites with a complete set of stimulation rates from 200 to 800 pulses/s. This allowed us to fit this representative sample of stimulation sites with a regression analysis for comparison to the estimated 45-ms time constant of decay suggested by Nichols and Sparks (1995). The regression fit an exponential curve to the data
<IT>y</IT><IT>=amplitude ∗ </IT>[<IT>1−exp</IT><SUP><IT>−</IT>((<IT>interval+delay</IT>)<IT>/tau</IT>)</SUP>] (1)
where amplitude was the average asymptotic movement size for all seven stimulation sites, interval is the time period between movements measured as the time between peak eye speeds (Breznen et al. 1996), and delay and tau are the parameters of the decay. The regression settled to the following values: amplitude = 4.9 ± 0.14°; delay = 26.6 ± 1.13 ms; and tau = 18.7 ± 2.06 ms (means ± SD) with a correlation coefficient of r2 = 0.859.

To confirm this surprisingly low value of tau, we sought to apply a similar analysis individually to all stimulation sites. However, unlike the experiments of Nichols and Sparks (1995) and Kruskov and Robinson (1995), we had no explicit experimental control over the inter-saccadic intervals. Our analysis was dependent on whatever intervals occurred for the range of stimulation frequencies for the step input, which were often relatively long (>50 ms) and of limited range. We found 41 stimulation sites that produced a sufficient number of movements with inter-movement latencies down to <50 ms. Two examples of the fitting procedure are shown in Fig. 3, A and B, one example from each subject. The example in Fig. 3A was selected specifically because it closely approximated the 45-ms value estimated by Nichols and Sparks. The second example in Fig. 3B was selected as a more typical example of the entire sample for its value of tau (12 ms) and its variance. Histograms of the calculated values of tau from all 41 samples are shown in Fig. 3C. Clearly, most estimates of tau were below 25 ms. The mean and standard deviation of tau from this analysis was 26 ± 27.4 ms; the median value was 16 ms. The same parameters for the delay parameter was 27 ± 7.8 ms and 28 ms. The mean and standard deviation of the correlation coefficient for the regressions was r2 = 0.883 ± 0.062. The issue of fitting these data with a static time constant is considered in further detail in DISCUSSION.



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Fig. 3. Curve fitting for the time constant, tau, of a hypothetical leaky integrator. A: 1 example from a stimulation site that yielded a time constant of 49 ms. B: another example from a site that yielded a time constant of 12 ms. C: histogram of the distribution for values of tau.

Figure 4 shows the characteristic movement directions and amplitudes for all of the stimulation sites. Because stimulations were made in the right superior colliculus, all movements were directed through some angle toward the left. As is characteristic of collicular stimulation, all movements were conjugate and affected little by the originating eye position. Additionally, the direction for each subsequent movement was always the same as the characteristic movement for that stimulation site. To maximize use of the recording range, most stimulations began with the eyes directed >= 15° to the right. Figure 4, bottom, provides a frequency histogram of the movement amplitudes for the entire sample. Movement amplitudes ranged from a minimum of 0.5° to a maximum of 15.4°. The majority of the stimulation sites produced movement sizes between 1 and 4° of rotation (mean = 3.2°), which routinely allowed for the investigation of six or more sequential movements per stimulation over a time span of >= 300 ms.



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Fig. 4. Distribution of characteristic movements. Top and middle: the direction and amplitude of the characteristic movement for each stimulation site is shown as a line from the origin of the polar plot to the average spot of the endpoint of the first movements in the response to a 400-pulse/s STEP function. The relatively smaller movements (<4°) are shown at the top; larger movements (4-16°) are illustrated in the middle panel on an expanded amplitude scale. Bottom: histogram shows the distribution of movement amplitudes from the entire data set.

"Main sequence" analysis

Figure 5 plots the peak eye speed as a function of movement amplitude, the so called "main sequence" analysis, for large numbers of stimulated movements from each subject. Normal saccades are known to fall along a standard curve (the main sequence) that is characteristic for the saccadic modality of eye movement. Data from the step response for each subject are shown in Fig. 5, A and D. So that we could perform statistical regressions, the data were fit with a simple power function (---)
peak eye speed=<IT>K</IT>(<IT>movement amplitude</IT>)<SUP><IT>P</IT></SUP> (2)
For comparison, each graph also includes the plot from the corresponding monkey for the normal main sequence (- - -) obtained similarly from voluntary saccades. For clarity, the constituent data points for the voluntary saccades are not shown. While there was considerable overlap between voluntary and stimulated movements, inspection reveals that the majority of data points from the STEP response fell slightly below the main sequence curve. The regression curve for the stimulation data also tended to lie below the main sequence curve. One of the two free parameters of the regression curves (K, P) was significantly different between each pair of curves (t-test, P < 0.05).



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Fig. 5. Main sequence analysis. The peak eye speed is plotted as a function of movement amplitude for large numbers of movements stimulated with the STEP function (A and D), the FM function (B and E), and the STEP + FM function (C and F). Left: data from monkey 148; right: data from monkey 987. Each data set is fit by a power function (---), which can be compared with the main sequence from a sample of normal, voluntary movements (- - -) for each subject. Values for the regression curves are given in Table 1.

We also show main sequence curves for the movements created during the FM (Fig. 5, B and E), and the STEP + FM stimulation (Fig. 5, C and F). Table 1 compiles the values of the parameters for the regressions by monkey and by stimulation function. As in the preceding text, one of the parameters for the regression curves (K, P) was significantly different from the normal main sequence (P < 0.05) for each condition.


                              
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Table 1. Main sequence, regression parameters

Frequency response

Because of system transients, the frequency response reaches steady-state only after a period of three to four time constants. For the saccadic system, this would be at times >200 ms of stimulation. Due to the finite range of ocular movements and the 40° range of the recording system, it was not practical in our experiments to exclude the initial 200 ms of stimulation. For example, the largest movements within our database (15.4°) would exceed the recording range within only three movements in a time period of <300 ms (see Fig. 6B). However, by using relatively small movements, we were able to study long sequences of eye movements over time periods even longer than 1,000 ms. Thus our analysis included the behavior from the beginning of the stimulation for all sites and included at least an equal period of presumably steady-state behavior (>200 ms) for most of the sites.



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Fig. 6. Examples of the frequency response. Representative examples of the frequency response at FM values of 20 cycles/s (top), 10 cycles/s (middle), and 1 cycle/s (bottom). A: stimulation site 829 from monkey 148. B: stimulation site 1003 from monkey 987. Each panel shows the stimulus function as instantaneous frequency of stimulation pulses (Stim Freq, pulses/s), the eye position (Eye Pos), and the Eye Speed as a function of time on the abscissa (100 ms/div). Scale bar = 10° (Eye Pos) and 100°/s (Eye Speed).

Two examples of the frequency response, one from each subject, are shown in Fig. 6. Figure 6B was selected specifically because it was the largest characteristic movement from our sample. Figure 6A demonstrates how smaller movements allow the analysis of a longer sequence of many stimulated movements. The top panels show the results of FM stimulation at 20 cycles/s. As predicted, the saccadic circuit was entrained to the FM cycle, producing one movement per cycle. The middle panels of each column show the response to an input of 10 cycles/s. At this FM, the saccades were also entrained to the input frequency. However, note that the amplitudes of the movements, and consequently their peak velocities, were generally larger than those at an FM of 20 cycles/s, even though the peak stimulation frequency was the same. Finally, the bottom panels show the monkeys response to an input FM of 1 cycle/s. At this input frequency, the stimulation produced one characteristic saccade followed by a series of small, rapid movements similar to that seen for a step response.

Entrainment

Figure 7, A and B, show the entrainment function for each of the stimulation sites in Fig. 6 by plotting the response frequency (inverse of the inter-movement interval) as a function of the input frequency (FM) for all the input frequencies tested at that stimulation site. Figure 7C compiles the data points from all stimulation sites at all input frequencies, thus showing the overall entrainment for the entire data set. Entrainment is reliable for input values from 5 up to 20 cycles/s where the distribution of data fell very densely along the unit slope. This represents entrainment of one movement per stimulation cycle. Data samples that fell near the slope of 1:2 or 1:3 demonstrate examples where the subject produced only one response for two or three input cycles, respectively; so called periodic failures. Data points above the 2:1 slope (- - -) represent movements that were doublets, triplets, or higher for each input cycle. The unstable behavior at very low input modulation, as shown at the bottom of Fig. 6, produced the scatter of data points well above the - - - at low input frequencies.



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Fig. 7. Entrainment functions plotted as the output frequency (1/inter-movement interval) as a function of input frequency (FM, cycles/s). A: data from same stimulation site as Fig. 6A (stimulation site 829). B: data from same stimulation site as Fig. 6B (stimulation site 1003). C: data from all stimulation sites in both subjects (n = 9,184). 1:1 = unit slope of 1-to-1 entrainment; 1:2 = 1-in-2 periodic failures; 1:3 = 1-in-3 periodic failures; 2:1 = doubling slope and higher multiples (above - - -). To facilitate the viewing of overlying data points,  are plotted with a small amount of random jitter. For high-frequency responses (>15 movements/s), the sampling interval for eye position (2 ms) causes a noncontinuous appearance to the distribution of data.

Resonance

At 22 of the stimulation sites, we evaluated resonance of the saccadic circuit by plotting the average peak eye speed of all stimulus-induced movements as a function of the FM input frequency at closely spaced intervals through the range of 0.25-25 cycles/s. These data included both the first characteristic movement for the stimulation site and the various-sized subsequent movements during the sustained stimulation. Because eye speed is well correlated (Cullen and Guitton 1997) with the firing rate of the brain stem burst neurons, the peak eye speed is a good measure of the actual neuronal output of the circuit. To compare the different peak eye speeds from different sites (due to the different sizes of movements), we normalized the data to the peak eye speed of the characteristic saccade from that site. Our standard STEP input was 400 pulses/s, which was used at all but one site that had data only from 200 pulse/s. However, for completeness, we also compared the normalization to STEP frequencies of 200 pulses/s (RMS time-average of a half-wave rectified sinusoid) and 600 pulse/s STEP response (thus accounting for the possibility that we could have produced large movements simply by having higher peak stimulation rates). Two stimulation sites that lacked 600-pulses/s reference data and two sites that lacked 200pulses/s data could not be included in those respective comparisons.

In Fig. 8, the column on the left plots the average peak eye speed as a function of FM input frequency (cycles/s) for each of the 22 stimulation sites tested, normalized to the characteristic movement from the 400-pulse/s STEP. The data from the different sites are grouped into three categories as described in the following text. For the majority of the stimulation sites (13/22), there were systematically peaked curves for the range of input modulations from 0.25 up to 25 cycles/s (Fig. 8, A and B). Half of the curves (11/22) had the shape represented in Fig. 8A with a peak value in the range of 4 and 10 cycles/s. For these curves, the peak eye speed ratio varied from 0.94 to 2.04 with a mean of 1.36 ± 0.313, which is significantly >1.0 (t = 3.761, P < 0.004). Two other stimulation sites (Fig. 8B) clearly had peak eye speed peaks at FM values between 10 and 20 cycles/s with peak ratios of 1.57 and 2.28. The rest of the stimulation sites (9/22) shown in Fig. 8C yielded curves that showed no systematic peak. The average normalized peak eye speed for these sites ranged between values of 0.71 and 1.42 with a mean of 1.06 ± 0.240. The distribution of values from this group of 9 sites was not significantly different from 1.0 (t = 0.731).



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Fig. 8. Resonance. Normalized peak eye speed as a function of input frequency (FM, cycles/s). Left: data from individual stimulation sites are shown normalized to the characteristic movement from the 400-pulses/s STEP (with the exception, *, of the 1 site for which only the 200-pulse/s STEP was available). Data points from each stimulation site are connected by lines through the average values at each input frequency tested. Right: graphs show the average curves for the corresponding data from the left column, normalized for the 400 pulses/s STEP (- - -), 200 pulses/s STEP (---), and 600 pulses/s STEP ( · · · ). A: stimulation sites with peak input values below 10 cycles/s input functions. B: stimulation sites with peak input values above 10 cycles/s. C: stimulation sites with no clear single input peak. The 1 site marked with * had no normalization data available for 400-pulses/s STEP; this curve was normalized to the 200-pulse/s STEP.

Figure 8, right, shows the average curve for all the corresponding sites from each group to the left (400 pulse/s, - - -), and for normalization to 200 pulses/s (---) and 600 pulses/s ( · · · ). There were no substantial changes in the curves among the three different normalization factors.

Values of normalized peak velocity greater than 1.0 represent stimulation parameters that produced super-normal movements at that stimulation site. Nine of 11 sites in Fig. 8A and both of the sites in Fig. 8B demonstrated super-normal peak eye speeds that were input-frequency dependent.

Factors determining the shape of these curves are complex. With input modulations 3-5 cycles/s, the entrainment broke down (see Fig. 7), and there was a combination of large first movements and various numbers of smaller doublets, triplets, or sustained and unstable oscillations (see Fig. 6). As the input modulations approached values near 20 cycles/s, the period between entrained movements diminished to 50 ms. This is within the range of the system time response that has been estimated to lie at <= 60 ms (Breznen and Gnadt 1997; Nichols and Sparks 1995). Thus at the higher input frequencies, the circuit's feedback integrator begins to limit the size of sequential movements that progressively diminishes the peak eye speeds at higher input frequencies.

Interference patterns

If the periodic movements of the STEP and the FM stimulations are due to cyclic oscillations in the brain circuit, we should expect phase- and frequency-dependent interference patterns during a STEP + FM pattern of stimulation. An intuitively simple example of periodic interference is to consider the case of one oscillation that is half the frequency of the other. In this case, each period of the slower oscillation will alternately add to and subtract from each cycle of the faster oscillation. Thus this predicts a regular alternation between large and small movements. Figure 9 shows an example of this from one stimulation site. The top panel shows the "staircase" movements and the corresponding eye speeds produced by a 400-pulses/s STEP input. Following the first characteristic movement, subsequent movements are at first diminished in size with recovery of size and speed as the period between sequential movements increased slightly. The slight asymptotic buildup of movement size by the fourth movement was characteristic of near threshold stimulations for the STEP response in this subject.



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Fig. 9. Alternate wave cancellation for the STEP + FM input functions. Top: an individual example as a function of time (left) and an average response as a function of sequential movements (right) for the STEP function of 400 pulses/s from stimulation site 418. Middle: similar data from the STEP + FM input function at the same site with the FM phase set to 0. Bottom: similar data with FM phase adjusted to 180°. For the middle and bottom panels, the FM frequency was set to 10 cycles/s, which was approximately half of the frequency of movements stimulated by the STEP function alone (the damped natural frequency). Left: the stimulus input functions as instantaneous frequency, the eye position trace (Eye Pos), and the eye speed. Right: the average peak eye speed (±SE) as a function of sequential movement. Eye Pos scale bar = 5°; eye speed scale bar = 100°/s.

Figure 9, middle, shows the pattern of eye movements to a 10 cycles/s FM added to the STEP input (STEP + FM) with the phase of the FM adjusted to 0°. Note that the alternating pattern of movement amplitudes is easily discriminated in the eye speed profiles. To illustrate the reliability of this effect, the graph to the right plots the mean ± SE peak eye speeds for all the stimulations for this combination of FM frequency and phase. The peak eye speeds are plotted as a function of the movement sequence (1st, 2nd, 3rd, etc.), which can be compared with the similar graph the STEP response alone.

If this effect is indeed an interference phenomenon, we should be able to reverse the pattern of size alternations by shifting the FM phase by one-half cycle. This reversal of pattern is shown in Fig. 9, bottom, as an individual trial and as the mean peak eye speeds. Note that the second, third, and fourth movements alternate in completely opposing patterns when the two input phases differ by 180°.

Interference phase

According to the theory of colliding circuit oscillations, the exact pattern of amplitude modulation is dependent on the phase between the two oscillating processes. Because the FM input pattern is sinusoidal, this predicts that the amplitude modulation of each sequential movement should change sinusoidally as we systematically shifted the input phase through 360°. For the special case of input frequencies that are an integer fraction of the damped natural frequency (e.g., 1/2, 1/3, 1/4), the maximum interference will alternate with each second, third, or fourth sequential movement, respectively.

Figure 10 shows another example of this phase dependence. This example has an FM frequency of 10 cycles/s that is close to one-half of the damped natural frequency for the STEP response. Like the example in Fig. 9, the first four or five movements cycled through an alternating pattern of movement amplitudes for each input phase.



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Fig. 10. FM phase and the STEP + FM input functions. Left: graph shows the average profile of movement amplitudes as a function of movement sequence at each of 8 different input phases. The FM input frequency was 10 cycles/s at stimulation site 420. Right: the same data from the 1st, 3rd, 5th, and 7th movements are shown with regression to a sinusoidal function of input phase as described in the text. The sinusoidal component was significant (P < 0.05) for movements of sequence 1-6 and not significant for sequence = 7.

To test for the sinusoidal phase-dependent AM, we fit the movement amplitudes at each movement sequence using a regression of the form
amplitude=<IT>K</IT><IT> ∗ </IT>[<IT>sin </IT>(<IT>FM phase+&thgr;</IT>)]<IT>+</IT><IT>C</IT> (3)
The term K is the coefficient for the magnitude of the AM; theta  is the value of the phase shift, and C is a constant that gives the average movement amplitude for all phases at that movement sequence. The sinusoidal modulation was considered statistically significant if the 95% confidence interval for the estimate of K excluded the value of zero. For illustration, examples of the fitting procedure for the first, third, fifth, and seventh movements are shown in Fig. 10, right. The regressions accounted for roughly 90% of the sample variance (mean r2 = 0.893 ± 0.032) and interference patterns from the first through sixth movements were reliably modulated (P < 0.05). If the input frequency was exactly half a stationary damped natural frequency, the value of theta  would alternate exactly by 180° from one movement to the next. The average phase shift and within-sample standard deviation for these data (excluding sequence = 7) were 187 ± 18°.

Besides the intuitively obvious alternate wave pattern of the 1:2 ratio between FM frequency and damped natural frequency, we were able to obtain many other reliable patterns of interference using different combinations of FM frequency and phase. By way of example, Fig. 11 shows several patterns from other stimulation sites at various input frequencies and phases.



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Fig. 11. Interference patterns for 4 representative examples at different input frequencies for the STEP + FM stimulation. Each graph plots the average movement amplitude as a function of movement sequence at each input phase tested. FM = 5 is from stimulation site 416; FM = 12.5 is from site 1014; FM = 15 is from site 401; and FM = 25 is from site 421. Similar to Fig. 11, each movement sequence was submitted to a sinusoidal regression. Regression results are summarized in Table 2 as the proportion of significant sinusoidal fits out of the movement sequences for each input frequency for all the STEP + FM stimulation sites.

The example for FM = 5 of Fig. 11 demonstrates an interference pattern that is close to 1/4 that of the circuit's damped natural frequency. The shifting phase from one movement to the next can be seen graphically as the systematic ripples of troughs and peaks that run diagonally across the surface of the plotted lines. Figure 11 also provides examples taken from other stimulation sites at FM values of 12.5, 15, and 25 cycles/s. Each data set exhibits its own pattern of phase-related beating according to the relative ratios of the FM frequency and the oscillation frequency from the STEP response.

To get a population measure of the reliability for the effect of input phase, we made the same sinusoidal fit as illustrated in Fig. 10 to the AM for each movement sequence at each input frequency for 19 stimulation sites where we had complete data sets for four or more phases. Table 2 shows the number of significant sinusoidal fits as a fraction of the total number of sequential movements at that site and FM frequency. Of the total data set, all 19 stimulation sites had beating patterns that produced significant sinusoidal fits for one or more of the sequential movements. This effectively excludes the null hypothesis that the AM was random and supports the premise that the beating patterns should be sinusoidal.


                              
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Table 2. Phase interference, summary of significant sinusoidal regressions


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Systems engineering considerations predict three general patterns of behavior in response to a frequency modulated input to the saccadic circuit: entrainment, resonance, and interference patterns. Because we analyze the frequency response in comparison to the step response, we also provide some new analyses from the step response that provide control data and address questions about the time response of the saccadic circuit.

Entrainment

We have shown that the saccadic circuit in monkeys entrains well to cyclic, sinusoidal input at the superior colliculus for input frequencies between 5 and 20 cycles/s. Below 5 cycles/s, the circuit increasingly becomes unstable and responds with rapid low-amplitude saccades similar to the step response (which is effectively the limit case of 0 cycles/s). Above input frequencies of 20 cycle/s, the circuit increasingly exhibits periodic failures of one movement response for two or more input cycles. Thus the biological circuit is effectively band-pass limited between 5 and 20 cycles/s of input. Obviously, we should expect entrainment from any causal relationship between collicular stimulation and movement of the eyes. However, several quantitative properties of the entrained movements and the entrainment are revealing.

The stimulated movements were conjugate and their dynamics conformed well to a saccade-like main sequence of eye speed profiles, regardless of which stimulation pattern was applied (Fig. 5). This means that the stimulation was engaging the saccadic circuit in a manner that was reasonably physiologic. Indeed, this demonstrates that the stimulation site must be prior to the "local" feedback mechanism. Stimulation at or after the feedback influence would override or bypass feedback effects and thus would more directly reflect the stimulation pattern in the output.

The circuit failures at low- and high-input frequencies can largely be attributed to influence from the nonlinear, latch-like behavior of the OPN. Between input frequencies of 5-20 cycles/s, we presume OPN are cyclically inhibited by the stimulation during each saccade-like movement. Once the OPN are turned off, the high-gain excitatory input to the brain stem burst neurons produce a burst pattern that moves the eyes in the high-speed saccadic profile, even as the output of the collicular stimulation may rise with a slower time course. The natural feedback mechanism distal to the superior colliculus ends the movement when the amplitude matches that characteristic for the site of stimulation in the collicular motor map. At FM inputs <3 cycles/s, the slowly modulated ON period of the input cycles hold the OPN in quiescence for a period of time that outlasts the initial movement. This allows the unlatched feedback circuit to cyclically reactivate as has already been described for the step response (Breznen and Gnadt 1997; Breznen et al. 1996). Thus the movement dynamics during both entrainment and the unstable "ringing" at low input frequencies conform to the saccade-like main sequence because they both engage the saccadic feedback circuit although by different trigger mechanisms.

The circuit can respond with high-frequency movements at rates well above 20 movements/s (see Fig. 7, <1 cycle/s input) (see also Breznen et al. 1996). However, the circuit is engaged quite differently during the periodic high-frequency input (>20 cycles/s). During the truncated negative half-phase of each input cycle (see Fig. 1), we presume the excitatory feedforward drive into the circuit is transiently removed and the inhibitory drive onto the OPN is withdrawn. When the following positive half-phase of the stimulation arrives, the circuit is in an unresponsive state that is refractory to renewed activation. Thus we suggest that the break down of entrainment at very low input modulations reveals the circuit dynamics in the absence of OPN influence; whereas the periodic failures at high-input modulations reveals the deactivation/reactivation dynamics of the OPN and the feedback decay dynamics.

We do note that the main sequence relationship for all the stimulated movements were slightly, but reliably, slower than normal voluntary saccades. This suggests that the collicular stimulations engage the saccadic circuit somewhat differently than volitional activation. One possibility is that electrical microstimulation recruits a smaller population of collicular output cells than does volitional activation. Another possibility is that the collicular stimulation fails to engage a parallel input from the frontal eye fields or some other structure that during normal behavior provides an additional drive. Yet another possibility is that the collicular stimulation engages a counteractive inhibitory drive that is not present in normal behavior. These studies cannot distinguish among these possible alternatives.

Resonance

As discussed in the INTRODUCTION, it is a natural property of negative feedback systems to resonate. However, the brain stem burst neurons obviously cannot express a negative firing rate and thus cannot realize the dynamic overshoot normally required for resonance (Jackson et al. 2001). Thus one might predict that the saccadic circuit should not resonate. For the 9 of 22 examples from our data where there was no frequency-dependent AM, this may indeed be the case. On the other hand, frequency-dependent AM seen in Fig. 8, A and B, are exactly the patterns predicted for small or moderate amounts of rebound overshoot, respectively (Jackson et al. 2001). It is not clear what caused this difference between stimulation sites. We could find no identifiable correlation with any parameters of the movements (e.g., size, direction, peak speed), parameters of the stimulation (e.g., stimulus current, stimulation depth) or of the behavioral task demands that could explain the difference between these groups. Nonetheless finding a resonance-like phenomenon in the saccadic circuit is highly provocative since it is an unusual circumstance to get super-normal movements from a single stimulation site in the superior colliculus.

For the possible biological mechanisms of this phenomenon, we can consider the potential sources of counterphase inhibition. One possibility would be cross-coupling from contralateral inhibitory burst neurons (Hikosaka and Kawakami 1977) for movements in the opposite direction. However, the eyes never reverse direction in these stimulations and the near normal, saccade-like velocity profiles argue against a substantial co-activation of antagonist movement directions. A more likely mechanism comes from consideration of the brain stem omnipause cells (OPN). If we assume that the OPNs are active during the truncated negative half-phase of the stimulation pattern, we can expect that they would produce counterphase inhibition in the brain stem burst neurons. While this sub-synaptic membrane hyperpolarization cannot be realized as a sub-zero firing rate, many neurons do exhibit postinhibitory rebound where the sudden release of hyperpolarization produces a rebound activation of a few action potentials (e.g., Ito and Oshima 1965). Presumably this would add a few extra spikes to the following positive half-phase of stimulation.

Thus perhaps the most biologically realistic mechanism to account for this finding would not come from a true system resonance but from a nonlinear subcellular rebound mechanism that simulates the modest resonance-like phenomenon. Interestingly, this same mechanism would presumably add one or two extra spikes to voluntary movements of all sizes and might account for the why it is so difficult to get saccadic models to simulate both large and small size movements. Although the form of the rebound phenomenon was different, we also note that Enderle and Engelken (1995) have suggested a postinhibitory rebound mechanism for pontine burst cell activity, and we know this process is expressed in the primate CNS (e.g., Ramcharan et al. 2000).

Interference patterns

We have now confirmed in monkeys our prediction for amplitude beating patterns from a combined step response and frequency response, including reversal of the pattern by a half cycle shift of input phase (Jackson et al. 2001). Periodic re-triggering of stereotypical saccades would not produce this result. We have shown that the phase and frequency dependence of the beating patterns are completely accounted for by simple interference mechanisms for colliding oscillations on a circuit downstream from the site of stimulation. Besides supporting our oscillation theory of operation for the saccadic circuit, this establishes that the circuit must experience a reasonable approximation of the FM pattern of stimulation and that the superior colliculus must not be within the lowest-level local feedback. If the FM of the STEP + FM patterns of stimulation were being substantially blunted, the resulting movements would have reverted to a constant STEP response. Instead, these amplitude modulations appear to reflect the moment-to-moment drive from competing inputs at the brain stem burst neurons. One of the inputs is derived from the modulated collicular output; the other comes from the recurrent, inhibitory influence of the circuit's feedback. Depending on the dynamic drive at the burst neurons, the resulting movement can be super-normal, sub-normal, or equal to the characteristic movement from that site. The alternative hypothesis that the amplitude modulation might be due to cyclic partial inhibition of the OPN would not seem consistent with the super-normal amplitudes or the sinusoidal phase-dependence (Fig. 10). Experiments are currently underway to test this alternative directly.

The issue of whether the superior colliculus is inside the amplitude-controlling feedback loop has at times been controversial. However, we point out that there is incontrovertible evidence that the superior colliculus is not a necessary element in the saccadic circuit (Schiller et al. 1980). Moreover, our data from the step response (Breznen et al. 1996) have offered results that are not consistent with having the superior colliculus within the lowest-order local feedback. If the stimulations in the superior colliculi were at, or distal to, the point where the feedback closes, we would predict entirely different results. For example, let us consider the results of microstimulation at points arguably more distal in the circuit. Applying long trains of stimulation pulses in the region of the pontine burst neurons produce constant-velocity, ramp-like, conjugate movements (Cohen and Komatsuzaki 1972). This indicates that the stimulation charged the output integrator without direct influence from feedback inhibition, presumably because the microstimulation overrode the feedback influence that would have occurred at, or prior to, the point of stimulation. On the other hand, brief stimulation of pontine burst cells in many cases can be compensated by interaction with voluntary saccades (Brown and Glimcher 2000; Sparks et al. 1987), which indicates that these stimulation-induced effects are available to the feedback mechanism. Thus unlike collicular stimulation, the pontine burst cells appear to be both inside the feedback loop and fairly directly connected to the output. To look even more distal in the circuit, consider stimulation of the circuit at the ocular nerves (Cooper and Eccles 1930). Stimulating at this point produces uncompensated monocular eye movements with first-order dynamics, indicating that this point in the circuit is distal to the conjugate mechanisms, the neural integrator and the local feedback loop.

Thus our finding of predictable periodic amplitude modulation for the STEP + FM stimulation lends further support to the hypothesis that the superior colliculus is not within the lowest level feedback of the saccadic circuit. We point out that while our results do indicate that the mechanism controlling the amplitude of the movements must lie distal to the superior colliculus, this does not contradict the possibility that the superior colliculus might receive some form of dynamic movement signals during the movement (e.g., Keller and Edelman 1994; Keller et al. 2000; Soetedjo et al. 1999).

System time response

Another sometimes controversial issue in oculomotor physiology has been whether the accumulation of activity in the feedback loop during a movement is reset nearly instantaneously or over a slower time course. Using "collisions" of closely spaced movements, several labs (Breznen et al. 1996; Kustov and Robinson 1995; Nichols and Sparks 1995) have shown that each saccade can produce residual effects on subsequent movements over a time course of several tens of milliseconds. When subsequent saccades follow an earlier saccade with a latency <100 ms, the second movements are systematically hypometric. This was taken as evidence that there was residual activity in the feedback mechanism at the time of the second saccade, thus arguing for a relatively slow reset for the feedback mechanism under natural circumstance. Nichols and Sparks (1995) suggested that the physiological discharge process was well described by an exponential decay with a time constant of ~45 ms and that the decay began at the completion of each movement. Others have suggested that the decline in feedback activity begins during the movement (Breznen and Gnadt 1997; Schlag et al. 1998). Furthermore we have argued that the circuit's time response is not static but instead decreases dynamically as a function of the circuit's own activity (Breznen and Gnadt 1997).

Thus to reconcile these findings, we fit a sample of our data for an exponential regression similar to that described by Nichols and Sparks (1995). Using a simple exponential regression, we found time constants with a median value of 16 ms and a mean of 26 ms, which is much shorter that the 45 ms estimated by Nichols and Sparks (1995). In fact, inspection of Figs. 2C and 3, A and B, suggests that the data points decline by a function steeper than exponential. This would be the expected result for lower values of a time "constant" at shorter inter-movement intervals where the circuit is being maximally driven. This is consistent with our analytical considerations of the step response (Breznen and Gnadt 1997) that the gains within the circuit, and thus the effective time constants (the system time response) must change dynamically as a function of the circuit's own activity. We suggest that the initial state of the feedback circuit at rest has a time constant close to 60 ms that adjusts downward depending on the dynamics of the ongoing movements. During the rapid multiple movements produced by high levels of microstimulation, the apparent time constant can decrease to values as low as 10 ms (Fig. 3). We have hypothesized that these gain dynamics could occur due to synaptic adaptation and/or spike frequency adaptation (Breznen and Gnadt 1997; Jackson et al. 2001). Certainly both of these mechanisms are common in vertebrate central nervous systems (e.g., Honig et al. 1983; Lansner et al. 1998). This prediction awaits experimental verification at the cellular level.

Assumptions and limitations of the microstimulation technique

Interpretation of these data is, of course, critically dependent on an understanding of the assumptions of the method and is constrained by the limitations of the microstimulation technique in general and in how we use it in particular.

Microstimulation recruits volleys of action potentials by direct depolarization of the excitable membrane, and the usual assumption is that axons have the lowest threshold for activation (Rank 1975; Rattay 1998). Thus synaptically mediated influences on excitability are largely superseded by the microstimulation technique at the site of stimulation. We have experimental verification of this from consideration of the results of the STEP + FM response (see preceding text). Presumably, our stimulation currents of 10 to 50 µA recruit axons for distances up to ~0.5 mm radius (Gamlin et al. 1989). However, we extrapolate from data of Glimcher and Sparks (1993) that the same self-normalizing recruitment of collicular neurons that occurs for voluntary movements must also occur for electrically stimulated saccades. Therefore we assume that intra-collicular mechanisms recruit a population of neurons across the tectal motor map with a diameter on the order of 2-3 mm. This assumption is basically consistent with the main sequence findings (Fig. 5).

We were careful to use instantaneous frequencies of pulses that were physiologically reasonable (<600 pulses/s). We must consider, however, the possibility that collicular output did not faithfully follow the input stimulus functions. It would be exceedingly difficult to confirm this by recording from identified output neurons while stimulating with trains of pulses. On the other hand, our results from the STEP + FM stimulation does prove that the temporal pattern of the collicular output makes predictable differences in the details of movement dynamics. An important aspect of this conclusion is that it makes unlikely the possibility that our microstimulation results can be explained by intra-collicular waves of activity (e.g., Munoz and Wurtz 1995; Munoz et al. 1991; Optican 1995). Especially, during the step response and low frequency FM, we can be confident that we would be disrupting intra-collicular waves by a sustained output from one site within the collicular motor map.

Regarding neuronal recruitment by microstimulation, we can be certain that the stimulation did not recruit only the output neurons of the superior colliculus. On the other hand, we can reasonably assume that collicular saccade-related outputs were effectively driven by the microstimulation. Our main sequence results are consistent with reasonably physiologic recruitment of the postcollicular saccadic circuit and inconsistent with recruitment of nonsaccadic subsystems, like smooth pursuit (Krauzlis et al. 1997; Missal et al. 1996) or collicular outputs that bypass the OPN latch (Grantyn et al. 1996). In fact, we assume that microstimulation recruits all functional types of cells within the radius of influence, including both the collicular saccade-related burst neurons and the so called build-up neurons. Thus these data are not consistent with the suggestion that functional differences between these two cell types are critical to determining the dynamics of saccadic movements (Munoz and Wurtz 1995; Optican 1995). A similar conclusion has been reached by other investigations as well (Anderson et al. 1998; Quaia et al. 1999).

To keep the multiple movements of these studies within the recording and oculomotor ranges, we had to limit this investigation to saccades of <= 15° from the anterior 1/3 of the collicular motor map. This has the advantage of testing the collicular stimulations in the range of gaze shifts where eye movements predominate over head contributions (Freedman et al. 1996). However, we have no reason to believe that the principles of collicular involvement for gaze control would be fundamentally different in this small to medium range than it would be for larger movements. Until more is known about the downstream mechanisms for coordination of eye and head components of gaze control, we address here only the saccade-specific issues that are undoubtedly concentrated in the more rostral colliculus.

Finally, these experimentally induced movements would be uninterpretable if contaminated with voluntary movements. First, we excluded trials from analysis that included the occasional, but obvious, interceding movements of wildly different direction, and we confirmed reliability using many repetitions of the same pattern of stimulation. The interference patterns from the STEP + FM stimulations were far more regular and in faster sequence than would be possible from usual behavioral response latencies, and they exactly matched the predictions from traditional systems control considerations of the input functions.

Thus we have considered that these microstimulation studies do not recruit brain circuits in the exact patterns that occur naturally. However, the fact that we can use microstimulation to engage brain circuits experimentally in systematic and predictable ways proves its usefulness as an experimental tool. The more we can produce patterns of eye movements in monkeys that are predicted by systems response considerations, the more we are fortified about the validity of these underlying assumptions.

Comparison of the empirical data to systems control models

First, we have shown that all movements produced by electrical microstimulation in the superior colliculus of the monkey can be accounted for by engaging the traditionally recognized saccadic control circuit and do not require nontraditional theories regarding engagement of smooth pursuit mechanisms (Krauzlis et al. 1997; Missal et al. 1996) or direct recruitment of ocular motor neurons (Grantyn et al. 1996). The progressive nature of the movements is consistent with the consensus schema of the saccadic circuit as displacement controller (e.g., Arai et al. 1994; Breznen and Gnadt 1997; Gancarz and Grossberg 1998; Jürgens et al. 1981; Lefèvre and Galiana 1992; Massone 1994; Moschovakis 1994; Optican 1995; van Opstal and Kappen 1993) and is not consistent with an end position controller that would drive the eyes to specified orbital positions (Robinson 1975). Quaia et al. (1999) have recently suggested a somewhat different form of the feedback mechanism. Their model incorporates a "spatial displacement integrator" through the cerebellum that serves the role of dynamic feedback. However, their model does not explicitly calculate dynamic motor error. Instead it "chokes off" a generalized directional drive from the colliculus when feedback from the displacement integrator matches the desired change in position. Despite this conceptual difference, the relatively longer delay of collicular output through the cerebellum, compared with a faster more direct path onto the brain stem burst cells, should produce some component of second-order dynamics similar to that of traditional saccadic feedback controllers (Breznen et al. 1997; Jackson et al. 2001).

Our findings argue that the superior colliculus acts as an input to a feedback mechanism that closes the low-level feedback loop distal to the superior colliculus (e.g., Breznen and Gnadt 1997; Gancarz and Grossberg 1998; Jürgens et al. 1981; Moschovakis 1994; Quaia et al. 1999), and contradicts suggestions that the superior colliculus is part of the "local" feedback mechanism (e.g., Arai et al. 1994; Lefèvre and Galiana 1992; Optican 1995; van Opstal and Kappen 1993; Waitzman et al. 1991). Of the collicular feedforward models, four have been shown to produce "staircase" saccades to prolonged stimulation: the MSH displacement controller of Moschovakis (1994), the modified Jürgens model of Breznen and Gnadt (1997), the FOVEATE model of Gancarz and Grossberg (1998), and the cerebellar model of Quaia et al. (1999). The MSH model was later modified by Bozis and Moschovakis (1998) to mimic the empirical findings of smaller movements with shorter inter-movement times (Breznen et al. 1996; Kustov and Robinson 1995; Nichols and Sparks 1995). The system dynamics model of Breznen and Gnadt (1997) also captured this behavior and suggested the adaptive mechanisms that would produce both the decreasing amplitudes and the decreasing movement intervals. Because the FOVEATE model (Gancarz and Grossberg 1998) includes the OPN as part of the feedback reset mechanism, it produces smooth movements at high stimulation rates due to tonic inhibition of OPN. However, it fails to produce the decreasing amplitudes and movement intervals at intermediate stimulation rates. The model of Bozis and Moschovakis (1998) does not produce individual movements 3.6° (where movements begin to run together) because it also incorporates the OPN as an obligatory part of the feedback reset. Instead these investigators invoked an extra-saccadic mechanism (Grantyn et al. 1996) to explain the relatively smooth movements seen at highest levels of stimulation. This explanation, however, stands in contradiction to the absence of an impulse response from superior colliculus stimulation (see discussion in the preceding text), which demonstrates that there is not a strong functional output of the superior colliculus that bypasses the inhibition of the OPN. Analysis for the dynamics of repeating movements to sustained collicular stimulation for the model of Quaia et al. (1999) is not currently available.

Our systems response data are not compatible with an "instantaneous" reset of the feedback circuit following each movement. Instead it appears that each movement produces residual effects on subsequent movements over a time period of up to several tens of milliseconds (Kustov and Robinson 1995; Nichols and Sparks 1995). Our data do not corroborate other labs' findings, which failed to confirm this slow time course of reset using behaviorally induced short-latency movements (Goossens and Van Opstal 1997) or mid-flight saccadic interruptions from brief stimulation of pause neurons (Keller et al. 1996). In consideration of this, we and others have made simulations (Bozis and Moschovakis 1998; Jackson and Gnadt 1998) that replicate the mid-flight interruption experiments despite the inclusion of a slowly decaying reset of the feedback signal. It turns out that the mid-flight interruptions are adequate tests of feedback reset dynamics only if the input drive to the circuit (e.g., from activity within the superior colliculus) is constant, which of course is not a valid assumption for volitionally induced movements (e.g., Anderson et al. 1998; Waitzman et al. 1991; also see Schlag et al. 1998).

Our data predict that the OPN are tonically inhibited during prolonged activation of the collicular output, such as during the step response (Breznen et al. 1996) or very slow FM (Fig. 6). Similarly, we predict that the OPN are tonically inhibited during the interference patterns produced for the STEP + FM response. Preliminary data appear to confirm this for the step response (Reusser et al. 1996). If this hypothesis is true, then we conclude that the OPN must play a permissive role for the saccadic circuit and not serve as an active reset. According to this theory, the natural slow decay of feedback activity would be masked by two events during volitional activation. For one, the normal re-activation of OPN at the end of the movement clamps the burst neuron output to quiescence distal to, and independent from, the feedback reset. At the same time, the excitatory drive from descending inputs like the superior colliculus is withdrawn by the naturally declining time course of activity (e.g., Waitzman et al. 1991). Only during the experimental manipulations provided by microstimulation (Breznen et al. 1996; Kustov and Robinson 1995; Nichols and Sparks 1995) (Figs. 2 and 3) is the time course of the feedback dynamics revealed. Thus these data are incompatible with models that incorporate the OPN as part of an obligatory reset mechanism (e.g., MSH model of Moschovakis 1994; Gancarz and Grossberg 1998) and instead support models that incorporate the OPN as a kind of permissive side loop of the circuit (e.g., WOMW model of Moschovakis 1994; Breznen et al. 1997; Quaia et al. 1999).


    ACKNOWLEDGMENTS

Many thanks go to J. Beyer, M. Fucito, and L. Corrigan for technical assistance and to J. Ganz for data analysis. We appreciate the original insights from B. Breznen in earlier collaborations for the step response.

This work was supported by National Eye Institute Grant EY-08217 to J. W. Gnadt and National Research Service Award fellowships (T32 NS-07371) to M. E. Jackson and O. Litvak.


    FOOTNOTES

Address for reprint requests: J. W. Gnadt (E-mail: jgnadt{at}sunysb.edu).

Received 27 July 2000; accepted in final form 27 April 2001.


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ABSTRACT
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0022-3077/01 $5.00 Copyright © 2001 The American Physiological Society