Cerebellar Flocculus and Paraflocculus Purkinje Cell Activity During Circular Pursuit in Monkey

H.-C. Leung,1 M. Suh,2 and R. E. Kettner3

 1Department of Diagnostic Radiology, Yale University School of Medicine, New Haven, Connecticut 06520;  2Department of Biomedical Engineering, Northwestern University, Evanston, Illinois 60208; and  3Department of Physiology and Neuroscience Institute, Northwestern University Medical School, Chicago, Illinois 60611


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Leung, H.-C., M. Suh, and R. E. Kettner. Cerebellar Flocculus and Paraflocculus Purkinje Cell Activity During Circular Pursuit in Monkey. J. Neurophysiol. 83: 13-30, 2000. Responses from 69 Purkinje cells in the flocculus and paraflocculus of two rhesus monkeys were studied during smooth pursuit of targets moving along circular trajectories and compared with responses during sinusoidal pursuit and fixation. A variety of interesting responses was observed during circular pursuit. Although some neurons fired most strongly in a single preferred direction during clockwise (CW) and counterclockwise (CCW) pursuit, others had directional preferences that changed with rotation direction. Some of these neurons showed similar modulation amplitudes during CW and CCW pursuit, whereas other neurons showed a preference for a particular rotation direction. Response specificity also was observed during sinusoidal pursuit. Some neurons showed responses that were much stronger during centrifugal pursuit, others showed a preference for centripetal pursuit, and still others showed responses during both centripetal and centrifugal motion. Differences in preferred response direction were sometimes observed for centripetal versus centrifugal pursuit. CW/CCW and centrifugal/centripetal preferences were not explained by a breakdown in component additivity. That is, modulations in firing rate during pursuit along a circular trajectory equaled the sum of modulations during horizontal and vertical sinusoidal components as well as for diagonal components. Instead all responses were well fit by a model that expressed the instantaneous firing rate of each neuron as a multilinear function of the two-dimensional position and velocity of the eye. This model generalized well to performance at different sinusoidal frequencies. It did somewhat less well for responses during fixation, suggesting some separation in the neural mechanisms of dynamic and static positioning. The model indicates that position sensitivity accounted for ~36% of the modulation during circular pursuit, and velocity sensitivity accounted for ~64%. When position and velocity sensitivity vectors were aligned, responses were simpler and modulations were similar during CW versus CCW pursuit. In contrast, when these vectors pointed in different directions, response complexity increased. Nonaligned position and velocity influences tended to reinforce during circular pursuit in one direction and to cancel each other during pursuit in the opposite direction. They also tended to produce response differences during centripetal versus centrifugal sinusoidal pursuit. The distinct roles played by position and velocity in shaping Purkinje cell responses are compatible with the control signals required to generate smooth pursuit along circular and other two-dimensional trajectories.


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

The floccular and parafloccular regions of the cerebellum play an important role in controlling smooth pursuit eye movements (Zee et al. 1981). Previous studies of single-neuron responses during smooth pursuit have concentrated on its role in controlling eye movements during one-dimensional (1D) ramp, step-ramp, sinusoidal, square, and trapezoidal target motions. Many of these studies have used target motions restricted to the horizontal axis (Lisberger and Fuchs 1978). Some have studied a combination of horizontal and vertical target motions (Lisberger et al. 1994; Stone and Lisberger 1990), whereas others also have included diagonal motions (Krauzlis and Lisberger 1996; Miles et al. 1980; Noda and Suzuki 1979).

The experiments described in this paper extend this work to two-dimensional (2D) target motions. One goal was to determine how neural responses during 2D pursuit relate to responses from the same cell during 1D pursuit. We were interested particularly in whether the smooth pursuit system satisfies component additivity. That is, does a neuron's response during 2D tracking equal the summation of responses obtained during 1D component motions that combine to create the 2D motion? Establishing component additivity is important because it simplifies data collection and data analysis and facilitates modeling of the pursuit system. For example, it implies that the response of the pursuit system during a general 2D motion can be characterized by its responses during the horizontal and vertical components of that 2D motion.

Circular target motions were selected for study because they are simple 2D trajectories that are radially symmetric. This allowed direct comparisons of component additivity for both horizontal/vertical (0°/90°) and diagonal/diagonal (45°/135°) coordinate axes. Combinations of horizontal and vertical sinusoids produce clockwise (CW) rotation when horizontal motion leads vertical motion by 90° and counterclockwise (CCW) rotation when vertical motion leads horizontal motion by 90°. Similarly, combinations of sinusoids on diagonal/diagonal axes (45°/135°) with phase shifts of ±90° produce the same CW and CCW motions.

Interestingly, some neurons showed stronger modulations in firing rate for CW pursuit than for CCW pursuit. Similar preferences for CCW pursuit also were observed. These responses were more complex than one would expect based on simple extrapolations from responses during 1D pursuit. We had expected that each neuron would have a single preferred direction related to its direction of maximal velocity sensitivity and that responses during circular pursuit would be related to the component of circular pursuit aligned with this preferred velocity direction. This single-direction hypothesis predicts that modulation amplitudes will be the same for CW and CCW circular pursuit because both have component motions of equal amplitude on any single axis.

Instead, response patterns were related to both the position and velocity of the eye based on responses during circular pursuit at a single frequency, responses during sinusoidal pursuit at several frequencies, and responses during fixation. It was hypothesized that response complexity was produced by interactions between influences related to eye position and eye velocity acting in different directions. This idea was tested quantitatively with a model that characterized the firing rate of each neuron as a function of the 2D position and velocity of the eye. The model provided a good fit for responses during sinusoidal and circular pursuit as well as good estimates of modulation differences during CW and CCW circular pursuit.


    METHODS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Animals

Two adult male rhesus monkeys (Macaca mulatta; 4-8 kg) were cared for and housed by the Northwestern University Center for Experimental Animal Resources (CEAR) according to Principles of Laboratory Animal Care (National Institutes of Health publication No. 86-23, revised 1985). The monkeys were checked daily by a certified veterinarian, and care was provided if necessary. Although the monkeys did not receive water outside of the experimental room on week days, they did obtain as much water as they were willing to drink during each experimental session. On a typical day the monkeys received from 500 to 900 ml of liquid during experimental tasks. This is a large volume of water, and well within the normal range of liquid intake for such a small animal. When neural recording failed, the monkey still was allowed to perform the task until he received his standard amount of water, or infrequently, was given supplemental water to satiation. Free water was given on weekends. Supplemental foods and toys were used to improve the well being of the monkeys.

Surgery

Sterile surgery was performed under deep anesthesia induced by sodium thiamylal (20 mg/kg iv) and maintained with halothane (1%) administered via an endotracheal tube. An oval of skin was resected and the underlying cranium scraped clean of periosteal tissue. A stainless steel socket used to fix the head during the experiment was positioned forward on the head so that a recording chamber could be placed above the cerebellum after training was complete. An eye coil was implanted under the conjunctiva of one eye (Judge et al. 1980; Robinson 1963) with its connecting wires led under the skin to an electrical plug positioned near the head fixation socket. The exposed bone, fixation socket, and eye-coil plug then were covered with a smoothed layer of dental acrylic anchored with stainless steel screws drilled into the skull. A recording chamber was implanted in a later surgery to reduce the chance of infection during a period of several months when training and initial behavioral experiments were performed to establish optimal behavioral parameters. A trephine hole 20 mm in diameter was made over the cerebellum that left the dura intact. A stainless steel recording cylinder of 20 mm ID was placed over the opening and held in place by dental acrylic. The chamber was filled with sterile saline combined with a dilute antibiotic solution and sealed with a sterile cap. For 1 wk after surgery, the monkey was given daily doses of cephalothin (15 mg/kg im), and an antibiotic salve was applied topically to prevent infection. Analgesia was maintained for 4 days by injections of buprenorphine HCl (0.01 mg/kg im) and later with topical applications of lidocaine until the incision had healed.

Apparatus

Eye position was monitored by a magnetic search coil system (CNC Engineering). Horizontal eye position was monitored using phase differences relative to a reference coil mounted 3 cm from the eye coil based on a rotating electromagnetic field vector (Collewijn et al. 1975). Vertical eye position was based on amplitude changes in induced eye-coil current (Robinson 1963); these were not corrected for cosine error because eye motion was less than ±10°. Eye position measurements were accurate to 15 min of arc for the range of eye movements studied. The pursuit stimulus was a laser spot (0.7° diam, luminance 92 cd/m2) back-projected onto a tangent screen that measured 60 by 60° located 40 cm from the monkey's eyes. Spot motion was controlled by a pair of servo-controlled mirror galvanometers (General Scanning, Watertown, MA) that provided output signals proportional to mirror position. Mirror and eye position signals to the computers were amplified and filtered with 8-pole low-pass 500-Hz Bessel filters. Galvanometer control signals from the computer were amplified and filtered with 8-pole low-pass 100-Hz Bessel filters. One computer was used to generate analogue waveforms (1,000 samples/s) and to deliver juice rewards when the eye was within the ±2° target window for a specified time. A second computer collected six channels of analogue data (1,000 samples/s): horizontal and vertical eye position, horizontal and vertical target position, laser level, and reward pulses. It also stored the time of individual neural events with an accuracy of 0.1 ms.

Training, calibration, and behavioral paradigms

The monkeys were trained using positive reinforcement with fruit and juice rewards. Training proceeded at a slow rate so that the animals never were frustrated. Initially, larger error windows and shorter fixation times produced reward delivery, but the monkeys rapidly developed accurate pursuit behavior. The monkeys were trained for several weeks before neural recording started until stable and consistent performance was obtained. Laser-spot and eye-position signals were calibrated each day both statically (horizontal, vertical, and diagonal locations at 0, ±5, and ±10°) and dynamically (5 and 10° circles). This was done relative to lines on a moveable screen that was in contact with the viewing screen during calibrations and then moved away during the experiment. Calibration gain was very stable and there were only slight changes in offset each day.

CIRCLES AND SINUSOIDS TASK. Neural responses were recorded during smooth pursuit along six waveform trajectories (5°, 0.6 Hz): CW and CCW circles, and sinusoids along four axes that provided information about pursuit in eight directions (0-315° in steps of 45°). Trials began when the monkey fixated a stationary target light at the start of the trajectory for 1,000 ms. The monkey was required to track the target within a ±2° error window for five complete repetitions of each waveform trajectory. If his performance was accurate, he received a liquid reward during each successful waveform cycle. If an error (error rate <5%) occurred during a waveform presentation, the monkey did not receive a reward and the waveform was repeated. Trajectories were selected randomly without replacement until all six had been presented in two blocks of trials.

FIXATION TASK. Neural responses were recorded during fixation at a center point and eight peripheral points 5° from the center in directions ranging from 0 to 315° in steps of 45°. A trial began after center fixation was maintained for 1,000 ms. The target then jumped to one of the eight peripheral fixation points for 1,000 ms. The monkey received a reward after each successful fixation defined as maintenance of eye position within ±2° of the target during a 600-ms period beginning 300 ms after the target had moved. A complete trial consisted of five successful cycles of center to peripheral fixation. Cycles were repeated if there were errors (error rate <5%). Fixation points were selected randomly without replacement until all eight had been presented in one or two blocks.

FREQUENCY TASK. Neurons were studied during horizontal and vertical sinusoidal pursuit at six frequencies (0.2-1.2 Hz) using a modulation amplitude of 5°. Trial initiation, reward delivery, and trial completion after five error-free cycles were as described earlier (CIRCLES AND SINUSOIDS TASK). The frequency of each trial was selected randomly without replacement until all six frequencies had been presented in two blocks.

Physiological recording, localization of recording sites, and euthanasia

On each day of recording, the head was attached to the primate chair, and a sterile plastic insert with a grid of drilled holes (Crist et al. 1988) was inserted into the recording chamber. A guide tube preloaded with an electrode was inserted into one of the grid holes, and the electrode was lowered with a microdrive to the region of study. The microelectrodes were glass-coated platinum-iridium wires (1-2 µm tip diameter, 1-2 MOmega impedance) optimized for extracellular recording from Purkinje cells. Only stable single waveforms that were clearly separable from background activity were selected for further study. These initial decisions about waveform separation were verified later by analyzing stored waveforms using spike-sorting displays and algorithms (DataWaves). All neurons selected for this report showed clear separation in scatter plots of waveform parameters and in overlaid waveform traces. Neurons were identified as Purkinje cells based on a combination of factors including waveform shape, spike width, the presence of complex spiking patterns, extended recording times that generally exceeded 1 h, and maintained recording while moving the electrode 50 µm. A single microelectrode penetration was generally made each day. Recording times were limited to 3-5 h and continued for 4-6 mo in each monkey.

A three-dimensional (3D) map of each recording site was constructed during the course of the experiments that was later overlaid on an anatomic reconstruction of the flocculus and paraflocculus. Anterior-posterior and lateral-medial locations were measured relative to the 1-mm spacing of the grid system that held the guide-tube/electrode assembly. Depths relative to the bottom of the grid were obtained from the microdrive scale. After the completion of recording, the same grid was used to localize 12 marking lesions (30 µA direct positive current for 30 s) that bracketed the recording sites of interest. Gliosis was allowed to develop for 1 wk. The animals then were sedated with ketamine hydrochloride (10 mg/kg im), killed by an overdose of pentobarbital sodium (100 mg/kg iv), and perfused transcardially with normal saline followed by paraformaldehyde (10.0%). The brain was cut in 50-µm sections, mounted on slides, and stained with thionin. Representative slides that included the lesion sites were digitized using a high-resolution computer scanner. A custom computer program was used to align recording- and lesion-site maps with these anatomic scans using rotation and translation in 3D space. The results are summarized in Fig. 18 where recording sites are projected into two or three representative sections evenly spaced within the recording area.

Data analysis

All neurons described in this report showed stable firing characteristics based on raster displays similar to those shown in Figs. 1-5. They also showed clear modulation during pursuit defined by a significant (P < 0.001) cosine regression fit of Eq. 1 and a modulation-to-baseline ratio in excess of 0.2 during CW or CCW pursuit. Well-isolated and stable neurons sometimes were collected that were not modulated during circular pursuit; these neurons were not included in this report. Eye and target data were digitally smoothed using a quadratic filter (Press et al. 1992) that maintained components <10 Hz and preserved higher-order moments. Eye velocity and acceleration data were obtained by quadratic differentiation. Discrete neural event times were converted to neural response rates sampled at 1,000 Hz using an inverse interval algorithm: firing rate equaled the inverse of the time interval between the spike immediately before and after each sample time. Saccades were deleted automatically by computer by removing data when the eye and target differed by >1° in position, 10°/s in velocity or 100°/s2 in acceleration along horizontal or vertical axes. Data with rectification nonlinearities during near zero firing (<5 spikes/s) were also deleted. These deletions were verified by visual inspection.

Analyses typically were based on data from 10 repetitions of all waveforms, although data loss sometimes reduced this number to 8. The only exception was for the fixation task; only 5 repetitions initially were collected for the first monkey. This number later was increased to 10 repetitions in the second monkey when the importance of eye position influences became apparent. In all analyses left-hemisphere data have been converted to right-hemisphere format so that spatial angles of 0, 90, 180, and 270° indicate ipsilateral, upward, contralateral, and downward directions, respectively. For the figures and their description in the text, ipsilateral is to the right and contralateral is to the left.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Responses during circular pursuit

During circular and sinusoidal pursuit, a variety of responses were observed for 69 neurons in two monkeys that showed significant modulation in firing rate during circular pursuit (see METHODS). A model that expressed firing rates in terms of position and velocity sensitivities accounted for all these responses. We first describe the phenomena that motivated this model with four examples. These neurons do not represent distinct classes of neurons but instead illustrate the range of a response continuum.

Responses from a "velocity/position" neuron (Fig. 1 ) were explained by a dominant sensitivity to downward eye velocity and a much weaker sensitivity to downward eye position. Raster/histogram displays at the top of this figure show a strong and consistent modulation in activity that was approximately sinusoidal. Modulation amplitudes were strongest during vertical pursuit, minimal during horizontal pursuit, and approximately equal during CW and CCW pursuit. The dependence on downward eye velocity is apparent in the 2D plots at the bottom of the figure. Firing increased in the direction of downward eye velocity during both CW and CCW pursuit, as well as during outward (centrifugal) pursuit along a center-to-down trajectory and inward (centripetal) pursuit along an up-to-center trajectory. A weak dependence on downward eye position was indicated by activity during inward pursuit at downward positions even though the target was moving with a slow upward velocity.



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Fig. 1. Velocity/position cell that fired maximally during pursuit in the downward direction with a much smaller dependence on eye position. Top: 6 responses during clockwise (CW) and counterclockwise (CCW) circular pursuit and sinusoidal motion along horizontal, vertical, and 2 diagonal axes. Numbers in parentheses indicate the initial and secondary directions of sinusoidal motions initiated at the center position. Each panel displays: horizontal eye (---) and target ( · · · ) traces (positive = ipsilateral), similar traces for vertical motion (positive = up), raster displays of firing times on individual repetitions, and a comparison of the average firing rate for the neuron (histogram) and the model (---). A two-dimensional (2D) format is used to display the same data at the bottom of the figure. Here, responses are shown for the same neuron during CW and CCW circular pursuit and during outward (Out) centrifugal and inward (In) centripetal sinusoidal pursuit. Radius of each circle corresponds to the average firing rate at a particular point along a trajectory. right-arrow, direction of target motion.

The "position/velocity" neuron illustrated in Fig. 2 had responses that were influenced by both eye position and eye velocity. Raster/histogram plots indicated consistent variations in firing rate that were similar in amplitude during circular pursuit and showed a preference for horizontal pursuit. Interestingly, the direction of maximal response was different for CW compared with CCW pursuit. There was also a strong preference for outward pursuit and a much weaker activation during inward pursuit. These responses can be explained by sensitivities to both leftward position and leftward velocity. Maximal responses to the top left in the 2D plot of CCW pursuit reflect position influences that were maximal along the left of the trajectory and velocity influences that were maximal along the upper part of the trajectory. A rotation of the maximal response during CW pursuit reflected a change in the point of maximal leftward velocity from upper to lower points on the trajectory. Similarly, a preference for outward pursuit was explained by the combined action of position and velocity influences that reinforced to the left and canceled to the right. During inward pursuit, these two influences were always in opposite directions resulting in low firing rates in all directions.



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Fig. 2. Position/velocity cell that fired maximally at leftward (contralateral) positions with a somewhat weaker velocity preference for pursuit in the leftward direction. See Fig. 1 and the text for more detail.

This neuron's sensitivity to eye position also was observed during fixation. Responses were recorded during fixation at a central point and at eight peripheral points in equally spaced directions. The raster/histogram plots in Fig. 3 show responses at leftward and rightward fixation points that produced the largest and smallest firing rates, respectively. The tuning function in the center shows average firing rates in these two fixation directions as well as the six other directions that were studied. It indicates that firing rate was maximal during leftward fixation, minimal during rightward fixation, and had intermediate values in other directions.



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Fig. 3. Responses from the position/velocity cell depicted in Fig. 2 during the fixation task indicate a preference for leftward (contralateral) positions. Left and right: repeated responses during a 1,000-ms period of fixation at screen center followed by a 1,000-ms period after the target had jumped 5° to leftward and rightward fixation positions, respectively. Center: position tuning function for average firing rate at all 8 fixation positions with left-arrow  pointing in the preferred fixation direction. Its length corresponds to the average firing rate of 28 spikes/s in that direction. See Fig. 1 and the text for more detail.

Responses from a "CW" neuron were more complex (Fig. 4). One striking feature was a stronger modulation during CW pursuit and a weaker modulation during CCW pursuit. Another interesting finding was a tendency for directional preferences to be stronger during inward pursuit than for outward pursuit. This is opposite to what was observed for the position/velocity neuron in Fig. 2. These responses suggest that this neuron had position and velocity sensitivities that pointed in opposite directions: a leftward sensitivity for position and a rightward sensitivity for velocity. During inward motion, these two sensitivities combined to produce maximal responses during left-to-center motions. During outward motion, these two influences were in opposite directions, and their combined action produced similar responses in all directions. This combination of position and velocity influences also explained differences in CW compared with CCW pursuit, but these interactions are best visualized using a computer simulation that will be presented later.



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Fig. 4. CW cell that showed its strongest modulations during CW circular pursuit. This neuron had position and velocity sensitivity vectors that pointed in nearly opposite directions (see Fig. 1 and the text for more detail).

Responses from a "CCW" neuron (Fig. 5) were even more complex. For CCW trajectories there were strong increases in activity during pursuit in the upward direction. This contrasted with an almost complete absence of response modulation during CW pursuit. Raster/histogram and 2D displays of responses during sinusoidal pursuit indicated clear modulations in all directions. This neuron also was modulated during both inward and outward sinusoidal pursuit although preferred activation directions were maximal to the right during outward pursuit and strongest to the upper left during inward pursuit. Simulations presented later will show that these response patterns result from rightward position and upward velocity sensitivities that are orthogonal.



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Fig. 5. CCW cell that showed its strongest modulations during CCW circular pursuit. This neuron had position and velocity sensitivity vectors that were approximately orthogonal (see Fig. 1 and the text for more detail).

The finding that responses were sometimes stronger during circular pursuit in one direction was quantified by fitting a cosine function
<IT>R</IT>(<IT>t</IT>)<IT>=</IT><IT>B</IT><IT>+</IT><IT>M</IT><IT> cos </IT>(<IT>2&pgr;</IT><IT>ft</IT><IT>+&phgr;</IT>) (1)
to each neuron's response during CW and CCW pursuit. Here B is the average firing rate, M is the modulation amplitude, f is the frequency of target rotation, and phi  is the phase relative to the start of target motion. Computed values of M during CW and CCW pursuit are shown in Fig. 6 for the entire population of 69 neurons. Differences in response modulation during CW versus CCW pursuit also were evaluated statistically using chi 2 tests. Data from nine neurons that showed significant differences (P < 0.01) are plotted as open circle  in Fig. 6. It is clear that many neurons showed similar response modulations during CW versus CCW circular pursuit including those cells showing the strongest modulations during circular pursuit. However, others showed significant differences in response modulation for the two directions of rotation.



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Fig. 6. Comparisons between responses during CW versus CCW circular pursuit. Horizontal and vertical coordinates of each point are the modulation amplitudes, MCW and MCCW, for CW and CCW circular pursuit. Diagonal line indicates equal modulation. Points above the diagonal indicate stronger modulation during CCW circular pursuit and points below it correspond to neurons with stronger modulation during CW pursuit. open circle , cells with statistically different modulations (P < 0.05, chi 2 test) for the 2 pursuit directions; , cells with nonsignificant differences.

Component additivity

When two component motions A and B sum to create a composite motion A + B, the pursuit system exhibits component additivity if its response during the composite motion equals the sum of its responses during the components tracked individually. For example, horizontal (h) and vertical (v) sinusoidal motions that are 90° out of phase sum to create a circular (c) motion. Component additivity implies that Rh (t) + Rv (t) = Rc (t) where Rh (t), Rv (t), Rc (t) represent the firing rates along horizontal, vertical, and circle trajectories at time t. Similarly, 45° diagonal (d45) and 135° diagonal (d135) sinusoidal motions sum to create a circular motion with Rd45 (t) + Rd135 (t) = Rc (t).

Component additivity was observed for all of the neurons studied. This is illustrated in Fig. 7 for the four neurons described in the preceding text. Each set of three traces in this figure depicts the modulation in firing rate relative to baseline during circular tracking (thick lines), summed horizontal and vertical sinusoidal tracking (medium lines), and summed tracking along 45 and 135° diagonals (thin lines). In each case, a good correspondence among the three traces indicates component additivity. In particular, additivity held for the CW and CCW neurons. chi 2 tests indicated nonsignificant (P > 0.94) differences in modulation profile during circular and summed component pursuit (CW vs. h v, CW vs. d45 + d135, CCW vs. h + v, CCW vs. d45 + d135) for all 69 neurons.



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Fig. 7. Component additivity for CCW and CW pursuit indicated by the similarity of overlaid plots of firing rate modulation during circular pursuit (cir, thick line), the sum of modulations during horizontal and vertical pursuit (h + v, medium line), and the sum of modulations during 45 and 135° diagonal pursuit (d + d, thin line). Data come from the velocity/position, position/velocity, CW, and CCW cells described in Figs. 1, 2, 4, and 5. Left: modulations related to 1 cycle of CCW pursuit; right: modulations related to CW pursuit.

In Fig. 8, the modulation and phase during circular pursuit are compared with the modulation and phase of summed responses during pursuit along horizontal and vertical components () or along diagonal 45 and 135° components (down-triangle). Each summation is based on standard rules for combining sinusoids of different amplitude and phase. The tendency for points to lie along the diagonal in each scatter plot indicates good component additivity across the population. Thus component modulations sum to create equal modulations during CW and CCW pursuit for some neurons and unequal modulations for others. This is true independent of any specific model. However, component additivity does suggest the use of a class of multilinear models that includes the model described in the following text.



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Fig. 8. Additivity tests for the entire population of 69 neurons. Point near a diagonal indicates component additivity because modulations during circular pursuit equaled the summed response during horizontal plus vertical () or 45 plus 135° diagonal (down-triangle) pursuit. Left: firing rate modulations in spikes/s; right: response phases in degrees.

Model of instantaneous firing rate during pursuit

A model was used to quantify the relationship between neural firing rate and the motion of the eye. This model tested the hypothesis that a neuron's firing rate at time t, R(t), is a multilinear function of the 2D position and velocity of the eye. The model equation had three mathematically equivalent forms
<IT>R</IT>(<IT>t</IT>)<IT>=</IT><IT>&bgr;+&rgr;</IT><SUB><IT>h</IT></SUB><IT>P<SUB>h</SUB></IT>(<IT>t</IT>)<IT>+&rgr;<SUB>&ugr;</SUB></IT><IT>P</IT><SUB><IT>&ugr;</IT></SUB>(<IT>t</IT>)<IT>+&ngr;</IT><SUB><IT>h</IT></SUB><IT>V<SUB>h</SUB></IT>(<IT>t</IT>)<IT>+&ngr;<SUB>&ugr;</SUB></IT><IT>V</IT><SUB><IT>&ugr;</IT></SUB>(<IT>t</IT>) (2a)

=&bgr;+<B>&rgr;</B><IT>·</IT><B>P</B>(<IT>t</IT>)<IT>+</IT><B>&ngr;</B><IT>·</IT><B>V</B>(<IT>t</IT>) (2b)

=&bgr;+‖<B>&rgr;</B><IT>‖‖</IT><B>P</B>(<IT>t</IT>)<IT>‖ cos </IT>(<IT>&thgr;</IT><B>&rgr;</B><IT>−&thgr;</IT><SUB><B>P</B>(<IT>t</IT>)</SUB>)<IT>+‖</IT><B>&ngr;</B><IT>‖‖</IT><B>V</B>(<IT>t</IT>)<IT>‖ cos </IT>(<IT>&thgr;</IT><B>&ngr;</B><IT>−&thgr;</IT><SUB><B>V</B>(<IT>t</IT>)</SUB>) (2c)
Because a neuron cannot have a negative firing rate, R(t) was set to zero if it took on a negative value.

The "component form" of the model (Eq. 2a) expressed firing rate in terms of eye motion relative to a specific 2D coordinate system. In this instance, a horizontal/vertical coordinate frame was used to specify the horizontal and vertical components of position [Ph(t), Pv(t)] and velocity [Vh(t), Vv(t)] at time t, where beta , rho h, rho v, nu h, and nu v were parameters estimated using multilinear regression techniques. This form of the model shows the very simple linear relationship between firing rate and increases in each component of eye motion. It also makes clear why the model satisfies component additivity: components on different axes combine via simple addition. This equation does not suggest a preference for a particular coordinate frame. A similar model equation can be written for other coordinate systems including the diagonal/diagonal system associated with axes at 45 and 135°. Thus the eye motion data and the model could just as easily have been collected and modeled in a diagonal coordinate frame by suitable rotations of the galvanometer and reference-coil axes.

The "vector form" of the model (Eq. 2b) is the same equation expressed using vector notation with vectors in bold type. The top of Fig. 9 shows how the position vector, P(t) = [Ph(t), Pv(t)], and the velocity vector, V(t) = [Vh(t), Vv(t)], specify the position and velocity of the eye at time t. Component parameters also were reexpressed as vectors so that rho  = (rho h, rho v) and nu  = (nu h, nu v) and multiplication was via the vector dot product (rho  · P = rho h Ph + rho v Pv). Illustrations in Fig. 9, bottom, show the parameter vectors calculated for the CCW cell (Fig. 5) and their relationship to the position and velocity vectors (top).



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Fig. 9. Examples of the vectors used in Eq. 2. Top: position and velocity vectors, P(t) and V(t), specify target motion at time t along a circular trajectory. Bottom left: how position sensitivity vector, rho , determines the influence of position via vector projection. Bottom right: how velocity sensitivity vector, nu , determines the influence of velocity.

The "polar-coordinate form" of the model (Eq. 2c) defines each vector by its length and direction. For example, rho  is defined by its length |rho | = (rho h2 + rho v2)1/2 and its angle, theta rho , with respect to the axis of the polar frame. This form of the equation illustrates two important properties of the model. First, it shows that cosine tuning is a natural consequence of the model. Second, it indicates how position and velocity influence firing rate via vector projection.

The vector rho  was called the position sensitivity vector and its direction, theta rho , the preferred position direction. On the basis of Eq. 2b, the influence of position on firing rate at point P(t) equals the vector dot product of rho  and P(t). Geometrically, this calculation corresponds to multiplying the length of P(t) by the length of the projection of the vector rho  onto the axis of P(t). This is illustrated in Fig. 9, bottom left, for the CCW cell. When rho  is aligned with P(t), its projection is maximal, when rho  is perpendicular to P(t), its projection is zero, and when rho  and P(t) point in opposite directions, its projection is negative. Equivalently (see Eq. 2c), the influence of position at P(t) equals the length of rho  times the length of P(t) times the cosine of the angle between rho  and P(t).

For similar reasons, nu  was called the velocity sensitivity vector and its direction, theta nu , was called the preferred velocity direction. It determines the influence of velocity via projection as illustrated in Fig. 9, bottom right. In this instance, the influence of velocity was stronger than the influence of position. It is important to remember that the relative influences of position and velocity will change over time as the eye-position and eye-velocity vectors P(t) and V(t) change with respect to rho  and nu . Thus the influence of position will exceed that of velocity when P(t) is more nearly aligned with rho , and when V(t) is less aligned with nu .

Statistical evaluation of the model

Model parameters were fit using standard multilinear regression techniques (Draper and Smith 1981, Press et al. 1992). Examples of the model's prediction of firing rate over time compared with actual firing levels are shown at the bottom of each panel in Figs. 1, 2, 4, and 5. For neurons studied in the circles and sinusoids task (n = 69), regression fits of Eq. 2 were always statistically significant (RTot2 = 43.5 ± 19.5%, P < 0.001) and lack-of-fit tests were always nonsignificant (P > 0.05). Although small deviations from model fits were sometimes observed (Fig. 5), these trends were not systematic and did not suggest a need for transformations of the data.

Other models also were tested. These models will be identified by their use of position (P), velocity (V), and acceleration (A) terms. Standard statistical approaches were used to select the PV model (Eq. 2) as the most appropriate for explaining the data collected in the present experiments. This choice was based on a lack of change in total R2 values when acceleration terms were added to the PV model to create the PVA model (RTot2 = 43.5 ± 19.4%). The PVA model also produced small partial R2 values for acceleration (0.7 ± 2.4%) compared with position (7.2 ± 9.7%) and velocity (21.5 ± 22.0%). In addition, position and velocity parameters (rho x, rho y, nu x, nu y) obtained for the PV model and the PVA model were not statistically different (P > 0.19, t <1.5, diff <0.06 ± 0.35). The VA model (RTot2 = 38.4 ± 19.0%) produced smaller total R2 values than either the PV model or the PVA model for all 69 neurons and many (16/69) showed significant lack of fit. Other neurons (53/69) showed nonsignificant lack of fit based primarily on the velocity term in the VA model. This is supported by the relatively good performance of the V model (RTot2 = 34.4 ± 19.4%). The improved performance of the VA model over the V model resulted from a weak correlation between eye position and eye acceleration that caused inappropriate loading of acceleration parameters when position terms were absent. This is demonstrated by the nearly identical performance of the PA model (RTot2 = 23.8 ± 15.3%) and the P model (RTot2 = 23.5 ± 15.3%) and the relatively reduced performance of the A model (RTot2 = 19.8 ± 15.0%).

Comparison with responses in the frequency task

More detailed evaluations of the model were conducted for 20 neurons studied in both the circles and sinusoids task and the frequency task. Again, fits of Eq. 2 based on the frequency task were significant (RTot2 = 34.4 ± 11.2%, P < 0.001) and lack-of-fit tests were nonsignificant (P > 0.05) for every neuron. Although this task allowed clearer distinctions among position, velocity, and acceleration influences, the PV model (Eq. 2) still produced the most parsimonious description of the data. The addition of an acceleration term in the PVA model produced little change in fit (RTot2 = 34.8 ± 10.9%) with small partial R2 values for acceleration (0.7 ± 0.8%) compared with values for position (5.2 ± 10.3%) and velocity (11.3 ± 10.1%). Again, differences in position and velocity parameters obtained for the PV model and the PVA model were not statistically different (P > 0.83, t < 0.25, diff < 0.07 ± 0.13). The VA model (RTot2 = 29.4 ± 8.5%), PA model (RTot2 = 21.1 ± 10.3%), P model (RTot2 = 17.8 ± 10.7%), V model (RTot2 = 24.2 ± 9.1%), and A model (RTot2 = 13.1 ± 5.8%) were associated with reduced performance.

These model fits were based on a combination of data collected at several frequencies. To test the stability of the model in estimating position and velocity sensitivities across frequencies, regressions were recomputed at each frequency used in the frequency task. Figure 10 shows results from the 20 neurons studied. Here each line represents data from one neuron, and the flatness of each line indicates the near constancy of position and velocity sensitivities at frequencies from 0.2 to 1.0 Hz. These values were also very similar to the final point in each line that represents the "combined" sensitivity based on data at all frequencies. Analysis of variance tests confirmed that average parameter values at each frequency were not statistically different (P > 0.95, F(4, 95) < 0.17) and that there were no systematic errors in estimating sensitivities at different frequencies. These results also argue against the presence of strong acceleration influences that would have increased more rapidly with frequency than either position or velocity influences and resulted in systematic deviations from flatness in these curves.



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Fig. 10. Constancy of position and velocity sensitivities as a function of frequency. Each line shows model parameters fits (rho x, rho y, nu x, nu y) based on data from 1 neuron during either horizontal or vertical sinusoidal motion. Points labeled by frequencies (0.2-1.0 Hz) show parameter fits based on data at a single frequency. Final point in each curve labeled "combined" represents the parameter fit for the neuron based on data at all 5 frequencies.

To test the generality of the model in fitting data from different tasks, model parameters computed for the circles and sinusoids task were compared with parameters based on the frequency task. The constancy of position and velocity sensitivities is illustrated in Fig. 11 for three neurons studied first in the circles and sinusoids task, later in the fixation task (see next section) and then in the frequency task. For these three neurons, differences in position direction (black tuning functions) were 9, 6, and 28° for the two pursuit tasks, whereas differences in preferred velocity direction (gray tuning functions) were only 9, 8, and 11°.



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Fig. 11. Position (back line) and velocity (gray line) tuning curves in rows A-C show results from 3 single neurons studied during the circles and sinusoids task, the fixation task, and the frequency task. Notice the consistency of position and velocity tuning for the 2 pursuit tasks, and more variability in the position sensitivity obtained for pursuit vs. fixation.

Figure 12, A and B, shows parameter comparisons for all 20 neurons studied in the circles and sinusoids and frequency tasks. Although there is some variability, points tend to lie near diagonals that represent parameter equality. Population statistical tests indicated that parameters differences were not statistically different for position magnitude (P = 0.96, t = 0.05, diff = 0.0 ± 2.2), position direction (P = 0.38, t = -0.90, diff = -7.6 ± 32.8), velocity magnitude (P = 0.58, t =0.57, diff = 0.3 ± 2.3) and velocity angle (P = 0.72, t = 0.36, diff = 2.2 ± 27.0).



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Fig. 12. Comparisons of model (Eq. 2) parameters obtained for the circles and sinusoids (Cir) task, the fixation (Fix) task, and the frequency (Freq) task. Points near each diagonal indicate similar parameters for the 2 tasks being compared. Magnitude refers to the modulation in spikes/s in the preferred position (, |Mpos| = M|rho |) or velocity (open circle , |Mvel| = 2pi fM|nu |) direction during sinusoidal motion (M = 5°, f = 0.6 Hz). Angle refers to the direction of a neuron's preferred position (angle  rho ) or velocity (angle  nu ) direction in degrees. Here, 360° was added to some small angles to facilitate comparisons at the 0°/360° transition. Some parameters did not reach levels of statistical significance in regression fits. Nonsignificant magnitudes are plotted in A and C, but nonsignificant angles have been set to 0 and appear along the axes of B and D.

Comparison with responses in the fixation task

For 51 neurons studied in the circles and sinusoids task, the dependence of firing rate on eye position was independently evaluated in the fixation task. Figure 3 shows the position dependence of the position/velocity neuron during changes in fixation. Comparisons between model fits and average firing rate histograms are shown at the bottom of the right and left panels. For the population of neurons studied, regression fits of fixation task data were always statistically significant (R2 = 18.2 ± 16.2%, P < 0.001) without significant lack of fit (P > 0.05). Nonlinearities corresponding to slow declines ("slides") in firing rate during long periods of fixation were not required for good fits. This may result from a relatively shorter fixation time than used by others.

Correspondences and differences in the position sensitivities measured for pursuit and fixation tasks are illustrated in Fig. 11 for three neurons each studied in all three tasks. The neuron in Fig. 11A had similar preferred position directions of 118, 117, and 124° in the three tasks. Figure 11B shows a neuron with similar position preferences of 259 and 250° for the two pursuit tasks but a different preference of 221° for the fixation task. Data from the CW cell (Fig. 11C) showed similar preferred position directions of 184 and 212° during the pursuit tasks but showed a large change to 112° during the fixation task. Some of this change can be attributed to a decline in tuning amplitude during the fixation task.

Results from the entire population of neurons studied (Fig. 12, C and D) confirm that parameters obtained for the fixation task and the circles and sinusoids task were similar for some neurons and somewhat different for others. Across the entire population of neurons, parameter differences were not significantly different for either position magnitude (P = 0.50, t = 0.68, diff = 0.4 ± 4.2) or position direction (P = 0.75, t = -0.32, diff = -3.1 ± 58.7). However, the standard deviations of these differences (4.2 spikes/s and 58.7°) were about twice as large as those obtained in comparisons of frequency-task and circles-and-sinusoids-task parameters (2.2 spikes/s and 32.8°). In particular, some neurons showed a statistically significant position dependence exclusively during the circles and sinusoids task (n = 9), whereas others showed a position dependence exclusively during the fixation task (n = 3). Nonsignificant direction parameters have been set to zero and plotted on the axes of Fig. 12D.

Directional tuning for position, velocity, and pursuit

Directional tuning functions for position, velocity, and pursuit were derived from the model by integrating firing rate along a sinusoidal trajectory from screen center to the end of target motion in a particular direction. These tuning functions take a relatively simple form during sinusoidal pursuit. The position tuning function, Tpos(theta ), gives the change in firing rate due to position in direction theta  by
<IT>T</IT><SUB><IT>pos</IT></SUB>(<IT>&thgr;</IT>)<IT>=‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖ cos </IT>(<IT>&thgr;−&thgr;</IT><SUB><B>&rgr;</B></SUB>) (3)
where Mpos = Mrho is a vector that points in the preferred position direction, theta rho , with a length equal to the modulation in firing rate in that direction for motions of modulation amplitude, M. Similarly, the velocity tuning function, Tvel(theta ), describes the change in firing rate due to velocity by
<IT>T</IT><SUB><IT>vel</IT></SUB>(<IT>&thgr;</IT>)<IT>=‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖ cos </IT>(<IT>&thgr;−&thgr;</IT><SUB><B>&ngr;</B></SUB>) (4)
where Mvel = 2pi fMnu is a vector in the preferred velocity direction, theta nu , with length equal to the increase in firing rate in that direction. Finally, the pursuit tuning function, Tpur(theta ), describes the total change in firing rate from baseline during pursuit due to both position and velocity by
<IT>T</IT><SUB><IT>pur</IT></SUB>(<IT>&thgr;</IT>)<IT>=</IT><IT>T</IT><SUB><IT>pos</IT></SUB>(<IT>&thgr;</IT>)<IT>+</IT><IT>T</IT><SUB><IT>vel</IT></SUB>(<IT>&thgr;</IT>)<IT>=‖</IT><B>M</B><SUB><IT>pur</IT></SUB><IT>‖ cos </IT>(<IT>&thgr;−&thgr;<SUB>pur</SUB></IT>) (5)
Here Mpur = Mpos + Mvel is a vector pointing in the preferred pursuit direction, theta pur, with length equal to the change in firing rate in that direction.

Examples of pursuit tuning functions for the four example neurons (Figs. 1, 2, 4, and 5) are illustrated in Fig. 13. In all instances there was a close correspondence between model predictions (dashed lines) and empirical measurements (solid lines) of pursuit tuning. Model and empirical preferred pursuit directions were highly correlated (r = 0.98, slope = 1.00, intercept = -3.7°) and not significantly different (P > 0.25, t = 1.1, diff = 3.2 ± 23.2) for the 69 neurons studied. Also shown are the Mpos (thin white arrow), Mvel (thin dark arrow), and Mpur (thick dark arrow) vectors that define these functions. As expected, tuning functions for the velocity/position neuron (Fig. 1) indicate a dominant role for velocity and a weaker influence from position. Pursuit tuning was therefore similar to velocity tuning. Position had a slightly stronger influence for the position/velocity neuron (Fig. 2), but velocity influences were almost as strong. Pursuit tuning for this neuron reflected the combined influence of position and velocity. Mpos, Mvel, and Mpur vectors for both of these neurons were in close alignment. The CW (Fig. 4) and the CCW neurons (Fig. 5) had position and velocity sensitivity vectors with different preferred directions. For these neurons, the pursuit-tuning vector Mpur was obtained by taking the vector sum of Mpos and Mvel.



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Fig. 13. Tuning functions for the velocity/position, position/velocity, CW, and CCW cells described in Figs. 1, 2, 4, and 5. Left: computed (dashed lines) and observed (thick lines) pursuit tuning functions are in good agreement. Right: Mpos (small open arrowhead) and Mvel (small filled arrowhead) sum to create Mpur (large filled arrowhead with thicker line). They define a parallelogram with area related to the difference in modulation during CW vs. CCW circular pursuit.

Distributions of preferred position, velocity, and pursuit directions (Fig. 14, right) indicate the existence of preferred-direction vectors in all quadrants with a slight preference for ipsilateral (0°) and down (270°) directions, particularly for velocity. Comparisons among preferred position, velocity, and pursuit directions (Fig. 14, left) indicate a close correspondence among preferred directions for pursuit, position, and velocity for some neurons, and clear differences for other neurons particularly for position and velocity.



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Fig. 14. Left: scatter plots comparing preferred directions for position and velocity, position and pursuit, and velocity and pursuit. Right: distributions of Mpos, Mvel, and Mpur, the preferred tuning vectors for position, velocity, and pursuit.

Figure 15 quantifies the relative strengths of position and velocity influences for all 69 neurons using two measures. Left compares position and velocity partial R2 values based on regression fits of Eq. 2. Average partial R2 values were 11.9 ± 14.7% for position and 21.4 ± 21.9% for velocity. Right compares the position and velocity modulation amplitudes during circular pursuit. The average position modulation amplitude (|Mpos|) was 6.7 ± 6.8 spikes/s and the average velocity modulation amplitude (|Mvel|) was 11.9 ± 13.1 spikes/s. A comparison of these two averages indicates that 36% of the average firing rate can be attributed to the influence of position and the remaining 64% to velocity. Both show that both position and velocity played significant roles in determining firing rate during circular pursuit although velocity had a stronger influence.



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Fig. 15. Scatter plots showing the relative influence of eye position vs. eye velocity in determining responses during circular pursuit. Left: percentage partial R2 values for position vs. velocity. Right: modulation amplitudes during circular pursuit due to position (|Mpos|) and velocity (|Mvel|).

These values are based on pursuit at 0.6 Hz along trajectories with modulation amplitudes of 5°, values associated with the mid- to upper-range of accurate pursuit behavior. It should be emphasized that the relative influence of position and velocity depend on the velocity, amplitude, and frequency of target motion. The model predicts that the ratio of modulations during sinusoidal or circular pursuit due to position and velocity will be
‖<B>M</B><SUB><IT>pos</IT></SUB><IT>‖/‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖=‖</IT><B>&rgr;</B><IT>‖/2&pgr;</IT><IT>f</IT><IT>‖</IT><B>&ngr;</B><IT>‖</IT> (6)
As expected, this equation indicates that the relative roles played by position and velocity are related to the lengths of rho  and nu , the position and velocity sensitivity vectors of the neuron. In addition, the influence of position will tend to be dominate at lower frequencies when velocities are lower, and the influence of velocity will tend to dominate at higher frequencies when velocities are higher.

Model description of rotation direction preferences

Figure 16 shows computer simulations that provide an intuitive explanation of how the model combined position and velocity influences to produce responses during circular pursuit. Responses from the velocity/position cell (Fig. 1) were produced by a dominant sensitivity to downward eye velocity that caused the neuron to fire at leftward locations during CCW pursuit and at rightward locations during CW pursuit. A weak sensitivity to downward eye position caused responses during circular pursuit to rotate slightly toward downward locations.



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Fig. 16. Graphic explanation of how the model explains CW vs. CCW modulation differences for the velocity/position, position/velocity, CW, and CCW cells described in the text. In each panel, the total response during circular pursuit (left) equals responses due to position (middle) and velocity (right) influences. Curved arrows indicate the direction of circular pursuit for all 3 plots. Straight arrows indicate the direction of position and velocity sensitivities. The radius of each small circle represents firing rates above (open circles) and below (filled circles) zero at points along the eye trajectory. Firing rates due to position and velocity are each added to half the baseline level so that their sum produces the full baseline level for the total response. Contralateral is to the left and ipsilateral is to the right.

The position/velocity cell (Fig. 2) had more complicated firing patterns. Although its position and velocity sensitivities were aligned, interactions between them were stronger because they were nearly equal in amplitude. Its position sensitivity caused firing increases at leftward locations for both rotation directions. In contrast, its velocity sensitivity increased firing rates along the upper trajectory during CCW pursuit and the lower trajectory during CW motion. Their combination resulted in a shift in the point of maximal modulation so that increases were maximal along the top left part of the CCW trajectory and along the bottom right part of the CW trajectory. Even so, modulation amplitudes were again similar during CCW and CW pursuit because position and velocity influences overlapped in a similar fashion during the two directions of circular pursuit.

The CW cell (Fig. 4) showed interactions between position and velocity input that produced stronger modulations in firing rate during CW rotation and weaker modulations during CCW rotation. Position and velocity responses tended to reinforce during CW pursuit and to cancel during CCW pursuit. This cell did not show the largest differences in CW and CCW pursuit because its position and velocity sensitivities were in almost opposite directions.

The CCW cell (Fig. 5) showed larger differences between CW and CCW pursuit than the CW cell because its preferences for position and velocity were nearly perpendicular. The influence of eye position produced maximal firing to the right during both CW and CCW pursuit. In contrast, the influence of eye velocity generated maximal increases to the right during CCW pursuit and to the left during CW pursuit. When summed, these two influences strongly reinforced each other during CCW pursuit and nearly cancelled each other during CW pursuit.

The modulations during CW and CCW pursuit predicted by Eq. 2 are
<IT>M</IT><SUB><IT>CW</IT></SUB><IT>=</IT>(<IT>‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖<SUP>2</SUP>+‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖<SUP>2</SUP>+2‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖ sin </IT>(<IT>&thgr;</IT><SUB><B>&rgr;</B></SUB><IT>−&thgr;</IT><SUB><B>&ngr;</B></SUB>))<SUP><IT>1/2</IT></SUP> (7)

<IT>M</IT><SUB><IT>CCW</IT></SUB><IT>=</IT>(<IT>‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖<SUP>2</SUP>+‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖<SUP>2</SUP>−2‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖ sin </IT>(<IT>&thgr;</IT><SUB><B>&rgr;</B></SUB><IT>−&thgr;</IT><SUB><B>&ngr;</B></SUB>))<SUP><IT>1/2</IT></SUP> (8)
where MCW and MCCW are the modulation amplitudes during CW and CCW pursuit. The dual roles of amplitude and direction become more apparent when Eqs. 7 and 8 are squared and then subtracted so that
<IT>M</IT><SUP><IT>2</IT></SUP><SUB><IT>CW</IT></SUB><IT>−</IT><IT>M</IT><SUP><IT>2</IT></SUP><SUB><IT>CCW</IT></SUB><IT>=4‖</IT><B>M</B><SUB><IT>pos</IT></SUB><IT>‖‖</IT><B>M</B><SUB><IT>vel</IT></SUB><IT>‖ sin </IT>(<IT>&thgr;</IT><SUB><B>&rgr;</B></SUB><IT>−&thgr;</IT><SUB><B>&ngr;</B></SUB>) (9)
The term on the right side of Eq. 9 can be identified as four times the area of the parallelogram formed by the vectors Mpos and Mvel. Parallelogram areas for the four example neurons are shown in Fig. 13. As expected CW and CCW modulations are equal when the length of either Mpos or Mvel is small (Fig. 13A) or when rho  and nu  are aligned (theta rho  - theta nu  = 0°, Fig. 13B). Differences in modulation increase until a maximum is reached when rho  and nu  are orthogonal (theta rho  - theta nu  = ±90°, Fig. 13D); modulation differences then decline as the angle between rho  and nu  increases (Fig. 13C) and then return to zero when rho  and nu  point in opposite directions (theta rho  - theta nu  = 180°). Specific conditions must occur for modulation to be completely eliminated in one rotation direction: Mpos and Mvel must be equal in length and orthogonal (Fig. 13D). Figure 17 quantifies these ideas for the entire population of 69 neurons. It indicates that predictions of the model based on Eq. 9 are in good agreement with the differences in modulation observed experimentally. This plot also shows the relatively wide and continuous distribution of rotation preferences across the population of neurons studied.



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Fig. 17. Comparisons between experimental and model estimates of differences in CW and CCW modulation. On the basis of Eq. 9, an index of modulation differences in firing rate was defined by the quantity (MCW2 - MCCW2)1/2, whereas estimates of this index derived from the model are based on the quantity [4 |Mpos| |Mvel| sin (theta rho  - theta nu )]1/2. The square root is used so that values are expressed in units of spikes/s. Values near the diagonal line indicate equality between experiment and model.

Anatomic location of neurons

The anatomic locations of the 69 neurons studied are shown in Fig. 18. We found responsive neurons in four regions: the flocculus, the ventral paraflocculus, the lobulus petrosus, and the dorsal paraflocculus (Larsell 1970). The posterolateral fissure was used to mark the division between the flocculus and the ventral paraflocculus and the lobulus petrosus was used to indicate the transition from ventral paraflocculus to dorsal paraflocculus. Most of the neurons were localized to the flocculus and the ventral paraflocculus with a smaller number found within the base of the lobulus petrosus adjacent to the ventral paraflocculus and within the dorsal paraflocculus. There was no obvious correspondence between a neuron's directional properties and its anatomic location, although there was a tendency for neurons with similar response properties to be in adjacent locations. For example, a clear line of horizontal sites is shown in Fig. 18A.



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Fig. 18. Recording sites projected onto representative horizontal sections through the flocculus (fl), ventral paraflocculus (vpfl), lobulus petrosus (pt), and dorsal paraflocculus (dpfl). Posterolateral fissure (plf) separates the flocculus from the ventral paraflocculus. Lobulus petrosus lies between the dorsal and ventral paraflocculi. Three sections from the 1st monkey summarize dorsal (A), intermediate (B), and ventral (C) levels, and 2 sections from the 2nd monkey show dorsal (D) and ventral (E) levels. F: location of A in relation to other brain structures. Adjacent sections in A-C and D and E are separated by 2 mm and share the orientation indicated in F. Magnification for A-E is indicated by the scale in C. Symbols represent the location of neurons with horizontal (open circle ), vertical (), and orthogonal (triangle ) sensitivity vectors.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
REFERENCES

Purkinje cells in the flocculus and paraflocculus showed a variety of responses during circular pursuit. Some showed relatively simple responses that were maximal for a particular direction of pursuit and showed similar amplitudes of modulation during CW compared with CCW pursuit. Others showed more complicated responses including preferences for a particular direction of circular pursuit or preferences for either inward or outward target motion. A simple model accounted for all the responses that were observed. It expressed firing rate as a multi-linear function of eye position and eye velocity sensitivities that sometimes pointed in different directions.

Component additivity

Component additivity was observed for all neurons studied. That is, neural modulation during circular pursuit equaled the sum of modulations for circle components. This suggests that the pursuit system can be characterized by its responses along only two axes. Possible choices include the horizontal/vertical and diagonal/diagonal axis pairs used in the present study. The results also demonstrate that it is not generally possible to evaluate a neuron's response on only one axis even if the axis is a diagonal axis because many neurons had position and velocity sensitivity vectors that were not aligned. For sinusoidal motions, this implies that the phase of a neuron's response can differ with spatial direction as well as its gain.

Theoretically, component additivity is compatible with the hypothesis that the pursuit system satisfies approximate spatial linearity, at least for the ranges of position and velocity values that were studied. This allowed modeling of the relationship between neural firing rate and eye motion using a relatively simple class of multilinear functions. Whether spatial linearity holds for a larger range of values will require additional experimentation. Linearity clearly will fail for very large values of target position and velocity, but it is also possible that pursuit behavior will break down at the point where linearity fails. Such a breakdown in pursuit performance was observed in the present studies. This suggests that Purkinje cell firing rates may be modeled adequately by a multilinear system for a "normal" range of velocities that allow accurate pursuit behavior with minimal saccadic intrusion. Pursuit behavior also shows good linearity for this range of target velocities during circular pursuit (Collewijn and Tamminga 1984; Deno et al. 1995; Kettner et al. 1996; Leung and Kettner 1997) although velocity gain declines and saccade frequency increases at higher velocities.

Model of neural firing rate that allows 2D interactions between position and velocity influences

Complex interactions between position and velocity influences with different spatial directions were simulated using a model that expressed the instantaneous neural firing rate in terms of the instantaneous position and velocity of the eye. The model quantified position and velocity influences during pursuit with two sensitivity vectors. A position sensitivity vector pointed in the preferred position direction of the neuron and had a magnitude equal to its position sensitivity in that direction. A velocity sensitivity vector had a magnitude equal the velocity sensitivity of the neuron in its preferred velocity direction. This approach represents an extension to 2D motions of 1D analyses of pursuit and ocular following in the flocculus and paraflocculus (De Zeeuw et al. 1995; Gomi et al. 1998; Lisberger and Fuchs 1978; Lisberger et al. 1994; Miles et al. 1980; Shidara et al. 1993; Stone and Lisberger 1990; Zhang et al. 1995), the vestibular nucleus (McFarland and Fuchs 1992; Scudder and Fuchs 1992; Stahl and Simpson 1995b; Tomlinson and Robinson 1984), and the abducens nucleus (Fuchs et al. 1988; Keller 1973; Stahl and Simpson 1995a).

The model produced good estimates of the instantaneous firing rate of each neuron during circular pursuit as well as during sinusoidal pursuit at several frequencies using a single set of baseline, position, and velocity sensitivity parameters. As expected, neurons with aligned position and velocity sensitivity vectors had simpler responses and showed similar modulations during CW and CCW circular pursuit. In contrast, neurons with nonaligned position and velocity sensitivity vectors of comparable amplitude showed responses that were more complicated including preferences for either CW or CCW pursuit. A natural consequence of the model is that position and velocity influences act via vector projection. This property produces cosine tuning for position, velocity, and pursuit.

Position and velocity influences in flocculus and paraflocculus

Our analyses indicate that the firing rates of most flocculus and paraflocculus Purkinje cells are correlated with both the position and velocity of the eye. Velocity played a somewhat stronger role (64%) than position (36%) in determining firing rate modulation during circular pursuit but did not play an exclusive role. Preferred position directions tended to be uniformly distributed throughout the 2D directional space as also reported by Noda and Suzuki (1979). Preferred velocity directions were concentrated in the ipsilateral and downward directions in agreement with previous studies (e.g., Krauzlis and Lisberger 1996; Lisberger and Fuchs 1978; Lisberger et al. 1994; Miles et al. 1980; Stone and Lisberger 1990).

Interestingly, position and velocity influences sometimes pointed in different directions. Zhang et al. (1995) also observed differences between the ON direction of eye position and the direction of eye velocity sensitivity in the vestibular nuclei of squirrel monkeys. Studies of neural responses during a variety of other motor behaviors also have observed different responses for different spatial directions of motion. These studies used optokinetic (Gomi et al. 1998; Shidara et al. 1993; Zhang et al. 1995), head rotation (Angelaki 1991; Baker et al. 1984; Schor and Angelaki 1992), and arm movement (Georgopoulos and Massey 1985; Kettner et al. 1988) paradigms.

For motion of constant amplitude, the model indicates that the relative influence of velocity versus position will be stronger during higher velocity pursuit and weaker during lower velocity pursuit. A tendency for position influences to dominate at lower velocities and velocity influences to dominate at higher velocities was previously reported by Noda and Warabi (1982). This idea also may account for the emphasis placed on velocity control in other reports. The peak velocity used for most stimuli in the present study was lower (19°/s) than peak velocities used by Lisberger and Fuchs (1978, 51°/s), Miles et al. (1980, 25°/s), Stone and Lisberger (1990, 31°/s), and Krauzlis and Lisberger (1996, 31°/s). These studies also tended to focus on "gaze velocity" Purkinje cells that may turn out to be more sensitive to eye velocity.

Some reports (Lisberger and Fuchs 1978: Lisberger et al. 1994; Miles et al. 1980; Stone and Lisberger 1990) show broad distributions of response phase for individual neurons. Although these distributions peak near eye velocity, many neurons have phases that either lead or lag eye velocity by as much as 90°. Phase lags are compatible with a combination of position and velocity coding, whereas phase leads generally have been associated with the combined influence of velocity and acceleration. However, at the target accelerations used in the present experiments, acceleration had only a weak influence on firing rate. In these paradigms, phase leads were produced instead by position and velocity influences that acted in different directions. This does not rule out a role for acceleration influences in other paradigms that require larger eye accelerations or that study pursuit initiation.

Position and velocity influences in pursuit behavior

Behavioral experiments indicate that accurate pursuit is generated using a combination of position and velocity control. Early experiments by Rashbass (1961) clearly established the role of target velocity in controlling pursuit using a step-ramp paradigm. Lisberger and Pavelko (1989) later varied step size while keeping ramp velocity constant and showed that initial target position also influenced pursuit albeit 20 ms after movement initiation. Other experiments have demonstrated that smooth eye movements are elicited by stabilized retinal-position errors created using off-fovea photoflash afterimages (Kommerell and Taumer 1972) or eye-motion feedback (Pola and Wyatt 1980). Subsequent experiments (Morris and Lisberger 1987) have demonstrated that both retinal-position and retinal-velocity errors produce changes in eye motion when the eye is pursuing a target immediately before image stabilization. Flocculus and paraflocculus sensitivities to both eye position and eye velocity are compatible with a role in generating this position/velocity control.

There is also evidence for differences in the control of fixation and pursuit. Robinson et al. (1986) stated that "fixation is not pursuit at zero velocity" based on the occurrence of oscillations after a step to constant velocity pursuit and the absence of oscillations after a similar step back to zero velocity (see also Luebke and Robinson 1988). Others have shown that target oscillations (Goldreich et al. 1992) and target perturbations (Schwartz and Lisberger 1994) have a smaller effect during fixation than during pursuit. Morris and Lisberger (1987) have demonstrated that stabilized retinal-position error plays a reduced role when the eye is fixating a target and a stronger role when the eye is pursuing a target. Interestingly, Krauzlis and Miles (1996) report that the behavioral context can also influence transitions to fixation. Offset oscillations were not observed on blocks of trials where the target always stopped but did occur when the target only sometimes stopped or changed direction or speed. These differences in fixation and pursuit may reflect the combined influence of different neural systems. Recently neural responses in the superior colliculus have been related to tracking errors during pursuit eye movements in addition to its responses related to saccades (Krauzlis et al. 1997). This suggests its possible role in pursuit eye movements in addition to those regions more commonly associated with pursuit. The present experiments suggest that some neurons in the flocculus and paraflocculus have different sensitivities to static position during fixation and dynamic position during pursuit. These differences may reflect a partial separation in the control of fixation and pursuit within these brain regions.

Coordination of position and velocity signals during complex motor control

At the level of the motoneuron, a combination of position and velocity signals acting in different directions generally are required to produce smooth eye motion. The only time position and velocity influences are aligned is during radial motions away from the origin (e.g., rightward motion to a rightward location). During radial motions toward the origin (e.g., leftward motion to a rightward location) position and velocity signals point in opposite directions. The position and velocity control signals required for circular motion are always at right angles to each other while they vary continuously in time. For example, during CW pursuit a combination of downward velocity and rightward position signals must grade smoothly into signals reflecting a combination of leftward velocity and downward position. These signals are reminiscent of our recordings in the flocculus and paraflocculus during pursuit: position and velocity influences are aligned for some neurons, nearly opposite for others, and approximately orthogonal for still others. Other position/velocity combinations are required for more complicated motions in 2D space.

All models of the pursuit system acknowledge the need to provide a coordinated combination of position, velocity, and acceleration signals to motoneurons. That said, most modeling approaches (e.g., Deno et al. 1989, 1995; Kettner et al. 1997; Krauzlis and Lisberger 1989, 1994; Krauzlis and Miles 1996; Robinson et al. 1986; Young 1971) have simplified the control problem by focusing on eye velocity and acceleration and assuming that position information is obtained by integration. This is mathematically defensible because a "perfect integrator" can compute current eye position from an initial eye position and a velocity history. Whether "neural integration" actually is used by the brain systems that control pursuit is an empiric question that has been discussed extensively (Fukushima et al. 1992; Robinson 1989). The presence of position and velocity signals in the flocculus and paraflocculus suggest that these regions could be involved in neural integration during pursuit.

It is also possible that the flocculus and paraflocculus directly generate the position and velocity signals that are used to control eye motion without the need for an independent neural integrator. This might be useful for the control of very slow eye movements where the neural integration of weakly modulated velocity signals could produce large errors. Under these conditions, position control becomes more appropriate than velocity control. The direct calculation of position information by the pursuit system also would increase its processing speed by eliminating any additional time required to perform neural integration. These signals could be combined downstream with signals from other systems to produce accurate pursuit over a wide range of velocities.


    ACKNOWLEDGMENTS

We thank A. Williams for help in data analysis and preparation of the manuscript and figures.

This work was supported by National Institutes of Health Grants P50 MH-48185 and T32 DC-00015-13, and by National Science Foundation Grant IBN-9723846.


    FOOTNOTES

Address for reprint requests: R. E. Kettner, Dept. of Physiology M211, Northwestern University Medical School, 303 E. Chicago Ave., Chicago, IL 60611.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received 5 October 1998; accepted in final form 30 August 1999.


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ABSTRACT
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