Aerospace Medical Research Unit, McGill University, Montreal, Quebec H3G 1Y6, Canada
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ABSTRACT |
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Sylvestre, Pierre A. and
Kathleen E. Cullen.
Quantitative Analysis of Abducens Neuron Discharge Dynamics
During Saccadic and Slow Eye Movements.
J. Neurophysiol. 82: 2612-2632, 1999.
The mechanics of the eyeball
and its surrounding tissues, which together form the oculomotor plant,
have been shown to be the same for smooth pursuit and saccadic eye
movements. Hence it was postulated that similar signals would be
carried by motoneurons during slow and rapid eye movements. In the
present study, we directly addressed this proposal by determining which
eye movement-based models best describe the discharge dynamics of
primate abducens neurons during a variety of eye movement behaviors. We
first characterized abducens neuron spike trains, as has been
classically done, during fixation and sinusoidal smooth pursuit. We
then systematically analyzed the discharge dynamics of abducens neurons
during and following saccades, during step-ramp pursuit and during high
velocity slow-phase vestibular nystagmus. We found that the commonly
utilized first-order description of abducens neuron firing rates
(FR = b + kE + r, where FR is firing rate, E and
are eye
position and velocity, respectively, and b,
k, and r are constants) provided an
adequate model of neuronal activity during saccades, smooth pursuit,
and slow phase vestibular nystagmus. However, the use of a second-order
model, which included an exponentially decaying term or "slide"
(FR = b + kE + r
+ uË
c
), notably improved our ability to describe
neuronal activity when the eye was moving and also enabled us to model
abducens neuron discharges during the postsaccadic interval. We also
found that, for a given model, a single set of parameters could not be
used to describe neuronal firing rates during both slow and rapid eye movements. Specifically, the eye velocity and position coefficients (r and k in the above models,
respectively) consistently decreased as a function of the mean (and
peak) eye velocity that was generated. In contrast, the bias
(b, firing rate when looking straight ahead) invariably
increased with eye velocity. Although these trends are likely to
reflect, in part, nonlinearities that are intrinsic to the extraocular
muscles, we propose that these results can also be explained by
considering the time-varying resistance to movement that is generated
by the antagonist muscle. We conclude that to create realistic and
meaningful models of the neural control of horizontal eye movements, it
is essential to consider the activation of the antagonist, as well as
agonist motoneuron pools.
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INTRODUCTION |
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To move the eye to a new position, the net force
generated by the extraocular muscles must compensate for the
passive restraining forces of the eyeball, extraocular
muscles, and supporting orbital tissues (i.e., the "oculomotor
plant"). Robinson (1964) characterized the oculomotor
plant during saccadic eye movements and found that the mechanics are
dominated by its viscoelastic properties. As a consequence of the
viscous drag opposing eye rotation, the extraocular muscles must
generate a burst of force (or "pulse") to produce rapid saccadic
eye movements. Furthermore, after a saccade, the muscles must generate
a tonic force (or "step") to counteract the elastic elements of the
orbital tissues and hold the eyeball stationary in the orbit. Finally,
during the transition from the pulse to the step, an exponential decay
in force offsets the slow viscoelastic properties of the orbital
tissues, thereby improving ocular stability.
Based on 1) his characterization of the oculomotor plant
mechanics, and 2) a linear approximation of the relationship
between motoneuron drive and resultant muscle force, Robinson
(1964) proposed a model of the net neural drive
(FRNET) during saccades
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(1) |
Robinson and colleagues showed that the "pulse-step" nature of
oculomotor (Robinson 1970) and abducens (Robinson
and Keller 1972
) neuron discharges during saccades could be
approximated using a first-order simplification of Eq. 1
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(2) |
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(3) |
A first-order model (Eq. 2) has also been commonly used to
describe neuronal discharges during slower eye movements
including fixation, smooth pursuit, vergence, and slow-phase vestibular nystagmus (Henn and Cohen 1973; Keller and
Robinson 1971
; King et al. 1994
; Robinson
and Keller 1972
; Skavenski and Robinson 1973
).
However, it has been suggested that some additional terms in Eq. 1 must be retained to describe the dynamics of abducens neuron
discharges. Keller (1973)
and Goldstein
(1983)
proposed adding an eye acceleration term, and a
c
term to Eq. 2, respectively. More recently, Fuchs et al. (1988)
and Stahl and
Simpson (1995)
used Eq. 2 to calculate eye position
(k) and eye velocity (r) sensitivities of
abducens neuron discharges during sinusoidal eye movements at different
frequencies. During such eye motion, the estimated eye position and eye
velocity sensitivities are only "apparent" because, for example,
they would also reflect neuronal sensitivities to acceleration and
jerk, respectively. Indeed, Fuchs and colleagues proposed that a
third-order simplification of Eq. 1 is required to describe
the frequency dependence of their calculated k and
r values. Stahl and Simpson (1995)
obtained a similar frequency dependence in their analysis but argued that a
second-order simplification of Eq. 1 is sufficient when the model's time constants are properly selected. The conclusions of both
studies were based on indirect estimates of the terms in Eq. 1 that were obtained by fitting the averaged r and
k coefficients. To date, no direct evaluation of each term
in Eq. 1 has been performed by fitting the discharges of
individual neurons.
Although the discharge of abducens nucleus (ABN) neurons has been
studied during rapid (e.g., saccadic) and slower (e.g., smooth pursuit
and slow-phase vestibular nystagmus) eye movements, a unifying
description of the agonist neuronal drive to the extraocular muscles
has not yet been reported. However, Robinson (1965)
demonstrated that the mechanics of the oculomotor plant are identical
for saccades and smooth pursuit. Consequently, in this study, we sought
to construct a mathematical model that best describes the input-output relationship between abducens neuron firing rates and eye movements during saccades as well as slower eye movements. We directly fitted neuronal discharges using an approach similar to that taken by Cullen and Guitton (1997)
, which allowed us to
objectively evaluate the relative importance of each term in Eq. 1 during paradigms for which the dynamic profiles of eye position,
velocity, and higher order derivatives differed.
Our results indicate that a second-order simplification of Eq. 1 provides an improved description of abducens neuron discharges when compared with that obtained with Eq. 2. However, the
coefficients that were estimated for this higher order model varied as
a function of both the mean and peak eye velocities generated during
the different behavioral paradigms. Although these trends are likely to
reflect, in part, nonlinearities that are intrinsic to the extraocular
muscles (Barmack 1977; Close and Luff
1974
; Collins 1971
; Goldberg et al.
1998
; Shall et al. 1996
), we suggest that our
results are also consistent with the relative change in active force
that is generated by the antagonist muscle during slow
versus rapid eye movements. We conclude that to create realistic models of oculomotor control, future work should consider the activation of
antagonist as well as agonist motoneuron pools.
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METHODS |
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Two rhesus monkeys (macaca mulatta) were prepared for chronic extracellular recording using aseptic surgical techniques. All procedures were approved by the McGill University Animal Care Committee and were in compliance with the guidelines of the Canadian Council on Animal Care.
Surgical procedures
The animals were preanesthetized using ketamine hydrochloride (12-15 mg/kg im). In addition, atropine sulfate (0.04 mg/kg im) and valium (1 mg/kg im) were administered to reduce salivation and provide muscle relaxation, respectively. Surgical levels of anesthesia were then achieved using isoflurane gas (2-3%, initially) inhaled through an endotracheal tube, and maintained for the duration of the surgery (0.8-1.5%). Heart rate, blood pressure, respiratory rate, and body temperature were monitored for the duration of the procedure.
To allow electrode access into the brain stem, craniotomies were
performed on both animals. A dental acrylic implant was fastened to
each animal's skull using stainless steel screws. A stainless steel
post, which was used to restrain the animal's head during the
experiment, and a stainless steel recording chamber, which was
positioned to provide access to the abducens nucleus region (posterior
angle of 30° and lateral angle of 30°), were cemented to the
implant. In addition, an 18-19 mm diam eye coil (3 loops of
teflon-coated stainless steel wire) was implanted in the right eye
behind the conjunctiva (Fuchs and Robinson 1966). After
the surgery, buprenorphine (0.01 mg/kg im) was utilized for
postoperative analgesia, and the antibiotic trimethyl sulfate (TMS;
24%, 0.125 ml/kg im, for 5 days) was administered to prevent
infection. Animals were given 2 wk to recover from the surgery before
any experiments were performed.
Data acquisition procedures
During each experiment, the monkey was comfortably seated in a
primate chair that was mounted on a vestibular turntable. The monkey's
head was restrained for the duration of the experiment, and the room
was dimly lit. Extracellular single-unit activity was recorded using
enamel insulated tungsten microelectrodes (7-10 M impedance,
Frederick Haer) as has been described elsewhere (Cullen and
Guitton 1997
). The abducens nucleus was identified on the basis
of an increase in background activity that occurred just below the
fourth ventricle. The simultaneous activity of the neurons in this
structure produced a characteristic "singing beehive" sound, which
was clearly related to ipsilaterally directed eye motion, when the
recorded activity was fed into an audio monitor (Fuchs and
Luschei 1970
; Robinson 1970
). Only units for
which this unique sound could be heard in the background were included in the present study. The location of each neuron was further confirmed
using three-dimensional reconstructions of electrode tracts; units that
were located in regions >0.5 mm from the estimated center of the
abducens nucleus were not included in the present study.
The abducens nucleus contains three classes of neurons: 1)
motoneurons (MNs) that project directly to the lateral rectus, 2) internuclear neurons (INNs) that project contralaterally
to the medial rectus division of the oculomotor nucleus, and
3) neurons that project to the cerebellum. It has been shown
that the signals carried by abducens MNs and INNs are qualitatively
similar during all types of eye movements (Delgado-Garcia et al.
1986a,b
; Fuchs et al. 1988
; Gamlin et al.
1989
). Due to the invasiveness of implanting an electrode in
the abducens nerve for antidromic activation (Delgado-Garcia et
al. 1986a
), and/or a recording electrode in the lateral rectus for spike-triggered averaging (Fuchs et al. 1988
), we
elected not to electrophysiologically identify MNs and INNs. Instead, we compared the discharge characteristics of the neurons in the present
study with those of the identified MNs and INNs in the study of
Fuchs et al. (1988)
, and estimated that our sample
contained roughly equal numbers of MNs and INNs (see Fig. 11 in
RESULTS). Floccular projecting neurons appear to constitute
only a small percentage of abducens nucleus neurons and are primarily
confined to the dorsal/rostral perimeter of the nucleus (Blanks
et al. 1983
; Langer et al. 1985
; Rodella
et al. 1996
). Therefore we estimate that our sample contained
only a small proportion, if any, of these units.
Unit activity, horizontal and vertical eye positions, target position,
and table velocity were recorded on digital audio tape for
later playback. The isolation of each unit was then carefully reevaluated off-line. An abducens neuron was only considered to be
adequately isolated when discrete action potential waveforms could be
clearly dissociated from the surrounding background activity during
saccades (Fig. 1) as well as during
fixation and smooth pursuit. During playback, action potentials were
discriminated using a windowing circuit (BAK Electronics)
following amplification and filtering of the recorded activity
(Cullen and Guitton 1997). Eye position, target
position, and table velocity signals were low-pass filtered at 250 Hz
(analogue 8-pole Bessel filter) and sampled at 1 kHz. Subsequent
analysis was performed using custom algorithms (Matlab, Mathworks).
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Behavioral paradigms
Both monkeys were trained to follow a target light for a juice
reward. The activity of abducens neurons was recorded during fixation,
saccades, smooth pursuit, and whole-body rotations in the dark that
elicited the vestibuloocular reflex (VOR). A target light (HeNe laser)
was projected, via a system of two galvanometer controlled mirrors,
onto an isovergence screen located 60 cm away from the monkey's head.
Target movements and on-line data displays were controlled using REX, a
QNX-based real-time acquisition system (Hayes et al.
1982).
Ipsilaterally and contralaterally directed saccades were elicited by
stepping the target between horizontal positions in blocked trials
(predictable, ±5, 10, 15, 20, 25, and 30°) and unblocked trials
(random, ±5, 10, 15, 20, 25 and 30°). In addition, saccades with
different starting positions, amplitudes, and velocity profiles were
obtained using the "barrier" paradigm in which a food target appeared unexpectedly on either side of an opaque screen facing the
monkey (Cullen and Guitton 1997).
Smooth pursuit eye movements were elicited using two different types of
target motion: 1) sinusoidal trajectories (40°/s peak velocity, 0.5 Hz) and 2) step-ramp trajectories
(Rashbass 1961). A step-ramp trial began when the animal
fixated a stationary target with its eye centered in the orbit (defined
as "primary position"). After a random fixation period (750-1,500
ms) the target was stepped toward either the left or the right, and
then began moving at a constant velocity (20, 40, or 60°/s) in the
direction opposite to that of the step. When the step size was properly
chosen, it was possible to obtain smooth eye movements that were not
preceded by corrective saccades.
Neuronal activity was also recorded during slow-phase vestibular
nystagmus. First, rapid manual rotations of the vestibular turntable
were utilized while the monkey sat in complete darkness to elicit
slow-phase velocities up to 200°/s (rapid VOR,
VORR) (Roy and Cullen 1998).
Second, a sinusoidal whole-body rotation paradigm in which the monkey
maintained fixation on a target that moved with the vestibular
turntable was used (VOR cancellation, VORC;
40°/s peak velocity, 0.5 Hz).
Analysis of abducens neuron discharges
Before analysis, recorded eye position signals were digitally
filtered at 125 Hz and digitally differentiated to produce eye velocity
profiles. A spike density function, in which a Gaussian function was
convolved with the spike train (standard deviation of 5 ms for saccades
and VORR, and 10 ms for fixation,
VORC, step-ramp, and sinusoidal smooth pursuit),
was employed to represent the neuronal discharges of abducens neurons
(Cullen and Guitton 1997; Cullen et al.
1996
).
ANALYSIS DURING FIXATION.
Only periods of steady fixation >100 ms in duration that did not
include the 50-ms period following or preceding a saccade, and for
which mean eye positions were within the linear range of the unit
(Robinson 1970) were included in the analysis. A
standard bivariate linear regression between mean firing rate and
fixation position was used to obtain the horizontal eye position
sensitivity (kFIX) and the resting
discharge at primary position (bFIX)
of each neuron.
DYNAMIC ANALYSIS DURING SACCADES.
We utilized a system identification technique that has been previously
developed for the analysis of horizontal inhibitory burst neurons
(IBNs) located in the paramedian pontine reticular formation
(Cullen et al. 1996; Cullen and Guitton
1997
). This method allowed us to determine how well different
models based on the dynamics of eye movement trajectories (models
M1-M9; Table 1) predicted abducens
neuron discharges during saccades. An advantage of using this approach
was that each sampled data point could be utilized in the analysis. For
example, a single saccade of 100 ms duration sampled at 1 kHz would
have provided 100 data points to the optimization algorithm. A dynamic
lead time value (td) was determined
for each abducens neuron (see RESULTS) and used in the
optimization of the models shown in Table 1.
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DYNAMIC ANALYSIS DURING POSTSACCADIC SLIDE.
Approximately 10 postsaccadic intervals were selected for each cell.
Intervals were chosen that 1) were exempt of artifacts and
2) followed saccades of amplitudes between 5 and 15°. Each interval spanned the period from saccade offset until 200 ms after saccade offset (equivalent to Phase 5 as described by Goldstein 1983). Parameters were estimated for models M3 and
M8 (Table 1). The same optimization techniques as described
in the DYNAMIC ANALYSIS DURING SACCADES section were used
to determine the best model fit. The initial conditions (ICs) for the
exponentially decaying term of model M8
(c
) were taken from the data (see
Cullen et al. 1996
), and the optimal lead time td was taken from the saccadic analysis.
DYNAMIC ANALYSIS DURING SMOOTH PURSUIT.
We investigated the ability of different dynamic models to predict the
activity of abducens neurons during sinusoidal smooth pursuit
(model M3; Table 1) and step-ramp pursuit (models
M1-M5, and M8; Table 1). Abducens neuron discharges
were first characterized during five or more cycles of sinusoidal
smooth pursuit that contained few saccades and for which pursuit gain
was >0.8. Only segments that did not include the 50-ms period
following or preceding a corrective saccade were included in the
analysis (Cullen et al. 1993).
DYNAMIC ANALYSIS DURING VORR. We also investigated the ability of models M1-M5 and M8 (Table 1) to predict the activity of abducens neurons during slow-phase vestibular nystagmus. Segments of slow phase VORR were chosen in which the peak velocities ranged from 50 to 200°/s. Segments of records spanning the interval 50 ms immediately preceding quick phases to 50 ms immediately following quick phases were excluded from the analysis. Each model fit was made to an ensemble of 20-40 VORR intervals. Eye movement traces for this analysis were offset by the optimal lead time (td) determined during saccades (see RESULTS).
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RESULTS |
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Neuronal database
A total of 46 isolated ABN neurons were analyzed during steady fixation, during saccades, immediately following saccades (postsaccadic intervals), and during sinusoidal smooth pursuit. Recordings from 37 of these neurons were obtained from monkey B, whereas 9 were obtained from monkey C. In addition, the firing rates of 25% of these neurons were also analyzed during pursuit of step-ramp target motion and during VORR.
Figure 2 shows the firing rate of a typical ABN neuron, unit B76_2. During fixation, this unit's tonic firing rate increased proportionally with ipsilateral eye position (Fig. 2A). It generated a burst of action potentials during ipsilaterally directed saccades ("ON direction") and ceased firing (paused) during contralaterally directed saccades ("OFF direction"; Fig. 2A, filled and open arrows, respectively). In addition, the modulation of this neuron's firing rate led ipsilateral eye position during sinusoidal smooth pursuit (Fig. 2B). Finally, this unit was unresponsive to head movements; during sinusoidal VORC paradigms, the residual modulation of this unit's firing rate could be accurately predicted based on the neuron's sensitivity to eye position and eye velocity (Fig. 2C).
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In our sample, all neurons had an ON direction response
similar to unit B76_2 during saccades (Fig. 2A,
filled arrows). Eighty-one percent of these neurons paused completely
for OFF direction saccades of all amplitudes (3°).
Unit B76_2, which was in this category, paused completely
during 5, 15, and 30° saccades (Fig.
3A). The remaining 19% of the
neurons in our sample ceased firing only during large saccades. Figure
3B shows the response of the unit for which we found the
least inhibition during OFF direction saccades (unit
B11_1). This neuron paused completely only for saccades >15°.
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Estimation of lead time
The time by which an ABN neuron's burst led saccadic eye
movements was calculated using two methods. In the first method, the
dynamic lead time by which an ABN neuron's firing rate
preceded saccadic eye movement onset,
td, was estimated using a first-order model (model M3; Table 1) (Cullen et al.
1996; Cullen and Guitton 1997
). Results are
shown in Fig. 4A (
), for
example unit B76_2. The lead time that provided the largest
VAF was defined as the optimal dynamic lead time,
td, and is shown by the thick arrow in
Fig. 4A (in this example,
td = 10 ms). We also attempted to compute the value of td using a
second-order model (model M4; Table 1). The results from
this analysis are also shown in Fig. 4A (
). The optimal
lead time computed with model M4 was 13 ms (small thin
arrow).
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Figure 4A clearly shows that model M3 was more
sharply "latency tuned" than model M4. In addition, to
our surprise, the sign of the acceleration term estimated using
model M4 was always negative at the optimal lead time.
Because a negative acceleration term is physiologically unrealistic, we
reestimated td using a version of
model M4 in which the sign of the acceleration term was
constrained to be nonnegative (model M4c; not in tables).
The results of this analysis are shown in Fig. 4B (thick
line). We found that because the value of the acceleration term in
model M4c consistently converged to zero at
td, the
td estimated using model
M4c was always identical to the value determined using model
M3. We also attempted to estimate the lead time with more complex
models. However, it became increasingly difficult to compute a reliable
estimate of lead time because the influence of this parameter was
obscured by the other parameters that were simultaneously optimized
(see Cullen et al. 1996). Accordingly, in the present
study, we utilized model M3 for the determination of
td. When the eye movement trajectories
were shifted by td, the main portion
of the ABN neuron burst was well aligned with the duration of the saccade.
Prior studies have estimated ABN neuron lead times by calculating the time interval between the occurrence of the first spike in the burst (as determined by visual inspection) and onset of saccadic eye movement. We also measured the lead times of our neurons using this method and found that the values obtained were significantly shorter than those provided by the dynamic analysis method (4.4 ± 1.2 versus 9.4 ± 1.9 ms, respectively; mean ± SE, Student t-test, P < 0.01). Across our sample of neurons, we observed a weak but significant linear relationship (R = 0.37, P < 0.01) between these two different estimates of lead time.
Figure 5 illustrates the distribution of
lead times that were obtained using both methods (black bars). For
comparison, the lead times obtained by Cullen and Guitton
(1997) for short lead inhibitory burst neurons (IBNs located in
the paramedian pontine reticular formation) are also shown (gray bars).
A difference of 2.4 ms was found when comparing the mean
td of short lead IBNs to that of ABN neurons
(Fig. 5A). In contrast, the lead times estimated for
short lead IBNs using the method of the first spike were on average 7.2 ms longer than those calculated using the same method for ABN neurons
(Fig. 5B). Short lead IBNs are known to project
monosynaptically to the abducens nucleus (Hikosaka et al.
1978
; Strassman et al. 1986a
,b
). Because a
monosynaptic connection is generally associated with <1.3 ms
processing delays, the 2.4 ms difference observed between the dynamic
latency estimates is more consistent with a monosynaptic projection
than is the 7.2 ms difference obtained using the first spike method.
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Metric analysis during fixation
Numerous studies have demonstrated that the mean firing rate of an
ABN neuron is well related to eye position during periods of steady
fixation. We verified this relationship for our sample of neurons.
Figure 6A shows the results
for unit B76_2. The inset illustrates two
fixation intervals that complied with our criteria (see
METHODS). Figure 6B shows the regression lines
plotted for each neuron in our sample. During fixation, the mean firing
rate was well correlated with eye position for all 46 neurons
[bFIX (y-intercept) = 97 ± 67 spikes/s, kFIX
(slope) = 5.2 ± 2.7 (spikes/s)/deg, and R = 0.80 ± 0.14]. In the present study, we made no attempt to
control for hysteresis in ABN neuron firing rates (Goldstein and
Robinson 1986). However, the strong correlations that we
obtained in our analysis of fixation suggest that the relative
contribution of hysteresis to the firing rate of ABN neurons was small.
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Metric analysis of saccades
Excitatory and inhibitory premotor burst neurons in the paramedian
pontine reticular formation (EBNs and IBNs, respectively) provide the
primary drive to ABN neurons during saccades. Previous studies have
demonstrated that the number of spikes (NOS) generated by EBNs and IBNs
is well correlated to saccade amplitude (Cullen and Guitton
1997; Scudder 1988
; Strassman et al.
1986a
,b
). A comparable analysis has never been performed on ABN
neurons, presumably because their inherent position sensitivity made it
difficult to isolate the "burst" component of their discharges.
By using dynamic analysis techniques, we were able to obtain an
objective estimate of each ABN neuron's position sensitivity during
saccades (see Dynamic analysis during saccades section below). We used model M3 (Table 1) to estimate the saccadic
eye position sensitivity of each neuron
(kSAC; as a convention, the k refers to the model parameter, and the subscript SAC
refers to the paradigm during which it was estimated). The neuronal
firing rate was first corrected by subtracting the time-varying eye
position contribution (kSACE), thus
unmasking the remaining burst signal. An eye position-corrected NOS
(NOSC) was then computed by multiplying the
residual firing rate (during saccades only) by the saccade duration. As
a control, we computed NOSC using a
kSAC value of 0 and found that
NOSC estimates differed from the measured NOS on
average by <1 spike (0.6 ± 0.6 spike).
The NOSC was well correlated with saccade
amplitude for our example neuron (Fig.
7A), unit B76_2.
Figure 7B shows the regression lines for our sample of
neurons (thin lines). The mean regression coefficient (R)
for our sample was 0.89 ± 0.06. The averaged slope obtained for
the NOSC versus saccade amplitude relationship
was 0.96 ± 0.46 spikes/deg, with an intercept of 3.3 ± 2.3 spikes (thick line). Cullen and Guitton (1997) reported
slightly lower correlation coefficients (R = 0.79), and
similar slopes (1.0 ± 0.5 spikes/deg) in a comparable analysis of
IBNs.
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We also estimated the relationship between peak eye velocity and peak
firing rate (corrected for eye position) during saccades. Figure
7C shows this relationship for our example neuron,
unit B76 2. Although we observed lower correlations
(R = 0.48 ± 0.19) than for the
NOSC analysis, 85% of our units had significant
relationships (P < 0.05) between their peak firing
rate and peak velocity. The mean slope was 0.27 ± 0.22 (spikes/s)/(deg/s), with a mean intercept of 242 ± 116 spikes/s.
Figure 7D shows the regression lines for the 39 ABN neurons
for which this relationship was significant. Removing the seven units
with nonsignificant correlations did not change the mean values
significantly (P > 0.05). The mean correlation for
this relationship was larger than that which has been reported
previously for primate IBNs (mean R = 0.39)
(Cullen and Guitton 1997). In addition, larger slopes
were obtained for IBNs [0.40 ± 0.20(spikes/s)/(deg/s)]
(Cullen and Guitton 1997
) than for the ABN neurons in
the present study.
Dynamic analysis during saccades
We constructed a set of models that allowed us to systematically investigate which terms from Eq. 1 are required to predict ABN neuron discharges during saccades (models M1-M5 and M8-M9; Table 1). We also investigated other models that have been suggested by prior analyses of ABN neurons and IBNs (models M6-M7; Table 1). The mean VAF and BIC values obtained during saccades, for each model, are provided in Table 1. The improvement in the mean VAF value relative to model M3 is also shown for each model. We chose model M3 as our reference because it represents the first-order model that is often used in the oculomotor literature to describe ABN neuron discharges (Eq. 2). Table 2 provides the mean parameters that were estimated for each model during saccades (associated ranges are shown in parentheses).
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Given that the saccadic burst of ABN neurons is thought to originate
principally from EBNs that are known to carry an eye velocity signal,
we determined the percentage of variance in an ABN neuron's discharge
that could be predicted by a model based solely on eye velocity
(model M1; Table 1). The negative VAF obtained for
model M1 (mean VAF = 1.1) indicates that the model fit was poor, and that in fact fitting a mean value through the firing
rate profiles would have provided a better fit. The addition of a bias
term (model M2; Table 1) resulted in a significant increase
in VAF (mean VAF = 0.33). Including a position term (model M3; Table 1) further increased the VAF (mean VAF = 0.59),
highlighting the need for such a term even when describing the
"pulse" portion of the neuronal drive. Note that the BIC values
decreased markedly as terms were sequentially added (compare
models M1-M3), confirming the importance of the position
and bias terms.
Model M3 is generally recognized as a good estimate of ABN
neuron activity during fixation, smooth pursuit, and saccades
(Fuchs and Luschei 1970; Keller 1973
;
Keller and Robinson 1972
). The fit of model
M3 to the firing rate of our example neuron during saccades is
shown in the top panel of Fig.
8. The thicker line represents the model
fit that is superimposed on the actual firing rate (shaded area). In
addition, the average parameters and VAF estimated across all neurons
are shown below the model fits.
|
The addition of an acceleration term to model M3 had little influence on our ability to predict ABN neuron discharges (model M4; Table 1). The mean VAF value of model M4 was increased by only 1%, and the BIC value was the same for both models (Table 1). Moreover, the value of the estimated acceleration term was most often (99%) negative and invariably very small (Table 2). The similarity between this model's ability to fit the data and that of model M3 is illustrated in Fig. 8 (compare the 2nd and top panels). We also found that the addition of a jerk term to model M4 was not warranted, because the mean VAF and BIC values obtained with model M5 were identical to those of model M4 (Table 1).
In the present study, we also investigated the usefulness of nonlinear
models. The first nonlinear model (M6; Table 1) included an
amplitude-dependent term (z). Prior analyses had
demonstrated that such a term is important for describing IBN
discharges during saccades (Cullen and Guitton 1997). We
therefore postulated that an amplitude-dependent term might also be
present on ABN neuron discharges. However, the addition of this term
only marginally improved our ability to fit ABN neuron firing rates
(VAF = 0.60 vs. 0.59, BIC = 7.5 vs. 7.6, M6 vs.
M3, respectively).
A second nonlinear model was tested that included a second- and
third-order nonlinearity
(r12 and
r2
3) in
addition to the terms of model M4 (M7; Table 1).
This model was included in our analysis to approximate the hypothesis
of Van Gisbergen and colleagues (1981)
that a nonlinear
function of eye velocity as well as an eye acceleration term are
necessary to accurately predict the firing rate of ABN neurons during
saccades (Eq. 3). Model M7 provided little
improvement in the mean sample VAF when compared with model
M3 (VAF = 0.62 vs. 0.59, BIC = 7.5 vs. 7.6, respectively). The small decrease in BIC for model M7 versus
M3 implies that the addition of nonlinear terms was
warranted. Nevertheless, the accompanying small increase in VAF
indicates that these terms have a relatively small influence on
neuronal discharge dynamics.
We also evaluated the importance of the c
in Eq. 1 for describing ABN neuron discharges during
saccades. Goldstein (1983)
first quantified this term in
the interval immediately following saccades. In our analysis, we
investigated whether such a transient term was important
during saccades. We first added this term to model
M3 (FR = b + kE + r
c
; not
in tables). This model yielded generally negative and invariably very
small estimates for cSAC (mean =
0.0003 ± 0.0008 ms). Note that during saccades, the shapes of
the firing rate and of the eye velocity profiles were very similar
(Fig. 8), and consequently,
(t)
Ë(t
td),
approximately. Hence it follows that adding a
c
term to model M3 roughly
simplifies to model M4 during saccades. In fact, the mean
VAF value obtained for this model (0.60) was equal to that obtained for
model M4. Furthermore, adding a
c
to model M3 resulted in a
relative increase in BIC (7.7 vs. 7.6, respectively), which confirms
that the simple addition of a
(t) term to
model M3 was not warranted.
We next tested a model that provided a second-order simplification of
Eq. 1, model M8. This model contained an
acceleration term in addition to a slide term. It provided a notable
increase in mean VAF when compared with model M3 (mean
VAF = 0.66 vs. 0.59, respectively), which was accompanied by a
relative decrease in BIC (mean BIC = 7.4 vs. 7.6, respectively).
As previously mentioned, (t)
Ë(t
td)
approximately, during saccades. However, although the dynamic profiles
of these two terms may be similar, it is clear that they are not
identical; the resultant increase in VAF obtained with the addition of
both acceleration and slide terms indicates that the dynamics of the
two terms interact in a synergistic manner to fit neuronal discharges.
The parameter values (b, k, and
r) obtained using model M8 did not differ
significantly from those estimated using model M3. Importantly, this model was the only one for which the estimates of the
acceleration term were in the ON direction of the unit (Table 2). The model fit to the firing rate of unit B76_2
using model M8 is illustrated in Fig. 8 (bottom
panel) for the same three example saccades that were used to
illustrate fits for models M3 and M4. Note that
the parameter estimates for the term obtained in the present study
(cSAC = 15 ± 16 ms) were
comparable to those previously calculated for short lead IBNs using a
similar model (cIBN = 19 ± 16 ms) (Cullen and Guitton 1997).
We also tested a variation of model M8 for which the ICs of
the c term were
estimated separately for each saccade rather than taken from the data
as they were in model M8 (Cullen et al. 1996
;
Cullen and Guitton 1997
). Although the mean sample VAF
increased dramatically for this model (mean VAF = 0.76; not in
tables), it is not likely that this model provides a physiologically
relevant description of ABN neuron discharges. This model often yielded
highly nonphysiological values for estimated parameters. For example,
large unrealistic values were generally estimated for the bias (mean
bias =
346 spikes/s). Similar results were obtained in a prior
analysis of IBN discharges (Cullen and Guitton 1997
).
We conclude that model M3 provided a good description of ABN
neuron discharges during saccades. Adding higher order derivatives of
eye position or nonlinear terms to this simple first-order model only
marginally improved our capacity to predict ABN neuron saccadic firing
rates. Finally, a second-order model that included a
c term (model M8)
provided the most accurate description of ABN neuron discharges of the
models that we tested.
A striking result of this analysis was that the dynamic eye position sensitivities of ABN neurons estimated using either model M3 or M8 during saccades were considerably smaller than the static values estimated during fixation (kSAC < kFIX, P < 0.05). In addition, the biases estimated during saccades were notably larger than those estimated during fixation (bSAC > bFIX, P < 0.01). To further emphasize this result, we tested a final model (model M9; Table 1). This model was similar to model M3, with the difference that its bias and position parameters were assigned the values estimated during fixation (i.e., bFIX and kFIX). The mean sample VAF provided by this model was very poor (Table 1); the value was comparable to those of our worst models (M1 and M2; Table 1).
Dynamic analysis of postsaccadic slide
We next analyzed the firing rate of ABN neurons during the 200-ms
time interval that immediately follows the end of a saccade. During
this interval, the eyes are immobile; however, the firing rate of ABN
neurons decays exponentially. Theoretically, the
c term of Eq. 1 would express
itself as an exponentially decaying term with a time constant of
c. To evaluate the importance of a
c
term during the postsaccadic interval,
and to characterize its time course, we estimated the parameters of two
models: model M3, and model M8 (Table 1). Note,
because
= 0 and Ë = 0 during the postsaccadic
interval, model M3 simplifies to FR = b + kE, and model M8 simplifies to FR = b + kE
c
.
Figure 9 shows the fits obtained using
these two models for our example neuron, unit B76_2.
Comparison of the top and second rows of Fig. 9
(models M3 and M8, respectively) clearly shows that an exponentially decaying term is required to properly model the
firing rate of ABN neurons during this postsaccadic interval (mean VAF:
0.49 vs. 0.92, M3 vs. M8). Figure
10 shows the distribution of time
constants (cPOST) for the 46 neurons
in the present sample (filled bars). The mean
cPOST obtained for our sample was
26 ± 20 ms. This value did not differ significantly from the
estimate obtained during saccades
(cPOST cSAC; P > 0.05). The
cPOST values obtained by
Goldstein (1983)
, who used a similar approach, are shown
in Fig. 10 for comparison (cPOST = 72 ± 28 ms, n = 14). Note that our estimated
cPOST values were significantly
smaller (P < 0.01) than those reported by
Goldstein (1983)
. We consider this difference in the
DISCUSSION.
|
|
Dynamic analysis during smooth pursuit
The firing rates of all 46 ABN neurons in our sample were modeled
during 0.5 Hz, 40°/s peak velocity sinusoidal smooth pursuit eye
movements using model M3. The average eye position
(kSP) and eye velocity
(rSP) sensitivities for our sample of
cells were 5.6 ± 3.5 (spikes/s)/deg and 1.3 ± 0.9 (spikes/s)/(deg/s), respectively (not in tables). The average bias
(bSP) was 108 ± 76 (spikes/s)/(deg/s). The position sensitivity values estimated during
sinusoidal smooth pursuit using model M3 were on average
slightly larger, although not significantly, than those estimated
during fixation (kSP kFIX). The same also applied to the
bias values estimated during the two behaviors
(bSP
bFIX). However, the eye position and eye velocity sensitivity values estimated during sinusoidal smooth pursuit were significantly larger than the values estimated during saccades (kSP > kSAC, P < 0.05; rSP > rSAC, P < 0.01). In
contrast, the bias values estimated during sinusoidal smooth pursuit
were significantly smaller than those estimated during saccades
(bSP < bSAC, P < 0.01).
For each unit in our sample, its eye position sensitivity during
fixation (kFIX) and its eye velocity
sensitivity during sinusoidal smooth pursuit
(rSP) was plotted as a function of eye
position threshold (Fig. 11,
A and B, respectively). Figure 11A
shows the regression line that Fuchs and colleagues
(1988) obtained for the relationship between eye position
sensitivities during fixation (kFIX)
and eye position thresholds for identified abducens MNs. Many of the
neurons in the present study were plotted in the vicinity of this line.
We also plotted each neuron's eye position threshold versus its eye
velocity sensitivities during sinusoidal smooth pursuit
(rSP; Fig. 11B). The broken
line in Fig. 11B is based on the data of Fuchs et al.
(1988)
; neurons that were located above and below this line
were, in general, INNs and MNs, respectively. We utilized this line to
estimate the percentage of MNs versus INNs in our sample (an approach
similar to that of Broussard et al. 1995
), and concluded
that our sample contained approximately 24 MNs and 22 INNs. We
investigated whether the discharges of putative MNs were better
described by dynamic models than the discharges of putative INNs. We
found that regardless of the paradigm that was utilized, a given model
described the discharges of both groups of neurons equally well.
|
The discharges of a subset of neurons in our sample (n = 11) were also analyzed during step-ramp pursuit. Step-ramp trajectories are well suited for this approach because they elicit smooth pursuit eye movements for which the eye acceleration profiles are distinguishable from the eye position profiles (Fig. 12). This contrasts with sinusoidal smooth pursuit, for which the eye position and eye acceleration profiles are exactly 180° out of phase.
|
During ipsilaterally directed step-ramp pursuit, the firing rate of ABN
neurons increased continuously (Fig. 12, A and C,
units B96_1 and B116_2, respectively). The neuronal
firing rate likewise decreased during contralaterally directed pursuit
(Fig. 12, B and D, units B96_1 and
B116_2, respectively), and often (for 40% of the units in
our sample) reached cutoff (when the unit stops firing action
potentials) during the initial acceleration interval. On average, those
cells that were driven into cutoff were silent for approximately
, 1/2, and
of the acceleration interval
duration for 20, 40 and 60°/s step-ramp trials, respectively. Figure
12D shows an example of a neuron (unit B116_2),
which demonstrated cutoff during 40°/s OFF direction pursuit.
The VAF values provided by models M1-M5 and M8 are shown in Table 3 for each of the three ramp velocities tested. Several conclusions can be made from these results. First, model M3 is the simplest model to provide an accurate description of ON direction firing rates (Table 3; Fig. 12, A and C). Recall that negative VAF values mean that the model fit provided by a given model was worse than simply fitting the data with a mean value. Second, the BIC values suggest that adding an acceleration term is not warranted, but that adding both acceleration and jerk terms marginally improved the model fit (models M4 and M5, respectively; Table 3). None of the model parameters differed significantly (P > 0.05) across the three step-ramp velocities tested (Table 4). We also tested the usefulness of model M8 during step-ramp pursuit. The improvements in VAF values obtained during step ramps using model M8 relative to M3 were similar to those observed during saccades (Table 3). The parameter values of b, k, and r estimated using model M8 did not differ significantly (P > 0.05) from those estimated using model M3. Furthermore, the time constant estimates (cSR) for the exponentially decaying term were smaller than those estimated during saccades (6 ± 20, 8 ± 17, and 8 ± 9 ms for 20, 40, and 60°/s velocity, respectively); however, these differences were not significant (P > 0.05).
|
|
We also tested the ability of model M3 to predict OFF direction discharges during step-ramp pursuit. Example model fits are shown in Fig. 12B for unit B96_1. This neuron was typical of the majority of ABN neurons in our sample (60%) that did not reach cutoff during the initiation of OFF direction step-ramp pursuit. Its discharge could be modeled as a mirror image of its ON direction responses, because the parameter values obtained during ON and OFF direction pursuit were very similar [82 vs. 83 spikes/s, bON vs. bOFF; 3.0 vs. 4.2 (spikes/s)/deg, kON vs. kOFF; 0.40 vs. 0.37 (spikes/s)/(deg/s), rON vs. rOFF]. Figure 12D shows an example of an ABN neuron, unit B116_2, that reached cutoff during OFF direction pursuit. As for unit B90_3, the ON and OFF direction responses were nearly mirror images while the unit was firing (thick black line). However, when the parameter values estimated before cutoff were utilized to predict the neuron's discharge after cutoff, unrealistic negative firing rates were obtained (thick gray line). Hence the OFF direction responses of 40% of the units in our sample could not be modeled as a mirror image of their ON direction responses for the duration of the acceleration interval.
We also compared the parameter values estimated during 40°/s
step-ramp pursuit (using model M3) to the values estimated
during other paradigms. Note that the saccadic data obtained for the subset of neurons that were analyzed during step-ramp pursuit were
included in Tables 3 and 4 to facilitate comparison. The bias and the
eye position sensitivity values estimated during step-ramp pursuit did
not differ significantly from the equivalent values estimated during
fixation (bFIX bSR;
kFIX
kSR). However, the values estimated
during step-ramp pursuit for all three parameters of model
M3 were found to differ significantly from the values estimated
during saccades (bSAC > bSR, P < 0.01;
kSAC < kSR, P < 0.01;
rSAC < rSR, P < 0.01). In
summary, we observed significant trends in the parameter values
estimated during different behavioral paradigms (i.e.,
bSR, bFIX < bSAC;
kSAC < kFIX < kSR;
rSAC < rSR). We will consider these
differences in the DISCUSSION.
Dynamic analysis during rapid VOR
In the previous sections, we have reported the results of our
analysis of ABN neuron spike trains during smooth pursuit and fast
saccadic eye movements. To our knowledge, there is no report that
describes the firing rate of ABN neurons during eye movements falling
in a "velocity gap" (~100300°/s) between these two different classes of eye movements. Here we have analyzed the activity of ABN
neurons during slow phases of VOR elicited by rapid whole-body rotations (VORR). The VORR
responses generated during this paradigm were of particular interest
because nonsaccadic eye velocities up to 200°/s could be achieved,
thus effectively bridging the velocities generated in smooth pursuit
and saccadic eye movement paradigms.
We analyzed the discharges of the same subset of neurons that were previously characterized during step-ramp pursuit. The ABN neuron firing rates increased during ipsilaterally directed VORR eye movements that were elicited by contralateral whole-body rotations (ON direction responses; Fig. 13A). In addition, all ABN neuron firing rates decreased, and most reached cutoff, during contralaterally directed VORR eye movements (Fig. 13B). The likelihood that a neuron would be driven into cutoff was roughly half way between that observed during 40°/s step-ramp pursuit (40%) and large saccades (100%). Specifically, we found that within the first 100 ms of VORR, 50% of the units were always driven into cutoff, 33% of the units were driven into cutoff >60% of the time, and the remaining 17% of the units were driven into cutoff <30% of the time.
|
As was the case for saccades and step-ramp pursuit, model M3 provided a good estimate of the neurons' firing rate during VORR (Table 3). The addition of acceleration or jerk terms did not improve the VAF values markedly (models M4-M5, Table 3). We found that the average parameter estimates during VORR tended to fall between those estimated during pursuit and saccades (i.e., bFIX, bVOR, bSR < bSAC; kSAC, kVOR < kFIX < kSR; rSAC < rVOR, rSR). However, except for a significant difference between the position sensitivity values estimated during VORR and step-ramp pursuit (kVOR < kSR, P < 0.05), the values estimated during VORR did not differ significantly from those estimated during either step-ramp pursuit or saccades. Representative fits provided by model M3 for ON direction VORR are shown in Fig. 13A, for unit B90_3. Figure 13B shows that, in general, only a fraction of the OFF direction discharges could be accurately predicted using a model estimated during ON direction VORR (thick black line).
We also tested the usefulness of model M8 during VORR. The results we obtained were consistent with those obtained during saccades and smooth pursuit: adding an eye acceleration and a slide term to model M3 improved our ability to fit ABN neuron discharges during VORR (model M8), and the parameter estimates of b, k, and r did not differ significantly from those obtained using model M3. Furthermore, the estimated time constants (cVOR = 19 ± 20 ms, Table 4) were not significantly different from those estimated during saccades.
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DISCUSSION |
---|
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---|
The oculomotor system, because of its relative simplicity, is well
suited to a modeling approach (see Robinson 1981a-c
;
Van Gisbergen and Van Opstal 1989
). Here, we compared
the ability of a series of linear and nonlinear eye movement-based
models that have been postulated to describe MNs firing rates. The
models that we tested were based on prior characterizations of
1) the oculomotor plant (Robinson 1964
,
1965
), 2) ABN neuron discharges (Fuchs
et al. 1988
; Goldstein 1983
; Goldstein
and Robinson 1986
; Keller and Robinson 1972
;
Stahl and Simpson 1995
; Van Gisbergen et al.
1981
), and 3) the upstream drive to ABN neurons
known to be carried by premotor burst neurons (Cullen and
Guitton 1997
).
Comparison with previous studies
In the present study, the dynamic lead time estimates were
significantly longer than those determined using the onset of the first
spike. Using the first spike method, we obtained a mean lead time of
4.4 ± 1.2 ms. While similar values were obtained by Keller
and Robinson (1972) (5.4 ms), longer values have been obtained
in other studies [7.0 ± 1.9 ms (Luschei and Fuchs
1972
) and 8.8 ± 1.0 ms (Van Gisbergen et al.
1981
)]. This interstudy variability suggests that it is
difficult to objectively identify the first spike in the saccadic burst
of tonically discharging neurons. Alternatively, our dynamic lead time
method provided objective estimates that were consistent with the
results of simian muscle activation studies. Following abducens nerve
stimulation, twitch times of 6.2 and 7.0 ms have been reported
(Fuchs and Luschei 1971
and Shall et al.
1996
, respectively). Furthermore, Miller and Robins
(1992)
have shown that eye movement lags the increase in
agonist lateral rectus activity by 2.6 ms. Together, these results
suggest a motoneuron lead time of ~9.2 ms, which is consistent with
our dynamic lead time estimates (mean 9.4 ms).
The results of our analysis of primate ABN neuron activity during
fixation and sinusoidal smooth pursuit were similar to those of
previous studies (Fuchs and Luschei 1970; Fuchs
et al. 1988
; Robinson and Keller 1972
).
Moreover, during the interval immediately following saccades, we
confirmed the importance of an exponentially decaying (or slide) term
in modeling ABN neuron firing rates that was originally quantified by
Goldstein (1983)
. This term was predicted by Robinson's
original characterization of the oculomotor plant and is now ubiquitous
to many current models of oculomotor plant dynamics (for example,
Fuchs et al. 1988
; Optican and Miles
1985
; Stahl and Simpson 1995
). In the present
study, our estimates of the time constant of this term were smaller
than those of Goldstein (compare the black and gray bars in Fig. 10).
The difference between the two studies could arise from a number of
factors: first, to compute firing rates, Goldstein used a method based
on interspike intervals that is inherently nonlinear, whereas we
utilized a spike density function that rises linearly with increasing
frequency (Cullen et al. 1996
). Second, we used dynamic
lead time estimates for our analysis, whereas Goldstein utilized
shorter lead time estimates that were calculated via the first spike
method. Finally, the amplitudes of the saccades that we analyzed were
smaller than those included in Goldstein's analysis (5-15 vs. 20°
saccades, respectively). These methodological differences most likely
explain the discrepancies in time constant measurements of the two studies.
Dynamics of ABN neuron discharges during saccades
Our results demonstrate that an adequate model of saccade-related
ABN neuron discharges requires, in its most simple form, a bias term,
an eye position term, and an eye velocity term (model M3;
Table 1). We showed that simply adding higher order derivatives of eye
movement (e.g., models M4 and M5; Table 1) to
this first-order model only marginally improved its ability to predict
neuronal discharges. In addition, we found that including nonlinear
terms in our models resulted in little improvement in our ability to fit ABN discharges (models M6 and M7; Table 1).
However, the addition of a slide term as well as an acceleration term
to model M3 markedly improved our ability to fit saccadic
discharges (model M8)
![]() |
(4) |
Dynamics of ABN neuron discharges during smooth pursuit and VORR
The discharges of a subset of ABN neurons were also analyzed during slower eye movements. Our main conclusion is that the same models (i.e., M3 and M8) that are useful for predicting ABN neuron firing rates during saccades also provide excellent descriptions of ABN neuron firing rates during slower eye movements. Although model M3 provided a good representation of neuronal activity, model M8 provided a 7% increase in VAF values during sinusoidal pursuit, 26, 15, and 3% improvements in VAF values for 20, 40, and 60°/s step-ramp pursuit, respectively, and a 38% improvement in VAF values for VORR. The acceleration terms provided by model M8 were always in the ON direction of the unit, and the estimates of the time constant of the slide term (cSP, cSR, and cVOR) were generally smaller, although not significantly, than those calculated during the postsaccadic interval (Table 4).
Variability of parameter estimates across paradigms
It is not possible to utilize the results from our analysis to generate a single linear transfer function relating agonist ABN neuron discharges to eye movement dynamics. Although the improvement in VAF provided by increasing model complexity relative to model M3 was comparable during saccades, smooth pursuit, and VORR, the coefficient values estimated during smooth pursuit and saccades differed significantly from each other, whereas the values estimated during VORR were located between the two. Figure 14 highlights the trends that we observed for parameters of model M8. The eye velocity (r) and eye position (k) coefficients decreased as peak and mean eye velocity increased (Fig. 14, A and B, respectively), whereas the biases (b) increased.
|
A similar relationship between model parameters and eye velocity has
been observed by Fuchs and colleagues (1988). They
reported that the values of the eye velocity coefficients estimated
during sinusoidal smooth pursuit (rSP)
decreased as a function of increasing pursuit frequency. However,
because the peak-to-peak amplitude of the target motion was kept
constant in their experiments, an increase in target frequency was
invariably accompanied by an increase in eye velocity. Once reanalyzed
as a function of eye velocity, the trends that they reported are
comparable to those described in the present study.
Implications for modeling the control of eye movements
Our results suggest that linear plant models are useful for
describing the discharges of oculomotor motoneurons. However, we found
that a single model equation could not be used to describe neuronal
discharges across different oculomotor behaviors. This result implies
that the model structure of M8 is not sufficient to generate
a general description of the relationship between ABN neuron discharges
and eye movements. To obtain an improved model, two approaches could be
used. One obvious approach would be to include nonlinear terms in the
model formulation. Prior analyses of extraocular motor-unit responses
to nerve stimulation have revealed that the MN-to-muscle transformation
is intrinsically nonlinear due to hysteresis, nonlinear summation,
saturation of motor-unit force, and muscle mechanical properties
(Barmack 1977; Close and Luff 1974
;
Collins 1971
; Goldberg et al. 1998
;
Shall et al. 1996
). However, many of these
nonlinearities are not consistent with the trends illustrated in Fig.
14. For example, the inverse relationship between a neuron's eye
velocity coefficient (r) and peak/mean eye velocity is in
the opposite direction from that which would be predicted based on
saturation and nonlinear summation properties of extraocular muscles.
Furthermore, it could be argued that attempts to formulate a general
description of the relationship between ABN neuron discharges and eye
motion are physiologically meaningless; such an analysis would
invariably under model the control of eye movements, because it ignores
the contribution of the antagonist MNs/muscle to the net force on the eye.
Based on these considerations, we suggest a second approach that
includes the relative contributions of the antagonist as well the
agonist muscles to each type of eye motion. The possible role of the
antagonist muscle in producing the trends shown in Fig. 14 can be best
understood by using the oversimplified system described by model
M1 as an example
![]() |
(5) |
The mechanics of the oculomotor plant are determined by the
viscoelastic properties of the agonist and antagonist muscles, as well
as by the surrounding orbital tissues (Collins 1971;
Robinson 1964
, 1981a
). During eye
movements, the active (i.e., contractile) elements of the
agonist muscle drive the rotation of the eye, whereas the inherent
passive viscoelastic properties of the extraocular muscles
and surrounding orbital tissues combine with the active viscoelastic properties of the antagonist muscle to oppose this movement.
Collins (1971) showed that the viscosity (the
resistance to movement) related to stretching a dissected extraocular
muscle (i.e., the antagonist muscle during an eye movement) varies
nonlinearly as a function of that muscle's stimulation frequency and
stretch velocity (see Figs.
15B and 16B). He
also demonstrated that the viscosity of the passive orbital tissues
remains roughly constant, even at saccadic eye velocities. We propose
that these observations can help explain the data in Fig. 14.
|
During saccadic eye movements, most antagonist extraocular MNs completely pause (Figs. 3 and 15A). It follows that shortly after saccade initiation, the viscosity of the antagonist muscle "jumps" from a high resting value to a much lower one (Fig. 15B, epochs 1 and 2). During the saccade, when the drive to the antagonist muscle is negligible and the eye velocity is large, the viscosity contributed by the antagonist muscle to opposing the eye movement is nearly equal to the passive viscosity of the muscle (Fig. 15B, epochs 2-4). At the end of the saccade, when the antagonist MNs resume firing, the viscosity of the antagonist muscle returns to a larger static value (Fig. 15B, epoch 5). Hence, during saccades, the eye movement is driven by the rate of contraction of the agonist muscle, but is opposed, albeit minimally, by the combined passive viscous properties of the antagonist muscle and orbital tissues.
During pursuit eye movements, which generate slower eye velocities, the changes in viscosity for the antagonist muscle are considerably less dramatic. Figure 16B illustrates the temporal progression of viscosity for an example of 40°/s step ramp. We show unit B96_2 because like the majority of neurons in our sample (60%), it continued to fire throughout OFF direction pursuit initiation (Fig. 16A). Hence, in contrast to saccades, the firing rate of this antagonist MN during step-ramp pursuit does not rapidly reach zero, but rather it decreases smoothly, more or less as a mirror image of the firing rate of agonist MNs. The remaining 40% of the neurons in our sample reached cutoff approximately midway through the analysis interval. Figure 16 shows that the antagonist muscle viscosity is much larger during the initial acceleration phase of pursuit (epochs 1 and 2) than during saccades (compare with Fig. 15B). To summarize, during the initiation of slow pursuit eye movements, an additional active viscosity that results from the contractile properties of the antagonist muscle, and which is minimal during saccades, combines with the passive viscous properties of the antagonist muscle and orbital tissues to oppose the agonist drive.
|
The arguments presented above are consistent with our finding that
r values were lower for faster eye movements given that saccadic eye movements encounter less viscous resistance than slower
smooth pursuit eye movements. The changes in b and
k values with increasing velocities are more difficult to
explain because these parameters are generally related to static
parameters. However, we suggest that similar principles that apply to
the r coefficient (i.e., dynamic changes in antagonist
muscle viscosity) may also apply to these parameters. For example, it
is conceivable that the stiffness of a muscle, which is likely to
affect the k coefficient, will vary with the strength of the
neural drive to this muscle during eye movement, as it does during
static conditions (Barmack 1976; Collins
1971
; Goldberg et al. 1997
; Shall and
Goldberg 1992
; Shall et al. 1996
). Indeed, our
current hypothesis is consistent with the recent findings of
Miller and Robins (1992)
. These investigators directly
measured the forces generated by the agonist and antagonist muscles in
the alert monkey and demonstrated that the agonist/antagonist force
ratio is greater for large than for small saccades. Nevertheless, future efforts aimed at describing the mechanical properties of the
extraocular muscles and the time-varying contribution of the antagonist
motor units during eye movements will be needed before a
realistic model of oculomotor control can be fully elaborated.
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ACKNOWLEDGMENTS |
---|
We are most grateful to J. E. Roy for contributing to the data collection and also for several valuable suggestions. We thank D. Guitton for many helpful comments on the manuscript and H. L. Galiana for several insightful discussions. We also thank M. Huterer and A. Dubrovsky for critically reading the manuscript and A. Smith, W. Kucharski, and M. Drossos for outstanding technical assistance.
This study was supported by the Medical Research Council of Canada (MRC) and by the National Science and Engineering Research Council of Canada (NSERC).
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FOOTNOTES |
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Address for reprint requests: K. E. Cullen, 3655 Drummond St., Rm. 1220, Montreal, Quebec, Canada.
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 4 March 1999; accepted in final form 7 July 1999.
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