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Journal of Medical and Biological Engineering, 30(1): 1-15
1
Review:
Behaviors, Models, and Clinical Applications of Vergence
Eye Movements
Yung-Fu Chen1
You-Yun Lee2
John L. Semmlow3
Tainsong Chen2
Tara L. Alvarez4,*
1
Department of Health Services Administration, China Medical University, Taichung 404, Taiwan, ROC
2
Institute of Biomedical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC
3
Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08854, USA
4
Department of Biomedical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA
Received 19 Nov 2009; Accepted 1 Feb 2010
Abstract
It is known that vergence allows the brain to perceive depth. Although convergence (inward turning of the eyes)
and divergence (outward turning of the eyes) movements utilize the same extraocular muscles and visual field,
emerging evidence supports that they are different neural control systems. Both adapt (the system’s ability to alter its
dynamics) and are correlated to phoria (the resting state of the visual system); however the behavior is different
depending upon the system. This review discusses the classical and new tools used to analyze vergence movements, the
neural control of each system, and its interaction with version. A review of the current models shows that new models
are needed to explain these recent behavior studies which will facilitate the understanding of vergence dysfunctions.
Eye movement from far to near (convergent movement) or from near to far (divergent movement) are performed
quickly and accurately. The convergence and divergence movements appear to be mediated by different neural control
processes. For example, while divergence can be faster at near, convergence can be faster at far. The vergence resting
level adapts to sustained stimuli and this adaptation can influence the dynamics of both systems. Movement dynamics
can also adapt as repetitive movements and alter the peak velocity of each system. However, typical vergence
movements in daily living rarely consist of pure symmetrical vergence movements but exhibit a combination of version
and vergence movements. Reviewing recent models shows that none adequately describe the influence of phoria,
adaptation, the differences between convergence and divergence control and its interaction with version. The
development of a new model is needed to describe the neural control of convergence and divergence taking into
account the influence of adaptation, phoria and its interaction with the version system. Once this model is developed, it
can yield more insight into abnormalities of the vergence system such as convergence and divergence insufficiencies or
excess which can develop from numerous neurological dysfunctions.
Keywords: Vergence, Convergence, Divergence, Phoria, Adaptation model, Saccade, Convergence insufficiency
1. Introduction
The oculomotor system is the simplest motor unit for
humans, yet it can move the eyes to attain visual information
with astonishing speed and accuracy. These two properties
have been modeled using vastly different control strategies [1].
The human eye has only three pairs of ocular muscles to move
the eyeballs to various angular positions and is much less
* Corresponding author: Tara L. Alvarez
Tel: +1-973-596-5272; Fax: +1-973-596-5222
E-mail: [email protected]
complicated than other motor units. There are several
significant advantages of studying eye movement control to
investigate the neural control of the brain [1]. First of all, it
lacks the mechanism of monosynaptic stretch reflexes which
are generally found in other motor units. Secondly, different
functions and anatomical substrates can be identified for
various types of eye movements. Thirdly, the pathology of the
affected areas can generally be distinguished and traced for
abnormal eye movements. Fourth, eye movements are
restricted to rotation in three planes, which enables the
possibility of precise recording for quantitative analysis.
Finally, eye movements can easily be performed without head
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J. Med. Biol. Eng., Vol. 30. No. 1 2010
motion lending itself as an ideal motor system to study using
functional imaging. Hence, quantitative analysis of eye
movements has been extensively applied in probing the
function of various brain areas and in diagnosing brain
abnormalities caused by injury or degeneration. Throughout
the day, saccades and vergence components are generally
intermixed in oculomotor movements. Saccades are frequently
found in vergence eye movements even under pure
symmetrical vergence stimuli.
The movements of the eyes are controlled by three pairs
of extraocular muscles: the medial and lateral recti, superior
and inferior recti, and superior and inferior oblique muscles.
The motor neurons that innervate the extraocular muscles are
found in the III (oculomotor), IV (trochlear) and VI (abducens)
cranial nerve nuclei. In order to acquire, fixate, and track a
visual stimulus, eye movements are generated voluntarily or
involuntarily to keep the visual stimulus in focus and on the
fovea. According to whether two eyes rotate in the same or
opposing directions, eye movements can be categorized as
versional or vergent. The eyes rotate conjugatively for the
former and disconjugatively for the latter. Version can be
further divided into gaze-holding and gaze-shifting eye
movements [2].
Three types of gaze-holding eye movements are the
vestibulo-ocular reflex (VOR), optokinetic nystagmus (OKN),
and visual fixation. Gaze-shifting eye movements are
classified as saccades and smooth pursuits. Properties and
functions of various types of eye movements are briefly
reviewed below. For more detailed information, please refer to
[1-3]. The VOR can be evoked when the head moves or rotates.
It is a reflexive eye movement to keep the image of the target
on the fovea by moving the eyes in the opposite direction to
head movement [3]. A visual stimulus is not necessary to elicit
VOR, hence it functions in total darkness or when the eyes are
closed. OKN is a rapid and small movement of the eyes which
is activated to stabilize images on the retina when tracking a
moving target. The eyes can see moving images clearly until
they are out of the visual field by generating OKN [3].
Fixation occurs when the eyes look at a stationary target,
which facilitates maintaining the object of interest with zero
velocity on the fovea. Hence, a clear image of the target is
perceived. Fixation was believed to be a pursuit eye movement
for a stationary target. However, recent research indicates that
the neural control mechanism is different between fixation and
smooth pursuit.
Three other small eye movements occur during fixation:
drift, tremor and microsaccades [2,3]. Saccadic movements are
conjugate (version) where the eyes move in tandem. These
movements are commonly done when reading a book. It is the
fastest type of eye movements with velocities reaching up to
700 deg/sec [1,3]. When tracking a continuous moving target,
the eyes perform smooth pursuit eye movements to keep the
image of the target located on the fovea [1].
Vergence, in contrast, is a disconjugative eye movement
which enables depth perception by using the medial and lateral
recti muscles to rotate the eyes inward (convergence) or
outward (divergence). For example, it is the eye movement
tracking system that a baseball batter uses when tracking a
fastball. Vergence has four major inputs which include
disparity, accommodative, proximal, and tonic vergence [3].
Disparity is the retinal difference between where a target is
projected onto the retinal and the fovea. Accommodative
vergence is driven by a blur response of the image because of
the change in focal length when looking at a visual target
located at difference distances of depth from the person. This
phenomenon is manifested by observing the nasalward
movement of a covered eye with the other eye looking at an
object moving from far to near. Proximal vergence is elicited
by the change in vergence angle caused by the perceived
nearness of an object. In the absence of the above three inputs
to the vergence system, tonic vergence is the resting level of
the vergence system in the absence of all visual stimulation.
Tonic vergence will be a convergent position based upon
midbrain stimulation to the vergence system. Phoria is similar
to tonic vergence but with phoria one eye has a visual stimulus.
Dissociated heterophoria or simply ‘phoria’ is the steady state
position of one eye that has no visual stimulus such as when it
is occluded while the other eye is fixated on a target located
along midline. The viewing eye can be fixated on a target
typically located at near (40 cm) or at far (6 m). The occluded
eye may maintain its position, rotate nasally, or rotate
temporally; these three possible positions are termed
orthophoria, esophoria and exophoria, respectively.
This paper discusses the state of the art in behavior
research, current models and the clinical application of
vergence eye movements. Within the behavioral research
section, current methodologies to quantify parameters of
vergence to attain a better understanding of neural control will
be reviewed. Recent behavioral findings include an
understanding of how convergence and divergence subsystems
differ in characteristics, how adaptation influences the neural
control of vergence adaptation, and how version interacts with
symmetrical vergence responses. The modeling section
reviews the current models, limitations and proposed future
direction of modeling. There are numerous clinical
applications that can benefit from a deeper understanding of
vergence eye movements. This paper will conclude by
suggesting future directions to advance the science of
oculomotor research.
2. Behaviors of vergence eye movements
2.1 Parameters for quantitative analysis to understand neural
control
Several parameters, including latency, peak velocity, and
duration, are used to analyze the dynamics of a vergence eye
movement. These parameters yield insight into the neural
control of how the brain visually acquires a new target located
at different depths. The latency of a vergence eye movement is
defined as the time interval between target appearance and
when the eye starts moving towards the target of interest,
Figure 1(a). As shown in Figure 1(b), vergence duration and
peak velocity are generally assessed from the velocity profile.
Models and Clinical Applications in Vergence
4.5
30
(b)
4
Velocity (deg/sec)
25
3.5
Amplitude (deg)
3
2.5
2
25
20
15
10
1.5
1
30
(c)
Peak velocity
Velocity (deg/sec)
(a)
3
5
20
15
10
5
0.5
0
0
-0.5
0
Duration
Latency
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
-5
0
0.2
0.4
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Time (sec)
0.8
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Time (sec)
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2
-5
-0.5
0
0.5
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1.5
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Amplitude (deg)
3
3.5
4
4.5
Figure 1. Typical (a) position and (b) velocity profiles of a vergence response and its (c) phase plot.
Vergence velocity is calculated by differentiating the position
profile. The vergence position and velocity responses are
typically analyzed using either the transient or steady state
portion of the response. During the transient portion, the
maximum velocity occurs. Peak velocity can be manipulated
through many different experimental conditions and is a
primary parameter routinely quantified in eye movement
research. The duration of a vergence is defined as the entire
time of the movement between when vergence is initiated to
when the eyes reach steady state. Latency, peak velocity and
duration yield important insights into the vergence system. For
example, the mean latency of disparity vergence is shorter than
accommodative vergence [3,4] showing these vergence inputs
are delayed compared to each other and hence should be
modeled using different pathways.
The main sequence is an established analysis which
measures the system’s first order dynamics. It quantifies the
relationship between duration and amplitude or between peak
velocity and amplitude using the phase plane plot shown in
Figure 1(c) [5]. For normal subjects, the duration is
approximately proportional to amplitude, whereas an
exponential curve can be fitted to describe the relationship
between peak velocity and amplitude [5]. The main sequence
is also used for the quantitative analysis of vergence eye
movements [5-9]. It is a plot of the magnitude of peak velocity
versus the response amplitude [5,6], and is particularly useful
for comparing the dynamics of a large number of eye
movements over a range of response amplitudes. It provides a
measurement of the equivalent first-order dynamics of the
response. The peak velocity increases as the vergence
amplitude increases. Additionally, it was reported that the
slope of main sequence is greater for convergence than
divergence [9]. Figure 2 demonstrates the main sequence of
divergence responses for 4 subjects stimulated by different
high-velocity ramps and 4° steps with various initial
positions [10].
The time constant of an exponential function fitted to a
position profile has also been commonly used for analyzing
vergence dynamics [9-12]. A fast vergence eye movement
tends to have a shorter time constant. Hence, convergence
which can be a faster system generally has a smaller time
constant than divergence. The main sequence and time
constant represent only the first-order behavior of a response
which does not describe its dynamics in detail, as shown in
Figure 1(c). An alternative second-order parameter can be used
by measuring the slope of the rising and falling portions of the
phase plot which correspond to the major and minor time
constants [13].
2.2 Differences between convergence and divergence
Many studies have compared convergence and divergence
behaviors reporting differences between the systems.
Divergence is a movement in the opposite direction of
convergence; yet, it is not merely negative convergence.
Several studies suggested that the dynamics of
convergence are faster than divergence [7,9,14,15] while other
studies reported that pure divergence and convergence have
similar velocity characteristics [8] or even that divergence is
faster than convergence [16]. For example, the peak velocity of
convergence was demonstrated to be twice as fast as
divergence [7]. However, other findings reported by Patel et al.
[16], believed that divergence was faster than convergence
because of the accommodation demand (held in 0 diopter) and
the vergence posture of their experimental paradigms. Alvarez
and colleagues [17] showed that divergence dynamics were
dependent on the stimulus initial position. Hence depending on
where the visual stimuli were located, divergence could be
faster, slower or approximately equal to convergence speed.
The systems also differ in their temporal properties.
Rashbass and Westheimer [18] reported that divergence
and convergence have similar latencies of 160–170 msec.
However, some studies reported that the latency of
convergence is less than divergence [9,12], while other reports
stated that the opposite is true [19-21]. Alvarez and colleagues
[17] found that the latency of divergence becomes shorter as
the target moves closer to the subject while this behavior was
not observed for convergence.
Other experimental paradigms also show differences
between convergence and divergence. During a gap paradigm,
a visual target is illuminated and then extinguished before the
new visual target is shown. These movements are postulated to
be faster than responses without a gap because the brain does
not need to disengage or release fixation of the initial stimulus.
Using a gap paradigm, researchers report that divergence can
demonstrate shorter latencies than non-gap responses; however,
convergence did not show temporal differences [22].
When investigating the adaptive effects of sustained near
convergence, nonlinear differences in an adaptive mechanism
is reported between convergence and divergence [16]. Patel and
colleagues [16] studied step changes in disparity after a
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J. Med. Biol. Eng., Vol. 30. No. 1 2010
(a)
2°/s ramp stimuli
4°/s ramp stimuli
6°/s ramp stimuli
10°/s ramp stimuli
4° step initial position=8°
4° step initial position=12°
4° step initial position=16°
4° step initial position=20°
(b)
(c)
(d)
Figure 2. Main sequence analysis of divergence responses for 4 subjects (a-d) stimulated by high-velocity ramps (2°/s, 4°/s, 6°/s, 8°/s, and
10°/s) and 4° steps with various initial positions (8°, 12°, 16°, and 20°/s) [10].
sustained 6° convergence task of 5, 30, 60, and 90 seconds.
Their results showed that the peak velocity of divergence
responses decreased significantly after 30 seconds or longer of
sustained convergence compared to only 5 seconds, while the
convergence dynamics were unchanged for all the exposure
durations. They conclude that the transient component of the
horizontal disparity system adapts nonlinearly and
independently for convergence and divergence [16]. Ying and
Zee [23] did not study disparity vergence dynamics but
systematically studied the passive decay of divergence from a
convergence stimulus of 30° after 4 seconds of fixation and
again after 36 seconds of fixation. The dynamics of the
divergence decay were faster after 4 seconds of fixation
compared to 36 seconds of fixation suggesting that sustained
convergence influences divergence decay dynamics [23].
Recently, Lee and colleagues [24] showed that divergence
dynamics were dependent on the adapted phoria position. As a
result of sustained convergence, they showed that divergence
peak velocity significantly changed. When the phoria became
more esophoric (near adapted), the peak velocity for the
divergence steps with an initial position of 16° decreased and
vice versa when the phoria was far adapted.
Convergence and divergence have been reported to have
different influences on saccadic movements during
saccade-vergence
interaction
studies
[25].
Vertical
saccade-vergence interaction show that convergence velocities
do not typically vary but divergence is dependent on the
upward or downward vertical saccadic movement [26].
Convergence and divergence also exhibit distinct
dysfunctions which are discussed in more detailed in the
section 4. Clinical Applications of vergence eye movements
[27]. Neurophysiologists have shown different cells encode
convergence and divergence [28-33]. Hence, differences
between the behaviors of the two systems should be
anticipated.
Despite these behavioral findings current vergence models
do not account for the differences between the convergence and
divergence systems.
2.3 Adaptation
The adaptation process, a type of motor learning, plays
an important role for the survival of different species. It can
be found in almost every major physiological system,
including the oculomotor system. From an engineering
viewpoint, adaptive control is simply a modification of a
system parameter irrespective of the time course of that
change or its aftereffect. Behaviors of long-term adaptation
had been well studied in several oculomotor systems, such as
VOR [34-37], saccade [38-40], and smooth pursuit [41-45].
The disparity vergence system also shows clear adaptive
behavior. For example, changes in tonic vergence or phoria
occur in response to sustained demand [46,47]. Because this
sustained demand is usually produced by prisms, this
adaptive modification is termed prism adaptation.
Behavioral studies have demonstrated that the different
oculomotor systems learn adaptation adjustments separately.
Schor et al. [48] found that the pursuit paradigm for pursuit
adaptation had negligible effect on the saccade system.
Additionally, saccadic adaptation did not affect vergence eye
movements. Furthermore, it was found that the vergence
adaptation mechanism had negligible effects in either
saccades or smooth pursuit [48].
Models and Clinical Applications in Vergence
2.3.1 Adaptation in disparity vergence
Several studies have demonstrated short-term
modification in disparity vergence eye movement. The first
short-term modification experiment compared 4 deg steps
observed alone and those observed with a step-ramp response
in a 1:5 ratio [49]. The results showed that vergence dynamic
can be increased depending upon the conditioning stimulation
used. Takagi et al. [50] used a double disparity step to induce
vergence adaptation, and found that the dynamic change of
vergence after adaptation is similar to that of saccadic and
pursuit system. Furthermore, Alvarez and colleagues [17]
showed that convergence and divergence dynamics could be
increased or decreased depending upon the experimental
paradigm.
Semmlow and Yuan [51] analyzed the change of the fast
and slow components of disparity vergence after adaptation
by using independent component analysis (ICA). ICA is a
blind source separation analysis which can untangle the
underlying transient (fast) and sustained (slow) components
within a vergence response. They reported that instead of
generating additional components, the adaptive process
modifies these two components (initial or transient and
sustained). The adapted responses showed larger initial
transient components and double-step behavior was found in
the sustained component [51].
2.3.2 Effect of phoria adaptation in vergence eye movement
In the absence of a binocular stimulus, the occluded eye
will decay toward its initial, resting position, namely the
hetereophoria or dissociated phoria level. Phoria adaptation,
also called prism adaptation, occurs in our daily lives while
we look at visual stimuli located at different depths [52]. It
has important clinical implications for maintaining binocular
vision [53], especially when performing near work [54]. The
existence of prism adaptation has been well documented
[52,55-59].
Phoria adaptation occurs when sustained convergence is
driven either with physical targets [23], a stereoscope [60,61] or
positive/negative lenses [62,63]. Phoria also changes with
orthoptics or vision rehabilitation that is routinely used to reduce
symptoms related to the visual stress of near work which is
common for those with convergence insufficiency [53].
A near sustained convergence fixation will cause the
phoria to become more esophoric (convergent) compared to
the baseline measurement [54,64-66]. Previous studies have
shown that after five minutes of adaptation negligible changes
occur in phoria level [67]. Similar findings have also been
demonstrated for the associated phoria [68] which manifests
as a small change in vergence error or fixation disparity under
binocular viewing conditions [55].
Some studies reported that phoria adaptation influence
vergence dynamics. Patel and colleagues [16] studied
vergence dynamics after a sustained 6 deg convergence task
of 5, 30, 60, and 90 seconds. They found that divergence peak
velocity decreased about 25% after 30 seconds sustained
convergence compared to 5 seconds, while no significant
change was found in convergence for all the exposure
durations. Hence, the transient component in vergence control
5
adapts and the adaptation is dependent on the direction of
vergence [16]. Ying and Zee [23] reported that the natural
divergence decay after sustained symmetrical convergence to
a 30 deg stimulus for 4 and 30 seconds. Short-term phoria
adaptation was observed for the long period convergence (30
seconds). Furthermore, the dynamics of divergence decay
changed and the change depended upon how long the
sustained convergence was maintained. Their results suggest
that the divergence decay dynamics is influenced by phoria
adaptation.
To extend the work done by Patel et al. [16], Lee and
colleague [24] systematically used slow phoria adaptation to
determine if this adaptation influences the dynamics of
transient disparity divergence eye movements that start at two
different initial positions (a near initial position of 16 degrees
and a further initial position of 4.5 degrees). It was found that
the divergence dynamics are influenced by phoria adaptation
and the change in dynamics was dependent on the sustained
convergence task. Furthermore, divergence dynamics were
influenced by initial target position and the phoria was
adapted using a sustained convergence task [24].
As will be discussed in the following modeling section,
vergence models do not adequately incorporate adaptation of
the transient component or phoria adaptation in their design.
2.4 Saccade-vergence interaction
Most eye movements involve a combination of version
(conjugate
movements)
and
vergence
(disconjugate
movements), hence saccades and vergence components are
typically intermixed in oculomotor movements. Interestingly,
saccades are frequently found in vergence eye movements even
when the stimulus is carefully aligned to be binocularly
symmetrical [8,9,15,69]. Semmlow et al. [70] found that most
responses to a pure vergence stimulus do contain saccades. For
more than half of their subjects, every response contained at
least one saccade. A similar finding has been reported by
Coubard and Kapoula [71] who found either horizontal or
vertical saccades in 84% of all vergence responses. Zee et al.
[15] reported that, in pure vergence, saccades occurred more
often during divergence and more frequently for large
amplitudes. For the vergence stimuli with wider range spanning
from 5–25°, Collewijn et al. [8] reported that pure vergence
was accompanied by saccades, especially for divergence. A
pure saccadic stimulus would elicit an initial divergence
followed by convergence response in addition to the saccadic
response for nearly all normal subjects.
Although pure vergence stimuli are rare in daily life, they
are easy to construct in the laboratory and have been used to
study the disparity vergence eye movement system. Such stimuli
can be presented using a stereo pair of images moving in equal
and opposite directions or by two targets placed at different
depths along the mid-sagittal (i.e., central or cyclopean) axis. In
the latter case, an accommodative (i.e., blur-driven) stimulus may
also drive the vergence response. These stimuli are often referred
to as pure vergence stimuli and might be expected to produce a
pure vergence response: one in which the two eyes rotate an
equal amount in opposite or disjunctive directions. Pure vergence
J. Med. Biol. Eng., Vol. 30. No. 1 2010
6
responses would follow along the mid-sagittal plane (or
cyclopean axis) and there would be no version component to the
response. However, careful and systemic behavioral studies show
different behaviors.
2.4.1 Version and vergence components
In the response to a pure vergence stimulus, the version
component should theoretically be equal to zero. In normal
subjects with good binocular vision, the net version component
must be very small by the end of the movement, since steady
fixations errors are close to zero [72]. There are two factors that
lead to version movements during a typical symmetrical
vergence response. First, vergence responses often contain
saccades [8,15,70,73,74] which clearly generate a version
movement. Even a small saccade will generate a substantial
deviation for the midline. Second, even if no saccades are
present, vergence movements in response to a symmetrical
vergence stimulus are often asymmetrical because one eye
often moves faster than the other in pure vergence responses
[69]. For example, Horng et al. [69] showed that during
vergence responses to symmetrical stimuli, one eye can
respond twice as fast as the other during the transient portion of
the response. During the initial response, a large difference in
amplitude was always observed between the two eyes and was
corrected later by a slow vergence or saccadic movement to
bring the eyes to their final symmetric position. Irrespective of
how the version component is generated, it must go to zero by
the end of the response in order to achieve accurate binocular
fixation.
When both vergence and version components are found, it
is widely held that asymmetries induced in the saccadic
response assist in moving the eyes disjunctively; that is, in
opposition [15,75-77]. Although the two oculomotor
components may act collaboratively, they still appear to be
independent [78]. The version and vergence components of an
intermixed response can be obtained from the following
equations [70]:
Avergence = AL − AR
(1)
Aversion = ( AL + AR ) / 2
(2)
where AL and AR indicate the amplitudes of the left and right
eyes, respectively.
2.4.2 The roles of saccade in saccade-vergence interaction
Responses from a symmetrical vergence stimulus often
contained saccades or asymmetries between the left and right
eye movements. Saccades occurring during the transient
portion of the movement actually generate more error because
the saccadic movement takes one away further away from the
target while bringing the other eye closer to the target. This
error can be corrected by either a corrective saccade or an
asymmetric vergence. Horng et al. [69] demonstrated that
sometimes saccades were initiated to correct the errors
produced by asymmetrical vergence. Semmlow et al. [70]
found that the initial saccade in the responses to pure vergence
stimuli usually increased the deviation from the midline.
Although most initial saccades produced error, all subjects had
some responses where the initial saccade reduced the midline
deviation by compensating for a vergence-induced midline
deviations. It was suggested that most error-inducing initial
saccades were the result of a monocular distraction produced
by transient diplopia of the vergence stimulus along with ocular
preference or dominance [70]. A similar suggestion has been
made by van Leeuwen et al. [74] who showed that subjects
without a strong monocular preference were much more likely
to produce saccade-free vergence responses to pure vergence
stimuli. In a few subjects, initial error-inducing saccades
brought the preferred eye closer to the target, which could lead
to faster recognition at the expense of delayed binocular vision.
There are four immediately apparent explanations for the
presence of those seemly unnecessary saccades [70]. First, they
could be used to bring one eye, likely the preferred or dominant
eye, more quickly to the target [74]. Embedded saccades might
delay the acquisition of full binocular vision, but having one
eye on target, particularly the preferred eye, may be sufficient
for visual recognition. Second, the transient diplopia induced
by a pure vergence stimulus produces a compelling saccadic
stimulus, particularly if there is a strong ocular preference for
one eye. Third, there is considerable evidence that saccades can
enhance vergence movements [8,76,77] through saccade-like
burst cell activity that appears to be integrated into the vergence
feedback [79,80]. Such enhancements to vergence may bring
both eyes more quickly to the target despite the symmetry error
caused by the saccade. Fourth, the saccades could be in
response to symmetry errors produced by asymmetrical
vergence. It was shown that major differences between the
speeds at which the two eyes move in a disparity vergence
response even in responses that are free of saccades [69].
Recent evidence presented that all of these mechanisms are
active in all subjects, at least in a few responses, but some
subjects exhibit a predominance of one or more of these
mechanisms [70].
2.4.3 Dual roles of error generation and correction for both
saccade and vergence
For symmetrical vergence stimuli, saccades may occur at
the beginning [15] or at the latter stage [69] and play dual roles
as either the distracters in increasing the errors or the correctors
in decreasing the errors. Asymmetric vergence, on the other
hand, may also act as dual roles for error generation or error
correction [70,81]. Irrespective of the motivation for
error-inducing saccades, some compensation must occur by the
end of the movement. While larger errors, particularly
conjugate errors, may be tolerated under certain conditions,
normal subjects can achieve highly accurate binocular fixation
with errors of minutes of arcs [72]. If the initial saccade in the
vergence response moves the eyes away from the midline, then
either a subsequent compensatory saccade, an offsetting
vergence asymmetry (or slow version), or both is required to
bring the eyes back to the midline [81].
As shown in Table 1, there are four possible scenarios for
the development of errors in vergence symmetry and their
compensation by combining vergence asymmetries with
saccades.
Models and Clinical Applications in Vergence
7
a
b
c
d
e
f
Figure 3. Responses from two subjects to a symmetrical convergent 4.0 deg. step stimulus. In both responses, an early saccade creates a
symmetry error. (a) and (d) Individual left (red dashed line) and right (blue solid line) eye movements showing the presence of
saccades along with the vergence response. Convergence is plotted upward for both eyes. (b) and (e) The version components with
the rightward movement is plotted positive and leftward movement is negative with zero at the mid-sagittal plane. (c) and (f) The
vergence components with convergence plotted upward [70].
a
b
d
e
c
f
Figure 4. Responses from two subjects to a symmetrical convergent 4.0 deg. step stimulus. In both responses, an early vergence asymmetry
creates a symmetry error. (a) and (d) Individual right (red) and left (blue) eye movements. (b) and (e) The version component. (c)
and (f) The vergence responses are again smooth due to either saccade cancellation (c) or the absence of saccades (f) [70].
Table 1. Sources of Symmetry Errors and Corrective Mechanisms
during Pure Vergence Stimuli.
Vergence Saccade
Error Generation
Error correction
Saccade
Vergence
Saccadic errors,
Saccadic corrections
Saccadic errors,
Vergence correction
Vergence asymmetry,
Saccadic correction
Vergence asymmetry,
Vergence correction
Figure 3 shows responses from two subjects to a
symmetrical convergence 4° step stimulus. In both responses,
an early saccade (asymmetry-inducing) creates a symmetry
error. Individual left (red dashed line) and right (blue solid
line) eye movements showing the presence of saccades along
with the vergence response. The vergence components are
obtained from Eq. (1), while the version components are
calculated according to Eq. (2). As shown in Figures 3(b) and
(d), the saccade-induced symmetry errors are corrected by a
slow version movement (vergence asymmetry) and a
corrective saccade, respectively. Responses with asymmetry
vergence which generates symmetry errors are shown in
Figure 4. In Figure 4(b), two saccades appear to be correcting
the symmetry error produced by an initial fast disparity
vergence asymmetry, while in Figure 4(e) the fast disparity
vergence asymmetry is corrected by a compensatory vergence
asymmetry.
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J. Med. Biol. Eng., Vol. 30. No. 1 2010
DI
VE
Target VT +
Position
VP -
Integrator
+ +
Delay
Eye Plant
Vergence
Position
Feedback
Figure 5. Continuous feedback model, in which VT, Vp, and VE indicate the target angle, desired vergence angle and disparity angle,
respectively. DI represents the derivative-integrative component added by Krishnan and Stark [83]. For Rashbass and Westheimer’s
model, Integrator: k/s, Delay: e-sτ, and Eye Plant: zero-order; for Krishnan and Stark’s model, Integrator (leaky): 10Ki/(10s+1), DI:
sKs/(s+10), Delay: e-0.16τ, and Eye Plant: 3rd order.
These recent behavioral findings of saccadic movements
within symmetrical vergence responses have yet to be
incorporated in vergence models, presumably because it is
more difficult to describe this complex behavior. However in
order to attain a better understanding of oculomotor control,
more sophisticated models that incorporate not only vergence
but its interactions with other systems is necessary because in
visual search humans do not perform simple vergence
movements but a combination of movements.
3. Models for vergence
Physiological modeling facilitates the understanding of a
system. Vergence eye movements were first modeled using a
simple feedback control system by Rashbass and Westheimer
[18]. Numerous models have been developed since their first
attempt but controversy still exists regarding the basic control
structure mediating this motor response. Models of vergence
control can generally be classified into three basic
configurations: continuous feedback, switched-channel with
feedback, and feedback with preprogrammed control [82].
Their limitations of describing new behaviors recently reported
in the literature are summarized here.
3.1 Continuous feedback model
As shown in Figure 5, the first model proposed by
Rashbass and Westheimer [18] used simple linear feedback
with a feedforward controller, consisting of an integrator with a
delay, and a zero-order eye plant with unity gain. A negative
feedback control system was used to continuously reduce the
vergence disparity (VE), defined as the difference between
target position (VT) and vergence position (VP). During an open
loop experiment, the feedback loop of vergence is ‘opened’ by
monitoring the current position of the eye to keep the error
(difference between the stimulus and the current eye position)
constant. Rashbass and Westheimer [18] used an open loop
experiment by applying a ramp stimulus while keeping the
disparity magnitude constant and concluded that the response
velocity was proportional to the input disparity magnitude one
reaction time (approximately 160 ms) before for small
disparities up to 0.2°. However, their model was unable to
sufficiently model phase from sinusoidal disparity stimuli
because the experimental data had shorter lags compared to
those predicted by the model. This model also has difficulties
in modeling step data that have faster dynamics because it
becomes unstable with faster control signals due to the
presence of long processing delays.
The next feedback model was developed by Krishnan and
Stark [83] in which a derivative-integral (DI) element to better
represent the transient response was added to the controller in
parallel to the integrator to support the sustained response
(Figure 5). In contrast to Rashbass and Westheimer’s model, a
leaky integrator and a third-order eye plant were used to
simulate vergence responses. Additionally, the integrator
sustained the response of disparity magnitude while the
derivative-integral portion was triggered based on disparity
change rather than disparity magnitude, which significantly
improved the phase characteristics of Rashbass and
Westheimer’s model. The model was later modified by Schor
[84] in which a threshold was added to trigger vergence change
where he referred to it as a dead zone and incorporated input
from the accommodative vergence system. Additionally, the
system consisted of a fast neural integrator and a slow neural
integrator to simulate the initial step and the steady state
portions of the response, respectively. One problem with these
models is that the steady state error or fixation disparity from
the model is much greater than those found experimentally
[85].
3.2 Switch-channel model with feedback
Pobuda and Erkelens [86] were the first to present a
model with parallel channels in the feedforward path. As
shown in Figure 6, they proposed the following hypotheses
(1) vergence disparity is processed through several channels
simulated by low pass filters, (2) the filters are sensitive to
ranges of disparity amplitude, (3) vergence loops have delays
ranging from 80 to 120 ms rather than the 160 ms as
previously reported, and (4) the vergence loop is not
sensitive to the disparity change. The criticism to the model
is that it shows responses taking significantly longer to
process disparities compared to those seen experimentally
[87]. Another model has expanded the switched-channel
model using neural network architecture [88]. The model
consisted of seven functional stages and incorporated a
variation of the switched-channel model into one of the
neural layers with velocity used to select the input channel
rather than disparity. The primary limitation of this model is
that it could not model the high-velocity step-like component
observed in faster ramps [87,89].
3.3 Preprogrammed feedback model
A different approach to vergence modeling uses a
preprogrammed open-loop element in conjunction with
feedback control as shown in Figure 7 [14]. This dual-mode
Models and Clinical Applications in Vergence
VT +
VE
RD1
LP1
RD2
LP2
SL
+
Is
+
+
+
Delay
Eye Plant
+
…..
VP
9
Vergence
Position
LP5
RD5
Figure 6. Block diagram of switch-channel model, in which RD, LP, Is, and SL represent range detector, low-pass filter, slow integrator, and
saturation limit, respectively. A pure delay with 100 ms and a second-order eye plant with time constants of 8 and 150 ms were used
for simulation.
Slow Component
VT +
VE
+
+
P
VP
Fast Component
VR
+
+
Eye Plant
Vergence
Position
Figure 7. Block diagram of dual-mode model which incorporates preprogramming with feedback control.
model consists of a rapid, preprogrammed, transient control
component followed by a much slower sustained component
guided by feedback to represent the speed and accuracy of a
vergence movement respectively [87,90,91]
There is considerable behavioral support for the
dual-mode theory [14,69,87,90-92]. Specific behavioral
evidence supporting a preprogrammed element within
convergence control began with Jones’ research. Jones [93]
showed that by stepping a non-fusible target (a vertical line
paired with a horizontal line), a transient vergence response
was generated. Under these conditions, the vergence system
cannot use an external visual feedback system since an error
signal is not generated. He then developed a theory that
convergence was composed of a fusion initiating and a fusion
sustaining component [93,94]. Similarly, Semmlow et al. [90]
reported that by presenting a step stimulus that would
disappear in a mere 50 or 100 ms, a transient convergence
response was generated. The preprogrammed component is
also observed in divergence control. Lee and colleagues [10]
found that divergence responses to disappearing step stimuli
(i.e. the 4° step visual stimulus were illuminated only for 100
ms) have similar first-order dynamic characteristics to pure
step responses. Since the preprogrammed component is mostly
elicited if the visual stimulus disappears in 100 ms, their
finding suggested the existence of a preprogrammed
component in divergence control as well. Furthermore, when
using an uncrossed stimulus for 200 msec, divergence was
observed for small step changes [95]. These behavioral
findings led to a model that could accurately simulate
responses to a variety of experimental responses such as a
pulse, step-pulse, ramp and sinusoidal stimuli.
3.4 Responses simulated using
switched-channel model
dual-mode
model
and
Recently research has begun to compare responses from
both the dual-mode model and the switched-channel model.
Both models were reconstructed with the same eye plant
described by Robinson et al. [96] to facilitate comparison
between the two model controllers. Figure 8 shows simulated
4° vergence step responses using the dual-mode and
switched-channel models. In the dual-mode model response,
the dashed line indicates the transient component while the
dotted line shows the sustained component. For the switched
channel model, the five dashed lines represent the signals
generated by each channel. Note that the time scales are
different between the models.
Both the dual-mode model and switched-channel model
are adjusted to give the best fit simulated response to the 4°
experimental responses for a slower response and a faster
response shown in Figures 9 and 10, respectively. As shown
in the figures, the model responses are very similar to
experimental responses indicating that the behaviors of slow
and fast vergence responses can be described by both
dual-mode and switched-channel models.
3.5 Limitations to current vergence models
Although both models can accurately simulate vergence
step responses to 4° symmetrical stimuli, both models have
significant limitations. These models represent divergence with
a negative sign and do not adjust parameters compared to
convergence. Only a few models have tried to incorporate
other factors that may influence the vergence system such as
the tonic and/or phoria level. Schor's model [97] uses a
recruitment mechanism that is an order of magnitude slower
than the transient component. As sustained fixation duration is
increased, the recruitment of neurons is greater, thereby
increasing the output of the sustained component and reducing
the drive from the transient component. Hung's model [98]
suggests a variable time-constant mechanism in which neurons
increase their time-constants proportionally to the sustained
fixation duration. In both models, the transient component is
considered to be non-adaptable. Furthermore, both models
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J. Med. Biol. Eng., Vol. 30. No. 1 2010
5
5
a
total model response
transient component
sustained component
4.5
b
4
4
CH1
3.5
3.5
3
3
4.5
total model response
components form different channels
Position (deg)
Position (deg)
CH2
2.5
2
CH5
2
1.5
1
1
0.5
0.5
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
CH4
2.5
1.5
0
0
CH3
0
0
0.5
1
1.5
2
Time (sec)
2.5
3
3.5
4
Figure 8. Model responses obtained from (a) dual-mode model (b) switched-channel model.
assume identical dynamic behavior during convergence and
divergence movements. The only model that could account
for near or far adaptation was proposed by Saladin [99]. This
model consists of separate sensorimotor pathways for
convergence and divergence where each pathway is similar
to Schor’s model. New vergence models need to account for
the differences between convergence and divergence and
adaptation. To model adaptation both adaptation to the
transient component through conditioning stimuli and phoria
adaptation should be included in the architecture.
As discussed in the behavioral review section, several
studies show the occurrence of saccades in symmetrical
vergence responses. None of the current models simulate the
occurrence of saccades in a vergence response. New models
should incorporate these behaviors to create a more realistic
model of the vergence oculomotor system. When such a
model is created more insights can be gained to understand
vergence dysfunctions. For instance, one potential reason to
use a saccadic response within a vergence movement is to
facilitate visual object recognition even if it is perceived
monocularly which may be a primary compensatory
mechanism in vergence dysfunctions.
4. Clinical applications of vergence eye movements
4.1 Cortical and subcortical studies of vergence oculomotor
movements
Numerous studies have established the neural circuit for
vergence that can serve as a solid foundation to better
understand vergence dysfunctions. One study states “specific
abnormalities on the eye movement examination may provide
clues to the underlying pathology, and suggest strategies for
treatment of a variety of neurological disorders” [100]. There
are many single cell or lesion studies on primates, human case
reports, and transcranial magnetic stimulation studies which
yield insights into the operation of the vergence neural circuit.
Regions of vergence neural circuit from the sensory input to
the motor output are summarized here.
Sensory visual pathway: Disparity tuned cells have been
described as being located in the primary visual cortex (V1,
V2, V3 and V3a) where cells are excitatory or inhibitory.
These cells encode for stimuli located at different depths
where investigators hypothesize this is the input signal to the
binocular vergence system [101].
Parietal lobe: Studies on primates show that the vergence
circuit does involve cortical areas within the posterior
parietal lobe (lateral intraparietal (LIP) area) that are broadly
tuned to have a preferred direction for targets closer or
farther away [102,103]. Using transcranial magnetic
stimulation over the posterior parietal cortex (PPC),
investigations conclude that the right PPC is involved in
fixation disengagement; whereas the left PPC is involved in
spatial selective mechanisms that concern targets that are
closer or near to the subject [104,105].
Frontal lobe: Single cell recordings from primates reveal a
distinct area within the bilateral frontal eye fields (FEF) that
is allocated for step vergence responses and is located more
anterior compared to the cells responsible for saccadic
signaling [106]. Other investigators have shown a separate
smoothly tracking area in the FEF for vergence [107].
Cerebellum: The vergence signal is present within the deep
cerebellar nuclei [108]. Cells within the fastigial oculomotor
region modulate their firing rates with a convergence
stimulus alone or with convergence and saccadic stimuli.
Inactivation of this region with muscimol produces a
decrease in convergence velocity and a “convergence
insufficiency” [33,109,110]. Investigators show that cells are
involved in vergence only movements and in vergence with
smooth pursuit[33,107]. Vergence activity has also been
reported in the ventral paraflocculus [111] and in the
posterior interposed nucleus [32]. The dorsal vermal outputs
are sent to the midbrain via the caudal fastigial nucleus [33].
Lesions to the cerebellar vermis VI/VII in primates show a
decrease in prism-induced phoria adaptation [109]. Recent
human case studies report vergence dysfunctions in those
with cerebellar lesions, particularly those within the
vermis [112].
Models and Clinical Applications in Vergence
11
Dual-mode model using Robinson’s plant
(a)
4
2
experimental
response
experimental
response
model
response
model
response
0
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
4
(c)
2
0
0
0.2
0.4
0.6
0.8
1
1.2
10
(b)
experimental
response
experimental
response
model
response
model
response
5
0
-5
0
0.2
0.4
0.6
Switched-channel model using Robinson’s plant
6
Velocity
(deg/sec)
Velocity (deg/sec)
Position(deg)
(deg)
Position
Velocity (deg/sec)
(deg/sec)
Velocity
Position(deg)
(deg)
Position
6
1.4
1.6
1.8
Time (sec)
2
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
10
(d)
5
0
-5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time (sec)
Figure 9. The position (left) and velocity (right) profiles of model simulations (solid line) and selected slow dynamic experimental vergence
responses (dashed line). The dual-mode model simulations using Robinson’s plant with experimental data are shown in (a) and (b).
The switch-channel model simulations using Robinson’s plant are shown in (c) and (d).
Figure 10. The position (left) and velocity (right) profiles of model simulations (solid line) and selected faster dynamic experimental vergence
responses (dashed line). The dual-mode model simulations with experimental data are shown in (a) and (b) and the switch-channel
model simulations using Robinson’s plant are shown in (c) and (d).
Brainstem and oculomotor neurons: Researchers have
recorded action potentials from cells in the midbrain and they
report different cells for convergence and divergence. Within
this population, some cells exhibit bursting behaviors which
they termed “velocity-encoding cells” while others have
burst-tonic behaviors termed “position-encoding cells”
[28,29,113]. Midbrain neurons discharge before vergence eye
movements [33]. The vergence signal is also present within
the nucleus reticularis tegmenti pontis [108]. Investigators
studying single cell recordings from the abducens and
oculomotor neurons show that separate cells are encoded for
saccadic versus vergence signals to the lateral and medial
recti muscles [28].
4.2 Clinical Disorders of Vergence Eye Movements
Vergence dynamics have been reported to decrease with
age [114]. This study reported that age-related effects in
transient (fast) vergence were observed via an increased
latency and decreased peak velocity and acceleration.
Sustained (slow) vergence also showed age-related effects
with a decrease in accommodative vergence velocity and an
increase in latency. In addition to aging, numerous clinical
case studies have been reported describing dysfunctions to
the vergence system.
Case studies show that a symmetric paramedian
thalamic infarction disrupts vergence eye movements due to
an interruption of the supranuclear fibers to midbrain
vergence neurons [115]. Furthermore, unilateral mediolateral
pontine infarctions impair ramp vergence tracking
movements but not step vergence movements, suggesting
that vergence signals are distributed in the pontine nuclei
[116,117]. Small midbrain infarctions [118,119], pontine
lesions [116,117], cerebellar lesions [113], Parkinson’s
disease [120,121], progressive supranuclear palsy [122] and
midbrain hemorrhage [123] have all been reported as causing
a convergence dysfunction.
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J. Med. Biol. Eng., Vol. 30. No. 1 2010
Patients with progressive supranuclear palsy made
smaller vergence under both asymmetric and symmetric
stimuli. The regression slopes of the main sequence were
also lower than control for both stimuli [122]. Lesions in
both the nucleus reticularis tegment pontis (NRTP) [116] and
the mediolateral pontis [117] showed deficits of slow
convergence and divergence, but not faster vergence stimuli
such as step stimulation. Similar to pontine lesions, fast
vergence was also affected for patients with acute cerebellar
lesions [112]. It was found that slow vergence gain to
sinusoidal stimuli, both in convergence and divergence, was
reduced significantly, while only the velocity of divergence
(but not convergence) to ramp stimuli was reduced [112].
Impairments to the dorsal vermis within the cerebellum of
primates resulted in a decrease in peak velocity and initial
acceleration of convergence but not divergence [33].
Kapoula and colleagues [124] reported that the latency
of saccadic and vergence movement was extended for a
young subject with manifest latent nystagmus (MLN) when
viewing binocularly (but not monocularly). Furthermore, it
was found that the latency of vergence is longer in children
with divergent strabismus [125] and with vertigo in the
absence of vestibular dysfunction [126]. Children with
vertigo in the absence of vestibular dysfunction also showed
poor accuracy, longer duration, and reduced speed in
vergence eye movement compared to normal controls [127].
Bucci et al. [128] found that the occurrence of express
latencies for divergence is significantly greater for children
with dyslexia.
In addition to abnormalities in dynamic disparity
vergence movements, (quantified via latency, peak velocity,
duration, and main sequence analyses) many reports
document abnormalities in the phoria adaptation of the
vergence system. Several previous researches reported that
aging [129] and dysfunction of the cerebellum [130,131]
result in the reduced ability to adapt vergence phoria.
Although Hain and Luebke [58] reported that horizontal
phoria adaptation is not affected due to cerebellar
dysfunction, several studies discovered that lesions in the
cerebellum result in a decrease or loss of phoria adaptation in
humans, both horizontally [130] and vertically [131]. Similar
reports have been documented in primates [109].
4.3 Vision rehabilitation
The brain receives most of its afferent information
through the visual system. Hence, vision dysfunction can
have devastating consequences on a person’s daily living
activities. Some visual dysfunctions can be treated with
glasses or contact lenses, while others can be remediated via
vision rehabilitation. Vision rehabilitation or vision training
utilizes repetition of eye movements over many sessions to
facilitate oculomotor movements through motor learning
speculated to evoke neural plasticity. It has been used to treat
many visual disorders such as amblyopia, double vision,
convergence insufficiency (CI), and reading disabilities.
Several visual stimuli (step or ramp) have been adopted to
treat patients with different kinds of vision disorders. For
example, vision rehabilitation using step or ramp stimuli was
reported to be able to enhance vision in CI patients. Kapoor
and Ciuffreda [132] reported that after a period of ramp
training, a target moving smoothly and gradually toward or
away from the subject, the patient’s ability to fuse a target as
well as to maintain that level of vergence had been improved.
Convergence insufficiency, a common disorder in the
visual system [132-136], has been diagnosed in 42% (68 out
of 160) of patients with traumatic brain injury (TBI) [137].
Patients are generally observed with receded near point of
convergence, large exophoria, and reduced vergence ranges
[132]. The near point of convergence (NPC) break is
quantified as the distance along a subject’s midline when the
subject begins to see the visual target double or the eye
breaks fusion. Recovery is the distance that is required
before the doubled object returns to a single image [138,139].
Scheiman and Gallaway reported in 2005 [138], a typical
NPC of 20 cm for TBI patients which can be reduced to
normal ranges of less than 5 cm after vision rehabilitation.
These patient’s symptoms include double vision, fatigue, and
blurred vision within a few minutes of near work such as
reading a book. Vision rehabilitation increases functionality
which is quantified by a closer near point of convergence,
larger vergence ranges and reduction in symptoms [139,140].
Yet none of these studies measured eye movements. To date,
no modeling study has been performed to suggest how
changing a parameter in a vergence model can create a
vergence oculomotor dysfunction or how vision
rehabilitation facilitates changing oculomotor control.
5. Conclusion and future direction for vergence
oculomotor research
Many new behaviors have recently been reported in
vergence including: (1) divergence and convergence having
different dynamic and temporal properties, (2) adaptation of
the transient dynamics, (3) phoria and its adaptation
influencing the neural control of convergence and divergence
which is different for each system, and (4) the occurrence of
saccades in symmetrical vergence responses where saccadic
responses should not be present because visual stimulation is
symmetrical and hence does not contain version stimulation.
The current models reviewed here (feedback with
preprogrammed control and switched channel feedback) do not
incorporate any of these recent behavioral findings.
Furthermore, numerous case reports and studies document
vergence dysfunctions such as a decrease in vergence velocity
and acceleration as well as increased latencies or durations to
acquire a new visual target. Dysfunctions can selectively
disrupt convergence or divergence and can also selectively
affect either the fast or slow portions of the vergence system
identified through responses to step or ramp stimuli. Yet
despite the prevalence of vergence dysfunction, no modeling
studies have been conducted to understand how each
dysfunction can be simulated to understand the change that
dysfunction is causing in the oculomotor control. Furthermore,
several studies suggest that vision rehabilitation can improve
patient’s symptoms, yet no modeling work has been conducted
Models and Clinical Applications in Vergence
to explain what changes in the oculomotor control to account
for the patient’s improvement resulting in a reduction of
symptoms. Future work should include improved models of
vergence oculomotor control to improve our understanding of
the basic science and clinical applications of vergence
dysfunctions.
Acknowledgments
Dr. Y. F. Chen was supported in part by China Medical
University (CMU96-145) and National Science Council of
Taiwan (NSC93-2213-E-212-038, NSC98-2410-H-039-003MY2). Dr. T. L. Alvarez was supported in part by a CAREER
award from the National Science Foundation (BES-0447713).
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