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 Tainsong Chen2

John L. Semmlow3 Tara L. Alvarez4,*

1Department of Health Services Administration, China Medical University, Taichung 404, Taiwan, ROC 2Institute of Biomedical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC

3Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08854, USA 4Department 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: tara.l.alvarez@njit.edu

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

J. Med. Biol. Eng., Vol. 30. No. 1 2010 2

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 3

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

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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

J. Med. Biol. Eng., Vol. 30. No. 1 2010 4

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°

(a) (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 5

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

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]:

RLvergence AAA −= (1)

2/)( RLversion AAA += (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 c

d e 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.

Error correction

Saccade Vergence

Err

or

Gen

erat

ion

Sac

cade

Saccadic errors,

Saccadic corrections

Saccadic errors,

Vergence correction

Ver

gen

ce

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.

J. Med. Biol. Eng., Vol. 30. No. 1 2010 8

Integrator Delay Eye Plant

VT VE

VP

+

-

++

Vergence

Position

Target

PositionIntegrator Delay Eye Plant

VT VE

VP

+

-

++

Vergence

Position

Target

Position

DIDI

FeedbackFeedback

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 9

LP1

LP2

LP5

Delay Eye Plant

++

+

RD1

RD2

RD5

Is

+

-

VT VE

VP

Vergence

Position

++

SL

…..

LP1

LP2

LP5

Delay Eye Plant

++

+

RD1

RD2

RD5

Is

+

-

VT VE

VP

Vergence

Position

++

SL

…..

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.

VP

+

-

Fast Component Eye PlantVR+

+

+

+

VEVT Vergence

Position

VP

+

-

Fast Component Eye PlantVR+

+

+

+

VEVT Vergence

Position

Slow ComponentSlow Component

PP

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 dual-mode model and

switched-channel model

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

J. Med. Biol. Eng., Vol. 30. No. 1 2010 10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

Position (deg)

total model response

transient component

sustained component

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

Position (deg)

total model response

components form different channels

CH3

CH4

CH5

CH1

CH2

a b

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

Time (sec)Time (sec)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

2

4

6Position (deg)

experimental response

model response

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5

0

5

10

Velocity (deg/sec)

experimental response

model response

(a) (b)

Vel

ocit

y (

deg

/sec

)

Posi

tion

(d

eg)

experimental responsemodel response

experimental responsemodel response

Switched-channel model using Robinson’s plant

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

2

4

6

Position (deg)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5

0

5

10

Velocity (deg/sec)

Vel

ocit

y (

deg

/sec

)

Posi

tion

(d

eg)

Time (sec)Time (sec)

(c) (d)

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.

J. Med. Biol. Eng., Vol. 30. No. 1 2010 12

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 13

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-039-003-

MY2). Dr. T. L. Alvarez was supported in part by a CAREER

award from the National Science Foundation (BES-0447713).

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