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\title{A Control Systems Model of Smooth Pursuit
Eye Movements with Realistic Emergent Properties}
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\author{R.~J.\ Krauzlis\\
S.~G.\ Lisberger\\
{\small \sl Department of Physiology and Neuroscience Graduate Program,}\\
{\small \sl University of California, San Francisco, CA 94143, USA}}
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\def\Communicate{David A.~Robinson}
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\markboth{R.~J.~Krauzlis and S.~G.~Lisberger}{A Control Systems Model of Smooth
Pursuit Eye Movements} %
\maketitle{krauzlis}
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\begin{abstract}
Visual tracking of objects in a noisy environment is a difficult problem that
has been solved by the primate oculomotor system, but remains unsolved in
robotics.  In primates, smooth pursuit eye movements match eye motion to target
motion to keep the eye pointed at smoothly moving targets.  We have used
computer models as a tool to investigate possible computational strategies
underlying this behavior.  Here, we present a model based upon behavioral data
from monkeys.  The model emphasizes the variety of visual signals available for
pursuit and, in particular, includes a sensitivity to the acceleration of
retinal images.  The model was designed to replicate the initial eye velocity
response observed during pursuit of different target motions.  The strength of
the model is that it also exhibits a number of emergent properties that are
seen in the behavior of both humans and monkeys.  This suggests that the
elements in the model capture important aspects of the mechanism of visual
tracking by the primate smooth pursuit system.
\end{abstract}

\section{Introduction}

Computer models have advanced our understanding of eye movements by providing a
framework in which to test ideas suggested by behavioral and physiological
studies.  Our knowledge of the smooth pursuit system is at a stage where such
models are especially useful.  We know that pursuit eye movements are a response
to visual motion and that they are used by primates to stabilize the retinal
image of small moving targets.  Lesion and electrophysiological studies have
identified several cortical and subcortical sites that are involved in pursuit
(see Lisberger et al.\ 1987 for review), but the precise relationship between the
visual motion signals recorded at these sites and those used by pursuit remain
unclear.

We present a model that was designed to replicate the monkey's initial eye
velocity response as a function of time for different pursuit target motions. 
The model emphasizes the variety of visual signals available for pursuit and
minimizes the computations done by motor pathways.  The structure of the model
is based upon behavioral experiments which have characterized how different
aspects of visual motion act to initiate pursuit.  In monkeys, the pursuit
system responds not only to the velocity of retinal images, but also to smooth
accelerations and to the abrupt accelerations that accompany the onset of target
motion (Krauzlis and Lisberger 1987; Lisberger and Westbrook 1985).  Therefore,
the model includes three parallel pathways that are sensitive to these three
aspects of visual motion.

\section{Structure of the Model}

Our model is drawn in figure~1.  Within each pathway, a time-delayed signal
related to the motion of retinal images ($\dot{T} - \dot{E}$, called retinal
``slip") is processed by a nonlinear gain element derived from our behavioral
experiments and a filter.  The first pathway is sensitive to slip velocity ($\dot{e}$)
and its gain is linear.  The second and third pathways are both sensitive to slip
acceleration ($\ddot{e}$), but in different ways.  The impulse acceleration
pathway is sensitive to the large accelerations that accompany step changes in
target velocity, but has a dead zone  in the gain element that renders it
insensitive to smaller accelerations.  The smooth acceleration pathway is
sensitive to gradual changes in image velocity.  The outputs of the gain
elements in each pathway are low pass filtered to produce three signals with
different dynamics ($\ddot{E}'v, \ddot{E}'i, \ddot{E}'a$) that are then summed
and integrated to give a command for eye velocity ($\dot{E}'$).  The integrator
makes our model reproduce the fact that the pursuit system interprets visual
inputs as commands for eye accelerations (Lisberger et
al.\ 1981).  Eye velocity ($\ddot{E}$) is obtained by passing the eye velocity
command through a low pass filter that represents the eye muscles and orbital
tissues.

\begin{figure}[b]%fig 1
\figspace{5.38in}{3.7in}
\caption{ Model of smooth pursuit eye movement system. Boxes contain transfer
functions expressed in Laplace notation.  Abbreviations: $\dot{T}$, target
velocity; $\dot{e}$, slip velocity; $\ddot{e}$, slip acceleration; $\ddot{E}'v$,
output of slip velocity pathway; $\ddot{E}'i$, output of slip impulse
acceleration pathway; $\ddot{E}'a$, output of slip acceleration pathway;
$\ddot{E}'$, eye acceleration command; $\dot{E}'$, eye velocity command;
$\dot{E}$, eye velocity.  Parameters used: $t$ = .065; $T_v = 0.030$; $T_i =
0.020$; $T_a = 0.010$; $T_p = 0.015$.  Functions for gain elements: Slip
velocity pathway: $y = ax$; $a = 8.3$.  Slip impulse acceleration pathway: for
$x > c$, $y = a \log(bx + 1)$; $a = 17500$, $b = .00015$, $c = 3000$.  Slip
smooth acceleration pathway: for $d > x > e$, $y = a \log(bx + 1)$; for $x <
e$, $y = (cx^2) a \log(bx + 1)$; $a = 28$, $b = .1$, $c = .0016$, $d = 500$, $e =
18.5$.  Equations given for impulse and smooth acceleration pathways apply for
$x > 0$.  For $x < 0$, equivalent odd functions are used.}
\end{figure}


The behavior of the model was refined by matching its performance under
open-loop conditions to that of the monkey.  We removed visual feedback by
setting the value of feedback gain to zero, and compared the model's output to
averages of the monkey's eye velocity in behavioral trials where visual feedback
was electronically eliminated for 200 ms (methods in Morris and
Lisberger 1987).  First, we stimulated the model with steps in target velocity,
which activate the slip velocity and impulse acceleration pathways.  We adjusted
the filters in these pathways to obtain the fit in figure~2A, where the rising
phase of the model's output (dotted lines) matches the rising phase of the
monkey's eye velocity (solid lines) during the open-loop portion of the monkey's
response. Then, we stimulated the model with steps in target acceleration, which
activates the slip velocity and smooth acceleration pathways, and adjusted the
filter in the smooth acceleration pathway (Fig.~2B).

\section{Emergent Properties of the Model} 

We tested the model by restoring visual feedback and providing steps of 
target
velocity in closed-loop conditions.  Figure 2C shows that the model matches the
rising phase of the monkey's response and makes a realistic transition into
steady-state tracking.  Like the monkey, it reaches steady-state velocity at
later times for higher target speeds.  The contribution of each pathway to the
initiation of pursuit can be assessed by setting its gain to zero, effectively
``lesioning" that limb of the model.  When the slip impulse acceleration pathway
is lesioned, the rising phase is delayed and more sluggish, but the transition
to steady-state tracking is unchanged (Fig.~2D, open arrow).  This pathway
contributes exclusively to the initial 25--50 ms of the response to steps in
target velocity and allows the model to reproduce the observation that the
earliest component of the pursuit response is sensitive to target direction, but
relatively insensitive to target speed (Lisberger and Westbrook 1985).  When the
smooth acceleration pathway is eliminated, the rising phase is unchanged, but
there is a large overshoot in the transition to steady-state tracking (Fig.~2D,
filled arrow).  Thus, the smooth acceleration pathway normally decelerates the
eye as eye velocity approaches target velocity.  This is an emergent property of
the model, since this pathway was tuned by adjusting its contribution to the
acceleration of the eye as shown in figure~2B.

\begin{figure}%fig 2
\figspace{5.38in}{3.17in}
\caption{Comparison of the eye velocity output from the model and the monkey. 
In all panels, dotted lines show the model's output and solid lines show the
monkey's eye velocity.  A: Steps in target velocity of 5, 15, and 25 d/s under
open-loop conditions.  The monkey's open-loop response lasts for the first 200
ms of his response.  B: Steps in target acceleration of 45, 64, and 120 d/s$^2$
under open-loop conditions.  C: Steps in target velocity of 5, 15, and 25 d/s
under closed-loop conditions.  D: Effect of lesioning either acceleration
pathway on the model's response to a step in target velocity of 20 d/s.  Open
arrow, model's response when gain in slip impulse acceleration pathway is set to
zero.  Closed arrow, model's response when gain in slip smooth acceleration
pathway is set to zero.}
\end{figure} 


The model rings at a relatively high frequency, 5 Hz, in response to steps in
target velocity (solid lines in figure~3A).  Similar oscillations are seen in the
behavior of both humans and monkeys (Goldreich and Lisberger 1987; Robinson et
al.\ 1986).  If the model is driven at this resonant frequency, the output lags
target velocity by 180 degrees (Fig.~3B, filled arrow), an effect that is also
seen in the monkey's behavior (Goldreich and Lisberger 1987).  The high
frequency properties of the model depend upon the presence of the smooth
acceleration pathway.  If this pathway is eliminated, the spontaneous
oscillations still occur, but now at only 1.6 Hz (Fig.~3A, open arrow), and the
phase lag in the response to sinusoidal inputs increases (Fig.~3B, open arrow).

\begin{figure}[b]%fig 3
\figspace{5.38in}{5.05in}
\caption{Ringing and frequency response of the model.  A: Solid lines, model's
closed-loop response to steps in target velocity of 10 and 20 d/s, viewed on
larger time scale than in figure~2C.  Dotted line, model's response to step in
target velocity of 10 d/s when gain in smooth acceleration pathway is set to
zero.  B: Driving the model with sine wave target velocity at 5.0 Hz under
closed-loop conditions.  Filled arrow, the intact model's response lags target
motion  by 180 degrees.  Open arrow, model's response when gain in smooth
acceleration pathway is set to zero.  Phase lag is now 325 degrees.}
\end{figure}

\section{Discussion}

An important property of our model is that it allows independent control over
the initiation and maintenance phases of pursuit.  The rising phase is
determined mainly by the slip velocity and impulse acceleration pathways.  The
steady-state behavior is determined primarily by the smooth acceleration
pathway.  Since the differentiator in the smooth acceleration pathway introduces
a phase lead, the steady-state behavior of the model has a higher frequency
response than the rising phase.  The exact frequency of ringing depends upon the
total delay around the smooth acceleration pathway.  For example, if the delay
in the visual input is increased by 30 ms, the model will ring at 3.8 Hz,
similar to what is normally observed in humans (Robinson et al.\ 1986) or in
monkeys when the delay in visual feedback is increased (Goldreich and Lisberger
1987).  The amount of ringing depends upon the gain element in the smooth
acceleration pathway.  Lowering the gain dampens or eliminates the ringing,
increasing the gain produces persistent ringing.  Such variations are also seen
on individual trials in the monkey's behavior.

The strength of our model is that it uses open-loop behavioral data to embody
the pursuit system's sensitivity to different aspects of visual motion.  Although
it was designed to replicate the dynamics of the initiation of pursuit, the
model also serendipitously solves a problem noted by Robinson (Robinson et al.\
1986), namely, that the rising phase of pursuit is sluggish compared to the
frequency of its ringing.  The emergence of realistic steady-state properties in
the model indicates that the visual elements in its pathways capture important
aspects of signal processing within the smooth pursuit system.  Our model does
not include a sensitivity to position errors, which can affect steady-state
tracking in monkeys (Morris and
Lisberger 1987).  It also does not include the topographic organization of
the visual system and therefore cannot reproduce the retinotopic deficits in the
initiation of pursuit or the directional deficits in the maintenance of pursuit
which are seen after lesions of cortical areas MT and MST (Dursteler et al.\
1987; Newsome et al.\ 1985).  However, it should be possible
to embed the dynamics of our model within a topographic structure that would
account for these effects.

\section*{Acknowledgments}
This research was supported by NIH Grants EY03878 and EY07058.

\begin{thebibliography}
\indent\indent
Dursteler, M.R., R.H.~Wurtz, and W.T.~Newsome. 1987. Directional Pursuit
Deficits Following Lesions of the Foveal Representation within the Superior
Temporal Sulcus of the Macaque Monkey. {\it J. Neurophysiol\/} {\bf 57},
1262--1287.

Goldreich, D.\ and S.G.~Lisberger. 1987. Evidence that Visual Inputs Drive
Oscillations in Eye Velocity during Smooth Pursuit Eye Movements in the
Monkey. {\it Soc. Neurosci. Abstr.} {\bf 13}, 170.

Krauzlis, R.J.\ and S.G.~Lisberger. 1987. Smooth Pursuit Eye Movements are
Not Driven Simply by Target Velocity. {\it Soc. Neurosci. Abstr.} {\bf 13}, 170.

Lisberger, S.G., C.~Evinger, G.W.~Johanson, and A.F.~Fuchs. 1981.
Relationship between Eye Acceleration and Retinal Image Velocity during Foveal
Smooth Pursuit Eye Movements in Man and Monkey. {\it J. Neurophysiol.} {\bf
46},  229--249. 

Lisberger, S.G., E.J. Morris, and L.~Tychsen. 1987. Visual Motion Processing
and Sensory-motor Integration for Smooth Pursuit Eye Movements. {\it Ann. Rev.
Neurosci.} {\bf 10}, 97--129.

Lisberger, S.G.\ and L.E.~Westbrook. 1985. Properties of Visual Inputs that
Initiate Horizontal Smooth Pursuit Eye Movements in Monkeys. {\it J. Neurosci.}
{\bf 5}, 1662--1673.

Morris, E.J.\ and S.G.~Lisberger. 1987. Different Responses to Small Visual
Errors during Initiation and Maintenance of Smooth-pursuit Eye Movements in
Monkeys. {\it J. Neurophysiol.} {\bf 58}, 1351--1369.

Newsome, W.T., R.H.~Wurtz, M.R.~Dursteler, and A.~Mikami. 1985. Deficits in
Visual Motion Processing Following Ibotenic Acid Lesions of the Middle Temporal
Area of the Macaque Monkey. {\it J. Neurosci.} {\bf 5}, 825--840.

Robinson, D.A., J.L.~Gordon, and S.E.~Gordon. 1986. A Model of the Smooth
Pursuit Eye Movement System. {\it Biol. Cybern.} {\bf 55}, 43--57.
\end{thebibliography}
{\flushleft
\rule{2in}{.1mm}\hfill\break
\footnotesize Received 15 August; accepted 1 October 1988.
\normalsize}
\endpaper{krauzlis}













































            
 






















































