Neural Networks 18 (2005) 213–224
www.elsevier.com/locate/neunet

A model of smooth pursuit in primates based onlearning the target dynamics
Tomohiro Shibataa,b,c,*, Hiromitsu Tabatad, Stefan Schaalb,e, Mitsuo KawatobaMetalearing and Neuromodulation, CREST, Japan Science and Technology Corporation, Kyoto, Japanb
ATR Computational Neuroscience Labs, Kyoto, Japanc
Graduate School of Information Science, Nara Institute of Science and Technology, Nara 630-0192, Japand
Graduate School of Medicine, Kyoto University, Kyoto, Japane
Computer Science and Neuroscience, University of Southern California, LA, USA
Received 28 August 2002; revised 7 January 2005; accepted 7 January 2005

Abstract
While the predictive nature of the primate smooth pursuit system has been evident through several behavioural and neurophysiologicalexperiments, few models have attempted to explain these results comprehensively. The model we propose in this paper in line with previous
models employing optimal control theory; however, we hypothesize two new issues: (1) the medical superior temporal (MST) area in the
cerebral cortex implements a recurrent neural network (RNN) in order to predict the current or future target velocity, and (2) a forward modelof the target motion is acquired by on-line learning. We use stimulation studies to demonstrate how our new model supports these hypotheses.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Oculomotor control; Medial superior temporal (MST) area; Predictive visual tracking; On-line fast learning; Forward model; Recurrent neuralnetwork

1. Introduction
Due to their narrow foveal vision, which has a viewing
angle of only a few degrees, primates have to move theireyes to acquire accurate information on small moving
targets in the environment. Smooth pursuit eye movements (SPEMs) perform this function and can achieve
remarkable performance. For example, humans canmaintain a velocity gain, i.e. the ratio of eye velocity totarget velocity, of one up to a target speed of about 208/s,
while monkeys have been reported to even exceed thisvalue. The accuracy of smooth pursuit is not only
confined to constant velocity targets but has also been
observed in periodic motion such as sinusoidal signals
with frequencies of less than about 1 Hz (Dallos & Jones,
1963; Stark, Vossius, & Young, 1962; Westheimer, 1954).
* Corresponding author. Address: Graduate School of Information
Science, Nara Institute of Science and Technology, 8916-5 Takayamacho, Nara 630-0192, Japan. Tel.: C81 743 72 5981; fax: C81 743 72 5989.
E-mail address: tom@is.naist.jp (T. Shibata).

0893-6080/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.neunet.2005.01.001

Even a phase lead of the eyes can sometimes be observedin such experiments. Due to the information processing
delays (e.g. 80–130 ms for human brain) in the visual
pathways, the experimentally observed high performanceof the smooth pursuit system cannot be achieved solely
with standard negative feedback methods based on visualerror signals. Under such delayed information processing,simple feedback control has a significant phase shift to thetarget signal, and thus some form of predictive control
must take place (Pavel, 1990).
Several experimental results have been reported that also
shed some light on the predictive nature of the smooth
pursuit system. Morris & Lisberger (1983, 1987) demonstrated that monkeys were able to execute smooth pursuitwith zero retinal slip by using a special target presentation
technique called ‘target stabilization’. It is also known
that monkeys can maintain smooth pursuit during blink
periods, i.e. a sudden disappearance of the target for a brief
moment (Churchland & Lisberger, 2000; Kawano, Shidara,
Watanabe, & Yamane, 1994; Newsome, Wurtz, &Komatsu, 1988; Pola & Wyatt, 1997; Sakata, Shibutani, &Kawano, 1983). Such predictive compensation has been
observed both in constant velocity and sinusoidally moving

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targets (Becker & Fuchs, 1985; Fukushima, Yamanobe,
Shinmei, Fukushima, 2002; Whittaker & Eaholtz, 1982).
Therefore, it seems clear that SPEMs are a key touncovering mechanisms for predicting the external world in
the primate brain. In place of previous models, this article
develops a SPEM model that employs a compact representation of the target motion that can be quickly learned in an
on-line fashion based on visual error signals. By taking
neuroanatomical findings into account, our model further
suggests that the medial superior temporal (MST) area has
the possibility of predicting target velocity using only
signals that originate from visual information, without
relying on efference copies of the oculomotor command or
proprioceptive feedback.

2. Model development
2.1. Previous models
Pioneering computational models for SPEMs (Robinson,
Gordon, & Gordon, 1986; Yasui & Young, 1975) attempted
to cancel out the feedback signal in order to enable highvelocity gain; their model works as a feedforward
controller. Fig. 1 describes the essence of Robinson’smodel. In this model, the feedback signal with a delay D1 is
precisely canceled out by a positive feedback loop with thedelay D1CD3. However, their feedforward pathway stillcontains a significant delay determined by D2 and t suchthat this model cannot achieve zero-lag tracking of asinusoidal signal. For instance, for a 1 Hz sinusoidal motion,
the eye movement lags the target by more than 200 ms.
Krauzlis and Lisberger proposed a model that employed
nonlinear filters for velocity and acceleration of the retinal
slip in their visual processing pathways (Krauzlis &Lisberger, 1994). Their model can reconstruct the SPEMs
in their initiation phase as effectively as Robinson’s model,
but it cannot achieve zero-lag tracking of a sinusoidal signal
similar to the Robinson’s model, because it is essentially afeedback control model.
From the control-theoretic point of view, it is essential tohave an internal representation of target motion that enables
prediction of the current or future target velocity based on
the past target state, e.g. position and velocity in the world
coordinates. Bahill and McDonald (1983) proposed atheoretically sound control model based on predictive

Fig. 1. A modified Robinson’s model (Robinson et al., 1986) describing theessential property of the control system. T_ is target velocity and E_ is eye
velocity. D1, D2 and D3 are delay times. Here, s denotes a Laplacetransformation operator. t is a time constant. K is a gain.

control. Their model reconstructed eye movement similar
to that of humans where a sinusoidal or parabolic signal waspresented periodically, but it required a priori knowledge ofthe target motion model. In contrast, our model below will
acquire such a model by learning taking known biologicalconstraints into account.
Another mechanism is a memory-based model that relieson the periodicity of the target by estimating the period,
memorizing the entire retinal error over one period, andusing the memorized retinal error to generate the appropriate oculomotor command (Banes & Grealy, 1992; Barnes& Wells, 1999). However, such a memory-based approachseems more artificial and less biologically plausible than thepredictive control approach because (1) it requires aperiodicity estimator that needs a specialized computation,and (2) at least in humans improvement in SPEM
performance occurs continuously; if such an improvement
is heavily dependent on the target periodicity, the improvement must be periodic. Thus, our work focuses on apredictive control model.
2.2. Neuroanatomy
From a neurophysiological point of view, the middle
temporal (MT) area and medial superior temporal (MST)
area seem to be intimately involved in smooth pursuit.
Moreover, neurophysiological results support the idea thatthe major computation for the smooth pursuit maintenancephase is performed in area MST(Dürsteler, Wurtz, &Newsome, 1987; Kawano, Shidara, Watanabe, & Yamane,
1994). Kawano, Shidara, Watanabe, and Yamane. (1984)
shed some light on the information processing taking placein the MST area. Roughly in the area that is now called
MST, they found a group of neurons that responded to gazevelocity, i.e. the velocity of the gaze in the world
coordinates. This result was supported by experiments inwhich the activities of the same neurons were very similar ineither pure eye tracking of a sinusoid or vestibulo-ocularreflex (VOR) cancellation, where a target was moving
sinusoidally and the head of the monkey was also externally
moved sinusoidally, requiring the monkey to keep its eyes
still and cancel the VOR in order to keep its gaze on thetarget. On the other hand, the same neurons had nodischarge properties when the monkey was oscillated
sinusoidally in the dark (a pure VOR task). Therefore,
activity in these visual tracking neurons seems to originate
from highly preprocessed visual inputs and not from efferent
or afferent signals (related to eye movement). This point is
crucial from the computational neuroscience point of view
because it suggests that the major computation for smooth
pursuit could be achieved in the desired trajectory spacewithin the MST areas, and not in the motor command space.
Cortical eye fields are also thought to be involvedin smooth pursuit (Tian & Lynch, 1996); in particular,
the frontal eye field (FEF) has been intensively studied
for smooth pursuit (Fukushima, Yamanobe, Shinmei,

T. Shibata et al. / Neural Networks 18 (2005) 213–224

Fukushima, Kurkin, & Peterson, 2002; Gottlieb, Macavoy,& Bruce, 1994; MacAvoy, Gottlieb, & Bruce, 1991). This
paper, however, is not concerned with the FEF due to thefact that the MST areas and the FEF have reciprocal
connections, and their properties appear similar (Fukushima, Yamanobe, Shinmei, Fukushima, Kurkin et al., 2002;
Newsome et al., 1988). At present, we do not have further
information, such as the simultaneously recorded physiological data of the MST and the FEF, that would demand amodification of our model.
2.3. Our modelFig. 2 depicts our conceptual control model of smooth
pursuit mapped onto neuroanatomically plausible pathways.
Without loss of generality, we treat only one rotational
degree to highlight the principles of our control system—
multiple degrees would require an appropriate replication of
the circuit. In Fig. 2, the retinal slip E_ is generated from thedifference between the eye velocity E_ and the target velocity_ i.e.
x,_e_ Z x_ K E:

(1)

Information on visual motion is processed hierarchicallyin the visual cortices, and finally reaches MT and MST for
further analysis. From here, it is conveyed to the brain stem
via DLPN and the cerebellum, thus allowing smooth pursuit
to be maintained.
The physiological network in Fig. 2 itself is not novel;
however, by assigning computations to its structure, a newview of smooth pursuit maintenance can be derived. This
circuit consists of two subsystems: (i) a recurrent neuralnetwork (RNN) mapped onto MST, which receives theretinal slip with delays and predicts the current target
motion, and (ii) an inverse dynamics controller (IDC) of theoculomotor system, mapped onto the cerebellum and thebrainstem.
The first subsystem is the key point of our model. It
receives retinal signals that convey the past target motionand eye motion and predicts the current target velocity. It is215

theoretically possible that one predicts the future target
velocity by observing the target state x at a time with theknowledge of the forward model of the target motion.
Because the brain cannot observe the target state directly, itneeds to estimate the state based on D-time delayed retinal
slip information. Eq. (2) represents the prediction of the_^ from an estimated past target
current target velocity xðtÞstate xðtKDÞ by the function f:_^ Z f ðxðt
 K DÞÞ;xðtÞ

(2)where f essentially corresponds to the D-time discretizedtarget dynamics. Applying f recurrently computes a
predicted future target state. Note that the information in
the state variables in xðtKDÞ should suffice for calculating_^xðtÞ:
Based on theory and experiment showing that the
cerebellum and brainstem together act as an inverse
dynamics controller of the oculomotor plant (Kawato,
1999; Shidara, Kawano, Gomi, & Kawato, 1993), we
assume that the IDC has the capability to cancel the
dynamics of the eye plant, and, for the moment, that this isdone perfectly, making it is valid to write_^_ Z xðtÞ:
EðtÞ

(3)

Under this assumption, neither the IDC nor the eye plantneeds to be detailed in Fig. 2.
The state estimator xðtKDÞ in Eq. (2) results from
 K DÞ ¼ xðt^ K DÞ þ Kðxðt K DÞ K xðt^ K DÞÞ:xðt

(4)

The first term of the right-hand side of this equation is thepredicted state at time ðtK DÞ, while the second term is theprediction error multiplied by a weight matrix K, whichcorrects the predicted state by means of the ðtK DÞ delayedsensory signal. Essentially, Eq. (4) is a part of the update
equations of a Kalman filter (see Discussion 4.4). Since weassume KZ1 throughout this paper, the essence of Eq. (4)
can be reduced to xðtKDÞZ xðtK DÞ, i.e. the simplest formof an estimator. Given that xðtK DÞ is not directly
observable, Eq. (4) is nevertheless needed to computexðtK DÞ by noting the following transformations based on
the assumption in Eq. (3) and the retinal slip definition in (1):_^ K DÞ C kðxðt
_^ K DÞÞ
_ K DÞ Z xðt_ K DÞ K xðtxðt_^ K DÞ C kðxðt
_ K DÞÞ
_ K DÞ K Eðt
Z xðt_^ K DÞ C keðt_ K DÞ
Z xðt_^ K DÞ C eðt_ K DÞ:Z xðt

Fig. 2. Conceptual control model mapped onto physiologically plausible
pathways. We assume that the recurrent network (RNN) exists in the higher
visual cortex areas and that the cerebellum and brainstem together form theinverse dynamics controller (IDC) of the eye. The delay element D
represents the total delay in the visual pathway. See text for otherabbreviations.

(5)The last equation assumes kZ1. xðtKDÞ is obtained by_
integrating xðtKDÞ (See Section 4.1 for a discussion on
biological plausibility of the positional signals). Thus,
state estimation can operate correctly based on the_^ and the delayed retinal slip signal
internal feedback of xðtÞ
_
eðtK
DÞ.

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The final issue for our model is how to acquire the target
dynamics. It has been found that the gain and phase of
biological smooth pursuit can be gradually improved bypresenting the target motion repetitively (Dallos & Jones,
1963). We regard such gradual improvement as evidence foron-line learning of the target dynamics, which can beformulated as follows.
At time t, the predictor can only see the delayed
estimated target state xðtKDÞ. By making the parameter
vector w in the target state predictor f explicit, the target
velocity prediction is_^ Z f ðxðt
 K DÞ; wðtÞÞ hf ðtÞ;
xðtÞ

(6)

where w needs to be adjusted by the learning system. Notethat the predictor predicts only the target velocity instead of
the complete target state in accordance with biologicalconstraints, i.e. that only velocity information is conveyed
in the visual pathway, and the command generated in eachoculomotor behavior pathway is equally interpreted as avelocity command. Let the loss function J be the simple
squared prediction error
1
1_ 2:_ K EðtÞÞ
_ 2 Z ðxðtÞ
JðtÞ Z eðtÞ
2
2

(7)

Employing Eqs. (3) and (6), a gradient descent learning
rule for w can be written as
dwi ðtÞ
vJðtÞ
vf ðtÞ
_
Z K3
Z3
eðtÞ;
dt
vwi
vwi

(8)

with 3 denoting the learning rate.

3. Simulation results
3.1. Simulation setup
In order to verify that our model can achieve smooth
pursuit with gain one and zero-latency, we conducted two
evaluations by simulation. In one setup, the input was aramp input with a constant velocity of 0.5 rad/s, and in theother setup, a sinusoidal input was chosen with a frequencyof 1.0 Hz. Note that the dynamics of these inputs is asecond-order linear system, which guarantees that thecurrent target velocity can be predicted by the past target
state, i.e. position and velocity.
Fig. 3 is the actual model used for simulations in controltheoretic detail. Since the eye plant of primates can beapproximated by a second-order linear dynamical system(Keller, 1973), a linear inverse dynamics model suffices for
feedforward control of the eye system. It receives as input
the desired eye angular position, velocity, and acceleration.
The dark shaded block in Fig. 3 corresponds to the RNN.
According to Yamamoto, Kobuyashi, Takemura, Kawano,and Kawato (2000), the eye velocity and accelerationcommands are generated by Purkinje cells, but the eye
position command is generated elsewhere, possibly in

Fig. 3. Second-order linear model. The dark shaded block corresponds to
the RNN in Fig. 2. Here, s denotes a Laplace transformation operator. In
this diagram, s stands for differentiation and 1/s for integration.

the brainstem. We also know that the activities of Purkinjecells in smooth pursuit originate in the activities of the MST
area. Thus, while the predicted target velocity andacceleration are calculated in the RNN, the predicted target
position is not.
To enable the RNN to learn the target dynamics ofconstant velocity and sinusoidal motion, a second-orderlinear system suffices to represent their dynamics. Theoretically, when the target has second-order linear dynamics,_ thexðtÞZ AxðtK DÞ, with the target position x, velocity x,_ T , and D-time discretized state
state vector xZ ½x x
transition matrix A. To predict the target velocity, only thesecond row of A is required, denoted as w:_^ Z wT xðt K DÞ:xðtÞ

(9)

It is important that the causality between x^ and E_ be
maintained so that Eq. (8) effectively decreases the error_
function J. Because the predictor cannot see eðtÞ;at time t,
we use Eq. (10), where in the right-hand-side, time is shiftedby D-time to the past.
dwi ðtÞ
vf ðt K DÞ
_ K DÞ
Z3
eðt
dt
vwi

(10)

In each simulation, we employed the recursive leastsquares algorithm (RLS) (Ljung & Soderstrom, 1986) for
learning because its fast and guaranteed convergencemimics the quick and high-gain learning of humans. See
Appendix for our use of RLS in detail.
Using MATLAB/SIMULINK, the simulation model wasbuilt as a continuous system. Only the predictor, a recurrent
network, was run with discrete update times, ranging
between 30 and 200 Hz to test the robustness of the system.
The delay in the visual pathway was set to values between
50 and 150 ms, corresponding to the spectrum of values
reported in the neurobiological literature. Within theseranges, our model worked well in all simulations, although
it was necessary to tune the forgetting factor in RLS as afunction of the predictor’s update frequency.
All of our learning experiments started from scratch, i.e.
with all initial states including the weights of the learning
system set to zero. Note that the learned result does not

T. Shibata et al. / Neural Networks 18 (2005) 213–224

a 100-ms delay, the discrete state transition matrix of a ramp
input is expected to be
!
1 0:1
AZ
;
0 1

depend on the initial states because our model is linear.
However, we chose zero because large initial values cause
the system to become temporarily unstable.
According to Keller (Keller, 1973), the eye plant wasmodeled as a second-order linear system (Eq. (11)) withinertia term M, viscous term B, and spring term K set to0.02, 0.08, and 4.14, respectively. The term u is the input
torque:M x€ C Bx_ C Kx Z u:

the converged weights around 0.02 and 0.98 in Fig. 4(D)were very close to these optimal values, i.e. 0 and 1 from thesecond row of A.
The results of the second experiment with sinusoidal target
motion at 1.0 Hz and an amplitude of 0.5 rad are illustrated
in Fig. 5. Again, smooth pursuit learning was successful
in neatly converging regression coefficients. Theory predicts
an optimal weight vector of [K3.6931 0.8090]T. The actual
acquired final weight vector is [K3,678 0.7899]T, which is
close to our expectations.
These results were acquired with one set of hyperparameters in RLS that we had finely tuned. However, one could
still claim that learning speed is much slower than in human,
and that the transient response at the stimulus onset is crude.
While our model is not designed to explain the initiation
phase of smooth pursuit, an appropriate prior in RLS, suchas constant velocity motion, can improve the initial andtransient response. Fig. 6 demonstrates the effect of theprior. In this case, the target motion was generated by thesame sinusoidal signal used in Section 3, but the initial
weights for the estimated target position and velocity were(11)

3.2. Results
Fig. 4 shows that smooth pursuit was successfully
accomplished in the case of the ramp input. Fig. 4(B)presents the time course of the eye velocity rapidly
approaching the target velocity, and (C) depicts the timecourse of the retinal slip, quickly reaching zero. Fig. 4(B)illustrates a ringing in the eye velocity, which is mainly
caused by the parameter set to enable fast learning. It is
known that the eye velocity during tracking of the ramp
motion typically exhibits such ringing, whose frequency isabout 3–5 Hz. However, our simulation gives only around2-Hz ringing and does not help to explain the biological
data. This is thought to be the limitation of a linear system,
and will be improved by introducing nonlinearity
as discussed in Robinson et al. (1986). In the case of

B

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Time [s]Fig. 4. Simulation results in the case of a constant velocity motion of the target. (A) Time course of the angular position of the visual target (dotted) and the eye(solid). (B) Time course of the velocity of the visual target (dotted) and the eye (solid). (C) Time course of the retinal slip. (D) Time course of the weight for the
position input (dotted) and the velocity input (solid) of the learning system.

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4

4

3

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velocity [rad/s]

A

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Fig. 6. Learning of the forward model of a sinusoidal signal with priorknowledge of the target motion. The dotted line shows the target motion.
The solid line was generated by our model with the initial weight vector[0 0]T. The dashed line resulted from our model with the initial weightvector [0 1]T, which means a constant velocity motion was assumed. The
thick solid line corresponds to the case where the initial weight vector was[K2 1]T, which is closer to the theoretically correct values.

2
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0Time [s]

4
3

C

4

1

2

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4

0

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2.5

Time [s]Fig. 5. Simulation results in case of a sinusoidal motion of the target.
(A) Time course of the velocity of the visual target (dotted) and the eye(solid). (B) Time course of the retinal slip. (C) Time course of the weight
for the position input (dotted) and the velocity input (solid).

varied. Fig. 6 shows that the initial response of weights set
to [0 0]T is very poor, while the other two types of weight
initialization provided a vivid initial response.
Fig. 7 demonstrates that our model can continue
generating proper eye velocity during blink periods, as
described in Section 2.3. In particular, Fig. 7(B) illustrates
the eye acceleration changes during the blink periods bymeans of the appropriate prediction based on the knowledgeof target dynamics. For these simulations, the knowledge ofthe target dynamics was given from the beginning.

While we have assumed that the inverse dynamics
controller (IDC) consisting of the cerebellum and the brain
stem is perfect, it is useful to consider what the recurrent
neural network (RNN) in MST would do in the case where
the perfect IDC assumption does not apply. For this purpose,
the key idea is to imagine that an imperfect IDC and the eyeplant together form a new plant (extended plant). In thiscase, the RNN has to learn a composite function that
includes the target dynamics and the dynamics of theextended plant in order to minimize the retinal slip. Given
that the extended plant in our example is at most secondorder linear, the second-order linear system of the RNN
should be able to compensate for IDC inaccuracies
perfectly. Figs. 8 and 9 present simulation results supporting
our expectation. In the case of Fig. 9(A), with respect to theactual target velocity, the amplitude of the predicted target
velocity is slightly greater and its phase is delayed, which istheoretically necessary to compensate for the imperfect
IDC, where the spring constant is reduced by 50%. In thecase of Fig. 9(B), since the viscous term of the IDC is 50%of the true value, the theoretically required advanced phaseand increased amplitude can be observed correctly in our
simulation. Fig. 9(C), to compensate for two imperfectparameters in the IDC, the amplitude of the predicted target
velocity significantly increases, and its phase is almost thesame as the actual target velocity. We also tested a casewhere the mass term was changed and observed no problem
in successful learning. Note that the purpose of theseexperiments is to relax the former assumption of the perfect
IDC. Removing the IDC causes SPEMs to be absent andlearning to break down; the IDC is downstream from thepredictor and is thus inherently necessary to move the eyes.

T. Shibata et al. / Neural Networks 18 (2005) 213–224

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Time [s]Fig. 7. Simulation results of blink experiments. Here, (A) and (B) show the time course of target velocity (dotted) and eye velocity (solid) when the input is aconstant velocity motion and a sinusoidal signal, respectively. The dark shaded blocks represent blink periods. The blink period was set to 500 ms in (A) and
100 ms in (B).

model can also maintain SPEMs during the target blinking.
We have also demonstrated that the learning predictor oftarget motion can be realized by the cortical recurrent neuralnetwork, assumed to exist in the MST area, even if there is noefference copy of the eye.
As our model primarily emphasized a control-theoretic
development, the following sections focus on its neurophysiological plausibility.

4. Discussion
We have presented a computational model employing a
predictive controller with fast learning of the target dynamics
enables zero-lag SPEMs. In our model, the representation of
the target motion is much simpler than the memory-basedmodel, and learning proceeds quickly, decreasing the retinal
slip without waiting for one period of the target motion. Our

B

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Time [s]Fig. 8. Simulation results in the case of imperfect inverse dynamics controllers. (A–C) show the results when the input is a sinusoidal signal, while (D)presents the results when the input is a ramp. Time course of the angular velocity of the visual target (dotted) and the eye (solid). (A) The spring termis reduced by 50%. (B) The viscous term is reduced by 50%. (C) Both the spring term and the viscous term are reduced by 50%. (D) Solid line is thetarget velocity. Thick, dotted, and dashed lines are the generated eye velocities when the spring term, the viscous term, and both terms are reduced by
50%, respectively.

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A

4

B

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velocity [rad/s]

velocity [rad/s]

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Time [s]Fig. 9. Simulation results in the case of imperfect inverse dynamics controllers. (A–C) show the results when the input is a sinusoidal signal, while (D) presents
the results when the input is a ramp. Time course of the angular velocity of the visual target (solid) and the predicted target (dashed). (A) The spring term isreduced by 50%. (B) The viscous term is reduced by 50%. (C) Both the spring term and the viscous term are reduced by 50%. (D) Solid line is the target
velocity. Thick, dotted, and dashed lines are the predicted target velocities when the spring term, the viscous term, and both terms are reduced by 50%,
respectively.

4.1. Use of estimated target position
In general, we assume the inverse model controller
implemented in the cerebellum and brainstem is perfect,
and that the forward model of the target motion is
generated in the RNN of the MST area. While the targetstate estimator in the RNN estimates both target positionand velocity (Fig. 3), usually only velocity is referred toin neurophysiological studies of smooth pursuit maintenance in MST, e.g. (Komatsu & Wurtz, 1988a; Newsome
et al., 1988). However, in the monkey MSTd area,
Squatrito and Maioli (1997) found eye-position modulated
neurons which were also activated during SPEMs as well
as neurons activated but not influenced by eye position
during SPEMs. DeSouza, Dukelow, and Villis (2002).
found eye-position modulation of the brain activities in
MTC in their human fMRI study. According to thesestudies in which a target position and an eye position were
not distinguishable during SPEMs, the use of a target
position in our model is not implausible.
Alternatively, two other implementations can beconsidered. One is for estimating the target velocity andacceleration instead of the target position and velocity.
This alteration would not harm our model since thisinformation still suffices to predict the current target
velocity and would also be neurophysiologically plausible(Takemura, Inoue, Gomi, Kawato, & Kawano, 2001). The
other possible method is to employ a tapped delay line
(TDL) model, where target velocities at several delay
periods are maintained and used to predict the current
target velocity. As benefits of this method, only velocity
signals are required, and it can theoretically represent
higher-order linear motion by simply increasing thenumber of taps. Although there is so far no direct
evidence whether such a TDL model is implemented in
MST, Bi and Poo (1999) demonstrated the possibility ofthe existence of a TDL in the brain through theirexperiments with cultured hippocampal neurons. Whatthey accounted for is that polysynaptic transmission
pathways combined with spike-timing-based synaptic
modifications provide a TDL.
If such a TDL is available in the MST area, an interesting
issue is to what degree higher-order linear dynamics of thetarget motion can be learned in humans. Constant velocity
or sinusoidal motion has been typically used for the
stimulus, i.e. motion generated by either a first- or secondorder linear system. However, we can provide a wide
variety of other stimuli from the viewpoint of linear

T. Shibata et al. / Neural Networks 18 (2005) 213–224

221

systems. For instance, Bahill and McDonald (1983) used aperiodic parabolic stimulus. They also reported that subjects
were able to track a cubic wave form with zero-latency. In
future work, we will conduct a systematic and comprehensive investigation of the abilities of the human smooth
pursuit with more complicated target motion stimuli, and wewill also monitor the course of learning.

little effect at around G100 ms. Although current knowledge of the STDP is insufficient for realizing the timealignment of 80–130 ms assumed in this paper, this idea
has potential because the temporal window of MST
neurons has not been studied yet, and the window contains
regional variabilities (Feldman, 2000; Song & Abbott,2001).

4.2. Initial response

4.4. Optimal estimation

We have employed the recursive least squares algorithm (RLS) for learning, since it is fast and guarantees
convergence. We hoped it would mimic the quick andhigh-gain learning of humans; however, as demonstrated
in Fig. 5, the initial response was poor and the converging
speed was slower than that in humans. Consequently, we
have conducted research to determine if an appropriate
prior in RLS could solve these problems. Fig. 6 shows thatthe prior can significantly improve performance, although
we might claim that the converging speed is still slower
than in humans. For instance, zero-latency tracking canoccur after only one quarter of a target motion cycle(Bahill & McDConald, 1983) if the subjects areexperienced with the target motion. To explain this fact,
Bahill and McDonald proposed a ‘menu selection’method, which enables human-like initial performance if
the brain selects the correct target dynamics model fromthe ‘menu’. In our model, this corresponds to the casewhere the prior weights in RLS completely or mostly
describe the target dynamics, obviously leading to a goodinitial response. In relation to this issue, it would beinteresting to investigate such issues as how long we canmaintain a model image, how learning another model
affects the current memory, and how and where the model
images are stored.

To date, although we have no strong evidence that MST
implements an optimal estimation algorithm such as RLS,
some supportive studies have been conducted toward
interpreting cortical processing in light of estimation theory.
In neurophysiology, as has been found for other corticalareas such as M1 (Georgopoulos, Schwartz, & Kettner,
1986; Todorov, 2000), there is evidence for population
coding in MT (Lisberger & Movshon, 1999) in addition to
MST (Takemura, Inoue, Kawano, & Miles, 2001). Computational neuroscientists have been attempting to understandpopulation coding from the viewpoint of optimal estimation(Pouget, Zhang, Deneve, & Latham, 1998; Rao, 1999; Rao& Ballard, 1999; Salinas & Abbort, 1995). Accompanied bybehavioral evidence, it has also been hypothesized thatinternal models are represented in a probabilistic way byneural population codes in order to carry out optimal
estimation for perception (Burgi & Yuille, 2000; Zemel,
Dayan, & Pouget, 1998) and control (Witney, Goodbody, &Wolpert, 2000).
Note that the model we propose already has a structure
similar to a Kalman filter, which is the most famous on-line
linear probabilistic minimum variance estimator. Like theKalman filter, the RNN consists of two steps: target-state
estimation and target-state prediction. Furthermore, theKalman filter maintains the confidence of estimation anduses the Kalman gain K in the target state estimation step.
Eq. (4) already includes this style of computation, so only
the derivation of K would need to be changed to conform to
the on-line Kalman filter equations.
A true Kalman filter implementation would give us the
additional benefit of improving our smooth pursuit model
as follows. In this paper, we derived a smooth pursuit
model based on behavioral and physiological experimental
results. However, our present model cannot reproduce one
experimental finding, i.e. the smooth pursuit blink. Ourcurrent model keeps predicting the target state forever
without retinal slip information after learning is completed, as shown in Fig. 7. To reproduce the gradualreduction in the eye velocity during the smooth pursuit
blink that was observed in biological data, two mechanisms could be considered, based on the common
assumption that somehow the confidence of the target
state is accessible. Since there is no visual information
available during blink, the first idea is that a part of thebrain monitoring the prediction confidence directly
suppresses the MST area (Taguachi, Tabata, Shibata,4.3. Learning and prediction
The learning rule in Eq. (10) is a form of Hebbianlearning. The right side consists of the multiplication of aquantity correlated with the estimated target state,_vf ðtK DÞ=vwi , and the retinal slip, eðtKDÞ. Thesequantities are available in the MST area. The mechanism
for keeping the information vf ðtÞ=vwi for a D-time could be
realized by a signal transduction in MST neurons. It has
been investigated whether this type of temporal window inthe synaptic plasticity is mediated by such a signaltransduction in Purkinje cells (Houk, Singh, Fisher, &Barto, 1990; Kuroda, Schweifhofer, & Kawato, 2001;
Wang, Denk, & Häusser, 2000). Moreover, spike-timing
dependent plasticity (STDP) in hippocampal neurons has
been reported for the temporal window (Bi & Poo, 1998).
The STDP reported has a narrow temporal window that ismost effective at around G20 ms for long-term potentiation (LTP) and long-term depression (LTD), and it
reaches half its effectiveness at around G40 ms, leading to222

T. Shibata et al. / Neural Networks 18 (2005) 213–224

Doya, & Kawato, 2002). For example, either the frontal
pursuit area pointed out by Tanaka et al. (Taguchi,
Tabata, Shibata, & Kawato, 2004; Tanaka & Lisberger,
2001), or the frontal eye field can be candidate areas forthis task. Alternatively, we could imagine that such
behavior during smooth pursuit blink could be accomplished by population dynamics in the MST area and theconversion of its output by the cerebellum (Yamamoto,
Kobayashi, Takemura, Kawano, & Kawato, 2002).
4.5. Role of subregions in MST
Cells in the MST area can mainly be divided into a dorsal
medial (MSTd) and a lateral ventral (MSTl) region(Komatsu & Wurtz, 1988a; Tanaka, Sugita, Moriya, &Saito, 1993). The MSTd cells respond best to a large moving
pattern rather than to small spots of light, while cells in
MSTl respond better to the small spots. Further differentialfunctional characteristics of cells in these two areas havebeen reported.
The MSTl neurons play significant roles in smooth
pursuit in both its initiation and maintenance phases throughlesions (Dürsteler & Wurtz, 1988) or electrical stimulation
studies (Komatsu & Wurtz, 1989). Their activities rapidly
attenuate during blink periods. Self-induced motion drives
their activities preceding the onset of SPEM (Erickson &Their, 1991). The responses of neurons in the MSTl changed
when a stimulus in the region surrounding the excitatory
center of the receptive field was added to a stimulus fallingin the center of the field, which could facilitate theperceptual segmentation of a moving object from its
background (Eifuku & Wurtz, 1998).
On the other hand, while the MSTd neurons maintaintheir activities during blink periods (Newsome et al., 1988),
lesion or electrical stimulation of MSTd would have little
effect on pursuit eye movement (Dürsteler & Wurtz, 1988;
Komatsu & Wurtz, 1989). The role of MSTd neurons forsmooth pursuit could at least be to suppress the optokinetic
nystagmus (OKN) response (Komatsu & Wurtz, 1988b).
We could relate the above knowledge to our model tosome extent: lesion or electrical stimulation studies haveshown cells in MSTl are crucial for smooth pursuit. They
seem suitable for tracking a small target due to theirpreference for the target size as well as their capability ofperceptual segmentation. Furthermore, the MSTl neurons
can be predictive. Thus, the RNN part of our model might beachieved by the MSTl neurons.
We could also incorporate MSTd into our model. Basedon lesion or electrical stimulation studies for MSTd, cells in
MSTd seem to be not essential for smooth pursuit. However,this property could correspond to the target state estimatorin our model. Namely, these disturbances may have been
filtered out by the MSTd neurons. These neurons have alarge receptive field, and they may integrate many MT and
MSTl neurons as well as MSTd neurons, which are suitablefor estimating the target state.

Appendix A. Fast learning by recursive least squares(RLS)
Originally, RLS was formulated as in Eqs. (A1) and(A2), where w is the vector of regression parameters to be
estimated, P is the inverted covariance matrix of the input
data, x is the input vector, y is the output, and y^ is thepredicted output. A forgetting factor l is discussed below.

1Pðt K DÞxðtÞxðtÞT Pðt K DÞPðtÞ Z Pðt K DÞ K;
(A1)ll C xðtÞT Pðt K DÞxðtÞ
wðtÞ Z wðt K DÞ CPðtÞxðtÞ
^ðyðtÞ K yðtÞÞÞ;l C xðtÞT PðtÞxðtÞ

^ Z wðtÞT xðtÞ:
yðtÞ

(A2)
(A3)

Eq. (A2) shows that normal RLS requires the presence ofa target output y in the update rules. As described in the
introduction, the predictor cannot see this target output
directly, but it can utilize the retinal signals as the prediction
error. Thus, for our smooth pursuit, Eq. (A2) needs to bemodified to:wðtÞ Z wðt K DÞ CPðtÞxðtÞ
_ C 1Þ:eðtl C xðtÞT PðtÞxðtÞ

(A4)

This strategy corresponds to training RLS on ‘fake’^_ Initially, these fake targets aretargets, i.e. yZ yðtÞC
e.
fairly distant from the true targets, such that RLS is actually
trained on incorrect data. For this purpose, one needs toforget data from early training, accomplished by theforgetting factor l, which lies in the [0, 1] interval. For
lZ1, no forgetting takes place, while for smaller values,
older values in the matrix P are exponentially forgotten.
Essentially, the forgetting factor ensures that the predictionof RLS is only based on 1/(1Kl) data points. This forgetting
strategy also enables the predictor to be adaptive to thechanges in the target dynamics.
Another important element of Eq. (A4) is that it
explicitly shows the requirement for the time alignment of
the predictor output and the error since the learning module_
cannot see eðtC1Þ at time t. Thus, all variables in Eq. (A4)need to be delayed by one time step, which requires thestorage of some variables for a short time in memory.

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