IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 19874
Time-Optimal Control of Saccadic Eye MovementsJOHN D. ENDERLE, MEMBER, IEEE, AND JAMES W. WOLFE
Abstract-A new theory describing the time-optimal control of sac-cadic eye movements is proposed based on Pontryagin's minimumprinciple and physiological considerations. The lateral and medial rec-tus muscle of each eye is assumed to be a parallel combination of anactive state tension generator with a viscosity and elastic element, con-nected to a series elastic element. The eyeball is modeled as a sphereconnected to a viscosity and elastic element. Each of these elements isassumed to be ideal and linear. The neuronal control strategy is shownto be a first-order time-optimal control signal. Under this condition,the active state tension for each muscle is a low-pass filtered pulse-stepwaveform. The magnitude of the agonist pulse is a maximum for sac-cades of all sizes and only the duration of the agonist pulse affects thesize of the saccade. The antagonist muscle is completely inhibited dur-ing the period of maximum stimulation for the agonist muscle. Hori-zontal saccadic eye movements were recorded from infrared signalsreflected from the anterior surface of the cornea and then digitized.Parameter estimates for the model were calculated by using a conju-gate gradient search program which minimizes the integral of the ab-solute value of the squared error between the model and the data. Thepredictions of the model under a time-optimal controller are in goodagreement with the data.
INTRODUCTIONS ACCADIC eye movements, among the fastest volun-
tary muscle movements the human body is capable ofproducing, are characterized by a rapid shift of gaze fromone point of fixation to another. Although the purpose forsuch an eye movement is obvious, that is, to quickly re-direct the eyeball to the target, the neuronal control strat-egy is not. For instance, does the word "quickly" in theprevious sentence imply the most rapid movement possi-ble, or simply a fast as opposed to a slow movement? Toreach a destination in minimum time, the input to the oc-ulomotor system must be bang-bang according to Pontry-agin's minimum principle, that is, the oculomotor systemis either maximally or minimally stimulated during thesaccadic eye movement. With this control strategy, sac-cade magnitude is affected only by the length of the timeintervals during which the system is maximally or mini-mally stimulated. The present investigation utilizes Pon-tryagin's minimum principle and system identificationtechniques to estimate muscle active state tensions duringhorizontal saccadic eye movements in order to better un-derstand the neuronal control strategy.
Manuscript received January 2, 1986. This work was supported in partby Brooks AFB, TX under Contract F33615-83-D-0603.
J. D. Enderle is with the Division of Bioengineering, Department ofElectrical and Electronics Engineering, North Dakota State University,Fargo, ND 58105.
J. W. Wolfe is with the Neurosciences Function, USAF School of Aer-ospace Medicine, Brooks Air Force Base, San Antonio, TX 78235.
IEEE Log Number 8611135.
To detail the neuronal control strategy, it is necessaryto understand the effect of the saccadic innervation signalson the oculomotor plant and the resultant eyeball re-sponse. Many investigators have extensively studied thesaccadic innervation signals which are described by pulse-step waveforms. At the start of a saccade, the agonistmuscle is strongly stimulated and the antagonist muscleis completely inhibited. After a brief time interval, this isfollowed by a decrease in agonist stimulation and an in-crease in antagonist stimulation to tonic levels necessaryto maintain the eyeball in its new position [l]-[3]. Collinsstates that the amplitude and duration of the saccadic in-nervation signal determines the magnitude of each sac-cade [4]. Specifically, he determined a logarithmic rela-tionship between innervation amplitude and saccademagnitude. Zee and his co-workers assumed that a localfeedback loop automatically controlled the amplitude andduration of the saccadic innervation signal [5]. The dif-ference between the internal representation of the presenteye position and the desired eye position determines thesaccadic innervation signal. Note that Zee et al. still hy-pothesize a pulse-step innervation signal, but base thepulse size and duration on a nonlinear velocity function.Bahill reports a nonlinear relationship between saccadeinnervation magnitude and saccade magnitude, and a lin-ear relationship between saccade innervation duration andsaccade magnitude [6]. If these authors are correct in theirassertion that saccadic magnitude is a function of both theduration and the amplitude of the innervation signal, andnot duration alone, then neuronal control does not operatewith a minimum-time strategy.While investigators have recorded the innervation sig-
nal from several types of motoneurons that drive the eye-ball during a saccade, they have not directly measured theactive state tensions responsible for this movement. Col-lins and his co-workers have measured the muscle tensionin vivo at the muscle tendon during unrestrained humaneye movement using a miniature "C" gauge force trans-ducer [7]. The active state tension, however, is distrib-uted throughout the muscle and cannot be directly mea-sured since it is modified by the viscoelasticity of themuscle. Very little is known about the dynamic activestate tensions generated in the antagonist-agonist musclepair during a saccade and their relationship to the saccadicinnervation signal. Naturally, under static conditions dur-ing fixation, the active state tensions are proportional tothe innervation signal, and a constant firing frequencyproduces a constant active state tension. During a sac-cade, the agonist active state tension changes rapidly, ris-
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 1987
ing in a matter of milliseconds to a new level approxi-mately tenfold higher than during fixation, and then fallingto the new fixation level. The proportional relationshipthat exists for innervation and active state tension duringfixation is not valid during a saccade due to the effect ofsaturation and filtering of the input signal. Thus, the exactshape of the input to the muscle is uncertain. The activestate tensions are typically modeled by low-pass filteringthe innervation signal [8]. While very little is known aboutthe activation and deactivation time constants due to lackof in vivo testing, Bahill has estimated their values to bebetween 0.2 and 13 ms based on the rise of the isometricforce during electrical stimulation [9].When artificially stimulated, oculomotor muscles do not
develop additional tension when stimulus frequency isabove 200 Hz [8]. The normal firing frequency for theagonist muscle innervation averages approximately 800Hz during the saccade [8]. Motoneurons fire at rates be-yond which the muscle can respond because the rate ofchange in both the antagonist and agonist muscle forcessignificantly contribute to driving the eyeball to its desti-nation [8], [10], [11].Robinson presented data that seem to contradict the pre-
vious relationship of amplitude and duration with saccademagnitude [8]. From Fig. 4 in Robinson, note that theagonist motoneuron burst peaks at the same amplitude andthen drops to a constant level during the saccade regard-less of the size of the retinal error [8]. Since the moto-neurons fire well above 200 Hz for the initial pulse phaseof the trajectory regardless of the amplitude of the sac-cade, only the duration of the agonist pulse is a functionof the saccade displacement. Under these conditions, theeyeball appears to be driven to its destination in mini-mum-time for saccades of all sizes.
Other researchers have investigated the neuronal con-trol strategy of saccadic eye movements using optimalcontrol theory [12], [13]. While these investigators con-cluded that each saccadic eye movement is driven toachieve final eye position in minimum-time, they reporteddifferent neuronal control strategies. Clark and Stark pos-tulated second-order time-optimal control signals, but ob-served a first-order time-optimal control simulation solu-tion when using the same model in both analyses [12].While never completely rectifying these differences, Clarkand Stark concluded that the saccadic eye movement neu-ronal control strategy is first-order time-optimal. Clarkand Stark did not give switch-time details or comment onthe pulse magnitude-saccade magnitude relationship.Lehman and Stark, however, reported a second-ordertime-optimal controller using a simplified saccadic eyemovement model which excluded the activation and deac-tivation time constants [13]. Further, Lehman and Starkindicated that they were unable to solve the optimalityproblem using a saccadic eye movement model with ac-tivation and deactivation time constants. Lehman andStark also reported agonist pulse magnitude as a functionof saccade amplitude, which violates the bang-bang con-troller. Because of these inconsistencies in the literature,
it seems appropriate to reexamine the neuronal controlstrategy of saccadic eye movements.
This paper presents an original optimal control inves-tigation of horizontal saccadic eye movements based onPontryagin's minimum principle with a linear oculomotormodel in which activation and deactivation time constantsare explicitly included. Based on the optimality solution,it is shown that horizontal saccadic eye movement neu-ronal control is a first-order time-optimal control signal.The concepts underlying this hypothesis are 1) the agonistpulse is maximum regardless of the amplitude of the sac-cade, and 2) only the duration of the agonist pulse effectsthe size of the saccade. The antagonist muscle is assumedto be completely inhibited during the period of maximumstimulation for the agonist muscle. Furthermore, higherorder signals are found not to be time-optimal. A quan-titative analysis of saccadic eye movement data is alsopresented to support the hypothesis that the saccadic neu-ronal control mechanism operates to achieve final eye po-sition in minimum-time under a first-order controller.Thus, a consistent first-order time-optimal neuronal con-trol strategy is demonstrated which also agrees with ex-perimental data analysis.
TIME-OPTIMAL NEURONAL CONTROL STRATEGYThe hypothesis that the eyeball is driven to its destinationin minimum-time for saccades of all sizes is investigatedusing optimal control theory based on the minimum prin-ciple of Pontryagin with a linear oculomotor model. Re-cently, Enderle et al. presented a model of horizontal eyemovements, illustrated in Fig. 1, that is appropriate fortheoretical investigations involving the dynamics of sac-cadic eye movements [11]. This model is a modificationof the linear homeomorphic model by Bahill et al., anexcellent and accurate general model for analyzing sac-cadic eye movements [14]. In order to state the optimalcontrol problem to be solved, (13) of Enderle et al. [11]is written in terms of state variables with the agonist andantagonist active state tensions explicitly included as fol-lows.
05 = agonist active state tension06 = antagonist active state tensionn, = agonist neurological controln2 = antagonist neurological controlPi = mechanical components of the oculomotor
systemKst = sum of series and length tension elastic elementsBag = agonist viscosity element
Bant = antagonist viscosity elementTag = agonist time constantTant = antagonist time constant
( = a constant based on mechanical components ofoculomotor system.
Note that the filter time constants Tag and Tant in (1d)-(lf)are, in fact, functions of time; that is,
where ti are the switching times of the controller, and
(1 t 2- tjU(t - ti) =
t0t < ti.
Note that Tac in (2a) does not necessarily equal Trac in (2b)and that Tde in (2a) does not necessarily equal Tde in (2b).However, for purposes of this investigation and untilphysiological evidence proves otherwise, we will assumeequality. Due to physiological constraints, the agonist andantagonist neurological control must satisfy
0 c ni ' nmax for i = 1, 2. (3)
Equation (1) is written in matrix form as
H=AO +Bn (4)where
0 1 0 0 0
O 0 1 0 0
O O 0 1 0
-PO -P1 -P2 -P3 6(Kst -BantlTag)
O 0 0 lITag
O 0 0 0 0
0 -
0
0
-bBaglTant
0
1 /Tant -
0
0'
0
- (Kst- Bag!T.t)0
-1 /rant
n = [kn]Ln=
0 =
02
0304A =
040506_
0
0
0
bBantl/Tagi/tag
0O
45
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 1987
It should be clear that the system matrix A is stepwisetime-varying due to the time constants. The hypothesizedtime-optimal neural control strategy is to choose the con-trol n(t) to transfer H(0) according to 0 = AO + Bn to thedestination D so that the functional
(tf 6
J(n) = dt + Z G(OQ(tf)-Di)2
= tf + h(O(tf )) (5)is minimized, where the terminal time tf and 0(tf ) are un-specified. G is a weighting vector which determines thenearness of 0(tf ) to D.
TIME-OPTIMAL CONTROL SOLUTIONThe time-optimal control solution is investigated using
the minimum principle of Pontryagin. The minimum prin-ciple states that the time-optimal input to the model mustbe bang-bang. Using the standard approach to the opti-mality problem, the gradient or steepest descent method,it is impossible to solve for the switch-times for the sixth-order oculomotor model. Details of the gradient methodare provided in the Appendix. Instead of simplifying themodel in order to determine the optimal control, as didLehman and Stark [13], we decided to solve for the op-timal switch-times directly.
Direct Optimal Switch-Time EvaluationThe difficulties associated with the gradient method are
avoided by directly evaluating the optimal switch-timesfor the minimum-time controller based on the works ofPierre, Lee, and Smith [16], [18], [19]. First, using Pon-tryagin's minimum principle, this system's minimum-timecontroller is of the bang-bang type. Due to physiologicalconsiderations, the agonist neurological control is fullystimulated at the start of the saccade and the antagonistneurological control is completely inhibited. At eachswitch-time, the controllers exchange values. Thus, allthat is necessary to solve this problem is to specify theminimum-time controller switch-times. Since the oculo-motor system is a time-varying system, the direct evalu-ation procedure in solving for the optimal switch-times byPierre [16], Lee [18], and Smith [19] is not suitable sincethe commutativity condition is not satisfied [20]. Fortu-nately, differential equation (13) of Enderle et al. [11],which describes saccadic eye movements. with a bang-bang controller, is readily solved using classical tech-niques. Thus, by avoiding direct evaluation of the statetransition matrix, the direct evaluation computational pro-cedure can be modified appropriately with the saccadiceye movement model solution-to yield the optimal switch-times.The saccadic eye movement model is solved via super-
position by incorporating the bang-bang neurologicalcontrollers directly in the agonist and antagonist activestate tensions and treating the active state tensions as in-puts. That is, separate solutions are found for the tensionsoperating between the time intervals 0 to ti, tl to t2,
and then combined to yield the complete solution. Notethat the maximum number of switch-times must be deter-mined iteratively since the system is time-varying. Forone switch-time, the low-pass filtered pulse-step wave-forms which describe the agonist and antagonist activestate tensions are
Fgo = initial magnitude of the agonist active state ten-sion
FP = pulse magnitude of the agonist active state ten-sion
Fgs = step magnitude of the agonist active state tensionF,, = initial magnitude of the antagonist active state
tensionF,s = step magnitude of the antagonist active state ten-
sion.
The agonist and antagonist active state tensions along withthe corresponding neurological control signals are illus-trated in Fig. 2.The particular solution is given by
ENDERLE AND WOLFE: TIME-OPTIMAL CONTROL OF SACCADIC EYE MOVEMENTS
AGONIST MUSCLE
Fp-
Fgs i
Fgo
00 t
ANTAGONIST MUSCLE
Fto
Fts
00 tl
The only unknowns in (18) are t1 and tf. Equation (18) issolved by using a first-order exponential Taylor series ap-proximation iterative linearization technique. First, as-sume that tf is known. Next, substitute the truncated ex-ponential Taylor series approximation
exp (aitl+ l) = exp (ait4) exp (ai(tJ+1 -tJ
(19)= exp (ait-)(1 + ai(tl+l -t))into (18) for exp (aitl), which yields
+ K42 exp (a4(t - t))} U(t- ti) (16)where Kij are the constants determined from the system'sinitial conditions and particular solution, and ai are theeigenvalues.
Thus, the complete solution is
f(t) = Oh(t) + Op(t) (17)
The initial conditions are specified with the system at restat primary position (looking straight ahead), that is, 0(0)= 0. The extension to cases with more than one controllerswitch is determined in a similar manner.
In general, the minimum-time controller is specified byselecting t1, t2, , t, so that tf > t, is minimum and0(tf D. Consider the case in which one switch is re-
quired for the saccade eye movement model in [0, tf].
where to is the initial guess value of the switch-time and(j + 1)st iterates of tj +I are calculated from the jth it-erates of t4.The procedure of specifying the minimum-time con-
troller begins by fixing tf and using the linear approxi-mation of (18) to solve for t'+1 and iterating until thedesired degree of accuracy is achieved. Next, this pro-cedure is repeated after decreasing tf until the smallest tfis found. Finally, the previous procedure is repeated afterincreasing the number of switching times until the small-est tf is found. For multiple switching times, the linearapproximation of (17) is found by substituting a truncatedTaylor series exponential approximation for each un-known switch-time tk
exp (aitJk+l) = exp (aitk)(1 + ai(tk+1 - tk) (21)
where k = 1, * * *, n. The tf and corresponding tl, *tn are the desired results which specify the minimum-timecontroller.
Applying this procedure to the saccadic eye movementmodel using the parameter values described in Enderleand Wolfe resulted in a time-optimal control with oneswitch-time [17]. Simulations with two switch-timesyielded the identical value of tf with the one switch case,and t1 equaled t2. Extensions to additional switchings didnot seem warranted based on these results and physiolog-ical evidence to the contrary.
Presented in Fig. 3 are the optimal control results whichillustrate the nonlinear saccade magnitude and switch-timerelationship. The function is monotonically increasingwith an inflection point at approximately 100. For saccademagnitudes less than approximately 10°, the function is
47
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VQL. BME-34, NO. 1, JANUARY 1987
SACCADE MAGNITUDE IN DEGREES
20 r
15
10
5
0.010 0.020 0.030 0.040 0.050SWITCH-TIME IN SECONDS
Fig. 3. A diagram of the nonlinear saccade magnitude-switch-time rela-tionship for Fp = 1.0, Tac = 4 ms, and rde = 5ms.
(a) (b)
0
£ 00 0
i0.)nh
DisplacementDegrees
(d)
Ea
00
3 0
DisplacementDegrees
DisplacementDegrees
DisplacementDegrees
.00
Fig. 4. Diagram illustrating the effect of the activation time constant on
the switch-time for first-order time-optimal neurological control signals.(a) Tac = 0.004 (b) -ac = 0.007 (c) Tac = 0.010 (d) rTac = 0.013, and "de
= 0.005.
concave upward. For saccade magnitudes greater than ap-proximately 100, the function is concave downward. In-terestingly, most naturally occurring saccades occur withmagnitudes less than 150 [RI], the region in which thefunction is concave upward. Time-optimal control resultsare presented in Fig. 4 for various values of Tac and Tde =
0.005. These results were verified by simulating saccadiceye movements with the IBM continuous system model-ing program for both the one- and two-switch cases usingperturbations about the optimal value.
EXPERIMENTAL METHODS
The validity of the theoretical predictions determinedin the last section are further investigated by analysis on
saccadic eye movement data. Data were collected fromsubjects seated before a target display of seven small redlight-emitting diodes (LED), each separated by 5°. Thesubject's head was restrained by a bite-bar. The subjectwas instructed to follow the "jumping" target whichmoved from the center position to one of the other LED's,
0.E ,, a
0 aC00=0c1
(c)
c£c0003:0(0)
48
ENDERLE AND WOLFE: TIME-OPTIMAL CONTROL OF SACCADIC EYE MOVEMENTS
and then returned to the center position. The subject firstobserved 14 5° target movements, then 14 100 targetmovements, and finally, 14 15° target movements. Thesubject was allowed to rest after each set of 14 targetmovements. The left-right ordering of the target move-
ments, as well as the time interval between target dis-placements, were randomized. Data were only recordedfor the initial displacement from the center position. Hor-
izontal eye movements were recorded from the right eye
using an infrared signal reflected from the anterior surfaceof the cornea-scleral interface with instrumentation de-scribed by Engelken et al. [22]. Signals for bilateraltracking were digitized using the analog/digital converterof the DECLAB PDP 11/34 computer and stored in diskmemory. These signals were sampled at a rate of 1000samples per second for I S.
EXPERIMENTAL RESULTSData analysis is carried out in both the time and fre-
quency domain as described by Enderle and Wolfe [17].First, the data are analyzed using the two-point centraldifference method [23] to obtain estimates of velocity andacceleration. Then, using the time domain results, the dataare analyzed using the system identification technique toobtain estimates for model parameters and inputs for thesaccadic eye movement model.
Two-Point Central Difference Method ResultsThe results of the data analysis on one of the three sub-
jects tested using the two-point central difference methodare illustrated in Fig. 5 [17]. Velocity estimates are com-
puted with a step size of 3 and a sampling interval of 1ms. The time interval from the start of the saccade to thetime at peak velocity tin showed marked variation withinthe 5, 10, and 150 target movements. The time at peakvelocity should not be interpreted as a switching time. Infact, theoretical predictions indicate that peak velocity oc-
curs after the switching time for small eye movements andbefore the switching time for large eye movements [17].The range of variation on tmi within each target movementis approximately constant and independent of the size ofthe target displacement. Since the actual mechanical ele-ments of the oculomotor system are not changing for sac-
cades of the same size, the input to the oculomotor systemmust be responsible for the differences in saccade dynam-ics. As illustrated by Enderle and Wolfe, the only param-eter capable of changing the time to peak velocity is a
variable activation time constant [17]. Increasing the ac-
tivation time constant predominantly increases the time topeak velocity, while slightly reducing the peak velocity.These results are indicative of a random or variable acti-vation time constant acting independently of saccademagnitude. Next, peak velocity varies greatly for all tar-get displacements within the 5, 10, and 150 target move-
ments. Peak velocity is influenced by the filter time con-
stants, the magnitude of the agonist pulse, as well as theswitching time [17]. The most dominant factor affectingpeak velocity, however, is the size of the agonist pulse
Fig. 5. Two-point central difference estimates of peak velocity and timeto peak velocity as a function of saccade magnitude. Velocity estimatesare computed with a step size of 3 and a sampling interval of 1 ms.
magnitude. Increasing the size of the agonist pulse mag-nitude directly increases the peak velocity. This impliesthat the peak velocity variability apparent in the centraldifference results is primarily due to the magnitude of theagonist pulse, and that agonist pulse magnitude is a ran-dom variable, independent of the size of the target dis-placement. Note that the random behavior of these param-eters is not a function of fatigue because we havepurposely kept the recording sessions short to avoid fa-tigue [24]. The observation that both tmy and peak velocityexhibit random behavior is also supported by the mainsequence diagram [25] and data presented by Bahill et al.[24].
System Identification Technique ResultsParameter estimates and inputs for the model of the oc-
ulomotor system are found using the system identificationtechnique as described by Enderle and Wolfe [17]. Allparameters were estimated with a conjugate gradientsearch program which minimizes the integral of the ab-solute value of the error squared between the frequencyresponse and the data for the first saccadic eye movement.Great care was taken in evaluating the initial parameterestimates since large differences from the true values couldcause the estimation routine to converge to suboptimal andnonphysiologically consistent results. Published physio-logical data are used to estimate all of the initial parameterestimates [7], [26]. The initial estimates for the agonistand antagonist active state tension including time con-
49
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 1987
14
1 3
1 2
11
a 9a 8z- 7Z 6wa
504 4,. 3
a 2
1
0
O.0.047 0.694 0.1'41 0.1'89 0.2'36 0.283 0.330
TIME IN SECONDS
Fig. 6. Saccadic eye movement in response to a 150 target movement.Solid line is the prediction of the saccadic eye movement model with thefinal parameter estimates computed using the system identification tech-nique. Dots are the data.
stants, agonist pulse magnitude, agonist and antagoniststep magnitude, and pulse duration are based on physio-logical conditions and experimental data [17]'. Peak ve-locity, time to peak velocity, and duration of the saccadedetermine initial estimates for Fp, tl, 'rag, and Tant. Finaleye position determines initial estimates for the agonistand antagonist step magnitude. Final parameter estimatesand inputs'for the model of the- oculomotor system arefound using the system identification technique.As detailed in Enderle and Wolfe, a close agreement
between the predicted saccadic eye response and the datais reported for all of the target displacements for each ofthe three subjects tested [17]. Figs. 6-8 show the systemidentification technique results for a 150 target move-ment. Using final parameter estimates from the estimationroutine, these results are simulated with the saccadic eyemovement model in (1) and first-order control signals onCSMP. A close fit between the model prediction and thedata is seen in Fig. 6. Figs. 7 and 8 further illustrate theaccuracy of the final parameter estimates by noting the fitbetween the simulation results with the two-point centraldifference estimates for velocity and acceleration. Thestrong correlation between the model predictions of po-sition, velocity, and acceleration with position data andvelocity and acceleration central difference estimates val-idate the system identification technique parameter esti-mates. The accuracy of these results are typical for alltarget movements for each subject.
Displayed in Fig. 9 are the estimates of agonist pulsemagnitude as a function of displacement for the three sub-jects tested. The estimated agonist pulse magnitudeshowed more variation within each target movement thanbetween target movements. One pronounced feature evi-dent from this graph is the apparent lack of a strong re-lationship between agonist pulse magnitude and saccadeamplitude related by other investigators [4]-[6]; in fact,
Fig. 7. Velocity estimates for the saccadic eye movement illustrated inFig. 6. The solid line is the saccadic eye movement model velocity pre-diction using the final parameter estimates computed using the systemidentification technique. The dots are the two-point central differenceestimates of velocity computed with a step size of 3 and a sampling in-terval of 1 ms.
Fig. 8. Acceleration estimates for the saccadic eye movement illustratedin Fig. 6. The solid line is the saccadic eye movement model accelerationprediction using the final parameter estimates computed using the systemidentification technique. The dots are the two-point central differenceestimates of acceleration computed with a step size of 4 and a samplinginterval of 1 ms.
pulse magnitude evidently does not depend on saccadesize, consistent with a time-optimal control. Under a timeoptimal control, the magnitude of the agonist pulse shouldbe a maximum regardless of the size of the saccade. Onlythe duration of the agonist pulse affects the size of thesaccade. Fig. 10 compares'the average agonist pulse mag-nitude of one of the three subjects tested to the agonistpulse magnitude according to Bahill [6]. As indicated ear-lier, had the agonist pulse magnitude estimates displayed
50
ENDERLE AND WOLFE: TIME-OPTIMAL CONTROL OF SACCADIC EYE MOVEMENTS
Fig. 9. System identification technique estimate of agonist pulse magni-tude as a function of saccade magnitude for the three subjects tested.Figs. 6-8 correspond to agonist pulse magnitudes of subject (a).
AGONIST PULSE MAGNITUDE IN NEWTONS
2.0 r-
1.0
0.5
5 10 15SACCADE MAGNITUDE IN DEGREES
Fig. 10. Diagram comparing the agonist pulse magnitude as a function ofsaccade magnitude as predicted by a first-order time-optimal control sig-nal (dashed line) and ac6ording to Bahill [6] (solid line). Note that thefirst-order time-optimal agonist pulse magnitude is the average value fromsubject (a).
any dependence on saccade magnitude, as in Bahill's pre-diction, the controller would not have been time-optimal.Interestingly, Lehman and Stark use an agonist pulsemagnitude that is a function of amplitude, a controller that
they acknowledge as violating a bang-bang or optimalcontroller [13].
DISCUSSIONThe objective of this study was to investigate the hy-
pothesis that the saccadic neuronal control mechanism op-
erates to achieve final eye position in minimum-time forsaccades of all sizes. Our study of the theoretical time-optimal control of saccadic eye movements comple-mented our experience in parameter estimation using thesystem identification technique. The results of both anal-yses support the hypothesis that saccadic eye movementsare driven by first-order time-optimal control signals.The theoretical time-optimality solution using Pontry-
agin's minimum principle and a sixth-order horizontalsaccadic eye movement model (in which the activationand deactivation time constants are explicitly included),showed that the neuronal control is a first-order time-op-timal control signal. The optimal control results indicatethat saccade magnitude varies nonlinearly with pulsewidth. Furthermore, the direct optimal switch-time eval-uation results indicate control signals with more than one
switch-time are not time-optimal for the oculomotor sys-
tem. According to the time-optimal control results, theactive state tension for each muscle is a low-pass filteredpulse-step waveform. The magnitude of the agonist pulseis a maximum constant for saccades of all sizes and onlythe duration of the agonist pulse effects the size of thesaccade. During the period of maximum stimulation forthe agonist muscle, the antagonist muscle is completelyinhibited. Following the pulse phase, an exponential de-crease in agonist stimulation and an exponential increasein antagonist stimulation to tonic levels maintain the eye-
ball in its new position. Thus, under this control, the eye-
ball is driven to its final position in minimum-time forsaccades of all sizes.The first-order time-optimal neurological control sig-
nals are supported through experimental recordings of thesaccadic innervation signals as well [2], [3], [8]. Theseexperimental recordings indicate a clear relationship be-tween the pulse duration of the agonist saccadic innerva-tion signal and the size of the saccade. The agonist mo-
toneurons fire maximally during the pulse phase of thetrajectory regardless of the saccade amplitude.
Experimental findings by Enderle and Wolfe alsoclearly support the control of saccadic eye movements bya first-order time-optimal control signal [17]. The modelused is a modification of the sixth-order linear homeo-morphic horizontal saccadic eye movement model whichhas been extensively verified for saccades of all sizes [14].The active state tension for each muscle is assumed to bea low-pass filtered pulse-step waveform consistent withthe first-order time-optimal control signal. All of the pa-rameters of the model are established from recently pub-lished physiological data. Using system identificationtechniques to estimate oculomotor and active state tensionparameters gave simulation results which closely matchthe measured saccadic eye movement data for target dis-
(a) -1.5
z
z 1.0* w
; * D-.0 , ,I-
* z
* - 0.5
w
0)
-20 -15 -10 -5 0 5 10 15 20DISPLACEMENT (DEGREES)
(b) _- 1.5
cn
z0
w:. *. . z- -1.0.
* aw
4 0.5
wU)
0.
-20 -15 -10 -5 0 5 10 15 20DISPLACEMENT (DEGREES)
(c)-
1.5
z0
* 31.0.
z0
-0.5
-20 -15 -10 -5 0 5 10 15 20
DISPLACEMENT (DEGREES)
51
1.5 F
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 1987
placements from -15 to 15° [17]. Estimates of the ago-nist pulse magnitude do not appear to depend on saccadesize, consistent with the predictions of the theoreticaltime-optimal control signal. The magnitude of the agonistpulse is a maximum regardless of the saccade size andonly the size of the agonist pulse duration is used tochange the size of the saccade. The randomness observedin the agonist pulse estimates do not appear to depend onsaccade magnitude since the range of variation is approx-imately constant as a function of saccade magnitude. Ap-parently, the central nervous system's (CNS) saccadic re-sponse to a target movement is a bang-bang innervationcontroller signal that is affected by the internal CNS en-vironment, resulting in a variable agonist pulse magni-tude.
Clark and Stark first stated that the neuronal controlstrategy for human saccadic eye movements is time-op-timal based on experimental data analysis [12]. Using anonlinear model with activation and deactivation timeconstants, they analyzed three different sets of agonist-antagonist controller inputs. Based on a curve fitting in-vestigation matching model predictions to saccadic eyemovement data, Clark and Stark concluded that the bestresults are obtained with a first-order pulse-step neuronalcontroller. Based on their optimal control investigation,however, they reported a second-order time optimal con-trol signal. By reducing the order of the model from sixth-to fourth-order, eliminating the activation and deactiva-tion time constants, Clark and Stark's optimal control in-vestigation yielded a first-order time optimal control sig-nal, consistent with their experimental findings. Note thatClark and Stark did not solve for the switch-times in theiroptimal control analysis or comment on the pulse mag-nitude-saccade amplitude relationship.Lehman and Stark also investigated the neuronal con-
trol strategy for human saccadic eye movements [13]. Ap-plying Pontryagin's minimum principle on a linear model,which includes the activation and deactivation time con-stants, failed to give robust results. After reducing theorder of the model from sixth- to fourth-order, as Clarkand Stark did by eliminating the activation and deactiva-tion time constants, and applying Pontryagin's minimumprinciple, their analysis yielded a second-order time-op-timal control signal. In simulating saccadic eye move-ments, Lehman and Stark, however, assumed that the ag-onist pulse magnitude is a function of saccade magnitude,a c-ontroller that is not time optimal since it violates Pon-tryagin's minimum principle.
In omitting the activation and deactivation time con-stants in their optimal control investigation, but not intheir simulations, both Clark and Stark [12] and Lehmanand Stark [13] implicitly assume that the values of thetime constants are zero. However, there is abundant phys-iological evidence for including activation and deactiva-tion time constants in models of saccadic eye movements[8], [9]. Additionally, sensitivity analyses indicate thatboth of these time constants are important, but not dom-
TABLE ISACCADIC EYE MOVEMENT TIME-OPTIMAL CONTROLLER RESULTS WITH THE
ACTIVATION AND DEACTIVATION TIME CONSTANTS INCLUDED IN THEANALYSIS
Time-OptimalInvestigator Model Controller
Clark and Stark Sixth-order nonlinear Second-orderLehman and Stark Sixth-order linear Unable to
specifyEnderle and Wolfe Sixth-order linear First-order
TABLE IISACCADIC EYE MOVEMENT TIME-OPTIMAL CONTROLLER RESULTS WITHOUTTHE ACTIVATION AND DEACTIVATION TIME CONSTANTS INCLUDED IN THE
ANALYSIS
Time-OptimalInvestigator Model Controller
Clark and Stark Fourth-order nonlinear First-orderLehman and Stark Fourth-order linear Second-order
inant, control parameters [27], [28]. It therefore seemsimportant that these time constants be included in the neu-rological control investigation of saccadic eye move-ments.
Tables I and II summarize the theoretical time-optimalsaccadic eye movement control results reported here inthis paper, Clark and Stark [12], and Lehman and Stark[13]. Each of the investigators use a fourth-order oculo-motor plant in their analysis. The difference between thetables involves including the activation and deactivationtime constants in the optimal control analysis (which in-creases the order of the system from fourth- to a sixth-order), or not including the time constants. Furthermore,it should be noted that Lehman and Stark's model is lin-earized from Clark and Stark's model.In comparing the table listings, it is apparent that the
optimal controller for the nonlinear oculomotor system isdifferent from the optimal controller for the linear ocu-lomotor model. The differences in the linear and nonlineartime-optimal controllers are probably not attributable tothe linearization because of the high degree of accuracyexhibited by the linear model. Such differences might beattributed to the assumptions regarding the costate vari-ables P2 and p3 by Clark and Stark. Lehman and Starkreport that the costate variables are extremely sensitive tothe initial conditions, which can only be roughly esti-mated. Further, Lehman and Stark state that varying thecostate variable initial conditions results in different ordercontrollers. Thus, the theoretical time-optimal controllerspecified by Clark and Stark is probably in error due tothe lack of information about the costate variables initialconditions, and since it did not agree with their observedfirst-order time-optimal control simulation solution. Onlythe theoretical first-order time-optimal control results pre-sented here in this paper include the activation and deac-tivation time constants in the oculomotor system model,
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ENDERLE AND WOLFE: TIME-OPTIMAL CONTROL OF SACCADIC EYE MOVEMENTS
and agree with the experimental results of Clark and Stark,and Enderle and Wolfe [17].Our results did not support the claims by Bahill et al.
[29] and Lehman and Stark [13], [30], [31] describingdynamic overshoot by way of second-order time-optimalcontrol signals. Bahill et al. reports that dynamic over-shoot is quite capricious and approximately 70 percent ofhorizontal saccades have dynamic overshoot [29]. Leh-man and Stark [13], however, report that dynamic over-shoot occurs in only 5-10 percent of saccadic eye move-ments and Kapoula et al. [32] report only 13 percent withdynamic overshoot. Bahill et al. [29] and Lehman andStark [13], [30], [31] claim that dynamic overshoot iscaused by mismatched second-order time-optimal controlsignals. They state that after the first switch-time the ag-onist muscle is completely inhibited and the antagonistmuscle is maximally stimulated; dynamic overshoot oc-curs because of a mismatched antagonist pulse durationafter the first switch-time. Rather, we hypothesize thatsince the time-optimal control mechanism is first-order,then dynamic overshoot must be caused by unplannedhigher than tonic firing rates in the antagonist muscle dur-ing the initial phase of the step component of the controlsignal, a stochastic component of the first-order time-op-timal controller. We propose that the antagonist muscle isstimulated above tonic level right after the switch-time bya small and randomly occurring motoneuronal burst ofshort duration, but not maximally stimulated as Bahill etal. [29] and Lehman and Stark [13], [30], [31] claim.Further, since the incidence of dynamic overshoot is ran-dom and time dependent [29], the control signal compo-nent responsible for it is stochastic. Kapoula et al. giveevidence to support stochastic first-order time-optimalsaccadic control signals [32]. They indicate that there isno second-order controller pause seen in the firing raterecorded in the agonist motoneurons nor maximal firingrates recorded in the antagonist motoneurons right afterthe first switch-time. Moreover, they report a "tiny bumpin the discharge rate of the motoneurons of the antago-nist," occurring occasionally. Similar results are seen inFig. 4 of Robinson [8]; it appears that increased saccadicinnervation activity often has random and unplannedhigher initial firing rates, as seen in the agonist pulse andthe antagonist step components of the control signal. Sincethe motoneuronal burst activity in the agonist pulse isabove saturation, the higher firing rate does not effect thesaccade trajectory. However, the rate of change of themotoneuronal signal significantly effects the saccade tra-jectory [11]. Because the antagonist motoneuronal burston the step after the first switch-time is a stochastic wave-form, it is not possible to predict the waveform shape.Based on the electrophysiological evidence by Robinson[8] and the time-optimal control results presented here,we know that the antagonist muscle is certainly not max-imally stimulated and that the duration of the small burstis somewhat less than 5 ms. Consequently, we choose notto include this stochastic component in the deterministic
antagonist control because of the uncertainty of the antag-onist burst waveform and its small effect on the saccadetrajectory (less than 0.230 of the total saccade amplitude[32]). Certainly, further study is needed to more fully un-derstand this phenomenon. Therefore, we hypothesize thatthe saccadic eye movement system operates in responseto first-order time-optimal control signals; dynamicovershoot occurs as an unintentional and occasional phe-nomenon due to a small stochastic antagonist motoneu-ronal burst immediately after the switch-time.
CONCLUSIONIn summary, the results presented in this paper indicate
that the saccadic eye movement system operates under afirst-order time-optimal neuronal control strategy. Theseresults are supported from a theoretical optimal controlinvestigation and experimental findings. The concepts un-derlying this hypothesis are that the active state tensionfor each muscle is described by a low-pass filtered pulse-step waveform in which the magnitude of the agonist pulseis a maximum regardless of the amplitude of the saccade,and that only the duration of the agonist pulse affects thesize of the saccade. The antagonist muscle is completelyinhibited during the period of maximum stimulation forthe agonist muscle. Both the agonist and antagonist activestate tensions then respond exponentially to tonic levelsnecessary to maintain the final eye position. Each of thecontroller parameters exhibit random behavior from sac-cade to saccade. A sixth-order time-varying linear modelof saccadic eye movements is used in the application ofPontryagin's minimum principle to the optimality prob-lem. The simulation results, as reported by Enderle andWolfe, closely match the measured saccadic eye move-ment data for target displacements from -15 to 150 underthis time optimal hypothesis [17].
APPENDIXGRADIENT METHOD
The optimal control is determined using a gradient orsteepest descent method and the results of Berkovitz forincluding control constraints [15]. According to the min-imum principle, the Euler-Lagrange equations necessaryfor a time-optimal controller are
o =-= AO + Bnap
-aHao
where the Hamiltonian for this system is defined as
H = pT (AO + Bn).
(6)
(7)
(8)Equations (6) and (7) are usually known as the state andcostate equations, respectively. The optimal control is thatwhich minimizes H for all time in the interval (0, tf). Since
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-34, NO. 1, JANUARY 1987
the state variables are unspecified at tf [6] A. T. Bahill, "Development, validation and sensitivity analyses ofhuman eye movement models," CRC Crit. Rev. Bioeng., vol. 4, no.
dh(O(tf)) 4, pp. 311-355, 1980.a -p(tf) = 0 (9) [7] C. C. Collins, D. M. O'Meara, and A. B. Scott, "Muscle tension
during unrestrained human eye movements," J. Physiol., vol. 245,pp. 351-369, 1975.
or [8] D. A. Robinson, "Models of mechanics of eye movements," inModels of Oculomotor Behavior and Control, B. L. Zuber, Ed. Boca
p(tf) = 2G(O(tf) - D). (10) Raton, FL: CRC Press, 1981, pp. 21-41.[9] A. T. Bahill, Bioengineering: Biomedical, Medical, and Clinical En-
Furthermore, since the final time is unspecified gineering. Englewood Cliffs, NJ: Prentice-Hall, 1981.[10] G. Lennerstrand, "Motor units in eye muscles," in Basic Mecha-
-aJ nisms of Ocular Motility and Their Clinical Implications, G. Lenner-H(tf) = = - 1. (11) strand and P. Bach-y-Rita, Eds. Oxford, England: Pergamon, 1975,
a tf pp. 119-143.[11] J. D. Enderle, J. W. Wolfe, and J. T. Yates, "The linear homeo-
The gradient method, utilizing (6)-( 11), determines the morphic saccadic eye movement model-A modification," IEEEoptimal control. Whenever the corrected control violates Trans. Biomed. Eng., vol. BME-3 1, no. 11, pp. 717-720, 1984.
[12] M. R. Clark, and L. Stark, "Time optimal behavior of human sac-the control constraint, it is set to the appropriate boundary cadic eye movement," IEEE Trans. Automat. Contr., vol. AC-20,condition (that is, either 0 or nmax). Since this is an iter- pp. 345-348, 1975.ative method, a nominal control n(t) = 0 is first selected [13] s. Lehman and L. Stark, "Simulation of linear and nonlinear eye
movement models: Sensitivity analyses and enumeration studies offor all time, and then updated until the optimal control 15 time optimal control," J. Cybern. Inform. Sci., vol. 2, pp. 21-43,found. A fourth-order Runge-Kutta method is used for 1979.forward simulation of the state equations and backward [14] A. T. Bahill, J. R. Latimer, and B. T. Troost, "Linear homeo-morphic model for human movement," IEEE Trans. Biomed. Eng.,simulation of the costate equations. The control is up- vol. BME-27, no. 11, pp. 631-639, 1980.dated according to [15] L. D. Berkovitz, "Variational methods in problems of control and
programming," J. Math. Anal. Appl., vol. 3, pp. 145-169, 1961.A A adHa [16] D. A. Pierre, Optimization Theory with Application. New York:n = - k - (12) Wiley, 1969, pp. 277-280.
\afl [17] J. D. Enderle and J. W. Wolfe, "Frequency response analysis ofhuman saccadic eye movements," Comput. Biol. Med.., to be pub-where lished.
P46Ban /[18] E. B. Lee, "Mathematical aspects of the synthesis of linear, mini-aH (P4Bant + P5)/Tag mum response-time controllers," IRE Trans. Automat. Contr., vol.
an+P)Tn'[9 AC-5, pp. 283-289, 1960.-dn L( P45ag + P6)/Tant- [191 F. W. Smith, Jr., "Time-optimal control of higher-order systems,"IRE Trans. Automat. Contr., vol. AC-6, pp. 16-21, 1961.
and k is selected by a curve fitting technique to yield the [20] B. K. Kinariwala, "Analysis of time varying networks," IRE Int.least value of J [16]. The control is iteratively updated Convention Rec., pt. 4, pp. 268-276, 1961.
[21] A. T. Bahill, D. Adler, and L. Stark, "Most naturally occurring hu-until the change in J tends to zero.' ' .man saccades have magnitudes of 15 degrees or less," Invest.Utilizing parameter estimates described in Enderle and Ophthalmol., vol. 14, pp. 468-469, 1975.
Wolfe [17], gradient method simulations have not verified [22] E. J. Engelken, K. W. Stevens, J. W. Wolfe, and J. T. Yates, "Alimbus sensing eye movement recorder," USAF/SAM Tech. Rep. 29,that the modified linear homeomorphic model of saccadic 1984.
eye movements operates with a time-optimal controller. [23] A. T. Bahill and J. D. McDonald, "Frequency limitations and op-The robustness of the gradient simulation results are ex- timal step size for the two-point central difference derivative algo-
tremely sensitive to G. G determines the closeness with rithm with applications to human eye movement data," IEEE Trans.tremely sensitive to G. G determines the closeness with Biomed. Eng., vol. BME-30, no. 3, pp. 191-194, 1983.which the control drives the state at tf to the terminal state [24] A. T. Bahill, A. Brockenbrough, and B. T. Troost, "Variability andD, and directly affects the costate vector. The gradient development of a normative data base for saccadic eye movements,"D,and directly affects the costate vector. The gradient Invest. Ophthalmol. Vis. Sci., pp. 116-125, 1981.method is unable to solve for the endpoint conditions on [25] A. T. Bahill, M. R. Clark, and L. Stark, "The main sequence, a toolp, so a minimum-time controller cannot be specified. for studying human eye movements," Math. Biosci., vol. 24, pp.These findings are consistent with those reported by Leh- 194-204, 1975.man and Stark [131] [26] D. A. Robinson, D. M. O'Meara, A. B. Scott, and C. C. Collins,man and Stark [13]. "Mechanical components of human eye movements," J. Appl. Phys-
iol., vol. 26, pp. 548-553, 1969.REFERENCES [27] M. R. Clark and L. Stark, "Sensitivity of control parameters in a
model of saccadic eye tracking and estimation of resultant nervous[1] D. A. Robinson, "The mechanics of human saccadic eye move- activity," Bull. Math. Biol., vol. 38, pp. 39-57, 1976.
ment," J. Physiol., vol. 174, pp. 245-264, 1964. [28] F. K. Hsu, A. T. Bahill, and L. Stark, "Parametric sensitivity anal-[2] - , "Oculomotor unit behavior in the monkey," J. Physiol., vol. ysis of a homeomorphic model for saccadic and vergence eye move-
33, pp. 393-404, 1970. ments," Comput. Progr. Biomed., vol. 6, pp. 108-116, 1976.[3] A. F. Fuchs and E. S. Luschei, "Firing patterns of abducens neurons [29] A. T. Bahill, M. R. Clark, and L. Stark, "Dynamic overshoot in
of alert monkeys in relationship to horizontal eye movements," J. saccadic eye movements is caused by neurological control signal re-Neurophysiol., vol. 33, pp. 382-392, 1970. versals," Exp. Neurol., vol. 48, pp. 107-122, 1975.
[41 C. C. Collins, "The human oculomotor control system," in Basic [30] S. L. Lehman and L. Stark, "Multipulse controller signals. II TimeMechanisms of Ocular Motility and Their Clinical Implications, G. optimality," Biol. Cybern., vol. 48, pp. 5-8, 1983.Lennerstrand and P. Bach-y-Rita, Eds. Oxford, England: Perga- [31] -, "Multipulse controller signals. III Dynamic overshoot," Biol.mon, 1975, pp. 145-180. Cybern., vol. 48, pp. 9-10, 1983.
[5] D. S. Zee, L. M. Optican, J. D. Cook, D. A. Robinson, and W. K. [32] Z. A. Kapoula, D. A. Robinson, and T. C. Hain, "Motion of the eyeEngel, "Slow saccades in spinocerebellar degeneration," Arch. Neu- immediately after a saccade,'" Exp. Brain Res., vol. 61, pp. 386-394,rol., vol. 33, pp. 243-251, 1976. 1986.
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ENDERLE.AND WOLFE: TIME-OPTIMAL CONTROL OF SACCADIC EYE M
John D. Enderle (S'75-M'80) was born inQueens, NYin 1953. He received tbe B.S., M.E.,and Ph.D. degrees in biomedical engineering, andthe M.E. degree in electrical engineering fromRensselaer Polytechnic Institute, Troy, NY, in1975, 1977, 1980, and 1978, respectively.
After completing his Ph.D. studies, he becamea Senior Staff Member at PAR Technology Cor-poration, Rome, NY from 1979 to 1981. In 1981,he joined the Faculty of North Dakota State Uni-versity, Fargo, where he is now an Associate Pro-
fessor in the Department of Electrical and Electronics Engineering and Di-rector of the Division of Bioengineering. His research interests includemodeling physiological systems, system identification, signal processing,and control theory.
Dr. Enderle is a member of the IEEE Biomedical Engineering Society,Board of Directors of the Rocky Mountain Bioengineering Symposium,Inc., Sigma Xi and Tau Beta Pi, and a Southeastern Center for ElectricalEngineering Education Fellow.
55
James W. Wolfe was born in Ludlowville, NY in1932. He received the B.A. degree from the Uni-versity, of California, Riverside, in 1963, and thePh.D. degree from the University of Rochester,Rochester, NY in 1966.
I From 1966 to 1968, he worked as a Physiolog-ical Research Psychologist at the U.S. ArmyMedical Research Laboratory, Fort Knox, KY.Currently, he is Chief of Neurosciences Functionat the USAF School of Aerospace Medicine,Brooks AFB, San Antonio, TX. His research in-
terests include electrophysiology and psychophysiology of the vestibularand cerebellar systems.
Dr. Wolfe is a member of the Aerospace Medical Association, BaranySociety, International Brain Research Organization, Sigma Xi, and a chartermember of the Society for Research in Otolaryngology.
